Propagation in a dielectric-linedcircular waveguide

E. H. Fooks

Indexing terms: Circular waveguides, Dielectric-loaded waveguides, Error analysis

Abstract: A model for dielectric-lined circular waveguide is developed for use in a feasibility study of trunkwaveguide communication systems. The model is designed so that a rapid calculation of the propagation co-efficients may be made over a wide range of typical waveguide parameters using an interactive minicomputerprogram. The theory, which is more general than previous theories for dielectric-lined circular waveguides,is derived making use of complex cutoff and propagation coefficients, and may be applied to most circularlysymmetric problems having two concentric dielectric materials. An error analysis of the model confirms itsusefulness in waveguide-system studies.

1 Introduction

The phase coefficient of a wave propagating along a di-electric rod contained axially in a circular waveguide hasbeen evaluated in the lossless case by Clarricoats1 usingapproximations of an exact equation. With the knowledgeof the fields in the waveguide, attenuation characteristicsare deduced from the induced current flowing in the wave-guide walls and the dielectric material. In this form ofsolution it is assumed that the losses are sufficiently smallfor the fields to be accurately represented by the loss-freeconditions. Following the work of Unger,2 solutions for adielectric-lined circular waveguide are given by Carlin andD'Agostino3 with attenuation coefficients being obtained inone of two ways: by the induced-current method, or by awall-impedance method, in which an approximate wall im-pedance is derived to replace the air-dielectric boundary.Whereas these approximations are quite accurate in theirown sphere, this paper considers a solution which is freefrom these limitations and may be applied to any circularlysymmetric waveguide system containing two concentricdielectric materials.

Karbowiak4 introduced the concept of a complex cutoffor transverse coefficient in a perturbation theory for im-perfect waveguides to satisfy the complex-impedanceboundary conditions at the waveguide walls. This approachhas been extended to give the general solution for propaga-tion in a circular waveguide filled with two coaxial dielectricmedia. By using Bessel functions with complex arguments,the complex transverse coefficients g and h in each dielectricregion are found, allowing for both the waveguide wallimpedance and the dielectric losses and leading directly tothe waveguide attenuation and phase coefficients.

The representation of a boundary by an equivalent wallimpedance is the only approximation made in the presenttheory. It is justified for a high-conductivity metal in whichthe normal component of the phase coefficient is verymuch greater than the value along the axis of the waveguide.

2 The two-dielectric waveguide

The propagation coefficients for modes in a circular wave-guide are determined for the structure illustrated in Fig. 1,

Paper T200M, first received 29th November 1977 and in revisedform 31st May 1978Dr. Fooks is with the School of Electrical Engineering, University ofNew South Wales, Kensington, N.S.W. 2033, Australia

where the relative permittivities are generally complex andthe waveguide wall at a radius r is characterised by circum-ferential and longitudinal surface impedances looking out-wards at that radius. In this paper, the two surface-impedance components for a high-conductivity metallicwall waveguide are assumed to have the same value, equalto the impedance presented to a normally incident planewave. The effects of the roughness and finish of the metalsurfaces are described by Benson.s The increased attenu-ation mav be accounted for by using reduced values of thetwo components of wall conductivity, and the change inphase coefficient may be ignored when compared with thechanges caused by the dielectric lining of the waveguide.

2.1 Special cases

The general solution may be used in the following specialcases by a suitable selection of the radii and relative per-mittivities:

(a) €i = e2 or rx - r2 for a uniformly filled waveguide(b)ri ^r2 and e! = 1 0 to give the properties of a

dielectric-lined circular waveguide2'3

(c) /"i « r2 and e2 = 1 0 to give the effect of small uni-form airgaps at the waveguide walls in the precision meas-urement of dielectric materials6'7'8

(d) r2 > rx and e2 = 1 0 give the modes of a shielded-dielectric-rod waveguide,1'9 whereas when r2 becomesmuch greater than rx the modes of an unshielded-dielectric-rod waveguide are given with increasing accuracy.

\

metal walldielectric lining , e2

Fig. 1 Circular waveguide filled with two dielectric media boundedby an anisotropic-surface-impedance wall

MICROWA VES, OPTICS AND ACOUSTICS, JUL Y1978, Vol. 2, No. 4 117

0308-6976/78/200M-0117 %!• 50/0

In this paper, case (b) is of particular importance as it will bemodelled so that quick accurate results may be obtained aspart of a programme of the computer-aided design of trunkwaveguide systems.

2.2 The field equations

For a circular waveguide of the type illustrated in Fig. 1,there will generally be hybrid modes of propagation exceptwhen there is a circular symmetry of the fields. However,as the thin dielectric lining perturbs the fields of a uni-formly filled waveguide, all modes in this paper will be re-ferred to in terms of their parent type.

In the inner region, r < rx,

E2 - E0Jm(gr)cos(m<l))

Hz = H0Jm{gr) sin (m0)

(1)

(2)

The term exp (jtot — yz) is assumed to multiply all the fieldcomponents. In the outer region, rx < r < r2,

Ez = {AJm(hr)+BYm(hr)}cos(m4>)

Hz = {CJmihr)+DYm(hr)}sin(m<l>)

where

AJEo = -

Hn —CJm(hri)+DYm(hri)

(3)

(4)

(5)

(6)

As the propagation coefficient in each region must be thesame

i.e.

g = [(ei-e2)kl+h2)2il/2

and

dg = -dhg

dy = -dh7

The circumferential magnetic field HQ is given by

1 ( . dEz ydHz-/we

or r 30

(7)

(8)

(9)

(10)

This field component is matched across the boundary atr = rx to give an equation of the form

anA

where

=

ai4D = 0 (11)

g

my

my

1 1

1-1g2 h2

(13)

(14)

(15)

For the Bessel functions, the number of primes denotes thenumber of differentiations with respect to the argument;

otherwise, a prime represents differentiation with respectto the transverse coefficient h.

Thus

dh[J'm(hrl)]=r1Jm(hrl)

' = d , \n - — (flu)

dh

(16)

(17)

The derivatives with respect to h of these four coefficients

jcoe2e0

h2

(B = J, Y)

Jmigri)

(18)

yB'm(hri)+~Bm(hri)g2 h

2myBm{hri)

(19)The circumferential electric field in the waveguide is given

(g,hf { r 30 dr )

This field component is matched across the air-dielectricboundary to give a second equation, similar to eqn. 11,but with coefficients a2k.

At the outer boundary, r = r2) the two conditions thathave to be satisfied are

z - - £2 H

(21)

(22)r=r,

These boundary conditions will lead to further equationshaving coefficients a3k and a4k, respectively.

2.3 So/u tion for propaga tion coefficien t

For the hybrid mode in the circular waveguide, equationsof the form

anA aj4D = 0 = 1, . . . , 4

(23)

are obtained. Provided that variables A, B, C and D arenonzero, this set of equations may be reduced to oneequation:

F{h) = 0 (24)

which has to be solved for the complex transverse coeffic-ient h so that the attenuation and phase coefficients may befound. For circularly symmetric modes, only {A, B) or(C, D) are present, giving a simplified form for eqn. 23. Asolution is obtained using the Newton-Raphson method, in

118 MICROWAVES, OPTICS AND ACOUSTICS, JULY 1978, Vol. 2, No. 4

which, given a function of h such that F(h) = 0, an im-proved solution is given by

/Z;+i = hi —i+l

(25)

The functions F(h) and F'(h) are calculated in terms of thea and a' coefficients. It is necessary to have at least 11-decimal-place accuracy for the Bessel functions with com-plex arguments in order that the small attenuation coeffic-ients of the low-loss modes have significance when evaluatedas part of the propagation coefficient.

The convergence for TE-modes is rapid, giving the resultafter about 4 iterations. For TM-modes, the choice of theinitial value for h is more critical as the dielectric-liningthickness tends to zero. If this initial value is not sufficientlyaccurate, it is found that there will be convergence to theTE-mode that has the same circumferential dependence andadjacent eigenvalue. It is normally sufficient to choose theinitial value such that

= [(€2-€l)k20+(Xmn/r1)

2)2-11/2 (26)

where Xm« is the appropriate root of the Bessel functionfor a uniformly filled circular waveguide.

2.4 The model

In deriving a model for dielectric-lined circular waveguides,the relative importance of several parameters has to beconsidered.

(i) The modesThe only modes that are included in the model are the low-loss TE01 mode, together with those modes that have sig-nificant interaction with it at bends, corners, tilts and offsetsin the otherwise perfect waveguide.10 The complete set ofnecessary modes is TEOi, TE02, TE03, TE12, TE13, TE14

and TM n .(ii) The frequency rangeFor this particular study of trunk waveguide communicationsystems, the frequency range is 30—70 GHz.

(iii) The waveguide radiusThe radius of the metal wall of the waveguide is fixed at0-025 m. This is the value that is used for all the results inthis paper, unless otherwise stated.

A data block is generated for each waveguide mode con-taining information for all combinations of the followingvariables:

(a) Frequency: 30-70GHz in steps of 10GHz(b) Relative permittivity: 2—3 in steps of 0-5(c) Dielectric thickness: 01— 0 6 mm in steps of 01 mm.

For each combination of the variables, three coefficientsare stored. These coefficients relate to

(i)Sifi, the real part of the eigenvalue, evaluated atthe air-dielectric interface and for a lossless waveguide

00^2^1> the imaginary part of the eigenvalue, evalu-ated at the air-dielectric interface, allowing for a dielectricloss tangent = 0-001, but with perfectly conducting wave-guide walls

(iii) g3 A"!, the imaginary part of the eigenvalue, evalu-ated at the air-dielectric interface with a perfect dielectricmaterial, but allowing for finite conductivity waveguidewalls with a conductivity of 5-8 x 107 S/m.

Data is also stored for the waveguide with zero-thicknessdielectric lining, and for other mode parameters as speci-fied later.

For any selection of frequency, relative permittivityand dielectric-lining thickness that falls within the above

ranges, 3-, 5- or 7-point Lagrangian interpolation formulasare used as appropriate to get the best results from thestored data. To allow for any practical combination ofloss tangent and wall conductivity, a composite complextransverse coefficient is derived from

= g2

/ 5-8 x 107 \

{conductivity I

[tan 5 \ t 5-8 x 107

0 001 conductivity

(27)

(28)

4 Propagation results

4.1 A ttenua tion charac teristics

Typical attenuation curves that illustrate the relative con-tributions of the finite wall conductivity (copper) loss andthe dielectric loss are shown in Figs. 2 and 3 for the TEOiand TMn modes, respectively. In each case, a relativepermittivity of 2-5 has been chosen for the dielectric layerthat lines the waveguide wall.

4.2 Phase charac teristics

Whenever there is a coupling mechanism between twomodes, the difference in the phase coefficients of the twocoupled modes is important. The results presented herecover the extremes of the frequency range of interest of30—70 GHz. As a basis for comparison, the phase coefficientof the TEOi mode in a lossless waveguide without dielectriclining is used, i.e. 0 = 609-79 radians/m at 30GHz and)3= 1459-06 radians/m at 70 GHz. The phase coefficientsfor the other modes are given in Table 1 as a percentagedifference from the phase coefficient for the TEOi mode.

70GHz

30GHz70GHZ50GHz

50GHZ

30GHz

00 0 1 0-2 0-3 0 4 0 5 0 6

dielectric thickness, mm

Fig. 2 TE0i-mode attenuation in a dielectric-lined circular wave-guide

a Wall conductivity = 5-8 X 1O7 S/mb Loss tangent = 0 0 0 1

0 0 0 1 02 03 04 0-5dielectric thickness, mm

TMl, -mode attenuation in a dielectric-lined circular wave-Fig. 3guide

a Wall conductivity = 5-8 X 107 S/mb Loss tangent = 0 0 0 1

MICROWA VES, OPTICS AND ACOUSTICS, JULY 1978, Vol. 2, No. 4 119

Table 1: The percentage difference between the phase coefficient ofa mode compared with the TE01 mode in a lossless air-filled circular

waveguide.

Mode 30 GHz 70 GHz

TE.3TEI4

TM..

- 7-7-21 -4- 30-13-4-31-2

0

- 1 - 2- 3 - 3- 0 - 4 5- 2 - 1- 4 - 7

0

It is seen from Fig. 4 that, for the important TE-modes, the eigenvalue is nearly constant, and so the percent-age difference in the phase coefficients does not changesignificantly for a dielectric-lining thickness up to 06mmin a waveguide operating in the 30-70 GHz frequencyrange. However, the dielectric-lining thickness does havea marked effect on the splitting of the mode degeneracybetween the TEOi and TMn modes. The percentage differ-ence between the phase coefficients of the two degeneratemodes is illustrated in Fig. 5 as a function of the lining

0 0 05 10

dielectric thickness, mm

Fig. 4 Eigenvalues for modes in a dielectric-lined circular wave-guide

1 '5

1 1 0

§ 0-5

0 0

30 GHz

0 0 0-1 0 2 0 3 0 4 0 5dielectric thickness, mm

0-6

Fig. 5 Splitting of the phase coefficients of the TE0l and TMnmodes

TE01 -mode phase coefficient:

Frequency Coefficient

GHz305070

rad/m609-8

1036-714591

thickness of the dielectric material, with a relative per-mittivity of 2-5 and at 30, 50 and 70 GHz.

5 Error analysis

The solution of the field equations is checked by choosingthe parameters of a general 2-layer coaxial dielectric-filledcircular waveguide in such a way that comparisons may bemade with previously known solutions. The followingspecial cases are included in the validation procedure, withcases (a)—(c) giving the solution for attenuation and phasecoefficients for a uniformly filled circular waveguide:

(a) the outer dielectric thickness tends to zero(b) both the dielectric materials have the same relative

permittivity but a variable dielectric interface radius(c) the dielectric interface radius tends to zero(d) the hybrid modes of a dielectric-rod waveguide in

free space, which are obtained by keeping the dielectric-air interface radius constant and increasing the radius ofthe metal wall until there are no longer any significantfields at the wall. Table 2 shows the transition between the

Table 2: The T E n to HE,, mode transition as the outer metal wallis removed from around the dielectric rod

Diameter ofmetal wall

mm5-16 08 0

100200400800

dielectric rod diameterrelative permittivity

loss tangentfrequency

Eigenvalueat r,

1-841181-785381-677731-631341 -599691-599151-59915

= 5-1 mm= 2 26= 000005= 30 GHz

Attenuation

dB/km3180225-3167-3148-51290128-1128-1

TEn dominant mode in a circular waveguide and the HEU

mode in a dielectric rod in free space. The parameters arechosen for direct comparison with the results of Reference9, which give an attenuation of 127 dB/km for the dielectric-rod waveguide.

5.7 Errors associated with the model

The use of an interpolation procedure naturally leads toerrors. An analysis of the errors is carried out to assess thereliability of the results obtained from the model.

5.7.7 The real part of the eigenvalue: The eigenvalue for theTEOi mode is the argument of the Bessel function at thewaveguide wall for a uniformly filled waveguide or at theair-dielectric interface for a dielectric-lined waveguide. Asthe eigenvalue decreases for both increasing dielectric thick-ness and increasing frequency, the error calculations aremade for the worst case of 70 GHz and a relative permitt-ivity of 3 0.

An exact solution is compared with the interpolatedvalues across the 0—0*6 mm dielectric thickness range. Theerror plot is shown in Fig. 6, in which a comparison is madewith the approximate error bounds for Lagrangian inter-polation.11

5.1.2 The imaginary part of the eigenvalue(a) Associated with wall lossesIt is seen from Fig. 2 that the loss associated with the finite

120 MICROWAVES, OPTICS AND ACOUSTICS, JULY 1978, Vol. 2, No. 4

wall conductivity for the TEOi mode is fairly constant at agiven frequency. The attenuation is derived directly from

y2 = (29)

where g is now a complex transverse coefficient equal tothe eigenvalue divided by the air-dielectric boundary radius.As an approximation it follows that

••{*%-&?*

ImQr)

(30)

(31)

whereto is the transverse coefficient for lossless conditions.Therefore, for this and other TE-modes, the imaginary partof the transverse coefficient follows the similar slow trendsof the attenuation coefficient and is suitable for interpol-ation.

OO25rLagrangian interpolation

01 02 0-3 0-4dielectric thickness, mm

0-5 0 6

Fig. 6 Eigenvalue interpretation error for the TEQl -mode com-pared with the approximate bounds of Reference 11

Increasing the thickness of the dielectric lining breaksthe degeneracy between the TEOi and TMn modes. How-ever, the increasing thickness decreases the attenuationcoefficient due to the finite wall conductivity for theTMn mode, and the imaginary part of the transverse co-efficient shows the same strong and variable curvature,which is not particularly suitable for interpolation, as seenfrom the higher errors for this mode in Section 5.2.

(b) Associated with dielectric lossesA typical format for the imaginary part of the transversecoefficient due to the finite dielectric losses and infinite-conductivity walls is given in Table 3 for the TEOi mode at40 GHz and with a relative permittivity of 2-5.

Table 3: The variation of the imaginary part of the eigenvalue withdielectric thickness

Thickness Imaginary part of eigenvalueX 108

mm00-10-20-30-40-50-6

0902

73-67258-4648-3

136702606-5

Interpolation directly on the numbers in Table 3 leads togross errors in the region of particular interest below 0-2mm dielectric-lining thickness. However, if the imaginarypart of the eigenvalue is represented by

Im(x) = (Af

where A, B and k are coefficients, t is the dielectric-liningthickness in mm and / i s the frequency in GHz, then it isseen from Table 4 that, using the values of the coefficientsfor the TE-modes as the dielectric thickness tends to zero,a modified set of numbers may be obtained for inter-polation. A typical set of values for the TEOi mode withparameters er = 2-5 and frequency = 40 GHz is given inTable 5.

Table 4: The mode coefficients for eqn. 32

Mode A,X 108

B,X 108

TM

, 2-24412 4-10273 5-9583, 7-51163 1-79014 0-6970

269-578

000000

791 -56

3331111

Table 5: The imaginary part of the eigenvalue, modified to allow forimproved interpolation properties

Dielectric thickness Modified imaginarypart of the eigenvalueX 10s

mm00-10-20-30-40-50-6

8-97590169-2099-570

10-13010-93612 067

(32)

After interpolation with respect to the dielectric thick-ness, the true imaginary part of the eigenvalue is the modi-fied eigenvalue multiplied by the dielectric thickness raisedto the appropriate power k. Normalisation with respect tothe other variables of eqn. 32 is not necessary.

5.2 An estimation of interpolation error using a sweepof parameters

The test on the accuracy of attenuation calculations followssimilar lines to Section 5.1.1. In this case the dielectricloss tangent and the wall conductivity are fixed, whereasthere is a sweep of the three variable parameters, frequency,relative permittivity and dielectric-lining thickness. The twoend points of the sweep are fixed on nodal values (30 GHz,2, 01 mm) and (60GHz, 3, 06mm). A linear variation ofthese parameters over the range does not pass through anyother nodes. The range is subdivided into 50 intervals anda comparison is made between the attenuation from inter-polation and the attenuation from the exact calculation.

The results are evaluated for the seven main modes andmay be used as a guide to the accuracy of attenuation cal-culations. The results for the other five TE modes aresimilar to those presented for the TEOi mode in Fig. 7.However, for the TMn mode, which has a rapid variationof attenuation with changes in dielectric thickness, theattenuation error is much larger, approaching 2% fordielectric thicknesses up to 0-4 mm.

5.3 The attenuation dependence

(a) On the dielectric loss tangentThe interpolation procedures for the imaginary part of theeigenvalue due to the dielectric losses are based on a loss

MICROWA VES, OPTICS AND ACOUSTICS, JULY 1978, Vol. 2, No. 4 121

tangent of 0-001. If a loss tangent other than this value isrequired, then the derived imaginary part of the transversecoefficient is multiplied by the ratio tan5/0-001. To testthe validity of this approach, a comparison is made in Table6 between the approximate formulation and the completesolution over a range of loss tangents. The percentage errorin the attenuation is negligible and may be ignored.

point represent a nodal point for the accurate stored modedata.

The attenuation errors due to the direct addition of theimaginary parts of the transverse coefficient are presentedas marked points on Fig. 7. The curve through these pointsmay be taken to represent the zero percentage error linefor Lagrangian interpolation for the TE01 mode.

0 2 r

-0-240 frequency,

GHz

20 25relative permittivity

3 0

01 0 2 03 04 05dielectric thickness, mm

0-6

Fig. 7 Attenuation error for the TE0l mode through a sweep ofparameters

Table 6: The attenuation error due to scaling for the dielectric losstangent for the TE01 mode at 50 GHz, er = 2-5, t = 0-3 mm.

Loss tangent Attenuation error

000010 0010010-1

0 0reference000010011

(b) On the wall conductivityThe interpolation procedures for the imaginary part of theeigenvalue due to the finite wall conductivity are based ona typical conductivity for copper of 5-8 x 107 S/m. If aconductivity other than this value is required, then theimaginary part of the transverse coefficient is scaled as de-pending on the inverse of the square root of the conduct-ivity. Table 7 verifies the validity of this approach over twoorders of magnitude for the wall conductivity.

Table 7: The attenuation error due to seating for wall conductivitywith tan 6 = 0, other parameters as for Table 6

Wall conductivity Attenuation error

S/m5-8 X 107

5-8 X 106

5-8 X 10s

reference00290-121

(c) On the interaction between the two loss mechanismThe imaginary parts of the transverse coefficient associatedwith the dielectric losses and the finite wall conductivityare added together, assuming that there is negligible inter-action between them. The fact that there is an interactionbetween the two loss mechanisms may be seen from thepoint at the high-frequency end of the curve for the atten-uation error for the TE01 mode with a sweep of para-meters (Fig. 7). The attenuation error should be zero at the60 GHz point on this curve if there is no dependencebetween the two types of loss, as the parameters at this

6 Conclusions

A modelling procedure for dielectric-lined circular wave-guides has been developed to give the propagation coeff-icients for modes that are of interest in trunk waveguidecommunication systems. An analysis of the errors associ-ated with the simple model has been presented. As thepropagation coefficients are derived by interpolation onstored data, a significant reduction in computation timehas been obtained compared with the computation time forthe rigorous calculation of the coefficients. The data forthe model is derived from a general solution for the com-plex propagation coefficients of hybrid modes in a cir-cular waveguide filled with two concentric dielectricmaterials. This method may be used for propagation alonga dielectric rod in free space as well as step-index opticalfibres by moving the metallic wall outwards until it nolonger has any effect on the particular mode solution.

7 Acknowledgments

The author wishes to thank Telecom Australia for theirsupport of this work as part of a study of transmissioncharacteristics in a trunk waveguide system, and to ack-nowledge the many useful discussions with Prof. A. E.Karbowiak and K. Poronnik during the course of the work.

8 References

1 CLARRICOATS, P.J.B.: 'Propagation along bounded and un-bounded dielectric rods. Part 2: Propagation along a dielectricrod contained in a circular waveguide', Proc. IEE, 1960, 107,pp. 170-176

2 UNGER, H.G.: 'Circular electric wave transmission in adielectric-coated waveguide', Bell Syst. Tech. J., 1957, 36,pp. 1253-1278

3 CARLIN, J.W., and D'AGOSTINO, P.: 'Normal modes in over-moded dielectric-lined circular waveguide', ibid., 1973, 52, pp.453-486

4 KARBOWIAK, A.E.: 'Theory of imperfect waveguides: theeffect of wall impedance', Proc. IEE, 1955, 102B,pp. 698-708

5 BENSON, F.A. (Ed.): 'Millimetre and submillimetre waves'(Iliffe, 1969), chap 14

6 COHN, S.B., and KELLY, K.C.: 'Microwave measurements ofhigh-dielectric-constant materials', IEEE Trans., 1966, MTT-14,pp. 406-410

7 HANFLING, J., and BOTTE, L.: 'Measurement of dielectricmaterials using a cut off circular-waveguide cavity', ibid.,1972, MTT-20, pp. 233-235

8 FOOKS, E.H.: 'Dielectric loss tangent measurements in an evane-scent mode coupled cavity'. Presented at the European Micro-wave conference, Brussels, Sept. 1973

9 RAVENSCROFT, I.A., and JACKSON, L.A.: 'Proposals for adielectric rod transmission system', Presented at the Europeanmicrowave conference, Brussels, Sept. 1973

10 KARBOWIAK, A.E.: 'Trunk waveguide communication' (Chap-man & Hall, 1965)

11 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of mathe-matical functions' (Dover, 1965)

122 MICROWAVES, OPTICS AND ACOUSTICS, JULY 1978, Vol. 2, No. 4