Research Article The Spiral Coaxial Cable · 2019. 7. 31. · ], and the channel waveguides...
Transcript of Research Article The Spiral Coaxial Cable · 2019. 7. 31. · ], and the channel waveguides...
Research ArticleThe Spiral Coaxial Cable
I M Fabbri
Department of Physics University of Milan Via Celoria 16 20133 Milan Italy
Correspondence should be addressed to I M Fabbri italomariofabbricrfmit
Received 18 September 2014 Revised 10 January 2015 Accepted 12 January 2015
Academic Editor Kamya Yekeh Yazdandoost
Copyright copy 2015 I M Fabbri This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A new concept of metal spiral coaxial cable is introduced The solution to Maxwellrsquos equations for the fundamental propagatingTEM eigenmode using a generalization of the Schwarz-Christoffel conformal mapping of the spiral transverse section is providedtogether with the analysis of the impedances and the Poynting vector of the line The new cable may find application as a mediumfor telecommunication and networking or in the sector of the Microwave Photonics A spiral plasmonic coaxial cable could beused to propagate subwavelength surface plasmon polaritons at optical frequencies Furthermore according to the present modelthe myelinated nerves can be considered natural examples of spiral coaxial cables This study suggests that a malformation of thePeters angle which determines the power of the neural signal in the TEM mode causes higherlower power to be transmitted inthe neural networks with respect to the natural level The formulas of the myelin sheaths thickness the diameter of the axon andthe spiral 119892 factor of the lipid bilayers which are mathematically related to the impedances of the spiral coaxial line can make iteasier to analyze the neural line impedance mismatches and the signal disconnections typical of the neurodegenerative diseases
1 Introduction
The coaxial cable invented by Heaviside [1] is a transmissionline composed of an inner conductor surrounded by aninsulating layer and an outer conducting shield The innerand the outer conductors share the same geometrical axis
Nowadays there exist several types of transmission linesthe coaxial cables the hollow waveguides (rectangular cir-cular elliptical and parallel plates [2 3]) the two wires [34] and the channel waveguides (buried strip-loaded ridgerib diffused and graded-dielectric index [5]) The two-wiretransmission line used in conventional circuits is inefficientfor transferring electromagnetic energy at microwave fre-quencies because the fields are not confined in all directionsthe energy escapes by radiation and it has large copper lossesdue to its relatively small surface area
On the other hand the larger surface area of the boundarymakes in general the waveguides more efficient with respectto the coaxials on reducing the copper losses
Dielectric losses in two wires and coaxial lines caused bythe heating of the insulation between the conductors are alsolower in waveguides The insulation behaves as the dielectricof a capacitor and the breakdown of the insulation betweenthe conductors of a transmission line is more frequently aproblem than is the dielectric loss in practical applications [3]
Waveguides are also subject to the dielectric breakdowncaused by the stationary voltage spikes at the nodes of thestanding waves
In spite of the dielectric in waveguides is air it causesarcing which decreases the efficiency of energy transfer andcan severely damage the waveguide
The spiral coaxial cable (SCC) discussed in this paper hasmany advantages with respect to the other types of trans-mission lines first of all the elm energy can be distributedefficiently over a larger area reducing all the undesiredaforementioned effects
The power handling versus frequency is consequentlyhigher for the metal spiral coaxial line (MSCC) with respectto the other lines
The Microwave Photonics (MWP) [6] a new disciplinethat joins together the radio-frequency engineering and opto-electronics represents the future in many civil and defenseapplications like speed of the digital signal processors cabletelevision and optical signal processing
The MSCC could become one of the key objects in thefield of MWP for its characteristics in terms of both powerhandling and energy transfer efficiency
Recently it has also been demonstrated experimentallythat metal coaxial waveguide nanostructures perform at
Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2015 Article ID 630131 18 pageshttpdxdoiorg1011552015630131
2 International Journal of Microwave Science and Technology
optical frequencies [7] opening up new research pursuitsunexpected in the area of nanoscale waveguiding fieldenhancement imaging with coaxial cavities and negative-index metamaterials [8]
Several different plasmonic waveguiding structureshave been proposed such as metallic nanowires [9 10]metal-dielectric-metal (MDM) structures [11] and metallicnanoparticle arrays [12] for achieving compact integratedphotonic devices
Most of these structures support a highly confined modenear the surface plasmon frequency [11] This study couldbecome the reference point to introduce the MSCC as avalid alternative to guide subwavelength surface plasmonpolaritons (SPPs)
The extraordinary transmission of light through an arrayof subwavelength apertures enhancement which arises fromthe coupling of the incident light with the SPPs through thesurface grating in metal film [13] could result particularlyefficiently on the spiral metal dielectric interface with peri-odic holes
High sensitivity spiral biosensors and spiral photonicintegrated circuits based on nonlinear surface plasmonpolariton optics [14 15] may be implemented
The aimof this pioneeringwork on the SCC is to representan initial landmark in the continuously growing sector of themicrowave research
The popularity of Video on Demand (VoD) and Overthe Top Technology (OTT) services to access high definitionvideos over home interconnected devices of the hybrid fibre-coax (HFC) networks is driving the research toward moreefficient and cost-effective cables Particularly the develop-ment of Converged Cable Access Platforms (CCAP) thatcombine video and data transmission supporting simultane-ous network access of multiple users over a single coaxialcable is flourishing and sustaining the demand for new highspeed transmission media
In a metallic guide the reflection mechanism responsiblefor confining the energy is due to the reflection from theconductors at the boundary [16] whose geometry is strictlyrelated to the propagating modes
Coaxial cables were designed to propagate high frequencyradio signals The principal constraints on performance of acoaxial are attenuation thermal noise and passive intermod-ulation noise (PIM)
In RF applications the wave propagates essentially in thefundamental transverse electric magnetic (TEM) mode thatis the electric and magnetic fields are both perpendicular tothe direction of propagation
In the ideal case the conductors can be considered to haveinfinite conductivity and the TEM eigenmode is the basicpropagating wave (see [17] page 110) along the transmissionline
Practical lines have finite conductivity and this results ina perturbation or change of the TEMmode (see [18] page 119)
Above the cutoff frequency transverse electric (TE) ortransverse magnetic (TM) modes [19] can also propagatewith different velocities within a practical cylindrical coaxinterferingwith each other producing distortion of the signal
The frequency of operation for a specific outer conductorsize is then limited by the highest usable cutoff frequencybefore undesirable modes of propagation occur
In order to prevent higher order modes from beinglaunched the radiuses of the coaxial conductors must bereduced diminishing the amount of power that can betransmitted
On the other hand at high frequencies it is impossible tomake the cylindrical coaxial line in the small size necessaryto propagate the TEMmode alone
The research described in this paper demonstrates thepropagation on the fundamental TEM wave along the idealMSCC
Since the mode of transmission on an ideal line is theTEM wave the relations for input impedance reflectioncoefficient return loss (RL) standing wave ratio (SWR) andso forth given afterward in the next sections are applicablein general to the spiral transmission lines (see [18] Chapter 3)
The metal double spiral coaxial cable or MDSCC result-ing from the superposition of two spiral conductors thatshare the same geometrical axis can be made multi-turnThe amount of heat generated by the losses for heating canbe distributed over a larger area and this would lower thetemperature and raise the reliability of the line
In fact operation at higher temperatures results in areduction in the life expectancy and reliability of the trans-mission line relative to the lower temperature performance
Applications like nanoscale optical components for inte-gration on semiconductor chips could benefit from thesecharacteristics of the MSCC
Where signal integrity is important coaxial cables areneeded to be shielded against radio frequency noise (RFnoise) The multiturn MDSCC is naturally shielded becausethe highest part of the elm energy can be distributed on theinner part of the cable which protects small signals frominterference due to external electric fields
A new class of spiral passive components computer-aided engineering (CAE) tools as well as electromagnetic(EM) simulators is required before new high-frequencyspiral RFmicrowave circuits will be implemented
The spiral geometry occurs widely in nature exampleslike the spiral galaxies are found at the universe level while themyelin bundles are common in the microcosm of the neuroncells
Recently a new spiral optical fibre has been proposedboth in the fundamental mode [20] and in the higher ordermodes [21] operation
Spirals are also of extreme interest to the field of the newmetamaterials and invisible cloaking [22]
Myelinated nerve fibers are micro-spiral coaxial cables(119892 ≪ 1) whose electric behaviour is still today described byneurophysiologists using W Thomsonrsquos (later known as LordKelvin) cable formula [23] of the 1860s which determinesthe velocity of the signal propagating in saltatory conduction[23 24]
Cable theory in neurobiology has a long history havingfirst been applied to neurons in 1863 byCMatteucci [25] whodiscovered that if a constant current flows through a portionof a platinum wire covered with a sheath saturated with fluid
International Journal of Microwave Science and Technology 3
extra-polar current can be led off which corresponds to theelectrotonic current of nerves
Since the 1950sndash60s myelinated nerves have been recog-nized to have a spiral structure and to behave like a high losscoaxial cable [26 27] with negligible inductance
The mathematical model presented in this paper can beused to refine the elm theory of the myelinated nerves bytaking into account their spiral geometry
In a coaxial guide the determination of the electromag-netic fields within any region of the guide is dependent upononersquos ability to explicitly solve the Maxwell field equations inan appropriate coordinate system [28]
Let us considerMaxwellrsquos equations
nabla times = minus119895120596120583
nabla times = 119895120596120598 119864
nabla sdot = 0
nabla sdot = 0
(1)
where the time variation of the fields is assumed to beexp(119895120596119905)
In view of the nature of the boundary surface it isconvenient to separate these field equations into componentsparallel and transverse to the waveguide 119911-axis
This is achieved by scalar and vector multiplication of (1)with 119890
119911 a unit vector in the 119911 direction thus obtaining
nablaperpsdot (119890
119911times
perp) = minus119895120596120583119867
119911
nablaperpsdot (119890
119911times
perp) = 119895120596120598119864
119911
(2)
nablaperp119864
119911minus120597119864
perp
120597119911= minus119895120596120583 (119890
119911times 119867
perp)
nablaperp119867
119911minus120597119867
perp
120597119911= 119895120596120598 (119890
119911times 119864
perp)
(3)
Since the transmission line description of the electromagneticfield within uniform guides is independent of the particularform of the coordinate system employed to describe the crosssection no reference to cross-sectional coordinates is madeon deriving the telegrapherrsquos equation [28 29]
Substituting (2) into (3) we obtain
120597119864perp
120597119911= minus119895119896120585 ( 120598 +
1
1198962nablaperpnablaperp) sdot (
perptimes 119890
119911)
120597119867perp
120597119911= minus119895119896120578 ( 120598 +
1
1198962nablaperpnablaperp) sdot (119890
119911times
perp)
(4)
Vector notation is employed with the following meanings forthe symbols
119864perp
= 119864perp(119909 119910 119911) = the rms electric field intensity
transverse to the 119911-axis119867
perp= 119867
perp(119909 119910 119911) = the rms magnetic field intensity
transverse to the 119911-axis
120578 = intrinsic impedance of the medium 1120578 = radic120583120598119896 = 120596radic120583120598 = 2120587120582 = propagation constant inmediumor the wavenumber (see [28] page 3)nablaperp
= gradient operator transverse to 119911-axis = nabla minus
119890119911(120597120597119911)120598 = unit dyadic defined such that 120598 sdot = sdot 120598 =
Equations (4) and (2) which are fully equivalent to theMaxwell equations make evident the separate dependenceof the field on the cross-sectional coordinates and on thelongitudinal coordinate 119911 The cross-sectional dependencemay be integrated out of (4) by means of a suitable set ofvector orthogonal functions provided they satisfy appropriateconditions on the boundary curve or curves 119904 of the crosssection
Such vector functions are known to be of two types theE-mode functions 1198901015840
119894defined by
1198901015840
119894= minusnabla
perpΦ
119894
ℎ1015840
119894= 119890
119911times 119890
1015840
119894
(5)
where
nabla2
perpΦ
119894+ 119896
10158402
119888119894Φ
119894= 0
Φ119894= 0 on 119904 if 1198961015840
119888119894= 0
120597Φ119894
120597119904= 0 on 119904 if 1198961015840
119888119894= 0
(6)
and the H-mode functions 11989010158401015840119894defined by
11989010158401015840
119894= 119890
119911times nabla
perpΨ
119894
ℎ10158401015840
119894= 119890
119911times 119890
10158401015840
119894
(7)
where
nabla2
perpΨ
119894+ 119896
101584010158402
119888119894Ψ
119894= 0
120597Ψ119894
120597119899= 0 on 119904
(8)
where 119894 denotes a double index and 119899 is the outward normalto 119904 in the cross-section plane
The constants 11989610158401015840
119888119894and 119896
1015840
119888119894are defined as the cutoff wave
numbers or eigenvalues associated with the guide crosssection
The functions 119890119894possess the vector orthogonality proper-
ties
∬1198901015840
119894sdot 119890
1015840
119895119889119878
perp= ∬119890
10158401015840
119894sdot 119890
10158401015840
119895119889119878
perp=
1 for 119894 = 119895
0 for 119894 = 119895
∬1198901015840
119894sdot 119890
10158401015840
119895119889119878
perp= 0
(9)
with the integration extended over the entire guide crosssection with surface 119878
perp
4 International Journal of Microwave Science and Technology
The total average power flow along the guide in the 119911direction is
119875119911=1
2Re(∬119864
perptimes 119867
lowast
perpsdot 119890
119911119889119878
perp) (10)
where all quantities are rms and the asterisk denotes thecomplex conjugate
In TEMmodes both 119864119911and119867
119911vanish and the fields are
fully transverse Their cutoff condition 1198962
119888= 0 or 120596 = 120573119888
(where 119888 is the speed of the light) is equivalent to the followingrelation [28]
perp=1
120578119890119911times
perp (11)
between the electric andmagnetic transverse fields where 120578 =radic120583120598 is the medium impedance so that 120578119888 = 120583 and 120578119888 = 1120598
The electric field perp
is determined from the rest ofMaxwellrsquos equations which read
nablaperptimes
perp= 0
nabla sdot perp= 0
(12)
These are recognized as the field equations of an equivalenttwo-dimensional electrostatic problem
Once the electrostatic solution perpis found the magnetic
field is constructed from (11)Because of the relationship between
perpand
perp the
Poynting vector 119878119911will be
119878119911=1
2Re (
perptimes
lowast
perp) sdot 119890
119911=1
120578
10038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
= 12057810038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
(13)
2 The Spiral Differential Geometry
For the MSCC structures it is difficult to construct solutionsfor Laplacersquos equation with polar or cartesian coordinates
The conformal mapping technique is a powerful methodfor solving two-dimensional potential problems andmappingthe boundaries into a simpler configuration for which solu-tions to Laplacersquos equation are easily found [17 18]
For the specific purposes of the MSCC the followingspiral coordinates based on a generalization of the Schwarz-Christoffelmapping (see appendix) are introduced
119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)
119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)
119911 = 119911
(14)
where 120579 120575 represent the spiral coordinates and 119892 gt 0
is a constant which characterizes the transformation (seeappendix)
As it can be seen in Figure 1 the equation 120575 = constrepresented by a vertical line in the 120575-120579 plane correspondsto a logarithmic spiral into the 119909-119910 plane and a constantcoordinate line of the spiral mapping
Observing (14) it appears clear that for 119892120579 minus 120575lowast
119892 rarr 0where 120575lowast is a constant the curve in the 119909-119910 plane locallyreduces (for |120579| ≪ 1 119892 ≪ 1) to an Archimedean spiral
The region between the two coaxial spirals maps into theregion inside the polygon bounded by the coordinate-lines120579 = 120579
1 120579 = 120579
2and 120575 = 120575
1 120575 = 120575
2 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) [18](see Figure 1(b))
It is also worth to observe that if 1205752minus120575
1= 2119902120587119892
2
(1+1198922
)119902 isin Z the two spirals 120575 = 120575
1 120575 = 120575
2are identical apart from
a shift of Δ120579119904= 2119902120587(1 + 119892
2
)We then require |120575
2minus120575
1| lt 2120587119892
2
(1+1198922
) in order to avoidcyclic spirals with 120575 gt 120575
2in the middle of the two with 120575 = 120575
1
and 120575 = 1205752
The differential one form of the spiral transformation ordual basis results in
119889119909 =120597119909
120597120575119889120575 +
120597119909
120597120579119889120579 +
120597119909
120597119911119889119911
119889119910 =120597119910
120597120575119889120575 +
120597119910
120597120579119889120579 +
120597119910
120597119911119889119911
119889119911 = 119889119911
(15)
The arc length 119889ℓ is given by
119889ℓ2
= 1198891199092
+ 1198891199102
+ 1198891199112
= 119892120575120575119889120575
2
+ 119892120579120579119889120579
2
+ 119892119911119911119889119911
2
(16)
where
ℎ2
120575= 119892
120575120575= 119890
(2(120575119892)minus2119892120579)
(1 +1
1198922)
ℎ2
120579= 119892
120579120579= 119890
(2(120575119892)minus2119892120579)
(1 + 1198922
)
ℎ2
119911= 119892
119911119911= 1 119892
120575120579= 119892
120575119911= 119892
120579119911= 0
(17)
are the components of themetric tensor and Lame coefficientsThe infinitesimal volume element is given by
119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)
1 + 1198922
119892119889120575 119889120579 119889119911 (18)
where the 119869 is the Jacobian of the spiral transformationLet us now define the spiral natural basis vectors 119890
120575 119890
120579 119890
119911
119890120575=120597
120597120575
= 119890(120575119892minus119892120579)
(1
119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890
119909
+ 119890(120575119892minus119892120579)
(1
119892sin (120575 + 120579) + cos (120575 + 120579)) 119890
119910
International Journal of Microwave Science and Technology 5
0
Conductor 1
Conductor 2
Region I
Region II
or
y
1205791
120579
1205792
1205751
1205752
x
1205751 minus2120587g2
1 + g2
∙
∙SI
SII
(a)
0
Region II
0
1205791
120579
1205792
120575
1205751205752 1205751
Φ
V0
∙
∙
polygonSchwarz-Christoffel
Region I
Con
duct
or 1
Con
duct
or 2
Con
duct
or 1
1205792 minus2120587
1 + g2
1205791 minus2120587g2
1 + g2
SI
SII
(b)
y
z
x
rarrdS120575z
rarrdS120579z
rarrdS120575120579
(c)
Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces
119890120579=120597
120597120579
= 119890(120575119892minus119892120579)
(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909
+ 119890(120575119892minus119892120579)
(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910
119890119911=120597
120597119911= 119890
119911
(19)
in terms of the cartesian basis vectors 119890119909 119890
119910 119890
119911
The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by
10038171003817100381710038171003817119889 119878
120575120579
10038171003817100381710038171003817= 119889119878
perp=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597120579
10038171003817100381710038171003817100381710038171003817119889120575 119889120579
= 119890(2120575119892minus2119892120579)
(1
119892+ 119892)119889120575 119889120579
10038171003817100381710038171003817119889 119878
120579119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120579times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890
(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579
10038171003817100381710038171003817119889 119878
120575119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890
(120575119892minus119892120579)
radic1 + 1198922
119892119889119911 119889120575
(20)
We then define the natural unitary spiral basis vectors
119890120575=119890120575
ℎ120575
119890120579=119890120579
ℎ120579
119890119911=119890119911
ℎ119911
(21)
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
2 International Journal of Microwave Science and Technology
optical frequencies [7] opening up new research pursuitsunexpected in the area of nanoscale waveguiding fieldenhancement imaging with coaxial cavities and negative-index metamaterials [8]
Several different plasmonic waveguiding structureshave been proposed such as metallic nanowires [9 10]metal-dielectric-metal (MDM) structures [11] and metallicnanoparticle arrays [12] for achieving compact integratedphotonic devices
Most of these structures support a highly confined modenear the surface plasmon frequency [11] This study couldbecome the reference point to introduce the MSCC as avalid alternative to guide subwavelength surface plasmonpolaritons (SPPs)
The extraordinary transmission of light through an arrayof subwavelength apertures enhancement which arises fromthe coupling of the incident light with the SPPs through thesurface grating in metal film [13] could result particularlyefficiently on the spiral metal dielectric interface with peri-odic holes
High sensitivity spiral biosensors and spiral photonicintegrated circuits based on nonlinear surface plasmonpolariton optics [14 15] may be implemented
The aimof this pioneeringwork on the SCC is to representan initial landmark in the continuously growing sector of themicrowave research
The popularity of Video on Demand (VoD) and Overthe Top Technology (OTT) services to access high definitionvideos over home interconnected devices of the hybrid fibre-coax (HFC) networks is driving the research toward moreefficient and cost-effective cables Particularly the develop-ment of Converged Cable Access Platforms (CCAP) thatcombine video and data transmission supporting simultane-ous network access of multiple users over a single coaxialcable is flourishing and sustaining the demand for new highspeed transmission media
In a metallic guide the reflection mechanism responsiblefor confining the energy is due to the reflection from theconductors at the boundary [16] whose geometry is strictlyrelated to the propagating modes
Coaxial cables were designed to propagate high frequencyradio signals The principal constraints on performance of acoaxial are attenuation thermal noise and passive intermod-ulation noise (PIM)
In RF applications the wave propagates essentially in thefundamental transverse electric magnetic (TEM) mode thatis the electric and magnetic fields are both perpendicular tothe direction of propagation
In the ideal case the conductors can be considered to haveinfinite conductivity and the TEM eigenmode is the basicpropagating wave (see [17] page 110) along the transmissionline
Practical lines have finite conductivity and this results ina perturbation or change of the TEMmode (see [18] page 119)
Above the cutoff frequency transverse electric (TE) ortransverse magnetic (TM) modes [19] can also propagatewith different velocities within a practical cylindrical coaxinterferingwith each other producing distortion of the signal
The frequency of operation for a specific outer conductorsize is then limited by the highest usable cutoff frequencybefore undesirable modes of propagation occur
In order to prevent higher order modes from beinglaunched the radiuses of the coaxial conductors must bereduced diminishing the amount of power that can betransmitted
On the other hand at high frequencies it is impossible tomake the cylindrical coaxial line in the small size necessaryto propagate the TEMmode alone
The research described in this paper demonstrates thepropagation on the fundamental TEM wave along the idealMSCC
Since the mode of transmission on an ideal line is theTEM wave the relations for input impedance reflectioncoefficient return loss (RL) standing wave ratio (SWR) andso forth given afterward in the next sections are applicablein general to the spiral transmission lines (see [18] Chapter 3)
The metal double spiral coaxial cable or MDSCC result-ing from the superposition of two spiral conductors thatshare the same geometrical axis can be made multi-turnThe amount of heat generated by the losses for heating canbe distributed over a larger area and this would lower thetemperature and raise the reliability of the line
In fact operation at higher temperatures results in areduction in the life expectancy and reliability of the trans-mission line relative to the lower temperature performance
Applications like nanoscale optical components for inte-gration on semiconductor chips could benefit from thesecharacteristics of the MSCC
Where signal integrity is important coaxial cables areneeded to be shielded against radio frequency noise (RFnoise) The multiturn MDSCC is naturally shielded becausethe highest part of the elm energy can be distributed on theinner part of the cable which protects small signals frominterference due to external electric fields
A new class of spiral passive components computer-aided engineering (CAE) tools as well as electromagnetic(EM) simulators is required before new high-frequencyspiral RFmicrowave circuits will be implemented
The spiral geometry occurs widely in nature exampleslike the spiral galaxies are found at the universe level while themyelin bundles are common in the microcosm of the neuroncells
Recently a new spiral optical fibre has been proposedboth in the fundamental mode [20] and in the higher ordermodes [21] operation
Spirals are also of extreme interest to the field of the newmetamaterials and invisible cloaking [22]
Myelinated nerve fibers are micro-spiral coaxial cables(119892 ≪ 1) whose electric behaviour is still today described byneurophysiologists using W Thomsonrsquos (later known as LordKelvin) cable formula [23] of the 1860s which determinesthe velocity of the signal propagating in saltatory conduction[23 24]
Cable theory in neurobiology has a long history havingfirst been applied to neurons in 1863 byCMatteucci [25] whodiscovered that if a constant current flows through a portionof a platinum wire covered with a sheath saturated with fluid
International Journal of Microwave Science and Technology 3
extra-polar current can be led off which corresponds to theelectrotonic current of nerves
Since the 1950sndash60s myelinated nerves have been recog-nized to have a spiral structure and to behave like a high losscoaxial cable [26 27] with negligible inductance
The mathematical model presented in this paper can beused to refine the elm theory of the myelinated nerves bytaking into account their spiral geometry
In a coaxial guide the determination of the electromag-netic fields within any region of the guide is dependent upononersquos ability to explicitly solve the Maxwell field equations inan appropriate coordinate system [28]
Let us considerMaxwellrsquos equations
nabla times = minus119895120596120583
nabla times = 119895120596120598 119864
nabla sdot = 0
nabla sdot = 0
(1)
where the time variation of the fields is assumed to beexp(119895120596119905)
In view of the nature of the boundary surface it isconvenient to separate these field equations into componentsparallel and transverse to the waveguide 119911-axis
This is achieved by scalar and vector multiplication of (1)with 119890
119911 a unit vector in the 119911 direction thus obtaining
nablaperpsdot (119890
119911times
perp) = minus119895120596120583119867
119911
nablaperpsdot (119890
119911times
perp) = 119895120596120598119864
119911
(2)
nablaperp119864
119911minus120597119864
perp
120597119911= minus119895120596120583 (119890
119911times 119867
perp)
nablaperp119867
119911minus120597119867
perp
120597119911= 119895120596120598 (119890
119911times 119864
perp)
(3)
Since the transmission line description of the electromagneticfield within uniform guides is independent of the particularform of the coordinate system employed to describe the crosssection no reference to cross-sectional coordinates is madeon deriving the telegrapherrsquos equation [28 29]
Substituting (2) into (3) we obtain
120597119864perp
120597119911= minus119895119896120585 ( 120598 +
1
1198962nablaperpnablaperp) sdot (
perptimes 119890
119911)
120597119867perp
120597119911= minus119895119896120578 ( 120598 +
1
1198962nablaperpnablaperp) sdot (119890
119911times
perp)
(4)
Vector notation is employed with the following meanings forthe symbols
119864perp
= 119864perp(119909 119910 119911) = the rms electric field intensity
transverse to the 119911-axis119867
perp= 119867
perp(119909 119910 119911) = the rms magnetic field intensity
transverse to the 119911-axis
120578 = intrinsic impedance of the medium 1120578 = radic120583120598119896 = 120596radic120583120598 = 2120587120582 = propagation constant inmediumor the wavenumber (see [28] page 3)nablaperp
= gradient operator transverse to 119911-axis = nabla minus
119890119911(120597120597119911)120598 = unit dyadic defined such that 120598 sdot = sdot 120598 =
Equations (4) and (2) which are fully equivalent to theMaxwell equations make evident the separate dependenceof the field on the cross-sectional coordinates and on thelongitudinal coordinate 119911 The cross-sectional dependencemay be integrated out of (4) by means of a suitable set ofvector orthogonal functions provided they satisfy appropriateconditions on the boundary curve or curves 119904 of the crosssection
Such vector functions are known to be of two types theE-mode functions 1198901015840
119894defined by
1198901015840
119894= minusnabla
perpΦ
119894
ℎ1015840
119894= 119890
119911times 119890
1015840
119894
(5)
where
nabla2
perpΦ
119894+ 119896
10158402
119888119894Φ
119894= 0
Φ119894= 0 on 119904 if 1198961015840
119888119894= 0
120597Φ119894
120597119904= 0 on 119904 if 1198961015840
119888119894= 0
(6)
and the H-mode functions 11989010158401015840119894defined by
11989010158401015840
119894= 119890
119911times nabla
perpΨ
119894
ℎ10158401015840
119894= 119890
119911times 119890
10158401015840
119894
(7)
where
nabla2
perpΨ
119894+ 119896
101584010158402
119888119894Ψ
119894= 0
120597Ψ119894
120597119899= 0 on 119904
(8)
where 119894 denotes a double index and 119899 is the outward normalto 119904 in the cross-section plane
The constants 11989610158401015840
119888119894and 119896
1015840
119888119894are defined as the cutoff wave
numbers or eigenvalues associated with the guide crosssection
The functions 119890119894possess the vector orthogonality proper-
ties
∬1198901015840
119894sdot 119890
1015840
119895119889119878
perp= ∬119890
10158401015840
119894sdot 119890
10158401015840
119895119889119878
perp=
1 for 119894 = 119895
0 for 119894 = 119895
∬1198901015840
119894sdot 119890
10158401015840
119895119889119878
perp= 0
(9)
with the integration extended over the entire guide crosssection with surface 119878
perp
4 International Journal of Microwave Science and Technology
The total average power flow along the guide in the 119911direction is
119875119911=1
2Re(∬119864
perptimes 119867
lowast
perpsdot 119890
119911119889119878
perp) (10)
where all quantities are rms and the asterisk denotes thecomplex conjugate
In TEMmodes both 119864119911and119867
119911vanish and the fields are
fully transverse Their cutoff condition 1198962
119888= 0 or 120596 = 120573119888
(where 119888 is the speed of the light) is equivalent to the followingrelation [28]
perp=1
120578119890119911times
perp (11)
between the electric andmagnetic transverse fields where 120578 =radic120583120598 is the medium impedance so that 120578119888 = 120583 and 120578119888 = 1120598
The electric field perp
is determined from the rest ofMaxwellrsquos equations which read
nablaperptimes
perp= 0
nabla sdot perp= 0
(12)
These are recognized as the field equations of an equivalenttwo-dimensional electrostatic problem
Once the electrostatic solution perpis found the magnetic
field is constructed from (11)Because of the relationship between
perpand
perp the
Poynting vector 119878119911will be
119878119911=1
2Re (
perptimes
lowast
perp) sdot 119890
119911=1
120578
10038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
= 12057810038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
(13)
2 The Spiral Differential Geometry
For the MSCC structures it is difficult to construct solutionsfor Laplacersquos equation with polar or cartesian coordinates
The conformal mapping technique is a powerful methodfor solving two-dimensional potential problems andmappingthe boundaries into a simpler configuration for which solu-tions to Laplacersquos equation are easily found [17 18]
For the specific purposes of the MSCC the followingspiral coordinates based on a generalization of the Schwarz-Christoffelmapping (see appendix) are introduced
119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)
119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)
119911 = 119911
(14)
where 120579 120575 represent the spiral coordinates and 119892 gt 0
is a constant which characterizes the transformation (seeappendix)
As it can be seen in Figure 1 the equation 120575 = constrepresented by a vertical line in the 120575-120579 plane correspondsto a logarithmic spiral into the 119909-119910 plane and a constantcoordinate line of the spiral mapping
Observing (14) it appears clear that for 119892120579 minus 120575lowast
119892 rarr 0where 120575lowast is a constant the curve in the 119909-119910 plane locallyreduces (for |120579| ≪ 1 119892 ≪ 1) to an Archimedean spiral
The region between the two coaxial spirals maps into theregion inside the polygon bounded by the coordinate-lines120579 = 120579
1 120579 = 120579
2and 120575 = 120575
1 120575 = 120575
2 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) [18](see Figure 1(b))
It is also worth to observe that if 1205752minus120575
1= 2119902120587119892
2
(1+1198922
)119902 isin Z the two spirals 120575 = 120575
1 120575 = 120575
2are identical apart from
a shift of Δ120579119904= 2119902120587(1 + 119892
2
)We then require |120575
2minus120575
1| lt 2120587119892
2
(1+1198922
) in order to avoidcyclic spirals with 120575 gt 120575
2in the middle of the two with 120575 = 120575
1
and 120575 = 1205752
The differential one form of the spiral transformation ordual basis results in
119889119909 =120597119909
120597120575119889120575 +
120597119909
120597120579119889120579 +
120597119909
120597119911119889119911
119889119910 =120597119910
120597120575119889120575 +
120597119910
120597120579119889120579 +
120597119910
120597119911119889119911
119889119911 = 119889119911
(15)
The arc length 119889ℓ is given by
119889ℓ2
= 1198891199092
+ 1198891199102
+ 1198891199112
= 119892120575120575119889120575
2
+ 119892120579120579119889120579
2
+ 119892119911119911119889119911
2
(16)
where
ℎ2
120575= 119892
120575120575= 119890
(2(120575119892)minus2119892120579)
(1 +1
1198922)
ℎ2
120579= 119892
120579120579= 119890
(2(120575119892)minus2119892120579)
(1 + 1198922
)
ℎ2
119911= 119892
119911119911= 1 119892
120575120579= 119892
120575119911= 119892
120579119911= 0
(17)
are the components of themetric tensor and Lame coefficientsThe infinitesimal volume element is given by
119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)
1 + 1198922
119892119889120575 119889120579 119889119911 (18)
where the 119869 is the Jacobian of the spiral transformationLet us now define the spiral natural basis vectors 119890
120575 119890
120579 119890
119911
119890120575=120597
120597120575
= 119890(120575119892minus119892120579)
(1
119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890
119909
+ 119890(120575119892minus119892120579)
(1
119892sin (120575 + 120579) + cos (120575 + 120579)) 119890
119910
International Journal of Microwave Science and Technology 5
0
Conductor 1
Conductor 2
Region I
Region II
or
y
1205791
120579
1205792
1205751
1205752
x
1205751 minus2120587g2
1 + g2
∙
∙SI
SII
(a)
0
Region II
0
1205791
120579
1205792
120575
1205751205752 1205751
Φ
V0
∙
∙
polygonSchwarz-Christoffel
Region I
Con
duct
or 1
Con
duct
or 2
Con
duct
or 1
1205792 minus2120587
1 + g2
1205791 minus2120587g2
1 + g2
SI
SII
(b)
y
z
x
rarrdS120575z
rarrdS120579z
rarrdS120575120579
(c)
Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces
119890120579=120597
120597120579
= 119890(120575119892minus119892120579)
(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909
+ 119890(120575119892minus119892120579)
(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910
119890119911=120597
120597119911= 119890
119911
(19)
in terms of the cartesian basis vectors 119890119909 119890
119910 119890
119911
The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by
10038171003817100381710038171003817119889 119878
120575120579
10038171003817100381710038171003817= 119889119878
perp=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597120579
10038171003817100381710038171003817100381710038171003817119889120575 119889120579
= 119890(2120575119892minus2119892120579)
(1
119892+ 119892)119889120575 119889120579
10038171003817100381710038171003817119889 119878
120579119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120579times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890
(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579
10038171003817100381710038171003817119889 119878
120575119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890
(120575119892minus119892120579)
radic1 + 1198922
119892119889119911 119889120575
(20)
We then define the natural unitary spiral basis vectors
119890120575=119890120575
ℎ120575
119890120579=119890120579
ℎ120579
119890119911=119890119911
ℎ119911
(21)
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 3
extra-polar current can be led off which corresponds to theelectrotonic current of nerves
Since the 1950sndash60s myelinated nerves have been recog-nized to have a spiral structure and to behave like a high losscoaxial cable [26 27] with negligible inductance
The mathematical model presented in this paper can beused to refine the elm theory of the myelinated nerves bytaking into account their spiral geometry
In a coaxial guide the determination of the electromag-netic fields within any region of the guide is dependent upononersquos ability to explicitly solve the Maxwell field equations inan appropriate coordinate system [28]
Let us considerMaxwellrsquos equations
nabla times = minus119895120596120583
nabla times = 119895120596120598 119864
nabla sdot = 0
nabla sdot = 0
(1)
where the time variation of the fields is assumed to beexp(119895120596119905)
In view of the nature of the boundary surface it isconvenient to separate these field equations into componentsparallel and transverse to the waveguide 119911-axis
This is achieved by scalar and vector multiplication of (1)with 119890
119911 a unit vector in the 119911 direction thus obtaining
nablaperpsdot (119890
119911times
perp) = minus119895120596120583119867
119911
nablaperpsdot (119890
119911times
perp) = 119895120596120598119864
119911
(2)
nablaperp119864
119911minus120597119864
perp
120597119911= minus119895120596120583 (119890
119911times 119867
perp)
nablaperp119867
119911minus120597119867
perp
120597119911= 119895120596120598 (119890
119911times 119864
perp)
(3)
Since the transmission line description of the electromagneticfield within uniform guides is independent of the particularform of the coordinate system employed to describe the crosssection no reference to cross-sectional coordinates is madeon deriving the telegrapherrsquos equation [28 29]
Substituting (2) into (3) we obtain
120597119864perp
120597119911= minus119895119896120585 ( 120598 +
1
1198962nablaperpnablaperp) sdot (
perptimes 119890
119911)
120597119867perp
120597119911= minus119895119896120578 ( 120598 +
1
1198962nablaperpnablaperp) sdot (119890
119911times
perp)
(4)
Vector notation is employed with the following meanings forthe symbols
119864perp
= 119864perp(119909 119910 119911) = the rms electric field intensity
transverse to the 119911-axis119867
perp= 119867
perp(119909 119910 119911) = the rms magnetic field intensity
transverse to the 119911-axis
120578 = intrinsic impedance of the medium 1120578 = radic120583120598119896 = 120596radic120583120598 = 2120587120582 = propagation constant inmediumor the wavenumber (see [28] page 3)nablaperp
= gradient operator transverse to 119911-axis = nabla minus
119890119911(120597120597119911)120598 = unit dyadic defined such that 120598 sdot = sdot 120598 =
Equations (4) and (2) which are fully equivalent to theMaxwell equations make evident the separate dependenceof the field on the cross-sectional coordinates and on thelongitudinal coordinate 119911 The cross-sectional dependencemay be integrated out of (4) by means of a suitable set ofvector orthogonal functions provided they satisfy appropriateconditions on the boundary curve or curves 119904 of the crosssection
Such vector functions are known to be of two types theE-mode functions 1198901015840
119894defined by
1198901015840
119894= minusnabla
perpΦ
119894
ℎ1015840
119894= 119890
119911times 119890
1015840
119894
(5)
where
nabla2
perpΦ
119894+ 119896
10158402
119888119894Φ
119894= 0
Φ119894= 0 on 119904 if 1198961015840
119888119894= 0
120597Φ119894
120597119904= 0 on 119904 if 1198961015840
119888119894= 0
(6)
and the H-mode functions 11989010158401015840119894defined by
11989010158401015840
119894= 119890
119911times nabla
perpΨ
119894
ℎ10158401015840
119894= 119890
119911times 119890
10158401015840
119894
(7)
where
nabla2
perpΨ
119894+ 119896
101584010158402
119888119894Ψ
119894= 0
120597Ψ119894
120597119899= 0 on 119904
(8)
where 119894 denotes a double index and 119899 is the outward normalto 119904 in the cross-section plane
The constants 11989610158401015840
119888119894and 119896
1015840
119888119894are defined as the cutoff wave
numbers or eigenvalues associated with the guide crosssection
The functions 119890119894possess the vector orthogonality proper-
ties
∬1198901015840
119894sdot 119890
1015840
119895119889119878
perp= ∬119890
10158401015840
119894sdot 119890
10158401015840
119895119889119878
perp=
1 for 119894 = 119895
0 for 119894 = 119895
∬1198901015840
119894sdot 119890
10158401015840
119895119889119878
perp= 0
(9)
with the integration extended over the entire guide crosssection with surface 119878
perp
4 International Journal of Microwave Science and Technology
The total average power flow along the guide in the 119911direction is
119875119911=1
2Re(∬119864
perptimes 119867
lowast
perpsdot 119890
119911119889119878
perp) (10)
where all quantities are rms and the asterisk denotes thecomplex conjugate
In TEMmodes both 119864119911and119867
119911vanish and the fields are
fully transverse Their cutoff condition 1198962
119888= 0 or 120596 = 120573119888
(where 119888 is the speed of the light) is equivalent to the followingrelation [28]
perp=1
120578119890119911times
perp (11)
between the electric andmagnetic transverse fields where 120578 =radic120583120598 is the medium impedance so that 120578119888 = 120583 and 120578119888 = 1120598
The electric field perp
is determined from the rest ofMaxwellrsquos equations which read
nablaperptimes
perp= 0
nabla sdot perp= 0
(12)
These are recognized as the field equations of an equivalenttwo-dimensional electrostatic problem
Once the electrostatic solution perpis found the magnetic
field is constructed from (11)Because of the relationship between
perpand
perp the
Poynting vector 119878119911will be
119878119911=1
2Re (
perptimes
lowast
perp) sdot 119890
119911=1
120578
10038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
= 12057810038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
(13)
2 The Spiral Differential Geometry
For the MSCC structures it is difficult to construct solutionsfor Laplacersquos equation with polar or cartesian coordinates
The conformal mapping technique is a powerful methodfor solving two-dimensional potential problems andmappingthe boundaries into a simpler configuration for which solu-tions to Laplacersquos equation are easily found [17 18]
For the specific purposes of the MSCC the followingspiral coordinates based on a generalization of the Schwarz-Christoffelmapping (see appendix) are introduced
119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)
119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)
119911 = 119911
(14)
where 120579 120575 represent the spiral coordinates and 119892 gt 0
is a constant which characterizes the transformation (seeappendix)
As it can be seen in Figure 1 the equation 120575 = constrepresented by a vertical line in the 120575-120579 plane correspondsto a logarithmic spiral into the 119909-119910 plane and a constantcoordinate line of the spiral mapping
Observing (14) it appears clear that for 119892120579 minus 120575lowast
119892 rarr 0where 120575lowast is a constant the curve in the 119909-119910 plane locallyreduces (for |120579| ≪ 1 119892 ≪ 1) to an Archimedean spiral
The region between the two coaxial spirals maps into theregion inside the polygon bounded by the coordinate-lines120579 = 120579
1 120579 = 120579
2and 120575 = 120575
1 120575 = 120575
2 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) [18](see Figure 1(b))
It is also worth to observe that if 1205752minus120575
1= 2119902120587119892
2
(1+1198922
)119902 isin Z the two spirals 120575 = 120575
1 120575 = 120575
2are identical apart from
a shift of Δ120579119904= 2119902120587(1 + 119892
2
)We then require |120575
2minus120575
1| lt 2120587119892
2
(1+1198922
) in order to avoidcyclic spirals with 120575 gt 120575
2in the middle of the two with 120575 = 120575
1
and 120575 = 1205752
The differential one form of the spiral transformation ordual basis results in
119889119909 =120597119909
120597120575119889120575 +
120597119909
120597120579119889120579 +
120597119909
120597119911119889119911
119889119910 =120597119910
120597120575119889120575 +
120597119910
120597120579119889120579 +
120597119910
120597119911119889119911
119889119911 = 119889119911
(15)
The arc length 119889ℓ is given by
119889ℓ2
= 1198891199092
+ 1198891199102
+ 1198891199112
= 119892120575120575119889120575
2
+ 119892120579120579119889120579
2
+ 119892119911119911119889119911
2
(16)
where
ℎ2
120575= 119892
120575120575= 119890
(2(120575119892)minus2119892120579)
(1 +1
1198922)
ℎ2
120579= 119892
120579120579= 119890
(2(120575119892)minus2119892120579)
(1 + 1198922
)
ℎ2
119911= 119892
119911119911= 1 119892
120575120579= 119892
120575119911= 119892
120579119911= 0
(17)
are the components of themetric tensor and Lame coefficientsThe infinitesimal volume element is given by
119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)
1 + 1198922
119892119889120575 119889120579 119889119911 (18)
where the 119869 is the Jacobian of the spiral transformationLet us now define the spiral natural basis vectors 119890
120575 119890
120579 119890
119911
119890120575=120597
120597120575
= 119890(120575119892minus119892120579)
(1
119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890
119909
+ 119890(120575119892minus119892120579)
(1
119892sin (120575 + 120579) + cos (120575 + 120579)) 119890
119910
International Journal of Microwave Science and Technology 5
0
Conductor 1
Conductor 2
Region I
Region II
or
y
1205791
120579
1205792
1205751
1205752
x
1205751 minus2120587g2
1 + g2
∙
∙SI
SII
(a)
0
Region II
0
1205791
120579
1205792
120575
1205751205752 1205751
Φ
V0
∙
∙
polygonSchwarz-Christoffel
Region I
Con
duct
or 1
Con
duct
or 2
Con
duct
or 1
1205792 minus2120587
1 + g2
1205791 minus2120587g2
1 + g2
SI
SII
(b)
y
z
x
rarrdS120575z
rarrdS120579z
rarrdS120575120579
(c)
Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces
119890120579=120597
120597120579
= 119890(120575119892minus119892120579)
(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909
+ 119890(120575119892minus119892120579)
(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910
119890119911=120597
120597119911= 119890
119911
(19)
in terms of the cartesian basis vectors 119890119909 119890
119910 119890
119911
The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by
10038171003817100381710038171003817119889 119878
120575120579
10038171003817100381710038171003817= 119889119878
perp=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597120579
10038171003817100381710038171003817100381710038171003817119889120575 119889120579
= 119890(2120575119892minus2119892120579)
(1
119892+ 119892)119889120575 119889120579
10038171003817100381710038171003817119889 119878
120579119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120579times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890
(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579
10038171003817100381710038171003817119889 119878
120575119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890
(120575119892minus119892120579)
radic1 + 1198922
119892119889119911 119889120575
(20)
We then define the natural unitary spiral basis vectors
119890120575=119890120575
ℎ120575
119890120579=119890120579
ℎ120579
119890119911=119890119911
ℎ119911
(21)
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
4 International Journal of Microwave Science and Technology
The total average power flow along the guide in the 119911direction is
119875119911=1
2Re(∬119864
perptimes 119867
lowast
perpsdot 119890
119911119889119878
perp) (10)
where all quantities are rms and the asterisk denotes thecomplex conjugate
In TEMmodes both 119864119911and119867
119911vanish and the fields are
fully transverse Their cutoff condition 1198962
119888= 0 or 120596 = 120573119888
(where 119888 is the speed of the light) is equivalent to the followingrelation [28]
perp=1
120578119890119911times
perp (11)
between the electric andmagnetic transverse fields where 120578 =radic120583120598 is the medium impedance so that 120578119888 = 120583 and 120578119888 = 1120598
The electric field perp
is determined from the rest ofMaxwellrsquos equations which read
nablaperptimes
perp= 0
nabla sdot perp= 0
(12)
These are recognized as the field equations of an equivalenttwo-dimensional electrostatic problem
Once the electrostatic solution perpis found the magnetic
field is constructed from (11)Because of the relationship between
perpand
perp the
Poynting vector 119878119911will be
119878119911=1
2Re (
perptimes
lowast
perp) sdot 119890
119911=1
120578
10038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
= 12057810038161003816100381610038161003816
perp
10038161003816100381610038161003816
2
(13)
2 The Spiral Differential Geometry
For the MSCC structures it is difficult to construct solutionsfor Laplacersquos equation with polar or cartesian coordinates
The conformal mapping technique is a powerful methodfor solving two-dimensional potential problems andmappingthe boundaries into a simpler configuration for which solu-tions to Laplacersquos equation are easily found [17 18]
For the specific purposes of the MSCC the followingspiral coordinates based on a generalization of the Schwarz-Christoffelmapping (see appendix) are introduced
119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)
119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)
119911 = 119911
(14)
where 120579 120575 represent the spiral coordinates and 119892 gt 0
is a constant which characterizes the transformation (seeappendix)
As it can be seen in Figure 1 the equation 120575 = constrepresented by a vertical line in the 120575-120579 plane correspondsto a logarithmic spiral into the 119909-119910 plane and a constantcoordinate line of the spiral mapping
Observing (14) it appears clear that for 119892120579 minus 120575lowast
119892 rarr 0where 120575lowast is a constant the curve in the 119909-119910 plane locallyreduces (for |120579| ≪ 1 119892 ≪ 1) to an Archimedean spiral
The region between the two coaxial spirals maps into theregion inside the polygon bounded by the coordinate-lines120579 = 120579
1 120579 = 120579
2and 120575 = 120575
1 120575 = 120575
2 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) [18](see Figure 1(b))
It is also worth to observe that if 1205752minus120575
1= 2119902120587119892
2
(1+1198922
)119902 isin Z the two spirals 120575 = 120575
1 120575 = 120575
2are identical apart from
a shift of Δ120579119904= 2119902120587(1 + 119892
2
)We then require |120575
2minus120575
1| lt 2120587119892
2
(1+1198922
) in order to avoidcyclic spirals with 120575 gt 120575
2in the middle of the two with 120575 = 120575
1
and 120575 = 1205752
The differential one form of the spiral transformation ordual basis results in
119889119909 =120597119909
120597120575119889120575 +
120597119909
120597120579119889120579 +
120597119909
120597119911119889119911
119889119910 =120597119910
120597120575119889120575 +
120597119910
120597120579119889120579 +
120597119910
120597119911119889119911
119889119911 = 119889119911
(15)
The arc length 119889ℓ is given by
119889ℓ2
= 1198891199092
+ 1198891199102
+ 1198891199112
= 119892120575120575119889120575
2
+ 119892120579120579119889120579
2
+ 119892119911119911119889119911
2
(16)
where
ℎ2
120575= 119892
120575120575= 119890
(2(120575119892)minus2119892120579)
(1 +1
1198922)
ℎ2
120579= 119892
120579120579= 119890
(2(120575119892)minus2119892120579)
(1 + 1198922
)
ℎ2
119911= 119892
119911119911= 1 119892
120575120579= 119892
120575119911= 119892
120579119911= 0
(17)
are the components of themetric tensor and Lame coefficientsThe infinitesimal volume element is given by
119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)
1 + 1198922
119892119889120575 119889120579 119889119911 (18)
where the 119869 is the Jacobian of the spiral transformationLet us now define the spiral natural basis vectors 119890
120575 119890
120579 119890
119911
119890120575=120597
120597120575
= 119890(120575119892minus119892120579)
(1
119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890
119909
+ 119890(120575119892minus119892120579)
(1
119892sin (120575 + 120579) + cos (120575 + 120579)) 119890
119910
International Journal of Microwave Science and Technology 5
0
Conductor 1
Conductor 2
Region I
Region II
or
y
1205791
120579
1205792
1205751
1205752
x
1205751 minus2120587g2
1 + g2
∙
∙SI
SII
(a)
0
Region II
0
1205791
120579
1205792
120575
1205751205752 1205751
Φ
V0
∙
∙
polygonSchwarz-Christoffel
Region I
Con
duct
or 1
Con
duct
or 2
Con
duct
or 1
1205792 minus2120587
1 + g2
1205791 minus2120587g2
1 + g2
SI
SII
(b)
y
z
x
rarrdS120575z
rarrdS120579z
rarrdS120575120579
(c)
Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces
119890120579=120597
120597120579
= 119890(120575119892minus119892120579)
(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909
+ 119890(120575119892minus119892120579)
(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910
119890119911=120597
120597119911= 119890
119911
(19)
in terms of the cartesian basis vectors 119890119909 119890
119910 119890
119911
The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by
10038171003817100381710038171003817119889 119878
120575120579
10038171003817100381710038171003817= 119889119878
perp=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597120579
10038171003817100381710038171003817100381710038171003817119889120575 119889120579
= 119890(2120575119892minus2119892120579)
(1
119892+ 119892)119889120575 119889120579
10038171003817100381710038171003817119889 119878
120579119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120579times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890
(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579
10038171003817100381710038171003817119889 119878
120575119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890
(120575119892minus119892120579)
radic1 + 1198922
119892119889119911 119889120575
(20)
We then define the natural unitary spiral basis vectors
119890120575=119890120575
ℎ120575
119890120579=119890120579
ℎ120579
119890119911=119890119911
ℎ119911
(21)
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
International Journal of Microwave Science and Technology 5
0
Conductor 1
Conductor 2
Region I
Region II
or
y
1205791
120579
1205792
1205751
1205752
x
1205751 minus2120587g2
1 + g2
∙
∙SI
SII
(a)
0
Region II
0
1205791
120579
1205792
120575
1205751205752 1205751
Φ
V0
∙
∙
polygonSchwarz-Christoffel
Region I
Con
duct
or 1
Con
duct
or 2
Con
duct
or 1
1205792 minus2120587
1 + g2
1205791 minus2120587g2
1 + g2
SI
SII
(b)
y
z
x
rarrdS120575z
rarrdS120579z
rarrdS120575120579
(c)
Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces
119890120579=120597
120597120579
= 119890(120575119892minus119892120579)
(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909
+ 119890(120575119892minus119892120579)
(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910
119890119911=120597
120597119911= 119890
119911
(19)
in terms of the cartesian basis vectors 119890119909 119890
119910 119890
119911
The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by
10038171003817100381710038171003817119889 119878
120575120579
10038171003817100381710038171003817= 119889119878
perp=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597120579
10038171003817100381710038171003817100381710038171003817119889120575 119889120579
= 119890(2120575119892minus2119892120579)
(1
119892+ 119892)119889120575 119889120579
10038171003817100381710038171003817119889 119878
120579119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120579times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890
(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579
10038171003817100381710038171003817119889 119878
120575119911
10038171003817100381710038171003817=
10038171003817100381710038171003817100381710038171003817
120597
120597120575times120597
120597119911
10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890
(120575119892minus119892120579)
radic1 + 1198922
119892119889119911 119889120575
(20)
We then define the natural unitary spiral basis vectors
119890120575=119890120575
ℎ120575
119890120579=119890120579
ℎ120579
119890119911=119890119911
ℎ119911
(21)
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
6 International Journal of Microwave Science and Technology
The usual unitary relations of orthogonality hold that is
119890120575= 119890
120579times 119890
119911 119890
120579= 119890
119911times 119890
120575 119890
119911= 119890
120575times 119890
120579
119890120575sdot 119890
120579= 0 = 119890
120575sdot 119890
119911= 119890
120579sdot 119890
119911= 0
(22)
In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892
2
minus 119870119892The radius vector in spiral coordinates becomes
119903 =119890(120575119892minus119892120579)
radic1 + 1198922
(119890120575minus 119892119890
120579) + 119911119890
119911 (23)
Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892
perp= minus1119892
It is also possible to define the orthogonal spiral coordi-nate mapping as follows
119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)
119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)
119911 = 119911
(24)
3 The TEM Mode for the Spiral Waveguide
Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]
Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is
119890minus2(120575119892)+2119892120579
1 + 1198922[119892
21205972
Φ
1205971205752+1205972
Φ
1205971205792] = 0 (25)
where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC
This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives
Φ isin C(0)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
capC(2)
[[1205751minus
21205871198922
1 + 1198922 120575
2] times (minusinfininfin)]
Φ isin C(0)
[[1205752 120575
1] times (minusinfininfin)]
capC(2)
[[1205752 120575
1] times (minusinfininfin)]
(26)
The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at
the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598
0120598119903is
shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =
21205871198922
(1 + 1198922
)The potentialΦ(120575 120579) is subject to the following boundary
conditions in the region I (see Figure 1)
Φ(1205751 120579) = 119881
0
Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)
(27)
and in the region II
Φ(1205752 120579) = 0
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 119881
0forall120579 isin (minusinfininfin)
(28)
1198810must be the same in both cases of (27) and (28) because
120575 = 1205751and 120575 = 120575
1minus 2120587119892
2
(1 + 1198922
) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls
By the method of separation of variable let Φ(120575 120579) beexpressed in product form as
Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)
Substituting (29) into (25) and dividing by 119877119875 give
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752+
1
119875 (120579)
1205972
119875 (120579)
1205971205792= 0 (30)
The two terms in (30) must be equal to constants so that
1198922
119877 (120575)
1205972
119877 (120575)
1205971205752= minus119896
2
120575 (31)
1
119875 (120579)
1205972
119875 (120579)
1205971205792= minus119896
2
120579 (32)
1198962
120575+ 119896
2
120579= 0 (33)
The general solution to (32) is
119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896
120579120579) (34)
Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896
120579
must be zero By (33) this implies that 119896120575must also be zero
so that (31) for 119877(120575) reduces to
1205972
119877 (120575)
1205971205752= 0 (35)
and so
Φ (120575 120579) = 119862120575 + 119863 (36)
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 7
The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]
Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I
Φ(1205751 120579) = 0 = 119862I1205751
+ 119863I
Φ (1205752 120579) = 119881
0= 119862I1205752
+ 119863I(37)
At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II
Φ(1205752 120579) = 119881
0= 119862II1205752
+ 119863II
Φ(1205751minus
21205871198922
1 + 1198922 120579) = 0 = 119862II (1205751
minus2120587119892
2
1 + 1198922) + 119863II
(38)
After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)
Φ (120575 120579) =119881
0
1205752minus 120575
1
(120575 minus 1205751)
region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575
1]
Φ (120575 120579) =119881
0
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
(120575 minus 1205751+
21205871198922
1 + 1198922)
region II 120579 isin [minusinfininfin] 120575 isin [1205751minus
21205871198922
1 + 1198922 120575
2]
(39)
The and fields can now be found using (5) and (39)
region I
perp= 119864
120575119890120575= minusnabla
perpΦ = minus
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810
1205752minus 120575
1
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1
119890120579
119867120575= 0
region II
perp= 119864
120575119890120575= minusnabla
perpΦ
= minus119892119890
(minus120575119892+119892120579)
radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575
=119892119890
(minus120575119892+119892120579)
120578radic1 + 1198922
1198810
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
119890120579
119867120575= 0
(40)
While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors
The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function
As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is
120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)
where 119899 equiv 119890120575is the normal to the spiral surface of the
conductors whilst I and II are the electric fields seen fromthe regions I and II respectively
According to (41) the electric charge distribution followsthe exponential electric field
The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions
At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is
119869119878tot
= 119899 times I minus 119899 times II (42)
The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is
119882119890=1
2int119878perp
1205981015840
sdot lowast
119889119878perp (43)
while circuit theory gives 119882119890= 119862
1015840
|1198810|2
4 resulting in thefollowing expression for the capacitance per unit length
1198621015840
=1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878
perp sdot
lowast
119889119878perp [Fm] (44)
As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips
The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors
The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579
1minus 120579
2| ≫ Δ120575) with |120579
1| |120579
2| not too high as in
the case of the myelin bundles so that the fringing fields canbe ignored [2]
Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown
Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
8 International Journal of Microwave Science and Technology
++
+
+
+
+
+
+
++
+
+
+
+
+
Inner conductor
Outer conductor
minusminus minus
minusminusminus
(a)
Outer conductor
Inner conductor+
+
+
+
+
+
+
+
+
+
+
+
+
minus
minus
minus
minusminus
minus
minus
minus
minus
minusminus
minus
(b)
Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)
The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by
1198621015840
1=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878I
I sdot lowast
I 119889119878perp =119892120598
1015840
(1205792minus 120579
1)
1205751minus 120575
2
1198621015840
2=
1205981015840
10038161003816100381610038161198810
10038161003816100381610038162int119878II
II sdot lowast
II119889119878perp
=119892120598
1015840
(1205792minus 120579
1minus 2120587 (1 + 119892
2
))
(1205752minus 120575
1+ 21205871198922 (1 + 1198922))
1205792gt 120579
1
(45)
Thus
1198621015840
tot = 1198621015840
1+ 119862
1015840
2
= 120598119892119882(1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
(46)
This value represents the capacitance 1198621015840
tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579
1minus 120579
2for the first greater
capacitor and 1205792minus 120579
1minus 2120587(1 + 119892
2
) for the smaller one (seeFigure 1)
If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible
In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field
The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575
and 119889119878II119911120575
are (see Figure 4) 119889ΦIII = IIIperp
sdot 119889 119878III119911120575
while the
total fluxes over the two spiral areas 119878I 119878II according to (40)are
ΦI = ΦII = 1198821198810
120583
120578 (47)
The fluxes per unit length are given by
Φ1015840
III =ΦIII
119882= 119871
1015840
1198680 (48)
Consequently
1198711015840
= 1198850
120583
120578 (49)
where 1198850and 119868
0are the impedances and current of the line
respectivelyAs it can be noted from (48) there is only one current 119868
0
flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length
(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]
119882119898=120583
2int119878perp
sdot lowast
119889119878perp (50)
Circuit theory gives 119882119898
= 1198711198682
04 in terms of the unique
current of the line 1198680and results from the sum of two
contributions119882119898= 119882
1+119882
2
Thus
1198711015840
=120583119885
2
0
1198812
0
(int119878I
sdot lowast
119889119878perp+ int
119878II
sdot lowast
119889119878perp) (51)
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 9
Conductor 1
Conductor 2
+
+
++
+ +
+
++
++
++
∙
∙
+
+
+
+
minus
minusminus
minusminus
minusminus
minus
Region I
Region II
120579
1205752
rarrn equiv e120575
1205751
Φ(1205752 120579) = 0
Φ(1205751 120579) = 0
12057511205751
1205752
1205752
SI
SII
(a)
+ + + + + + + + +
_ _ _ _ _ _ ___V0 V0
Q1 Q2
(b)
Inner conductor 2
Inner conductor 1
Outer conductor 1
y z
x
∙
∙
∙
∙
W
Outer conductor 2
JS2 int
JS1 int
JS2 out
JS1 out
(c)
Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC
Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives
1198711015840
=120583
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205752minus 120575
1+ 21205871198922 (1 + 1198922)
)
minus1
1198850=120578
119892sdot (
1205792minus 120579
1
1205751minus 120575
2
+1205792minus 120579
1minus 2120587(1 + 119892
2
)
1205752minus 120575
1+ 21205871198922(1 + 1198922)
)
minus1
(52)
According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges
synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations
From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface
Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]
Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
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Submit your manuscripts athttpwwwhindawicom
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Shock and Vibration
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2
minusI
WI
rarrBIIperp
rarrBIperp
dSII119911120575
dSI119911120575
∙
∙
(a)
Conductor 1
Conductor 2
W
minusI
∙
I
rarrBperp
dSperp = dSz120593
(b)
WConductor 1
Conductor 2
dSz120575
rarrBperp
minusI
I
(c)
Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC
metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis
All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper
4 The Spiral Transmission Line
A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits
Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors
For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are
related as follows
1198711015840
= 120583119885
120578
1198621015840
= 120598120578
119885
(53)
where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors
The equations of the ideal spiral transmission line [4]depicted in Figure 5 are
120597119881
120597119911= minus119871
1015840120597119868
120597119905minus 119877
1015840
119868
120597119868
120597119911= minus119862
1015840120597119881
120597119905minus 119866
1015840
119881
(54)
where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866
1015840 is the conductance per unit length of theline measured in [Sm]
The two equations (54) for 1198771015840
= 0 and 1198661015840
= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 11
L998400dz R998400dz
C9984001dz G998400
1dz G9984002dzC998400
2dzV
I I +120597I
120597zdz
V +120597V
120597zdz
dz
(a)
L998400dz R998400dz
V
I
C998400dz G998400dz
I +120597I
120597zdz
V +120597V
120597zdz
dz
(b)
Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)
variables [2] whose solutions are waves propagating alongthe ideal line with speed V
1205972
119881
1205971199112=1
V1205972
119881
1205971199052
1205972
119868
1205971199112=1
V1205972
119868
1205971199052 V =
1
radic11987110158401198621015840
(55)
Using the Fourier transform of the signals 119881 119868
119881 (120596) =1
2120587int
infin
minusinfin
119881 (119905) 119890minus119894120596119905
119889119905
119868 (120596) =1
2120587int
infin
minusinfin
119868 (119905) 119890minus119894120596119905
119889119905
(56)
The solution to (55) may be written in terms of exponen-tials
119881 = 119881+119890minus120574119911
+ 119881minus119890120574119911
119868 =1
1198850
(119881+119890minus120574119911
minus 119881minus119890120574119911
)
1205742
= minus1205962
1198711015840
1198621015840
(57)
If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885
119871at 119911 = 0 the reflection Γ and transmission 120591
coefficients will be
Γ =119881
minus
119881+
=119885
119871minus 119885
0
119885119871+ 119885
0
120591 =119881
119871
119881+
=2119885
119871
119885119871+ 119885
0
(58)
If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor
5 Waves in a Lossy Spiral CoaxialTransmission Line
Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain
1205972
119881
1205971199112= 119877
1015840
(1198661015840
119881 + 1198621015840120597119881
120597119905) + 119871
1015840
(1198621015840120597119881
120597119905+ 119862
10158401205972
119881
1205971199052)
1205972
119868
1205971199112= 119877
1015840
(1198661015840
119868 + 1198621015840120597119868
120597119905) + 119871
1015840
(1198621015840120597119868
120597119905+ 119862
10158401205972
119868
1205971199052)
(59)
By using the Fourier transform of the signals 119881 119868 weobtain
120574 = [minus1205962
1198711015840
1198621015840
+ 1198771015840
1198661015840
+ 119894120596 (1198771015840
1198621015840
+ 1198711015840
1198661015840
)]12
1198850= (
1198771015840
+ 1198941205961198711015840
1198661015840 + 1198941205961198711015840)
12
(60)
For most transmission lines the losses are very small that is119877
1015840
≪ 1205961198711015840 and 119866
1015840
≪ 1205961198621015840 a binomial expansion of 120574 then
holds
120574 ≃ 119894120596radic11987110158401198621015840 +1
2
radic11987110158401198621015840 (119877
1015840
1198711015840+119866
1015840
1198621015840) = 120572 + 119894120573 (61)
Thus the phase constant 120573 remains unchanged with respectto the ideal line
The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is
119875119888=119877
119878
2int119878120579119911
119869119878sdot 119869
lowast
119878119889119878
120579119911 (62)
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
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DistributedSensor Networks
International Journal of
12 International Journal of Microwave Science and Technology
where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors
In (62) 119877119904= 1(120590120575
119878) is the surface resistance of the
conductors where the skin depth or characteristic depth ofpenetration is defined as 120575
119878= radic2(120596120583120590)
The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598
1015840
minus 11989412059810158401015840 a
permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598
10158401015840
1205981015840
The shunt conductance per unit length 1198661015840 reported
in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is
119875119889=120596120598
10158401015840
2int119878Iperp
sdot lowast
119889119878perp+120596120598
10158401015840
2int119878IIperp
sdot lowast
119889119878perp (63)
The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave
119881 = 119881+(119890
minus120574119911
+ Γ119890120574119911
)
119868 =119881
+
1198850
(119890minus120574119911
minus Γ119890120574119911
)
(64)
The time-average power flow along the line at the point 119911 is
119875avg =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1 minus |Γ|2
) (65)
When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant
The return loss (RL) is
RL = minus20 log |Γ| [dB] (66)
A measure of the mismatch of a line is the standing waveratio (SWR)
SWR =1 + |Γ|
1 minus |Γ| (67)
At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is
119885in = 1198850
119885119871+ 119894119885
0tan 120574119897
119885119871minus 119894119885
0tan 120574119897
(68)
The power delivered to the input of the terminated line at119911 = minus119897 is
119875in =1
2
10038161003816100381610038161003816119881
0+
10038161003816100381610038161003816
2
1198850
(1198902120572119897
minus |Γ|2
1198902120572119897
) (69)
The difference 119875avg minus 119875in corresponds to the power lost in theline [2]
From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line
Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC
6 Single Spiral Coaxial Cable andthe Myelinated Nerves
The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor
The problem can be solved by using two independentstripes of the same single spiral surface with |120579
119891minus120579
119894| le 2120587 and
|1205791| |120579
2|not too high separated by a shiftΔ120575 = 2119899120587119892
2
(1+1198922
)
to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))
The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one
Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]
The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions
Φ(1205751 120579) = 0 = 119862
1198981205751+ 119863
119898
Φ(1205751+2119899120587119892
2
1 + 1198922 120579) = 119881
0= 119862
119898(120575
1+2119899120587119892
2
1 + 1198922) + 119863
119898
forall120579 isin [120579119894 120579
119891]
10038161003816100381610038161003816120579119894minus 120579
119891
10038161003816100381610038161003816le 2120587
(70)
Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form
Φ (120575 120579) = 1198810
1 + 1198922
21198991205871198922(120575 minus 120575
1) (71)
The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is
perp= 119864
120575119890120575= minusnabla
perpΦ =
119890(minus120575119892+119892120579)
radic1 + 1198922
1198921198810(1 + 119892
2
)
21198991205871198922119890120575
119864120579= 0
perp= 119867
120579119890120579=1
120578119890119911times 119864
120575119890120575= minus
119890(minus120575119892+119892120579)
120578radic1 + 1198922
1198810(1 + 119892
2
)
2119899120587119892119890120579
119867120575= 0
forall120579 isin [1205791198941
1205791198912
] 120575 isin [1205751 120575
1+2119899120587119892
2
1 + 1198922]
(72)
The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is
119876 = int119878119898
120590119889119878120579119911= 119882120598
1198810(1 + 119892
2
)
119899119892 (73)
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 13
Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models
Fibrediameter[119863]
Axondiameter
[119889]
119892mye 120598myeNumber oflamellae 119899
119897
Core-conductorcapacitance119862mye [34]
Single-coaxcapacitance 119862mye
Colersquosinductance119871mye [36]
Single-coaxinductance 119871mye
≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye
2120587
log(119863119889) 1205980120598mye
1 + 1198922
mye
2119899119897119892mye
120583mye
2120587log(119863119889) 120583mye119899119897
119892mye
1 + 1198922
mye
≃46119899119865119898
≃4119899119865119898
≃30119899119867119890119899119903119910
119898≃20
119899119867119890119899119903119910
119898
Since the potential difference between the two conductors isΔ119881 = 119881
0 the capacitance per unit length of the MSSCC with
119899 turns between the two spiral conductors takes the followingsimplified form
1198621015840
= 1205981 + 119892
2
119899119892 (74)
The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))
The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves
In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed
The diameter of the myelinated nerve fibre [35] growsaccording to the formula
119863 = 119889 + 2 times 119899119897times 119896
119897 (75)
where 119899119897is the number of lamellae bilayers 119896
119897is their average
width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre
Now using the formula of the spiral mapping we have
119889 = 2119890120575119898119892119898minus1198921198981205791198941
119863 = 2119890120575119898119892119898minus1198921198981205791198912
(76)
where 12057911989411198912
are the initial and final angles of the myelinsheaths and 120575
119898determine the lipidmembrane spiral contour
For 119892119898≪ 1 as in the case of the myelin the thickness of
the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903
119899= 119890
120575119898119892minus4119899120587119892
By taking as value of the thickness 119896119897≃ 119903
1minus 119903
0= 119903
0(1 minus
119890minus4119892119898120587
) ≃ 0018 120583m [35] we have
119892mye ≃1
4120587ln( 119889
119889 minus 2119896119897
) (77)
According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862
0+119862
1119889
By taking 4120587119899119897= 120579
1198941
minus 1205791198912
(each lipid bilayer consistsof two spiral turns 120579
1198941
≫ 1205791198912
) and using (76) we have thefollowing relation between the number of myelin lamellae 119899
119897
and the diameter 119889 of the axon
119899119897(119889) = Int 1
4120587119892119898
log [119862
0+ 119862
1119889
119889] (78)
which is confirmed by the statistics [35]In the case of the SCC we have
1198711015840
= 120583119899119892
1 + 1198922
1198850= 120578119899
119892
1 + 1198922
(79)
where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one
The transmitted power in SCC depends inversely on theimpedance of the line119885
0which is proportional to the 119892 factor
of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the
nerves including the inductive effects of the small membranecurrents
In Table 1 a comparison between the Cole and the SCCinductances is proposed
The expressions 1198771015840 and 1198661015840 for the SCC related to the
power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines
The inductance1198711015840
≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form
119881 = 12058221205972
119881
1205971199112minus 120591
120597119881
120597119905
120582 =1
radic11987710158401198661015840
120591 =119862
1015840
1198661015840
119879 =120591ℓ
2
1205822= 119877
1015840
1198621015840
ℓ2
(80)
where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
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Active and Passive Electronic Components
Control Scienceand Engineering
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Microwave Science and Technology
Table2Transm
issionparametersfor
theM
DSC
CMSSCC
the
cylin
dricalcoaxand
thep
arallelplatelines
Dou
bles
piralcoax
Sing
lespira
lcoax
Cylin
dricalcoax
Parallelplate
1205751
1205752
a 21
a 22
a 11
a 12
1205791 1205792
a 21
a 22
a 11
a 12
a
b
d
D
1198711015840
120583 119892
1
(((1205792minus
1205791)(1205751minus
1205752))+
((1205792minus
1205791minus
(2120587(1+
1198922)))(1205752minus
1205751+
(21205871198922(1+
1198922)))))
120583
119899119892
1+
1198922
120583
2120587
ln119887 119886
120583
119889 119863
1198621015840
1205981015840119892119882
(
1205792minus
1205791
1205751minus
1205752
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205752minus
1205751+
(21205871198922(1+
1198922))
)1205981015840(1+
1198922)
119899119892
1205981015840
2120587
ln119887119886
1205981015840119863 119889
1198771015840
119877119878
16119892radic1+
1198922
1
(1205792minus
1205791minus
(120587(1+
1198922)))2
((1(1205751minus
1205752))+
1(1205752minus
1205751)+
(21205871198922(1+
1198922)))2
times
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [
1
11988622
(
1
(1205752minus
1205751)2
+
119890(minus(2119892120587(1+1198922)))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988621
(
1
(1205752minus
1205751)2
+
1
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
+
1
11988612
(
1
(1205752minus
1205751)2
+
119890minus(4119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
minus
1
11988611
(
1
(1205752minus
1205751)2
+
119890minus(2119892120587(1+1198922))
(1205751minus
1205752+
(21205871198922(1+
1198922)))2)
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
119886119901119902=
119890(120575119901119892)minus119892120579119902
119901119902=
12
119877119878
81205872radic1+
1198922
times
[ [ [ [ [
1
11988611
minus
1
11988612
+
119890minus(21198991205871198922(1+1198922))
11988621
minus
119890minus(21198991205871198922(1+1198922))
11988622
] ] ] ] ]
119886119901119902=
119890
((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)
119901119902=
12
119877119878
2120587
(
1 119886
+
1 119887
)
2119877119878
119863
1198661015840
12059612059810158401015840119892(
1205792minus
1205791
(1205752minus
1205751)
+
1205792minus
1205791minus
(2120587(1+
1198922))
1205751minus
1205752+
(21205871198922(1+
1198922))
)
12059612059810158401015840
119892
(1+
1198922)
212058712059612059810158401015840
ln119887119886
12059612059810158401015840119863
119889
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
Control Scienceand Engineering
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
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Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 15
7 The Spiral Poynting Vector
On a matched spiral coaxial line the rms voltage 1198810is related
to the total average power flow 119875119911= (12) int
119878perp
times lowast
sdot 119890119911119889119878
perp
by
119875119911
=
1
2int
1205752
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
+int
1205751+21205871198922(1+119892
2)
1205752
int
1205792
1205791+2120587(1+119892
2)
times lowast
sdot 119890119911119889119878
perp
=1
2radic120598
120583119892119881
2
0(1205792minus 120579
1
1205752minus 120575
1
+1205792minus 120579
1minus 2120587 (1 + 119892
2
)
1205751minus 120575
2+ 21205871198922 (1 + 1198922)
)
double coax
1
2int
1205751+2119899120587119892
2(1+119892
2)
1205751
int
1205792
1205791
times lowast
sdot 119890119911119889119878
perp
=1
120578
1 + 1198922
2119892119899119881
2
0 single coax
(81)
where the infinitesimal cross section is 119889119878perpequiv 119889 119878
120575120579 of (20)
As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked
In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor
Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system
In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573
119901which represents a finicky control
of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level
8 Conclusions
In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC
A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of
the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation
The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line
Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone
The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles
The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields
The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength
Appendix
Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping
We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function
119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)
119891 (119911) = 1198600int
119911
1199110
1
120577119889120577 119860
0= 1 minus 119894119892 119911
0= 0 (A2)
where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula
119882(119911) = 1198600int
119911
1199110
119899
prod
119896=1
(120577 minus 120577119896)minus120572119896120587
119889120577 + 1198610
1198600 119861
0isin C
(A3)
because for 1205721= 120587 120577
1= 0 and 120572
119896= 0 forall119896 gt 1 120577
119896= 0 forall119896 ge 1
the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840
(119911) exists andit is independent of direction
For 119892 = 0 or 1198600isin R the spiral conformal mapping of
(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case
In terms of cartesian (119909 119910) or polar (119903 120593) coordinates
119911 = 119909 + 119894119910 = 119903119890119894120593
(A4)
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 International Journal of Microwave Science and Technology
Conductor 1
Conductor 2 Single spiral
1205751 +2120587g2
1 + g
1205751120579i
120579f
120579f
+
++
+
+
+
+
+
+
minus
minusminus
minus
minus
minus
minus
120579i
(a)
Conducting outer fluid(extracellular fluid)
Insulating layer
Conducting center(the axon)
(axon cell walls +myelin sheaths)
∙
∙
∙
∙
∙
∙∙
D
d
(b)
D2 d2
kl
minus
(c)
Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths
Substituting (A2) into (A1) we obtain
119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)
The value of the constant 119870 represents the phase of thetransformation and is related to 119911
0= 119890
minus119870In order to study the spiral coaxial cable a further
normalization of the angles 119906 and V is introduced
119906 + 119894V =1 + 119892
2
119892120575 + 119894 (1 + 119892
2
) 120579 (A6)
120579 120575 are the two normalized variables Using (A1) (A4)(A6) and
119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)
we obtain the direct complex spiral coordinate transforma-tion that is
119911 = 119890120575119892minus119892120579+119894(120575+120579)
(A8)
where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with
the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates
becomes
(119889ℓ)2
= |119889119911|2
= (119889119909)2
+ (119889119910)2
= (119889119903)2
+ (119903119889120593)2
(A9)
where
|119889119911|2
=10038161003816100381610038161003816119891
1015840
(119911)10038161003816100381610038161003816
minus2
|119889119908|2
(A10)
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Microwave Science and Technology 17
or in conformal coordinates
(119889ℓ)2
= |119904|2
((119889119906)2
+ (119889V)2) |119904| equiv1
10038161003816100381610038161198911015840 (119911)
1003816100381610038161003816
(A11)
where the scale factor is the inverse of the modulus of thederivative of the function that is
1198911015840
(119911) =1 minus 119894119892
119911 (A12)
Substituting (A6) into (A11) we have
(119889ℓ)2
= |119878|2
((119889120575
119892)
2
+ (119889120579)2
) (A13)
where
|119878| = (1 + 1198922
) |119904| (A14)
Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper
The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950
[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011
[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014
[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993
[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003
[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006
[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009
[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013
[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999
[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010
[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009
[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000
[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998
[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996
[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005
[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941
[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001
[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960
[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897
[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007
[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009
[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008
[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855
[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011
[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901
[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952
[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964
[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 International Journal of Microwave Science and Technology
[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993
[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962
[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976
[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999
[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982
[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009
[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983
[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968
[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949
[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005
[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971
[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995
[41] Z Nehari Conformal Mapping Dover New York NY USA1975
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
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