Circuit analysis of a parallel plate waveguide1182100/FULLTEXT01.pdf · goal and that was...
Transcript of Circuit analysis of a parallel plate waveguide1182100/FULLTEXT01.pdf · goal and that was...
INOM EXAMENSARBETE ELEKTROTEKNIK,AVANCERAD NIVÅ, 30 HP
, STOCKHOLM SVERIGE 2017
Circuit analysis of a parallel plate waveguide
NOJ KAZEMI
KTHSKOLAN FÖR ELEKTROTEKNIK OCH DATAVETENSKAP
Circuit analysis of a parallel plate waveguide
NOJ KAZEMI
Master of Science Thesis in Electrotechnical Theory and Design at the School ofElectrical Engineering Royal Institute of Technology Stockholm, Sweden,
September 2017
Abstract
The aim of this work was to model a four-port waveguide as a simple cir-cuit, by first starting with a two-port then a three-port waveguide. Duethat the work was based on Nathan Marcuvitz book Waveguide handbook,an analytical solution for the circuit parameters was desired. In order toobtain an analytical solution three methods were studied; the Variationalmethod, the Integral equation method and the Static method. Out of thesethree methods the latter was chosen, because its strength of simplifying theboundary conditions. The goal to model a four-port and a three-portedwaveguide was too complicated. This led to that the goal was changed totrying to get a higher accuracy on the existing circuit model for a two-portwaveguide, by solving an extension to the circuit parameter. This was donebecause Marcuvitz only treated the first two modes correctly and it was notclear if the circuit model was stable for the higher orders of Taylor series. Inthe end a circuit model for a waveguide with an iris that treats the first 16modes correctly was solved. By looking at the dispersive properties of thecircuit a comparison with simulation software CST Microwave Studio couldbe done, which resulted in that the circuit model gave good results up to2b/λ < 1. It was also showed that the accuracy was about the same as thecircuit model found in Waveguide handbook, but it can be mentioned thatthe accuracy is minimally better for the circuit model that was developed inthis work. Something that was discovered in this work is that the restrictionmentioned in Waveguide handbook for the case when the window is centeredis unreliable, it should be 2b/λ < 1. It also appeared that the circuit modelremained stable for higher orders of the Taylor series, in this case up to the16:th order.
Keywords: Waveguide, Discontinuities, Dispersion, ABCD matrix, Circuitmodel, E-mode functions, Variational Method, Integral Equation Method,Static Method, two-port, three-port, four-port
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Sammanfattning
Malet med detta arbete var att modellera en fyr-portars vagledare somen simpel elektrisk krets, genom att forst borja med en tva-portars sedantre-portars -vagledare. Detta arbete var baserat pa Nathan Marcuvitz bokWaveguide handbook, darav soktes det en analytisk losning for kretspara-metrarna. For att kunna fa en analytisk losning, studerades tre metoder;Variationsmetoden, Integralsekvationsmetoden samt den Statiskametoden.Av dessa tre metoder valdes den sistnamnda, pa grund av dess styrka medatt forenkla randvillkoren. Malet att modellera en fyr-portars samt en tre-portars vagledare var alldeles for komplicerat. Detta ledde till att maletandrades till att forsoka fa en hogre precision pa den befintliga kretsmodel-len for en tva portars vagledare, genom att losa ut flera termer till krets-parametern. Detta gjordes da Marcuvitz endast hanterade de tva forstamoderna korrekt, samt att det inte framgick ifall kretsmodellen ar stabil forhogre ordningar av Taylor serier. I slutandan lostes en kretsmodel for envagledare med en iris som hanterar de forsta 16 moderna korrekt. Genomatt kolla pa de dispersiva egenskaperna for kretsen, kunde en jamforelse medsimuleringsprogrammet CST Microwave Studio ske, dar slutsatsen blev attkretsmodellen gav goda resultat upp till 2b/λ < 1. Det visade aven sig attprecisionen var ungefar densamma som den kretsmodell som aterfinns i Wa-veguide handbook, men det kan namnas att precisionen ar minimalt battrefor den kretsmodell som togs fram i detta arbete. En sak som upptacktes un-der detta arbete var att restriktionen som namns i Waveguide handbook forfallet nar gapet for irisen ar centrerad stammer inte, den bor vara 2b/λ < 1.Dessutom visade det sig att kretsmodellen fortfarande var stabil for hogreordningar av Taylorserier, i detta fall upp till den 16:e ordningen.
Nyckelord: Vagledare, diskontinuiteter, Dispersion, ABCD Matris, Krets-modell, E-mods funktioner, Variationsmetoden, Integralekvations metoden,Statiska metoden, Tva-portar, Tre-portar, Fyr-portar
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Acknowledgements
I would like to thank my supervisors Oscar Quevedo-Teruel (Assistant Pro-fessor at Royal Institute of Technology) and Guido Valerio (Associate Pro-fessor at Universite Pierre et Marie Curie), for all their help, the greatpatience with me and an understanding for my sickness during this work. Iwould also like to thank my examiner Lars Jonsson (Professor at the RoyalInstitute of Technology). Then I would also give my thanks to my fam-ily and friends, especially my mother, who has been so supportive duringthis work and through my sickness. Then there is Erik Hallqvist, SebastianHakansson and Johan Linder, without you guys, studying at KTH wouldn’tbeen as fun and motivating.
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Contents
Abstract i
Sammanfattning ii
Acknowledgements iii
Contents iv
1 Introduction 1
2 Field theory to transmission line theory 22.1 Uniform waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Parallel plate waveguide . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Microwave Network 83.1 Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.1 Reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Two port equivalent circuit . . . . . . . . . . . . . . . . . . 12
3.2 ABCD Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Two port Capacitive Diaphragm 234.1 Circuit Model of parallel plate guide . . . . . . . . . . . . . . . . . 234.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Variational method . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Integral equation method . . . . . . . . . . . . . . . . . . . . . . . . 284.5 Static method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Results 39
6 Conclusion and Thoughts 476.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
References 52
A. More results 54
B. Transfer and Impedance parameters 69
C. Matlab function
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1 Introduction
It is possible to go from field theory for waveguides to simple circuit theory, byfinding a relation between electromagnetic fields in waveguides and currents, volt-ages in circuits. The advantage with obtaining a circuit model for a waveguideis that the simulation will be instant rather than long simulations in for exampleCST microwave studio, which means that it is rather efficient to use, because oftime and cost saved and also less power consumption. This will also be almost asaccurate as CST to a point where there will be some restrictions. Many types ofwaveguide discontinuities has been modeled as a circuit in Waveguide handbook byNathan Marcuvitz [1], but most of them doesn’t have a derivation. The goal withthis thesis was firstly to derive the circuit model for a two-port waveguide, to getan understanding on how the different methods described by Marcuvitz works.
The second goal was to derive the circuit model for a three-port waveguide(T-junction), which exist in the Waveguide Handbook [1], but is given withoutany derivations. The reason for this goal was that it was a step before the actualgoal and that was determining a circuit model for four-port waveguide. This canalso be seen as a eight port waveguide in joint. This type of waveguide was nevermodeled by Marcuvitz [1].
The main goal was to get an analytical solution of the circuit parameters for afour-port waveguide based on Waveguide Handbook by Nathan Marcuvitz [1] andobtain the dispersion properties for it. The dispersion properties will then tellwhich frequencies will propagate through the waveguide [2] and it can also be usedas a verification that the circuit model is correct when compared with CST. Thecircuit model can then be used to design 2D Ultrawideband lens, which can beused for example in communications [3].
The hard part of this should be to determine the three-port waveguide thenthe four-port should probably follow automatically. The reason for this is that onemore region will be added and the region in the junction will be open in one moredirection. This will leads to a change of the boundary condition in the joint ofthe waveguides and a addition of a boundary condition for the ”extra” waveguide,which is the same as the others.
Unfortunately the goals for the three-port and the four-port waveguide wasnever achieved, because of the complexity that occurs in these cases, which isdiscussed more in section 6.2. This lead to a change of a end goal, which was to doextension for the existing circuit parameter solution of a parallel plate guide witha iris, to see if it’s able to obtain higher accuracy than the existing circuit modelby Marcuvitz, which has only treated the first two modes correctly. The goal wasthen to get higher accuracy, by treating more modes correctly, in this case the first16 modes.
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2 Field theory to transmission line theory
2.1 Uniform waveguide
Sy
x
(a) Cross Sectional view
y
z
(b) Longitudinal view
Figure 1: Uniform waveguide
Uniform waveguides are defined such as that the cross section transverse to thepropagation are identical everywhere in the waveguide. By knowing this it issimple to determine the relation between the transverse fields with mode currentsand mode voltages. This will also lead to the transmission line equations with thehelp of some boundary conditions. Starting from Maxwell’s equations or actuallyMaxwell-Faraday law and Amperes Law in the frequency domain rather then time-domain,
∇× E = −jωµH = −jkηH (1)
∇×H = jωεE = jkη−1E (2)
The fields time variation is assumed to be ejωt, but is suppressed through out thederivations. Considering that the interesting part is the transverse fields, the fieldsshould be split into two parts, transverse and longitudinal parts,
E(x, y, z) = xEx(x, y, z) + yEy(x, y, z)︸ ︷︷ ︸Transverse
+ zEz(x, y, z)︸ ︷︷ ︸Longitudinal
= Et(x, y, z) + zEz(x, y, z)
(3)
The same should be done for the H-field and the nabla operator [4],
∇ = ∇t + zd
dz(4)
By using (3), (4) in Maxwell-Faraday’s law (1),
(∇t + z
d
dz
)× (Et + zEz) = −jkη (Ht +Hzz) (5)
2
From here it becomes obvious that,
∇tEz × z + zd
dz× Et = −jkηHt (6)
∇t × Et = −jkηzHz (7)
Doing the same procedure with the use of Ampere’s law in (2) leads to,
∇tHz × z + zd
dz×Ht = jkη−1Et (8)
∇t ×Ht = jkη−1zEz (9)
then by using cross product with z on each side in (6),(8) and dot product with zon each side in (7), (9) yields
dEt
dz= ∇tEz + jkηz×Ht,
dHt
dz= ∇tHz − jkηz× Et
jkηHz = ∇t · (z× Et), jkη−1Ez = ∇ · (Ht × z)(10)
By inserting the expression for Hz and Ez in respective parts for the derivative ofthe transverse fields, gives the transverse field to be,
dEt
dz= −jkη
(I +
1
k2∇t∇t
)· (Ht × z) (11)
dHt
dz= −jkη−1
(I +
1
k2∇t∇t
)· (z× Et) (12)
Where the I is the unit dyadic. The longitudinal fields are defined as,
jkη−1Ez = ∇t · (Ht × z) (13)
jkηHz = ∇t · (z× Et) (14)
The nabla operator which is the cross sectional dependence can be integrated outof (11) and (12) with the help of some vector orthogonal functions. These vectororthogonal functions should be proportional to itself when affected by the nablaoperator, as in (15) and they should also satisfy some boundary conditions.
∇t∇t · f ∝ f (15)
There are two such functions that satisfies (15), one of them are the E-modefunctions, which are defined as,
e′t = −∇tΦmn
h′t = z× e′t(16)
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Where the scalar function Φmn should satisfy these boundary conditions,
(∇2t + k′2c,mn)Φmn = 0
Φmn = 0 on S if k′c,mn 6= 0
∂Φmn
∂S= 0 on S if k′c,mn = 0
(17)
Where k′c,mn is the cutoff wavenumber for mn-mode. Then there also is the H-modefunctions, which are defined as,
e′′t = z×∇tΨmn
h′′t = z× e′′t(18)
with the boundary conditions
(∇2t + k′′2c,mn)Ψmn = 0
∂Ψmn
∂n′= 0 on S
(19)
There n′ is the outward normal to the surface S. As stated before the vectorfunctions are orthogonal and thus the E-mode and H-mode functions have theorthogonal properties,
∫ ∫e′ti · e′tjdS = δij =
∫ ∫e′′ti · e′′tjdS
∫ ∫e′ti · e′′tjdS = 0
(20)
With the help of these vector orthogonal functions, E-mode functions and H-modefunctions, the mode amplitudes for the voltages and currents can be expressed as,
V ′mn =
∫ ∫Et · e′t,mndS V ′′mn =
∫ ∫Et · e′′∗t,mndS
I ′mn =
∫ ∫Ht · h′t,mndS I ′′mn =
∫ ∫Ht · h′′∗t,mndS
(21)
By using these mode amplitudes for the voltages and currents, the orthogonalfunctions the transverse electric and magnetic fields can be expanded to,
Et =∑
V ′mne′t,mn +
∑V ′′mne
′′t,mn (22)
Ht =∑
I ′mnh′t,mn +
∑I ′′mnh
′t,mn (23)
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Now the longitudinal fields can be rewritten with the help of (11), (12), (16),(17),(18), (19), (22) and (23)
jkη−1Ez =∑
Imn(z)k′2c,mnΦmn (24)
jkηHz =∑
V ′′mn(z)k′′2c,mnΨmn (25)
To confirm all this, one can look at the transported power in the z-direction, byusing the surface integral over Poynting vector with (22) and (23) in (26),
Pz = Re
{∫ ∫(Et ×H∗t ) · zdS
}= Re
{∑V ′mnI
′∗mn +
∑V ′′mnI
′′∗mn
}(26)
Where the power is given by the product of all the mode amplitudes of the voltagesand currents, which it should. To get the standard transmission line equations theexpanded transverse fields in (22), (23) are used in (11), (12) with the definitionsfor the E-mode and H-mode functions and the corresponding boundary conditionsfor them.
dV ′mndz
= −j ηk
(k2 − k′2c,mn
)I ′mn = −jk′z,mnZ ′mnI ′mn
Z ′mn = ηk′z,mnk
, k′z,mn =√k2 − k′2c,mn
E-mode (27)
dV ′′mndz
= −jηkI ′′mn = −jk′′z,mnZ ′′mnI ′′mn
Z ′′mn = ηk
k′′z,mn, k′′z,mn =
√k2 − k′′2c,mn
H-mode (28)
dI ′mndz
= −jkη−1V ′mn = −jk′z,mnY ′mnV ′mn
Y ′mn = η−1k
k′z,mn, k′z,mn =
√k2 − k′2z,mn
E-mode (29)
dI ′′mndz
= −j η−1
k
(k2 − k′′2c,mn
)V ′′mn = −jk′′z,mnY ′′mnV ′′mn
Y ′′mn = η−1k′′z,mnk
, k′′z,mn =√k2 − k2c,mn
H-mode (30)
These are the standard transmission line equations for the E-mode and H-modefunctions, where kz,mn is the wavenumber in the z-direction and Zmn is the char-acteristic impedance for a certain mode mn. With these formulation it is possibleto go from field theory to transmission line theory, which could be used to modela certain waveguide to a simple circuit [1] [5].
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2.2 Parallel plate waveguide
x
zy bE
Figure 2: Parallel plate waveguide
The waveguide used in this work is a parallel plate guide as in Figure 2, where theheight of the waveguide is b. The results of this will be same as the rectangularwaveguide for the TE0n-modes. To get an expression of the field in this waveguidedepending on mode amplitudes currents and voltages, a scalar function needs tobe found, in this case the E-mode functions will be used. The problem with havinga parallel plate guide is the width of it is assumed to be infinite, which leads totwo types of characteristics of the modes. For simplicity it is assumed that thereare no variation in the x-direction but only in the y-direction, this yields that thescalar function to be used to derive the E-mode functions is [1],
Φ0n =
√ρnb
b
πnsin(nπy
b
)(31)
where,
ρn =
{ρn = 1 for n = 0
ρn = 2 for n 6= 0(32)
Where n = 0, 1, 2, 3, . . . , N . By using the scalar function (31) in (16) the E-modefunctions can be derived, then by using it in (22), (23) and (24). The E0n-mode
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field components can be found to be,
Ex = 0
Ey = −Vi(z)
√ρnb
cosnπy
b= Vi(z)e′t,0n(y)
Ez = −jηIi(z)nλ
2b
√ρnb
sinnπy
b
Hx = Ii(z)
√ρnb
cosnπy
b= Ii(z)h′t,0n(y)
Hy = 0
Hz = 0
(33)
To get the standard transmission line equation to find out the the characteristicimpedance and the propagation constant, then the field components is used in(11) or in (12), depending on whether the results should be the derivative of thevoltage or the current. The characteristic impedance and the propagation will bethe same either way. This then gives the standard transmission line equation tobe,
dVn(z)
dzy = −jkz
η
kIn(z)
√k2 −
(nπb
)2
︸ ︷︷ ︸=kz
y (34)
Where the m = 0 index has been omitted. It can then be rewritten to,
dVn(z)
dz= −jkzZnIn(z) (35)
and where the characteristic impedance,
Zn =1
Yn= η
kzk
=kzωε
=1
ωε
√k2 −
(nπb
)2(36)
Comparing the characteristic impedance with (27) it can be determined that thecut-off wavenumber for the waveguide is,
kc = nπ/b (37)
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3 Microwave Network
3.1 Equivalent Circuit
It has been derived before how the equivalent currents and voltages relate to thetransverse electric and magnetic fields. With these currents and voltages, theimpedance or admittance matrices can be used to describe how the ports for thewaveguide relate to each other for a microwave network. This type of represen-tation can be used to describe an equivalent circuit for a arbitrary microwavenetwork. In Figure 3 a N-port microwave network is illustrated, which is used togive an understanding how the currents and voltages relate to each other by theimpedance or the admittance.
V+1, I+1
V−1,−I−1
V +2 , I +2
V −2 ,−I −2
V+3, I+3
V−3,−I−3
V +N , I +NV −N ,−
I −N
S
T1
T2 T3
TN
Figure 3: N-port microwave network
In each terminal TN there is voltage and current going in and out of the terminal,there is then an incident and a reflected voltage and current. The total currentand voltage for the Nth terminal is defined as in (38) [6].
IN = I+N − I−NVN = V +
N + V −N(38)
The relation between these currents and voltages can be described by the impedanceor admittance matrix as in (39).
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V1V2...VN
=
Z11 Z12 . . . Z1N
Z21 Z22 . . . Z2N...
......
...ZN1 ZN2 . . . ZNN
I1I2...IN
I1I2...IN
=
Y11 Y12 . . . Y1NY21 Y22 . . . Y2N...
......
...YN1 YN2 . . . YNN
V1V2...VN
(39)
From these matrices it can be noted that the impedance matrix is just the inverseof the admittance matrix, Z = Y −1. The other thing that can be found from thisis that both the impedance and admittance is related to the total voltages andcurrent at the port. This means for example the impedance Z11 is the ratio ofV1 and I1 if the other ports are open circuits, which means that there isn’t anycurrent going to the other ports. The same goes for the admittance Y11, it is theratio between I1 and V1, when all the other ports are shorted, which means thereisn’t any voltage over the port [7]. As described in (40).
Zij = ViIj
∣∣∣Ik=0
Yij = IiVj
∣∣∣Vk=0
for k 6= j (40)
By having an open-circuit the impedance can be derived through the voltage ofthe ith port and by short-circuit the admittance can be found through the currentat port i.
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3.1.1 Reciprocal
The reciprocity theorem for circuits states that if there is an applied constantcurrent at a point A, then the measured voltage at a point B should be the sameas measuring the voltage at a point A, when the same constant current is appliedat a point B. By using a arbitrary network such as in Figure 3, then this can beshowed by using the Lorentz reciprocity theorem in (41).
∮
S1
Ea ×Hb · dS =
∮
S2
Eb ×Ha · dS (41)
This is valid when the region is source free i.e J1 = J2 = M1 = M2 = 0. S isthe closed surface along the network, the a, b notation is two independent sourcesfor the field in the network. As mentioned before the fields can be represented asvoltages and currents with the help of the E- and H-mode functions, for simplicityonly E-mode function is used and this gives (42)
Eia = Viaei Hia = IiahiEib = Vibei Hib = Iibhi
}i = 1, 2 (42)
Where the index i are the ports, which points where the transverse modal fields,the currents and voltages are coming from. Then for example E1b is the transverseelectric field at port 1 as the result of the source b. Using then (42) in (41) resultsin,
(V1aI1b − V1bI1a)∮
S1
e1 × h1 · dS = (V2bI2a − V2aI2b)∮
S2
e2 × h2 · dS (43)
en and hn is the transverse modal fields and it is then integrated over the surfacefor the network. This means that S1 and S2 is the cross-section at each of the portsand because the power transported through the network is the same, if transportedfrom port 1 to 2 or from port 2 to 1, the integral part becomes,
∮
S1
e1 × h1 · dS =
∮
S2
e2 × h2 · dS (44)
Which is the same as the first line in (20), this gives the integrals to become unity[7]. (43) is then reduced to,
V1aI1b − V1bI1a = V2bI2a − V2aI2b (45)
The arbitrary network was considered short circuited at every port with the ex-ception of two ports, by using the 2 × 2 impedance matrix the voltages can bewritten as currents instead,
V1 = Z11I1 + Z12I2
V2 = Z21I1 + Z22I2(46)
10
Using this in (45) gives,
(I1bI2a − I1aI2b) (Z12 − Z21) = 0 (47)
The solution for (47) is that Z12 = Z21 this because the currents can be any value,due to the sources a and b do not depend on each other, which leads that thesolution must be that Z12 = Z21. Then for a reciprocal system (48) applies,
Znm = Zmn (48)
Due to the fact that the impedance matrix is the inverse of the admittance matrixthe same applies there,
Ynm = Ymn
11
3.1.2 Two port equivalent circuit
The equivalent circuit for a two port network can be derived through the impedanceor the admittance matrix, by either short circuit or open circuit one port dependingon an impedance or admittance representation. A way to get an equivalent circuitis by constructing either a T -circuit or a π-circuit, by constructing a π-circuit for atwo port network the admittance representation is used. For example in Figure 4the π-circuit can be seen,
I1Y2
Y1 Y3
I2
+
−V1
+
−V2
Figure 4: π-circuit
The voltages and currents have been defined in such a way that the admittancematrix will have a positive off-diagonal. To derive the relation of the π-circuit inFigure 4 to the admittance matrix, (49) will be used, which comes from (39), whenN = 2 (i.e. two port).
I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2(49)
It is easy to find the relation between the admittance matrix and the π-circuit, byshorting one port (no voltage over the shoring port) at a time the solution can befound [8],
Y11 =I1
V1
∣∣∣∣∣V2=0
= Y1 + Y2 Y12 =I1
V2
∣∣∣∣∣V1=0
= Y2
Y21 =I2
V1
∣∣∣∣∣V2=0
= Y2 Y22 =I2
V2
∣∣∣∣∣V1=0
= Y2 + Y3
(50)
Using then (50) the admittance in the π-circuit can be rewritten in terms of thevalues in the admittance matrix, which gives Figure 5,
12
I1Y12
Y11 − Y12 Y22 − Y12
I2
+
−V1
+
−V2
Figure 5: π-circuit
By finding the relation between the admittance matrix and the π-circuit in Fig-ure 4, it became apparent that the network is reciprocal due to that Y12 = Y21,which is seen in (50). If the network is also symmetrical, then, Y11 = Y22 applies[9].
13
3.2 ABCD Matrix
A B
C D
I1
+
−V1
I2
+
−V2
Figure 6: Two-port network
A B
C D
I1
+
−V1
I2
+
−V2
A′ B′
C ′ D′+
−
I3
+
−V3
Figure 7: Two connected two-port networks
It has been discussed earlier about the impedance and admittance representationof a network with N-ports, but microwave networks is often built on a couple oftwo port networks. This makes it easier to use another type of representation,which is the ABCD matrix. With the ABCD matrix, every two port networkcan be described by it, by having a system of connected two port networks andmultiplying every ABCD matrix for each two port network there will be one ABCDmatrix that will describe the whole microwave network. In Figure 6 is a two-portnetwork where the voltages and currents are described by (51),
[V1I1
]=
[A BC D
] [V2I2
]
V1 = AV2 +BI2
I1 = CV2 +DI2
(51)
If it then is a cascade for two two-port networks as in Figure 7, then the networkcan be described by just multiplying the two different ABCD matrices to get anew matrix that describes the whole network, which gives (52)
[V1I1
]=
[A BC D
] [A′ B′
C ′ D′
] [V3I3
]=
[A B
C D
] [V3I3
](52)
This can be done, because of the current that flows out of the first network flowsin the next network, the same goes for the voltage that is over the output of the
14
first network, is the same voltage that goes over the input for the next network[10].
I1 I2Z
+
−V1+
− V2
Figure 8: Circuit with a impedance in series
in Figure 8 there is a circuit with an impedance in series and finding the relationto the ABCD matrix is done by using (51). By short circuit or open circuit eachof the ports, the parameters for the ABCD matrix can be derived [11],
A =V1V2
∣∣∣∣I2=0
(Open ciruit) ⇒ V1 = V2 A = 1
C =I1V2
∣∣∣∣I2=0
(Open ciruit) ⇒ I1 = 0 C = 0
B =V1I2
∣∣∣∣V2=0
(Short ciruit) ⇒ I1 = I2 =V1Z
B = Z
D =I1I2
∣∣∣∣V2=0
(Short ciruit) ⇒ I1 = I2 C = 1
Using this in matrix form gives (53),
[V1I1
]=
[1 Z0 1
] [V2I2
](53)
I1 I2
Z+
−V1+
− V2
Figure 9: Circuit with impedance in parallel
In Figure 9 there is an impedance in parallel and solving the ABCD parametersis done in the same way as before. Remembering the relation between admittance
15
and impedance is Z = 1/Y , then the ABCD matrix for this kind of circuit is asin (54)
A =V1V2
∣∣∣∣I2=0
(Open ciruit) ⇒ V1 = V2 A = 1
C =I1V2
∣∣∣∣I2=0
(Open ciruit) ⇒ V1 = V2 = I1Z C =1
Z= Y
B =V1I2
∣∣∣∣V2=0
(Short ciruit) ⇒ V1 = 0 B = 0
D =I1I2
∣∣∣∣V2=0
(Short ciruit) ⇒ I1 = I2 C = 1
[V1I1
]=
[1 0Y 1
] [V2I2
](54)
Now that the ABCD parameters are defined for both when there is an impedancein series and an impedance in parallel. It can then be used for various types ofcircuits to be described by the ABCD matrix.
I1 I2
Y2+
−V1+
− V2
Z1 Z3
Figure 10: T-circuit
In Figure 10 there is an T-circuit, this can easily be described by the ABCD matrix,by just looking at it as three separately two-port networks that are connected toeach other. Then by multiplication of the ABCD matrix for each of them, theparameters for the T-circuit can be defined. This is accomplished by using theABCD matrices that has been defined in (53) and (54). It is important that themultiplication of the ABCD matrix is arranged in the network order, which gives,
[V1I1
]=
[1 Z1
0 1
] [1 0Y2 1
] [1 Z3
0 1
] [V2I2
]=
[A BC D
] [V2I2
]
A = 1 +Z1
Z2
B = Z1 + Z3 +Z1Z3
Z2
C =1
Z2
D = 1 +Z3
Z2
16
I1Z2
Y1 Y3
I2+
−V1+
− V2
Figure 11: π-circuit
Instead of a T-circuit lets look at a π-circuit as in Figure 11. Using the sameapproach by looking at it as three separately networks that are connected, thenusing (53) and (54) gives the ABCD parameters,
A = 1 +Y3
Y2B =
1
Y2
C = Y1 + Y3 +Y1Y3
Y2D = 1 +
Y1
Y2
(55)
Now that it has been discussed how the different circuits relates to the ABCDmatrix, then its left to determine how the length of a network relates to the ABCDmatrix. In Figure 12 there is a circuit model of a transmission line, which will beused to derive how the length relates to the parameters.
Is I + ∆I
y∆x
I Ir
+
−Vs
+
−V + ∆V
+
−V
+
−Vr
z∆x
+ −∆V
∆x x
`
Figure 12: Circuit model of a transmission line with length `
In Figure 12, there is a long transmission line with the length ` and somewherethroughout this long line there is a small strip ∆x that is located a distance x fromthe receiving point r. Before the small strip the voltage and currents are V + ∆V ,I + ∆I. After the strip, the voltage is V and the current is I, which indicatesthat the voltage drop over the strip is ∆V and the current ∆I goes through the
17
admittance y∆x. Knowing this and by the use of Ohm’s law (56) is then derived.
∆V = Iz∆x ⇒ ∆V
∆x= Iz
∆I = (V + ∆V )y∆x = V y∆x+ ∆V y∆x ≈ V y∆x ⇒ ∆I
∆x= V y
(56)
Taking account that ∆x is small, then letting it go to zero gives,
dV
dx= Iz
dI
dx= V y
(57)
By differentiating the first part of (57) with respect to x and taking the other partof the equation in to account gives (58)
d2V
dx2− yzV = 0 (58)
The roots of (58) will be ±√yz and there will be unknown constants c1, c2. By
defining the characteristic impedance as Z0 =√z/y and the propagation constant
as γ =√yz, gives the solution for the voltage V and the current I (just differentiate
V with respect to x) to be [12],
V = c1eγx + c2e
−γx
I =c1
Z0
eγx − c2
Z0
e−γx(59)
The constants can be solved by setting x = 0, which would mean that V = Vr,I = Ir and it gives (60)
c1 =Vr + Z0Ir
2
c2 =Vr − Z0Ir
2
(60)
Now that the unknown constants c1 and c2 has been determined, the relationbetween the voltage and current at the starting point and receiving point can bederived. This is solved by letting x = ` in (59) (this gives that V = Vs, I = Is)and using (60) gives,
Vs = Vr cosh γ`+ IrZ0 sinh γ`
Is =VrZ0
sinh γ`+ Ir cosh γ`(61)
18
From here it is obvious what the ABCD parameters will be, as in (62) (remem-bering that 1/Z0 = Y0)
A = cosh γ` B = Z0 sinh γ`D = Y0 sinh γ` D = cosh γ`
(62)
Z0, γ
`
`
Figure 13: Simpler model of a transmission line with length `
Table 1. provides a summary of all the circuits with the corresponding ABCDparameters, that has been derived.
19
Circuit ABCD parameters
Z
`/2
A = 1 B = ZC = 0 D = 1
YA = 1 B = 0C = Y D = 1
Z1
Y2
Z3
`/2
A = 1 +Z1
Z2
B = Z1 + Z3 +Z1Z3
Z2
C =1
Z2
D = 1 +Z3
Z2
Z2
Y1 Y3
`/2
A = 1 +Y3
Y2B =
1
Y2
C = Y1 + Y3 +Y1Y3
Y2D = 1 +
Y1
Y2
Z0, γ
`
`
A = cosh γ` B = Z0 sinh γ`C = Y0 sinh γ` D = cosh γ`
Table 1: Circuits with corresponding ABCD parameters
20
3.3 Dispersion Relation
By looking at the dispersion relation it can be determined what frequencies willpropagate through a two-port network. This can be done by looking at a two-portnetwork as a infinite structure of identical two-port cells as in Figure 14.
A B
C D
A B
C D
In
+
−Vn
In+1
+
−Vn+1
In+2
+
−Vn+2
Figure 14: Periodic structure with identical cells described by the ABCD matrix
Each of these cells in the infinite structure can be described by using ABCDmatrices, this gives a relation for the currents and voltages as in (63).
[VnIn
]=
[A BC D
] [Vn+1
In+1
](63)
By using Floquets theorem, which states that the field at a point differs from oneperiod away with only a complex constant [13]. Due to the periodic structure, thedifference between each cell is just the propagation factor, which can be found in(64),
[VnIn
]=
[A BC D
] [Vn+1
In+1
]=
[Vn+1
In+1
]eγl (64)
Where γ = α + jβ, there α is the attenuation constant and β the phase constant(β` becomes then the electrical length). (64) can be rewritten as in (65).
([A BC D
]− Ieγl
)[Vn+1
In+1
]= 0 (65)
From (65) it can be determined that the eigenvectors are the currents and voltages,which means that the eigenvalue for the system is the propagation factor [14]. Theeigenvalues are solved by taking the determinant of the first part of (65) equal tozero, which becomes as in (65).
AD − (A+D)eγ` + e2γ` −BC = 0 (66)
The solution for the propagation factor is in (67). This is solved by knowing thatfor a reciprocal system AD −BC = 1 [15].
cosh γl =A+D
2(67)
21
Remembering that γ = α + jβ, where α was the attenuation constant and β thephase constant. Then for a lossless structure (α = 0) (67) can be written as in(68) [14].
cos βl =A+D
2(68)
For a lossless periodic structure the propagation constant, which is the same asthe electrical length, can be calculated for a unit cell as in (68), this gives thedispersion relation. This can be used to see which frequencies of a certain modecan propagate through a two-port system (e.g. two-port waveguide). The modescan easily be distinguished. The reason for this is that the dispersion relation forthe mode before becomes purely imaginary. An example of the dispersion relationwith separation of modes is in Figure 15, where only the real part is shown, becausethis is the part that propagates [4].
βℓ/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency
(GHz)
0.2
0.4
0.6
0.8
1
1.2
1.4
First mode
Second mode
Third mode
Figure 15: Example of the dispersion relation and the separation of modes
22
4 Two port Capacitive Diaphragm
4.1 Circuit Model of parallel plate guide
E
by
zd′
d
(a) Longitudinal view
E d′
d b
z
(b) Cross sectional view
Figure 16: Parallel plate waveguide with an iris
In this section the circuit parameter for a parallel plate guide with an obstaclethat forms a window with walls that has zero thickness, will be determined witha relation to the π-circuit. A illustration of the parallel plate guide is shown inFigure 16. To model this problem as a circuit is quite easy because the parallelplate guide is symmetric, which is evident due the geometry is exactly the sameon either side of z = 0. The parallel waveguide is also reciprocal, which meansthat Ymn = Ynm. Using the π-circuit from Figure 5, with the conditions that theparallel plate guide is symmetric yields to Figure 17 with the equation system in(69). The π-circuit was chosen instead of the T -circuit, because the obstacle in thewaveguide is symmetrical, which means that the π-circuit will also be symmetrical.
I1Y12
Y11 − Y12 Y11 − Y12
I2
+
−V1
+
−V2
Figure 17: Symmetric π-circuit
I1 = Y11V1 + Y12V2
I2 = Y12V1 + Y11V2(69)
23
For simplicity, the reference plane has been chosen to be at z = 0 (at the obstacle).Thus the transverse electric field, which is only composed by a y-component, whichis the same as the tangential electric field, should be continuous across the referenceplane, because the tangential component of the electric field is continuous [4]. Thetangential electric field has then its maximum in the aperture and the tangentialmagnetic field its zero. From (33), it is determined that the transverse electricand magnetic field are proportional to the voltage and the current respectively.Thus, the voltage should also be continuous across z = 0, which means that thecircuit should be a pure shunt capacitance, because if the voltage is continuousthen everything in the π-circuit is in parallel, this means that the admittance inseries is absent. It becomes a shunt capacitance because of the orientation thatthe transverse electric field has related to the obstacle [7]. Due to the fact that theplate guide is symmetric, half of the structure is sufficient enough to determinethe behavior of the field. Having this in mind, the shunt can be expressed as,
2I
V= 2(Y11 − Y12) = jB ⇒ I
V= j
B
2(70)
Where the equivalent circuit for this case can be reduced to as in fig. 18
I
jB
2Y0
+
−V
Figure 18: Equivalent circuit for half the structure
24
4.2 Boundary conditions
To solve the circuit parameter, the field equations need to be solved. This is solvedby using the boundary conditions of the tangential fields, which are the conditionsbelow:
Eta = 0, y = 0, b
Eta = 0, z = 0, 0 < y < d′, d+ d′ < y < b
Hta = 0, z = 0, d′ < y < d+ d′
In the region before the obstacle (z < 0) the TE and TM can be described as in(71). Only E-modes are used to describe this, because there aren’t any changesin the field structures in the x-direction and the magnetic field doesn’t have az-component. The E-mode function is given in (72), which comes from (33).
Et =∑
Vn(z)en(y)
Ht =∑
In(z)hn(y)(71)
Where the mode currents In and mode voltages Vn are unknown.
en(y) = en(y)y = −√ρnb
cosnπy
by = −hn(y)y
hn(y) = hn(y)x =
√ρnb
cosnπy
bx
(72)
where,
ρn = 1, n = 0
ρn = 2, n ≥ 1
By using the boundary conditions at the plane z = 0 and having in mind thatthe tangential fields is the same as the transverse fields, then (71) can be writtenas in (73). The reason for this is to solve the unknown mode currents and modevoltage.
∞∑
n=0
Vnen(y) = 0, at 0 < y < d′, d+ d′ < y < b, z = 0
∞∑
n=0
Inhn(y) = 0, at d′ < y < d+ d′ z = 0
(73)
25
With the use of (21), the voltage and current modes is defined as (74) and (75)respectively. The index t for the transverse fields is omitted from now on.
Vn =
∫
aper
E(y)en(y)dy, d′ < y < d+ d′, z = 0 (74)
In =
∫
obs
H(y)hn(y)dy, 0 < y < d′, d+ d′ < y < b, z = 0 (75)
From here on, the second equation in (73) will be used, as this problem will besolved through an aperture approach rather than an obstacle approach. The secondpart of (73) can be rewritten to,
I0h0(y) = −∞∑
n=1
Inhn(y) (76)
This can be rewritten by using the relations between the mode amplitudes Inand Vn, which can be written as in (77), the reason for this, is that the highermodes does not propagate. This means that the mode voltages and currents canbe described as lumped elements, remembering that the region is z < 0 whichreduces the relation to,
In = −YnVn, Vn = −ZnIn, n = 1, 2, 3... (77)
Then the characteristic impedance/admittance is the same as (36), which is,
Yn =1
Zn=
ωε√k2 −
(nπb
)2 =jωεb
nπ√
1−(2bnλ
)2 (78)
I0h0(y) =∞∑
n=1
YnVnhn(y) (79)
Then by the use of (74) and knowing that hn(y) = −en(y) gives,
I0h0(y) = −∫
aper
K(y, y′)E(y′)dy′ (80)
Where the mode current I0 and the electric field E(y′) are unknown and soughtafter. The kernel K(y, y′) is given by (81), which is known.
K(y, y′) =∞∑
n=1
Ynhn(y)hn(y′) (81)
From here, there are three different methods that can be used to solve (74), whichthen yields to a solution for the susceptance B, these are; the Variational method,the Integral equation method and the Static method.
26
4.3 Variational method
The variational method can be used to avoid solving the integral equation bythe use of a trial field. This expression is derived by using (80) and multiplyingit by E(y), then integrating over the aperture, which gives the solution for thesusceptance as [16],
jB
2=I0V0
=∞∑
n=1
Yn
∫E(y)E(y′)hn(y)hn(y′)dydy′
(∫E(y)h0(y)dy
)2 , d′ < y < d+ d′ (82)
Where a use of a trial field set as E(y) = 1,
jB
2=I0V0
=∞∑
n=1
Yn
(∫hn(y)dy∫h0(y)dy
)2
, d′ < y < d+ d′ (83)
This will yield to a rather rigorous approximation of the susceptance B. Thevariational method will not be more explained than this, because this methodwon’t be used in the following of this report, but is rather interesting to knowabout. It can be used in more detailed manner to get a better aproximation of thesusceptance B, rather than just solving the integral in (83).
27
4.4 Integral equation method
The integral equation method is used to solve the integral equation, which willlead to higher accuracy of the susceptance than the variational method describedeariler. But to use this method it is easier to use a static kernel Ks(y, y
′) ratherthan a dynamic kernel K(y, y′), where k2 → 0. To be able to use a static kernelinstead, (79) is rewritten as,
I0h0(y)−∞∑
n=1
Ynhn(y)Vn +∞∑
n=1
Ynshn(y)Vn =∞∑
n=1
Ynshn(y)Vn (84)
Yns =1
Zns=jωεb
nπ(85)
It becomes more convenient to express (84) as in (86), with In expressed as in (87).
I0h0(y) +∞∑
n=1
Inhn(y) = −∫
aper
Ks(y, y′)E(y′)dy′ (86)
In = (Yns − Yn)Vn, n > 0 (87)
Due to the linearity of the integral equation, the solution for E(y) can be writtenas in (88) [1]. This solution for the electric field is an assumption, where there aresome unknown partial fields E0(y), En(y) and unknown mode currents, that needsto be solved later on.
E(y) = I0E0(y) +∞∑
n=1
InEn(y) (88)
Inserting (88) into (86), will yield to (89), where in this case n = 0, 1, 2, 3, ....
hn(y) = −∫
aper
Ks(y, y′)En(y)dy′ (89)
From here the integral equation becomes straightforward to solve for the unknownpartial fields En(y), because all the other parameters are known in (89), but itbecomes more convenient to solve, if the y-domain which goes from d′ to d+ d′ istransformed to the θ-domain which goes from 0 to π, because of its dependencieson cosine. This means that a change of variable should be introduced, which leads
28
the static kernel Ks(y, y′) to be rewritten depending on the change of variable.
Then to solve the susceptance, (88) is inserted into (74), which yields,
V0 = Z00I0 +∞∑
n=1
Z0nIn
Vm = ZmnI0 +∞∑
n=1
ZmnImn
(90)
With the impedance parameters defined as,
Zmn = −∫
aper
En(y)hm(y)dy, m, n = 0, 1, 2, 3, .. (91)
Using (89) the unknown partial fields En(y) can be solved. The solution for En(y)can then be used in (91) to solve the impedance parameters. By the use of theimpedance parameters in (90) and the relation in (87), the susceptance can besolved through (92).
V0I0
=1
jB2
(92)
29
4.5 Static method
Rather than using the variational method or integral equation method, there isanother method, the static method. The static method is almost the same asthe integral equation method, but avoids the calculations of integrals by usingconformal mapping. This conformal mapping is the Schwarz-Christoffel mapping,which will be explained later on. Rather than using (88), which is proportional tothe current I0, it is better to be proportional V0, to be able to utilize the conformalmapping [1]. Therefore E0(y)→ E0(y), to able to use the proportionality to V0.
E(y) = V0E0(y) +∞∑
n=1
InEn(y) (93)
By using (74), it then follows
V0 = −V0∫
aper
E0(y)h0(y)dy
︸ ︷︷ ︸=−1
−∞∑
n=1
In
∫
aper
En(y)h0(y)dy
︸ ︷︷ ︸=0
(94)
and it also follows,
Vm = −V0∫
aper
E0(y)hm(y)dy −∞∑
n=1
In
∫
aper
En(y)hm(y)dy (95)
From here the transfer and impedance parameters are defined as,
Tm0 = −∫
aper
E0(y)hm(y)dy, Zmn = −∫
aper
En(y)hm(y)dy (96)
The network parameters can then be expressed as in (97)
I0 = Y00V0 −∞∑
n=1
T0nIn
Vm = Tm0V0 +∞∑
n=1
ZmnIn
(97)
by using the definitions of these network parameters with (86) and (96). Then(98) becomes apparent,
Y00h0(y) = −∫
aper
Ks(y, y′)E0(y′)dy′
−T0nh0(y) + hn(y) = −∫
aper
Ks(y, y′)En(y′)dy′
(98)
30
The static magnetic field can be determined from (98), where the expression needsto be expanded for when z < 0. By the use of a expanded solution for the Laplaceequation for that region, the static magnetic field becomes as (99) (for n = 0) and(100) (for n > 0) [1],
H0(y, z) = (Y00 − jωεz)h0(y)−∞∑
m=1
Tm0Ymshm(y)emπzb (99)
Hn(y, z) = −T0nh0(y) + coshnπz
bhn(y)−
∞∑
m=1
YmsZmnhm(y)emπzb (100)
In the expression for the static magnetic field, there are some unknown constantsthat can be solved through conformal mapping. This conformal mapping namedSchwarz-Christoffel mapping, is used to transform the geometry of the parallelplate waveguide to something simpler, where it is easier to solve the static magneticfield, due to simpler boundary conditions. This conformal mapping is used to finda solution for the potential between the plates through complex function theory,hence the proportionality to V0 in (93). In Figure 19 the conformal mapping ofthe ζ-plane to the ζ ′-plane is illustrated. The complex ζ-plane it transformed tothe upper half of the t-plane then transformed again to the ζ ′-plane.
d′
db
t = −∞
t =∞ t = 1
t = −1
t = t0
t = t1
(a) ζ = z + jy-plane
−∞ ∞t0 t1
(b) t-plane
b
t = −∞
t =∞ t = t1
t = t0
(c) ζ ′ = z′ + jy′-plane
Figure 19: Schwarz-Christoffel conformal mapping from ζ to ζ ′ plane
To use this conformal mapping, the ζ-plane is mapped to the t-plane by using(101). This is the definition for the Schwarz-Christoffel mapping, where α is theangle between the verticies and t are the relevant points used for the conformalmapping [17].
ζ = C
∫ ∞∏
n=0
(t− tn)αnπ−1 dt (101)
Mapping the ζ-plane to the t-plane, by using the definition of the Schwarz-Christoffelmapping, will lead to two unknown constants. These unknown constants are de-termined by knowing that when t = −1 then ζ = jb and when t = 1 then ζ = 0.
31
This gives that the constant C = b/π and L = 0, which leads to the mappingfunction to be,
ζ =
∫C√
(t+ 1)(t− 1)dt = C arccosh(t) + L =
b
πarccosh(t) (102)
t = coshπζ
b(103)
Now that the function for mapping the ζ-plane to the t-plane is determined, whichis the mapping from Figure 19a to Figure 19b, it is necessary to determine thepoints t1 and t0. These are found by using (103), which comes from (102), whent = t1 then ζ = jd′ and when t = t0 then ζ = j(d+ d′). This yields to,
t1 = cos
(πd′
b
), t0 = cos
(πd+ d′
b
)(104)
The mapping between the ζ-plane to the ζ ′-plane is found out by mapping theζ ′-plane to the t-plane then just setting them equal to each other. To do thisconformal mapping, (105) is used and then rewritten as in (106),
ζ ′ =
∫C√
(t+ t0)(t− t1)dt =
b
πarccosh
(2t
t1 − t0− t0 + t1t0 − t1
)(105)
t =t1 − t0
2cosh
(πζ ′
b
)+t0 + t1
2(106)
By using (103) with (106) and the expressions in (104), the relation between ζ ′-plane to ζ-plane is then derived to be,
ζ ′ =b
πarccosh
cosh
(πζ
b
)− A2
A1
(107)
With,
A1 =t1 − t0
2= sin
(πd
2b
)sin
(πd+ 2d′
2b
)
A2 =t0 + t1
2= cos
(πd
2b
)cos
(πd+ 2d′
2b
) (108)
To solve the static problem in the simpler transformed waveguide, then the solutioncorresponding to the boundary conditions needs to be found. These boundaryconditions for the stream function (i.e. static magnetic field) are as in (109),
u = 0, on z′ = 0du
dy′= 0 on y = 0, b (109)
32
The solution for the complex stream function (w = u+ jv) is evident and is [18],
w0 = C0ζ′, wn = Cn sinh
(nπζ ′
b
), n > 0 (110)
Where the unknown constants C0 and Cn needs to be determined later so theysatisfy (99), (100). Starting with the complex stream function w0 = C0ζ
′ and seewhat the complex stream function is for the actual guide by using (107),
w0 = u0 + jv0 = C0ζ′ = C0
b
πarccosh
cosh
(πζ
b
)− A2
A1
(111)
Which can be rewritten to,
w0 = C0b
π
(ln
(1
A1
)− πζ
b+ ln
[1
2
(1− 2A2e
πζb + e
2πζb +
√(1− 2A2e
πζb + e
2πζb
)2− 4A2
1e2πζb
)]) (112)
Then by using Taylor Series around the point zero (Maclaurin Series) for thesecond ln term in (112), where x = eπζ/b has been used, will lead to the series ofthe complex stream function,
w0 = C0b
π
(ln
(1
A1
)− πζ
b−2A2e
πζb + (1− A2
1 − 2A22)e
2πζb −
2
3A2(6A
21 + 4A2
2 − 3)e3πζb + ...
) (113)
To get the stream function u0 from the complex stream function w is pretty simple,because the stream function is just the real part of the complex stream function.The real part is found by remembering that ζ = z + jy, using this yields that thestream function is found to be,
u0 = C0b
π
(ln
(1
A1
)− πz
b− 2A2 cos
(πyb
)eπzb +
(1− A21 − 2A2
2) cos
(2πy
b
)e
2πzb −
2
3A2(6A
21 + 4A2
2 − 3) cos
(3πy
b
)e
3πzb + ...
)(114)
33
By comparing the stream function with the static magnetic field H0(y, z) in (99)and using the definition for Yns in (85), the constant C0 can found to be jωε/
√b.
From this it is simple to see what the admittance Y00 and the transfer coefficientsT0n should be,
Y00 = jωεb
πln
(1
A1
)
T10 =√
2A2
T20 =√
2(A21 + 2A2
2 − 1)
T30 =√
2A2(6A21 + 4A2
2 − 3)
(115)
For solving the static magnetic field when n = 1 (H1(y, z)), the complex streamfunction, w1 = C1 sinh(πζ ′/b) is used. This will yield that the complex stream
function in the actual waveguide with the use of x = eπζb is,
w1 = C1 sinh
(πζ ′
b
)= C1
√cosh2
(πζ ′
b
)− 1 =
C1
2A1x
√(2A2x− x2 − 1)2 − 4A2
1x2
(116)
or by doing a series expansion. But using Taylor series is quite problematic,because of the singularities that appear. Therefor Laurent series is used, whichanalyses functions near singularities [19]. However in this case the definition ofLaurent series isn’t necessary, because the square root term can be expanded inTaylor series. By just multiplying it with the term before, the series expansionwill be found as,
w1 =C1
A1
(−A2 +
eπζb + e−
πζb
2− A2
1eπζb − 2A2
1A2e2πζb − A2
1(A21 + 4A2
2 − 1)e3πζb + ...
)
(117)
Then, the stream function is found to be,
u0 =C1
A1
(−A2 + cosh
(πzb
)cos(πyb
)− A2
1 cos(πyb
)eπzb −
2A21A2 cos
(2πy
b
)e
2πzb −
A21(A
21 + 4A2
2 − 1) cos
(3πy
b
)e
3πzb + ...
)(118)
34
By comparing (118) with (100), the constant C1 is determined to be A1
√2/b. The
transfer coefficient T01 and the impedance parameters Zm1 on comparison are,
T01 =√
2A2
Z11 =A2
1
jωεb
π
Z21 =2A2
1A2
jωεb
2π
Z31 =A2
1(A21 + 4A2
2 − 1)
jωεb
3π
(119)
The next step is to determine when n = 2, which means that the complex streamfunction w2 should be C2 sinh(2πζ ′/b). However, solving this will lead to a minorproblem, which is that it will also contain the excitation of n = 1. Therefore,a linear combination between these two needs to be used to solve just for whenn = 2 [18]. This becomes apparent if the same steps as before is done. Thecomplex stream function, which will be called w′2 in this case, when n = 2 is,
w′2 = C2 sinh
(2πζ ′
b
)= 2C2 sinh
(πζ ′
b
)cosh
(πζ ′
b
)(120)
Then, to get the complex stream function for the actual guide, one needs to use(120) with the series expansion in (116) and (107). This gives the complex streamfunction to be,
w′2 =C2
A21
(−(A2
1 − 2A22 − 1)− 4A2 cosh
(πζ
b
)+ cosh
(2πζ
b
)− A4
1e2πζb + ...
)
(121)
By comparing this with (100), one observes that there shouldn’t be any coshπζ/bterm, this term exist due to excitation of n = 1. To get the excitation for only whenn = 2, the complex stream function should be as in (122). The linear combinationwas found by comparing (121) with (117).
w2 = C2
(sinh
(2πζ ′
b
)+
4A2
A1
sinh
(πζ ′
b
))(122)
This time the complex stream function should only be for the excitation n = 2.Then by using (122) with the series expansion in (117) and the series expansion
35
in (117), will yield to the complex stream function for the actual guide to be,
w2 =C2
A21
(−(A2
1 + 2A22 − 1) +
e2πζb + e−
2πζb
2− 4A2
1A2eπζb −
(8A21A
22 + A4
1)e2πζb −
4A21A2(2a
2 + 4b2 − 1)e3πζb + ...
)(123)
Then the stream function for the actual waveguide is,
u2 =C2
A21
(−(A2
1 + 2A22 − 1) + cosh
(2πz
b
)cos
(2πy
b
)− 4A2
1A2 cos(πyb
)eπzb −
(8A21A
22 + A4
1) cos
(2πy
b
)e
2πzb −
4A21A2(2A
21 + 4A2
2 − 1) cos
(3πy
b
)e
3πzb + ...
)
(124)
By comparing (124) with (100), when n = 2, the constant C2 is found to beA2
1
√2/b. The transfer coefficient and the impedance parameters are also found to
be,
T02 =√
2(A21 + 2A2
2 − 1)
Z12 =4A2
1A2
jωεb
π
Z22 =8A2
1A22 + A4
1
jωεb
2π
Z32 =4A2
1A2(2A21 + 4A2
2 − 1)
jωεb
3π
(125)
To get a more accurate result, up to n = 15 is going to be solved. This means thatthe transfer coefficients up to T15,0 = T0,15 need to be found and the impedanceparameters up to Z15,15. To solve then the complex stream function up to n =15, rather than doing the series expansion for every excitation mode, the seriesexpansion can just be solved for when n = 1, up to the 30th order (O(31)). Then
36
by using (126), the complex stream function for every excitation mode can besolved.
sinh
(nπζ ′
b
)= sinh
(πζ ′
b
) [(n−1)/2]∑
k=0
(−1)k(n− k − 1
k
)2n−2k−1 coshn−2k−1
(πζ ′
b
)
(126)
This expression is determined from the multiple angle formula for a sinus function[20], with the use of the relation between trigonometric functions and hyperbolicfunctions. [(n− 1)/2] is the floored value of (n− 1)/2. Then there is still the sameproblem with isolating the right excitation to get the correct complex stream func-tion. This is solved in the same manner as before, by finding a linear combinationbetween the modes, so one could eliminate the unwanted modes. This needs tobe done in the right order, which means first solving for when n = 2 with a linearcombination of when n = 1, to get the complex stream function for only n = 2.For solving when n = 3, a linear combination of n = 2 (where n = 2 is with alinear combination of n = 1) and n = 1 needs to be found. In (127), the linearcombination to solve for only n = 3 is determined to be,
w3 = C3
[sinh
(3πζ ′
b
)−M1
(sinh
(2πζ ′
b
)−M2 sinh
(πζ ′
b
))
−M3 sinh
(πζ ′
b
)] (127)
Where M1, M2 and M3 are constants that are used to isolate the wanted modes.By using the series expansion in (117) with (126) in (127), one should be able tosolve the complex stream function for the actual guide. Then, by comparing withthe static magnetic field in (100), leads to the solution of the transfer coefficientand the impedance parameters. The same procedure is done up to n = 15, by theuse of (97) leads to the system of linear equations,
I0 = Y00V0 − T10I1 − · · · − T15,0I15V1 = T10V0 + Z11I1 + · · ·+ Z1,15I15
... =...
V15 = T15,0V0 + Z15,1I1 + · · ·+ Z15,15I15
(128)
The first line in (128) is used to solve the susceptance B and the other equationsby the use of (87) is used to solve the parameter Vn/V0, this is solved by rewriting
37
the system of equations, which gives a 15× 15 matrix,
1− Z11(Y1s − Y1) . . . −Z1,15(Y15s − Y15)...
. . ....
−Z15,1(Y1s − Y1) . . . 1− Z15,15(Y15s − Y15)
V1V0...V15V0
=
T10...
T15,0
(129)
After solving the parameters Vn/V0, the susceptance B can then be solved by usingthe first equation in (128) and remembering that jB/2 = I0/V0,
B =2I0jV0
=2
j
(Y00 − T10(Y1s − Y1)
V1V0− · · · − T15,0(Y15s − Y15)
V15V0
)(130)
The susceptance B has been correctly treated up to the first 16th modes.
38
5 Results
`
E
b
d′
d
(a) Longitudinal view
E d′
d b
`
(b) Cross sectional view
Figure 20: Unit cell of a parallel plate guide with a length of `
In this section, results will be presented, such as the dispersion relation for the de-rived circuit model in (130) compared with simulated results from CST MicrowaveStudio and an already existing circuit model in (131) and (132) [1],
BM1 =4bYairλ
(ln
[csc
(πd
2b
)]+
R2 cos4(πd2b
)
1 +R2 sin4(πd2b
)+
1
16
(b
λ
)2(1− 3 sin2 πd
2b
)2
cos4πd
2b
) (131)
BM2 =4bYairλ
(ln
[csc
(πd
2b
)csc( π
2b(d+ 2d′)
)]+
2R1 cos2(πd2b
)cos2
(π2b
(d+ 2d′))
1 +R1 sin2(πd2b
)sin2
(π2b
(d+ 2d′))+
R2
(3 cos2
(πd
2b
)cos2
( π2b
(d+ 2d′))− cos2
(πd
2b
)− cos2
( π2b
(d+ 2d′)))2
)
(132)
where,
Rn =nπ
jωεb(Yn − Yns) =
1√1−
(2b
nλ
)2− 1 (133)
The susceptance BM1 is used when the window is centered, i.e. when d′ = (b−d)/2.To get the susceptance for when d′ = 0 then λ = λ/2 is used for BM1. Thesusceptance BM2 is used when the obstacles is asymmetrical. Only the two firstmodes have been treated correctly, but a variational extension has been made toget an approximation for the higher modes.
39
The dispersion relation was found by the use of ABCD matrices, but first it wasnecessary to define the circuit models as a periodic structure. By the use of Table 1,a unit cell of the circuit model with a length of ` can be constructed, which can beseen in Figure 21. The corresponding ABCD parameters gives then the relationbetween V1, I1 and V4, I4, seen in (134) .
jB+
−V1
I1
Z0, γ
`/2
`+
−V4
I4
Z0, γ
`/2
`/2
Figure 21: Circuit for the unit cell for the use of ABCD parameters
Where the susceptance B is interchangeable with the susceptance BM1 and BM1.
[V1I1
]=
[cosh γ`
2Z0 sinh γ`
2
Y0 sinh γ`2
cosh γ`2
] [1 0jB 1
] [cosh γ`
2Z0 sinh γ`
2
Y0 sinh γ`2
cosh γ`2
] [V4I4
](134)
By using (68), which is for a lossless system, the dispersion relation is found to be,
βc` = arccos
(cos β`− BZ0
2sin β`
)(135)
Where only the real part of (135) is interesting, because it is the part that propa-gates through the waveguide.
Figure 22: Model of the unit cell in CST Microwave Studio
In CST Microwave Studio, the dispersion relation was simulated by using theeigenmode solver with the appropriate boundary conditions. In Figure 22 is themodel of the parallel plate guide, which was constructed in CST. To get an accurateresult a more refined mesh were used in the simulations, the cells per wavelengthand cells per max model box edge used, was 40.
40
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45
CST
Extended
Simple
(a) CST simulation up to the fourth mode
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
30.2
30.4
30.6
30.8
31
31.2
31.4
31.6
31.8
32 CST
Extended
Simple
(b) At the third mode
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
44.9
45
45.1
45.2
45.3
45.4
45.5
45.6
45.7
45.8
45.9 CST
Extended
Simple
(c) At the fourth mode
Figure 23: Dispersion relation when b = 3.1 mm, d = 0.1 mm, d′ = 0 mm
41
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
CST
Extended
Simple
(a) CST simulation up to the fifth mode
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
60
60.5
61
61.5
62
62.5
63
63.5
64CST
Extended
Simple
(b) At the fifth mode
Figure 24: Dispersion relation when b = 2 mm, d = 0.25mm, d′ = 0.5 mm
42
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25 CST
Extended
Simple
(a) CST simulation up to the second mode
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
16
17
18
19
20
21
22
23CST
Extended
Simple
(b) At the second mode
Figure 25: Dispersion relation when b = 5 mm, d = 1.25mm, d′ = 1.875 mm
43
In Figure 23 to Figure 25 the results of the dispersion relation are represented,where the ”Extended” is referred to (135) with (130) and ”Simple” referred to(135) with either (131) or (132), depending on whether the window in the parallelplate guide is centered or on the bottom or neither of them. All of these caseshave the same length for the unit cell, which is ` = 10 mm. For the case whenb = 3.1 mm, d = 0.1 mm and d′ = 0 mm in Figure 23, which means that thewindow is at the bottom of the waveguide, then (131) with λ = λ/2 was used in(135). This lead to the dispersion relation to be as in Figure 23. There aren’tmuch difference between the ”Simple” and the ”Extended”. They both are closeto the simulated results, where the ”Extended” is a bit closer to the result of CSTin Figure 23c, but after the fourth mode both the ”Extended” and the ”Simple”are inaccurate. In the case where b = 2 mm, d = 0.25 mm and d′ = 0.5 mm, thedispersion relation is as in Figure 24. In this case the window is not centered orat the bottom of the waveguide, which means that ”Simple” is referred to the useof (132). Both the ”Extended” and the ”Simple” is close to the simulated resultsin CST, but the ”Extended” is a bit closer to the simulation than the ”Simple”,as seen in Figure 24b. Then in the last case where the window is centered thedispersion relation becomes as in Figure 25. Here ”Simple” is referred to (131)when finding the dispersion relation. In this case, there aren’t that much of adifference between the ”Extended” and the ”Simple”. As seen in Figure 25a, bothof the ”Extended” and ”Simple” becomes inaccurate after the second mode. Thereare more results for the dispersion relation results in Appendix: A.
Frequency (GHz)0 5 10 15 20 25 30 35 40 45 50
B/B
M1
1
1.002
1.004
1.006
1.008
1.01
1.012
1.014
1.016
1.018
Figure 26: The ratio between B and BM1
44
Frequency (GHz)0 10 20 30 40 50 60 70 80
B/B
M2
1
1.005
1.01
1.015
Figure 27: The ratio between B and BM2
Frequency (GHz)0 5 10 15 20 25 30
B/B
M1
1
1.0005
1.001
1.0015
1.002
1.0025
1.003
Figure 28: The ratio between B and BM1
45
The difference between the circuit models weren’t much, but the ”Extended” hasa bit more accuracy. This is seen in Figure 26 to Figure 28, which show the ratiobetween the susceptances for the different circuit models. The dimensions for theparallel plate guide are in the same order as in Figure 23 to Figure 25, but withoutthe length ` = 10 mm. The maximum frequencies have been set to the frequencywere the circuit models has a accurate result, accordingly with the simulations fromCST and the circuit models restricions. From Figure 26 it can be seen that thedifferences are small for low frequencies but increases as the frequency increasesto a point then it decreases again, but the difference is at the most just under2 %. The result from Figure 27 has the same tendencies as in Figure 26, but thedifference is lower, just under 1.5 %. In Figure 28 there isn’t much of a difference,the highest difference is under 0.3 %. This is also seen in Figure 25, there isn’tmuch of a difference for the dispersion relation between the circuit models.
46
6 Conclusion and Thoughts
6.1 Conclusion
In this work there were three different methods that were studied the Variationalmethod, the Integral equation method and the Static method, but in the endthe static method was chosen. This was chosen because it was easier to handlein mathematical tools such as Mathematica and of its strength by avoiding theintegral equations. The integral equation method is also a good method but can bea bit harder to handle in mathematical tools when finding the unknown impedanceparameters, because of the its structure of either side of the equal sign, which makesit harder to compare them. With the Static method it’s easier to compare, becausethe unknown parameters are just the constants before a polynomial, which is easierto handle. The Variational method won’t be as accurate as the other methods,due to the approximations that are made. That is why the Static method waschosen over the other methods, not only because of the structure, but also of thestrength that it has, by its use of conformal mapping to solve laplace equation insimpler geometry.
In the end, it is possible to get an analytic solution for circuit model of awaveguide discontinuity, but it has its restrictions. As seen in the results for thedispersion relation, it’s only plotted to a certain mode, because the susceptanceB becomes imaginary at higher frequency, which it shouldn’t become. This isbecause of (78), where if the 2b/nλ becomes greater then 1, then this part becomesimaginary. Therefore, for the lowest n value that is 1, the circuit model is restrictedto 2b/λ < 1. This is seen in Figure 29 when comparing with CST, the circuit modeldoesn’t give the same result at higher modes. The reason for this is that themodes that CST is gives are other type of polarization, in this case an orthogonalpolarization. The circuit model determined is only valid for a polarization andnot for orthogonal polarization or dual polarization. For example, if the heightof the parallel plate guide is set to 4 mm, then the circuit model is valid for afrequency less than 37.47 GHz. This is confirmed by Figure 29 and Figure 30a.When having the restriction in mind, this circuit model is very reliable and it haspretty good accuracy. One could just use the ”Simpler” circuit models instead ofthe ”Extended”, because the difference between them isn’t much, there may becases were there are a bit more difference than just 2 %, but it probably won’tbe much more than that. This means the circuit model that was determined inthis work isn’t more accurate than the existing ones. Regarding (131) when thewindow is centered, the restriction becomes b/λ < 1, however this is not reliable,because in the case for Figure 29, it is shown that the results are reliable up to37.47 GHz, but the restriction gives that it should be reliable up to around 75GHz, which it is not and also shown in Figure 30a. Therefore, the restriction for
47
(131) should be neglected and the correct restriction should be used, which is 2b/λ.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45
CST
Extended
Simple
Figure 29: Dispersion relation when b = 4 mm d = 2 mm, d′ = 1 mm
48
(a) Polarization of the electric field at a frequencyunder 37.47 GHz
(b) Polarization of the electric field at a frequency over37.47 GHz
Figure 30: Simulated polarization in CST when b = 4 mm, d = 2mm, d′ = 1 mm
49
One big advantage the circuit model has over CST Microwave studio, is that theresults can be found much faster. In CST Microwave studio, the simulation takessome time to do, depending on how refined the mesh is. For higher accuracy, amore refined mesh needs to be used. As an example, for the results in Figure 23it took CST about 45 minutes to solve for only 20 data points for each mode andfor the circuit model under 10 seconds for 10 000 data points. If one parameter ischanged, for example the height of the waveguide, the simulation in CST needs tobe done all over again. For the circuit model, it just takes seconds to get resultswhen the circuit parameter is defined. This shows that it will be more efficient touse the circuit model rather than CST, because of the reduced time and energyused for simulations.
The circuit model is then both reliable, when taking the restriction to account,and fast. There is one problem with CST at least for the 2014 version, which iswhen simulating for higher modes than the second mode, it can give unreliableresults. It seems to have a bug, which can be pretty frustrating, because of thisit’s hard to do parametric sweeps for different geometries without losing control.Endless time can be wasted because of this problem that CST has, and when thisoccurs the simulation needs to be restarted. In Figure 31 is a example of theproblem that often occurs in CST.
Something that was discovered in this work, was that the Static method wasstill stable up to the 16th mode, which wasn’t disclosed by Nathan MarcuvitzWaveguide handbook. The stability of the Static method was concluded by lookingif the impedance parameters and the transfer parameters were still reciprocal,which it is, at least up to Z14,15 = Z15,14 and T15,0 = T0,15.
0 20 40 60 80 100 120 140 160 180
0
5
10
15
20
25
30
35
40
Figure 31: Example of unreliable results from CST Microwave Studio 2014
50
6.2 Thoughts
E
(3)
(1) (4) (2) b
b′
(a) Side view of the waveguide
I1Y12 − Y13
Y11 − Y12 Y11 − Y12
I2
Y13
Y33 − Y13I3
+
−V1
+
−V2
+ −V3
(b) Equivalent circuit
Figure 32: Three port waveguide
The goal with this work was to find a circuit model for a four port waveguide, butstarting with a two port then three port waveguide. There came a stage were itwas too hard to solve a circuit model for the three port waveguide. Looking atthe three port waveguide in Figure 32a the fourth region in the waveguide is quiteproblematic, because the wave will propagate in two different direction, whichis hard to define and solve for later on. It’s also hard to find a good referenceplane in this case, which was easy for the two port case, because the referenceplane were set between the two regions. The other problem is to find a simplergeometry for the use of conformal mapping to make the boundaries easier withoutdisturbing the original boundaries. If a conformal mapping was to be found thenit would probably become an implicit function, which means that it would be hardto relate the actual waveguide to something simpler and it wouldn’t be possible touse a Taylor series expansion, to find the unknown coefficients of the hyperbolicfunctions. Then the use of Lagrange reversion theorem could be used, because itwould give a series for implicit functions. Then there’s an other problem, if the theother two problems were solved, how would it then relate to the circuit parametersin Figure 32b, because all these regions affect each other so it will become hard torelate the solution to the correct circuit parameter. It would be same for the fourport waveguide, but if the three port waveguide were to be solved, then it probablywould become easier to solve for a four port. This is not necessary true, but thehardest part would be to solve the fourth region in the three port waveguide,solving this could perhaps give the necessary knowledge to solve then for a fourthport waveguide.
51
References
[1] N. Marcuvitz, Waveguide handbook. Iet, 1951, no. 21.
[2] G. Valerio, Z. Sipus, A. Grbic, and O. Quevedo-Teruel, “Accurate equivalent-circuit descriptions of thin glide-symmetric corrugated metasurfaces,” IEEETransactions on Antennas and Propagation, vol. 65, no. 5, pp. 2695–2700,2017.
[3] O. Quevedo-Teruel, M. Ebrahimpouri, and M. Ng Mou Kehn, “Ultrawidebandmetasurface lenses based on off-shifted opposite layers,” IEEE Antennas andWireless Propagation Letters, vol. 15, pp. 484–487, 2016.
[4] S. J. Orfanidis, Electromagnetic waves and antennas. Rutgers UniversityNew Brunswick, NJ, 2002.
[5] K. Milton and J. Schwinger, Electromagnetic Radiation: Variational Methods,Waveguides and Accelerators. Springer Science & Business Media, 2006.
[6] R. E. Collin, Foundations for microwave engineering. John Wiley & Sons,2007.
[7] D. M. Pozar, Microwave engineering. John Wiley & Sons, 2009.
[8] R. Sorrentino and G. Bianchi, Microwave and RF engineering. John Wiley& Sons, 2010, vol. 1.
[9] S. P. Ghosh and A. K. Chakraborty, Network Analysis & Synthesis. McGraw-Hill Education (India) Pvt Limited, 2010.
[10] J. F. White, High Frequency Techniques: An Introduction to RF and Mi-crowave Design and Computer Simulation. John Wiley & Sons, 2004.
[11] J. Weiner and F. Nunes, Light-matter interaction: physics and engineering atthe nanoscale. OUP Oxford, 2012.
[12] L. L. Grigsby, Electric power generation, transmission, and distribution. CRCpress, 2012.
[13] D. A. Watkins, Topics in electromagnetic theory. Wiley, 1958.
[14] F. Martın, Artificial transmission lines for RF and microwave applications.John Wiley & Sons, 2015.
[15] C. Caloz and T. Itoh, Electromagnetic metamaterials: transmission line the-ory and microwave applications. John Wiley & Sons, 2005.
52
[16] R. E. Collin, Field theory of guided waves. McGraw-Hill, 1960.
[17] T. A. Driscoll and L. N. Trefethen, Schwarz-christoffel mapping. CambridgeUniversity Press, 2002, vol. 8.
[18] J. S. Schwinger and D. S. Saxon, Discontinuities in waveguides: notes onlectures by Julian Schwinger. CRC Press, 1968.
[19] G. B. Arfken and H. J. Weber, Mathematical methods for physicists interna-tional student edition. Academic press, 2005.
[20] E. W. Weisstein. (2017) Multiple-angle formulas. [Online]. Available:http://mathworld.wolfram.com/Multiple-AngleFormulas.html
53
A. More results
The number of modes refers to the simulated results of CST.
b = 4 mm, d = b/4 mm, d′ = 3b/8 mm,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35 CST
Extended
Simple
Figure 33: The first three modes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
30
31
32
33
34
35
36
37 CST
Extended
Simple
Figure 34: Third mode
54
b = 5 mm, d = b/4 mm, d′ = 3b/8 mm,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25 CST
Extended
Simple
Figure 35: The first two modes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
16
17
18
19
20
21
22
23CST
Extended
Simple
Figure 36: Second mode
55
b = 2 mm, d = 1mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60 CST
Extended
Simple
Figure 37: First five modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
60
61
62
63
64
65
66
67 CST
Extended
Simple
Figure 38: Fifth mode
b = 4 mm, d = 1mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30CST
Extended
Simple
Figure 39: First three modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
30.5
31
31.5
32
32.5
33CST
Extended
Simple
Figure 40: Third mode
56
b = 1 mm, d = b/8mm, d′ = b/4 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
120
140
160CST
Extended
Simple
Figure 41: The first ten modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
135
135.5
136
136.5
137
137.5
138
138.5
CST
Extended
Simple
Figure 42: Tenth mode
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
90
91
92
93
94
95
96 CST
Extended
Simple
Figure 43: Seventh mode
57
b = 2 mm, d = b/8mm, d′ = b/4 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
CST
Extended
Simple
Figure 44: The first five modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
60
60.5
61
61.5
62
62.5
63
63.5
64CST
Extended
Simple
Figure 45: Fifth mode
b = 3 mm, d = b/8mm, d′ = b/4 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45CST
Extended
Simple
Figure 46: The first four modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
45
45.5
46
46.5
47
47.5
48
CST
Extended
Simple
Figure 47: Fourth mode
58
b = 6 mm, d = b/8mm, d′ = b/4 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20CST
Extended
Simple
Figure 48: The first two modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20CST
Extended
Simple
Figure 49: Second mode
59
b = 10 mm, d = b/8mm, d′ = b/4 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7CST
Extended
Simple
Figure 50: First mode
b = 1.1 mm, d = 0.1mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
CST
Extended
Simple
Figure 51: The first ten modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
119
119.5
120
120.5
121
121.5CST
Extended
Simple
Figure 52: Ninth mode
60
b = 2.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
20
30
40
50
60
CST
Extended
Simple
Figure 53: First five modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
45
45.5
46
46.5
47
CST
Extended
Simple
Figure 54: Fourth mode
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
60
60.5
61
61.5
CST
Extended
Simple
Figure 55: Fifth mode
61
b = 3.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45
CST
Extended
Simple
Figure 56: First four modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
30.2
30.4
30.6
30.8
31
31.2
31.4
31.6
31.8
32 CST
Extended
Simple
Figure 57: Third mode
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
44.9
45
45.1
45.2
45.3
45.4
45.5
45.6
45.7
45.8
45.9 CST
Extended
Simple
Figure 58: Fourth mode
62
b = 4.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30CST
Extended
Simple
Figure 59: The first three modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
30
30.2
30.4
30.6
30.8
31
31.2CST
Extended
Simple
Figure 60: Third mode
b = 5.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2
4
6
8
10
12
14
16
18
20CST
Extended
Simple
Figure 61: First two modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
15.2
15.4
15.6
15.8
16
16.2
16.4
16.6
16.8
17 CST
Extended
Simple
Figure 62: Second mode
63
b = 8.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
CST
Extended
Simple
Figure 63: First two modes
b = 10.1 mm, d = 0.1mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
CST
Extended
Simple
Figure 64: The First mode
64
b = 1 mm, d = b/2mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
120
140CST
Extended
Simple
Figure 65: The first ten modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
135
136
137
138
139
140
141CST
Extended
Simple
Figure 66: Tenth mode
b = 2 mm, d = b/2mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
120
140CST
Extended
Simple
Figure 67: First ten modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
135
136
137
138
139
140
141CST
Extended
Simple
Figure 68: Tenth mode
65
b = 3 mm, d = b/2mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
5
10
15
20
25
30
35
40
45 CST
Extended
Simple
Figure 69: First four modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
45
45.5
46
46.5
47
47.5
48
48.5
49
49.5 CST
Extended
Simple
Figure 70: Fourth mode
b = 4 mm, d = b/2mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
5
10
15
20
25
30
35
CST
Extended
Simple
Figure 71: The first three modes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
30
31
32
33
34
35
36
CST
Extended
Simple
Figure 72: Third mode
66
b = 5 mm, d = b/2mm, d′ = 0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25CST
Extended
Simple
Figure 73: First two modes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
16
17
18
19
20
21
22
23CST
Extended
Simple
Figure 74: Second mode
67
b = 10 mm, d = b/2mm, d′ = 0 mm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
1
2
3
4
5
6
7
8
9
10
CST
Extended
Simple
Figure 75: The first mode
68
Y00=jωεb π
ln
(1 A1
)
T10=√2A
2
T20=−√2(−A
2 1−2A
2 2+1)
T30=−√2A
2(−
6A2 1−4A
2 2+3)
T40=−√2(−3A
4 1+(4−
24A
2 2)A
2 1−8A
4 2+8A
2 2−
1)
T50=√2A
2(30A
4 1+10
(8A
2 2−3)A
2 1+16A
4 2−
20A
2 2+5)
T60=−√2(−10A
6 1+(18−
180A
2 2)A
4 1−3(80A
4 2−48A
2 2+3)A
2 1−
32A
6 2+48A
4 2−
18A
2 2+1)
T70=√2A
2(140A
6 1+21
0(4A
2 2−1)A
4 1+28(24A
4 2−20A
2 2+3)A
2 1+64A
6 2−
112A
4 2+56A
2 2−7)
T80=−√2(−35A
8 1−80
(14A
2 2−
1)A
6 1−60
(56A
4 2−
24A
2 2+1)A
4 1−
16(112A
6 2−
120A
4 2+30A
2 2−
1)A
2 1−
128A
8 2+25
6A
6 2−16
0A
4 2+
32A
2 2−
1)
T90=√2A
2(630A
8 1+42
0(16A
2 2−
3)A
6 1+54(224A
4 2−
140A
2 2+15)A
4 1+36(128A
6 2−
168A
4 2+60A
2 2−
5)A
2 1+256A
8 2−576A
6 2+43
2A
4 2−
120A
2 2+9)
T10,0=−√2( −
126A
10
1−
350
(18A
2 2−
1)A
8 1−350(96A
4 2−32A
2 2+1)A
6 1−30(134
4A
6 2−
112
0A4 2+21
0A
2 2−
5)A
4 1−5(23
04A
8 2−
3584A
6 2+
1680A
4 2−
240A
2 2+5)A
2 1−
512A
10
2+1280A
8 2−11
20A
6 2+40
0A
4 2−50A
2 2+1
)
B. Transfer and Impedance parameters
69
T11,0=√2A
2
( 2772A
10
1+23
10(20A
2 2−
3)A
8 1+61
60(24A
4 2−12A
2 2+1)A
6 1+66(1920A
6 2−
201
6A4 2+560A
2 2−
35)A
4 1+22
(128
0A
8 2−
2304A
6 2+
1344A
4 2−
280A
2 2+15)A
2 1+102
4A
10
2−
2816A
8 2+28
16A
6 2−123
2A4 2+220A
2 2−11
)
T12,0=−√2( −
462A
12
1−
151
2(22A
2 2−1)A
10
1−
630
(440A
4 2−12
0A
2 2+3)A
8 1−
1120(528A
6 2−
360A
4 2+54A
2 2−1)A
6 1−
45(844
8A8 2−10
752A
6 2+
403
2A
4 2−
448A
2 2+7)A
4 1−
12(5632A
10
2−11
520A
8 2+80
64A
6 2−
224
0A4 2+210A
2 2−
3)A
2 1−
2048A
12
2+61
44A
10
2−6912A
8 2+35
84A
6 2−
840A
4 2+72A
2 2−
1)
T13,0=√2A
2
( 1201
2A
12
1+36
036
(8A
2 2−
1)A
10
1+27
30(528A
4 2−220A
2 2+15)A
8 1+3120
(704A
6 2−
616A
4 2+14
0A
2 2−7)A
6 1+78
0(140
8A
8 2−
2112A
6 2+1008A
4 2−
168A
2 2+7)A
4 1+26
(614
4A
10
2−
1408
0A8 2+11
520A
6 2−
4032A
4 2+560A
2 2−
21)A
2 1+40
96A
12
2−
133
12A
10
2+
1664
0A
8 2−
998
4A6 2+2912A
4 2−36
4A
2 2+13
)
T14,0=−√2( −
171
6A14
1−646
8(26A
2 2−1)A
12
1−970
2(208A
4 2−
48A
2 2+1)A
10
1−
1470(457
6A6 2−
2640A
4 2+33
0A
2 2−
5)A
8 1−
420(18
304A
8 2−
1971
2A
6 2+6160A
4 2−
560A
2 2+7)A
6 1−84
(36608A
10
2−63
360A
8 2+36
960A
6 2−840
0A
4 2+630A
2 2−7)A
4 1−
7(53248A
12
2−13
516
8A
10
2+
1267
20A
8 2−537
60A
6 2+1008
0A
4 2−67
2A
2 2+7)A
2 1−
8192A
14
2+2867
2A
12
2−
394
24A
10
2+268
80A
8 2−
9408A
6 2+1568A
4 2−98A
2 2+1
)
T15,0=√2A
2
( 5148
0A
14
1+60
060
(28A
2 2−3)A
12
1+16
632
(728A
4 2−
260A
2 2+15)A
10
1+34
650(832A
6 2−
624A
4 2+120A
2 2−
5)A
8 1+20
0(12
812
8A
8 2−
1647
36A
6 2+6652
8A
4 2−
9240A
2 2+31
5)A
6 1+18
0(4659
2A
10
2−9152
0A
8 2+6336
0A
6 2−18
480A
4 2+21
00A
2 2−
63)A
4 1+120
(716
8A
12
2−
1996
8A
10
2+21
120A
8 2−
105
60A
6 2+252
0A
4 2−252A
2 2+7)A
2 1+1638
4A
14
2−614
40A
12
2+921
60A
10
2−
7040
0A
8 2+28
800A
6 2−
6048A
4 2+
560A
2 2−
15
)
70
All
theZ-p
ara
meters
should
bedivided
withjωεb/π
Z11=A
2 1
Z12=4A
2 1A
2
Z13=3A
2 1
( A2 1+4A
2 2−
1)
Z14=8A
2 1A
2
( 3A2 1+4A
2 2−2
)
Z15=5A
2 1
( 2A4 1+3(8A
2 2−
1)A
2 1+16A
4 2−
12A
2 2+1
)
Z16=12A
2 1A
2
( 10A
4 1+4(10A
2 2−3)A
2 1+16A
4 2−16A
2 2+3
)
Z17=7A
2 1
( 5A6 1+10
(12A
2 2−
1)A
4 1+6(40A
4 2−20A
2 2+1)A
2 1+64A
6 2−
80A
4 2+24A
2 2−
1)
Z18=16A
2 1A
2
( 35A
6 1+20(14A
2 2−
3)A
4 1+6(56A
4 2−40A
2 2+5)A
2 1+64A
6 2−
96A
4 2+40A
2 2−
4)
Z19=9A
2 1
( 14A
8 1+35
(16A
2 2−
1)A
6 1+10
(224A
4 2−
84A
2 2+3)A
4 1+2(896A
6 2−840A
4 2+180A
2 2−
5)A
2 1+25
6A
8 2−44
8A
6 2+24
0A
4 2−
40A
2 2+1
)
Z1,10=20A
2 1A
2
( 126A
8 1+280
(6A
2 2−
1)A
6 1+14(288A
4 2−
160A
2 2+15
)A4 1+12
(192A
6 2−224A
4 2+70A
2 2−5)A
2 1+25
6A
8 2−
512A
6 2+
336A
4 2−80A
2 2+5
)
71
Z1,11=11A
2 1
( 42A
10
1+126(20A
2 2−
1)A
8 1+140(120A
4 2−36A
2 2+1)A
6 1+70(384A
6 2−28
8A
4 2+48A
2 2−
1)A
4 1+3(3840A
8 2−
537
6A6 2+
2240A
4 2−280A
2 2+5)A
2 1+102
4A
10
2−23
04A
8 2+17
92A
6 2−
560A
4 2+60A
2 2−
1)
Z1,12=24A
2 1A
2
( 462A
10
1+420
(22A
2 2−3)A
8 1+420
(88A
4 2−
40A
2 2+3)A
6 1+80
(528A
6 2−
504A
4 2+126A
2 2−7)A
4 1+(14080A
8 2−
2304
0A
6 2+12
096A
4 2−22
40A
2 2+105
)A2 1+2(512A
10
2−
1280A
8 2+11
52A
6 2−
448A
4 2+70A
2 2−3)
)
Z1,13=13A
2 1
( 132A
12
1+46
2(24A
2 2−
1)A
10
1+630
(176A
4 2−
44A
2 2+1)A
8 1+42
0(704A
6 2−440A
4 2+60A
2 2−
1)A
6 1+20(126
72A
8 2−
14784A
6 2+50
40A
4 2−
504A
2 2+7)A
4 1+3(22528A
10
2−
422
40A
8 2+268
80A
6 2−
6720A
4 2+560A
2 2−
7)A
2 1+409
6A
12
2−
112
64A
10
2+
11520A
8 2−
5376A
6 2+1120A
4 2−
84A
2 2+1
)
Z1,14=28A
2 1A
2
( 1716A
12
1+1848
(26A
2 2−
3)A
10
1+1386
(208A
4 2−80A
2 2+5)A
8 1+120
(4576A
6 2−
3696A
4 2+77
0A
2 2−
35)A
6 1+20
(1830
4A
8 2−
253
44A
6 2+110
88A
4 2−1680A
2 2+63)A
4 1+24(3328A
10
2−
704
0A8 2+528
0A
6 2−
1680A
4 2+21
0A
2 2−
7)A
2 1+4096A
12
2−
12288A
10
2+
140
80A
8 2−
7680A
6 2+20
16A
4 2−
224A
2 2+7
)
Z1,15=15A
2 1
( 429A
14
1+1716
(28A
2 2−
1)A
12
1+924(728A
4 2−
156A
2 2+3)A
10
1+462
(5824A
6 2−31
20A
4 2+36
0A
2 2−5)A
8 1+21
0(183
04A
8 2
−18
304A
6 2+52
80A
4 2−440A
2 2+5)A
6 1+4(5125
12A
10
2−82
368
0A
8 2+44
352
0A
6 2−
924
00A
4 2+6300A
2 2−
63)A
4 1+4(93
184A
12
2−
219
648A
10
2+1900
80A
8 2−
73920A
6 2+12
600A
4 2−
756A
2 2+7)A
2 1+163
84A
14
2−53
248A
12
2+67
584A
10
2−
4224
0A8 2+13
440A
6 2−
201
6A4 2+112A
2 2−
1)
72
Z22=2( A
4 1+8A
2 2A
2 1
)
Z23=12A
2 1A
2
( 2A2 1+4A
2 2−
1)
Z24=8A
2 1
( A4 1+(18A
2 2−
1)A
2 1+8A
2 2(2A
2 2−
1))
Z25=20A
2 1A
2
( 6A4 1+(32A
2 2−
6)A
2 1+16A
4 2−
12A
2 2+1
)
Z26=6A
2 1
( 5A6 1+8(20A
2 2−1)A
4 1+(400A
4 2−14
4A
2 2+3)A
2 1+8A
2 2(16A
4 2−16A
2 2+3))
Z27=28A
2 1A
2
( 20A
6 1+10
(20A
2 2−
3)A
4 1+4(72A
4 2−
40A
2 2+3)A
2 1+64A
6 2−80A
4 2+24A
2 2−1
)
Z28=16A
2 1
( 7A
8 1+5(70A
2 2−
3)A
6 1+10
(168A
4 2−48A
2 2+1)A
4 1+2(784A
6 2−
600A
4 2+90A
2 2−1)A
2 1+16A
2 2(16A
6 2−
24A
4 2+10A
2 2−
1)
)
Z29=36A
2 1A
2
( 70A
8 1+140
(8A
2 2−
1)A
6 1+2(1568A
4 2−700A
2 2+45)A
4 1+4(512A
6 2−
504A
4 2+120A
2 2−5)A
2 1+256A
8 2−
448A
6 2+24
0A
4 2−
40A
2 2+1
)
Z2,10=10A
2 1
( 42A
10
1+112
(27A
2 2−
1)A
8 1+35
(672A
4 2−
160A
2 2+3)A
6 1+8(537
6A
6 2−
336
0A4 2+420A
2 2−
5)A
4 1+(20736A
8 2−25
088A
6 2+
8400A
4 2−
720A
2 2+5)A
2 1+8A
2 2(256A
8 2−512A
6 2+33
6A
4 2−
80A
2 2+5))
Z2,11=44A
2 1A
2
( 252A
10
1+210(28A
2 2−
3)A
8 1+560(48A
4 2−18A
2 2+1)A
6 1+(34560A
6 2−28
224A
4 2+56
00A
2 2−
210
)A4 1+2(640
0A
8 2−
921
6A6 2+403
2A
4 2−56
0A
2 2+15
)A2 1+10
24A
10
2−230
4A
8 2+1792A
6 2−
560A
4 2+60A
2 2−
1)
73
Z2,12=24A
2 1
( 66A
12
1+42
(154A
2 2−
5)A
10
1+84(880A
4 2−
180A
2 2+3)A
8 1+14
0(1584A
6 2−
840A
4 2+90A
2 2−
1)A
6 1+5(422
40A
8 2−
43008A
6 2+
12096A
4 2−89
6A2 2+7)A
4 1+(6195
2A
10
2−
103
680A
8 2+564
48A
6 2−
1120
0A
4 2+63
0A
2 2−3)A
2 1+8A
2 2(512A
10
2−128
0A8 2+11
52A
6 2−
448A
4 2+70A
2 2−
3))
Z2,13=52A
2 1A
2
( 924A
12
1+92
4(32A
2 2−
3)A
10
1+42(4752A
4 2−
154
0A2 2+75)A
8 1+240
(1760A
6 2−12
32A
4 2+21
0A
2 2−7)A
6 1+20
(1548
8A
8 2−
19008A
6 2+705
6A
4 2−840A
2 2+21)A
4 1+2(36864A
10
2−
704
00A
8 2+460
80A
6 2−
1209
6A4 2+11
20A
2 2−21
)A2 1+40
96A
12
2−
11264A
10
2+
11520A
8 2−53
76A
6 2+11
20A
4 2−
84A
2 2+1
)
Z2,14=14A
2 1
( 429A
14
1+52
8(104A
2 2−
3)A
12
1+462
(1872A
4 2−
336A
2 2+5)A
10
1+16
80(228
8A
6 2−
105
6A4 2+99A
2 2−1)A
8 1+30
(201
344A
8 2−
17740
8A
6 2+43
120A
4 2−
2800A
2 2+21)A
6 1+16(2196
48A
10
2−316
800A
8 2+147
840A
6 2−
252
00A
4 2+126
0A
2 2−
7)A
4 1+(692
224A
12
2−
14868
48A
10
2+114
048
0A
8 2−37
632
0A
6 2+50
400A
4 2−
201
6A2 2+7)A
2 1+8A
2 2(4096A
12
2−
12288A
10
2+140
80A
8 2−76
80A
6 2+20
16A
4 2−
224A
2 2+7)
)
Z2,15=60A
2 1A
2
( 3432A
14
1+12
012(12A
2 2−
1)A
12
1+184
8(728A
4 2−
208A
2 2+9)A
10
1+46
2(915
2A
6 2−561
6A
4 2+840A
2 2−25
)A8 1+40(12
8128A
8 2−
1372
80A
6 2+44
352A
4 2−
4620A
2 2+10
5)A
6 1+4(605
696A
10
2−1006
720A
8 2+5702
40A
6 2−12
9360A
4 2+10
500A
2 2−
189)A
4 1+8(50
176A
12
2−
1198
08A
10
2+10
5600A
8 2−
42240A
6 2+756
0A
4 2−504A
2 2+7)A
2 1+1638
4A
14
2−53
248A
12
2+675
84A
10
2−
4224
0A
8 2+13
440A
6 2−
2016A
4 2+
112A
2 2−1
)
74
Z33=3( 4A
6 1−
(6−
48A
2 2)A
4 1+3(1−
4A2 2)2A
2 1
)
Z34=24A
2 1A
2
( 6A4 1+7(4A
2 2−
1)A
2 1+2(8A
4 2−6A
2 2+1)
)
Z35=15A
2 1
( 3A
6 1+(72A
2 2−6)A
4 1+4(44A
4 2−18A
2 2+1)A
2 1+64A
6 2−
64A
4 2+16A
2 2−
1)
Z36=12A
2 1A
2
( 60A
6 1+2(260A
2 2−
51)A
4 1+6(128A
4 2−76A
2 2+9)A
2 1+3(4A
2 2−
3)(1−
4A2 2)2)
Z37=21A
2 1
( 8A
8 1+20
(16A
2 2−
1)A
6 1+6(240A
4 2−
80A
2 2+3)A
4 1+(1408A
6 2−
112
0A4 2+216A
2 2−
7)A
2 1+25
6A
8 2−
384A
6 2+17
6A
4 2−
28A
2 2+1
)
Z38=48A
2 1A
2
( 70A
8 1+5(19
6A
2 2−
31)A
6 1+24(112A
4 2−
55A
2 2+5)A
4 1+2(928A
6 2−936A
4 2+26
0A
2 2−
19)A
2 1+4(64A
8 2−
112A
6 2+64A
4 2−
14A
2 2+1)
)
Z39=9A
2 1
( 70A
10
1+210
(20A
2 2−
1)A
8 1+240(126A
4 2−35A
2 2+1)A
6 1+2(2777
6A
6 2−
1848
0A4 2+288
0A
2 2−65
)A4 1+3(947
2A
8 2−11
648A
6 2+
4320A
4 2−
520A
2 2+11)A
2 1+3(1−4A
2 2)2(64A
6 2−96A
4 2+36A
2 2−1)
)
Z3,10=60A
2 1A
2
( 252A
10
1+14(372A
2 2−
49)A
8 1+28
(816A
4 2−
340A
2 2+25
)A6 1+6(499
2A
6 2−42
56A
4 2+980A
2 2−55
)A4 1+2(58
88A
8 2−
8576A
6 2+
4032A
4 2−
700A
2 2+35)A
2 1+102
4A
10
2−
2304A
8 2+18
56A
6 2−656A
4 2+100A
2 2−
5)
75
Z3,11=33A
2 1
( 72A
12
1+25
2(24A
2 2−
1)A
10
1+70
(912A
4 2−
216A
2 2+5)A
8 1+35(5376A
6 2−30
72A
4 2+40
0A
2 2−7)A
6 1+6(307
20A
8 2−32
256A
6 2+
1008
0A4 2−980A
2 2+15)A
4 1+8(7168A
10
2−
12096A
8 2+694
4A
6 2−
1610A
4 2+135A
2 2−
2)A
2 1+409
6A
12
2−
10240A
10
2+94
72A
8 2−
4032A
6 2+80
0A
4 2−
64A
2 2+1
)
Z3,12=24A
2 1A
2
( 277
2A12
1+42(189
2A
2 2−21
3)A
10
1+252
(2024A
4 2−
730A
2 2+45
)A8 1+60
(17952A
6 2−
132
72A
4 2+260
4A
2 2−
119)A
6 1+2(40
8320A
8 2−
5126
40A
6 2+2056
32A
4 2−
29680A
2 2+11
55)A
4 1+3(686
08A
10
2−
1318
40A
8 2+8985
6A
6 2−26
432A
4 2+322
0A
2 2−
117)A
2 1+6(1−
4A2 2)2(128A
8 2−
288A
6 2+21
6A
4 2−
58A
2 2+3))
Z3,13=39A
2 1
( 231A
14
1+924
(28A
2 2−1)A
12
1+504
(748A
4 2−15
4A
2 2+3)A
10
1+42
(387
20A
6 2−
1936
0A4 2+2160A
2 2−31
)A8 1+30(858
88A
8 2−
7884
8A6 2+
2128
0A4 2−1736A
2 2+21)A
6 1+24(6476
8A
10
2−950
40A
8 2+470
40A
6 2−9240A
4 2+630A
2 2−7)A
4 1+2(16179
2A
12
2−
34918
4A
10
2+27
6480A
8 2−
9945
6A6 2+16
240A
4 2−
1008A
2 2+11)A
2 1+163
84A
14
2−
4915
2A12
2+57
344A
10
2−
330
24A
8 2+985
6A
6 2−
1456A
4 2+88A
2 2−1
)
Z3,14=84A
2 1A
2
( 3432A
14
1+132
(988A
2 2−97
)A12
1+277
2(416A
4 2−
132A
2 2+7)A
10
1+30
(11897
6A
6 2−
7761
6A4 2+132
44A
2 2−
511)A
8 1+24
0(18
304A
8 2−
2024
0A6 2+70
84A
4 2−875A
2 2+28)A
6 1+12(179
712A
10
2−30
272
0A
8 2+17
952
0A
6 2−
453
60A
4 2+462
0A
2 2−
133)A
4 1+2(188
416A
12
2−
4515
84A
10
2+
40832
0A
8 2−
1747
20A
6 2+3628
8A
4 2−
3304A
2 2+91
)A2 1+16
384A
14
2−
5324
8A12
2+6860
8A
10
2−448
00A
8 2+157
44A
6 2−2912A
4 2+252A
2 2−7
)
Z3,15=15A
2 1
( 2288A
16
1+102
96(32A
2 2−1)A
14
1+277
2(2288A
4 2−
416A
2 2+7)A
12
1+46
2(8153
6A
6 2−
361
92A
4 2+352
8A
2 2−43)A
10
1+12
6(69
555
2A
8 2−
56742
4A
6 2+13
464
0A
4 2−946
0A
2 2+95)A
8 1+12(717
516
8A
10
2−
93350
40A
8 2+40
656
00A
6 2−693
000A
4 2+39
900A
2 2−
357)A
6 1+4(875
929
6A
12
2−
16693
248A
10
2+11
594
880A
8 2−
362
208
0A6 2+504
000A
4 2−
2570
4A
2 2+217)A
4 1+3(17367
04A
14
2−
457
932
8A12
2+466
329
6A
10
2−
2323
200A
8 2+
5913
60A
6 2−
725
76A
4 2+3472A
2 2−29
)A2 1+3(1−
4A2 2)2(4096A
12
2−
122
88A
10
2+1382
4A
8 2−71
04A
6 2+15
84A
4 2−
108A
2 2+1)
)
76
Z44=4
( 9A
8 1−(16−
288A
2 2)A
6 1+8(84A
4 2−
36A
2 2+1)A
4 1+64A
2 2(1−
2A2 2)2A
2 1
)
Z45=40A
2 1A
2
( 18A
6 1+6(28A
2 2−
5)A
4 1+(240A
4 2−
148A
2 2+15
)A2 1+64A
6 2−
80A
4 2+28A
2 2−
2)
Z46=24A
2 1
( 6A8 1+2(150A
2 2−
7)A
6 1+(1360A
4 2−
448A
2 2+11
)A4 1+(1312A
6 2−107
2A4 2+198A
2 2−3)A
2 1+8A
2 2(32A
6 2−48A
4 2+22A
2 2−3)
)
Z47=56A
2 1A
2
( 60A
8 1+10(92A
2 2−13
)A6 1+8(312A
4 2−
155A
2 2+12
)A4 1+(1728A
6 2−1776A
4 2+488A
2 2−27
)A2 1+256A
8 2−
448A
6 2+256A
4 2−
52A
2 2+2
)
Z48=16A
2 1
( 35A
10
1+20(126A
2 2−
5)A
8 1+5(3752A
4 2−
1000A
2 2+21
)A6 1+16
(2128A
6 2−144
0A
4 2+210A
2 2−3)A
4 1+8(220
8A
8 2−
2752A
6 2+
1020A
4 2−108A
2 2+1)A
2 1+64A
2 2(1−
2A2 2)2(8A
4 2−
8A2 2+1))
Z49=72A
2 1A
2
( 210A
10
1+280(17A
2 2−2)A
8 1+(210
56A
4 2−
868
0A2 2+550
)A6 1+8(3424A
6 2−
2968A
4 2+66
5A
2 2−
30)A
4 1+(1100
8A
8 2−
161
92A
6 2+763
2A
4 2−
1240A
2 2+43)A
2 1+2(512A
10
2−
115
2A8 2+928A
6 2−
320A
4 2+42A
2 2−
1))
Z4,10=40A
2 1
( 54A
12
1+14(378A
2 2−13)A
10
1+14
(4176A
4 2−
936A
2 2+17)A
8 1+2(86016A
6 2−
49280A
4 2+59
50A
2 2−75)A
6 1+(1681
92A
8 2−
1792
00A
6 2+5544
0A
4 2−
4800A
2 2+45
)A4 1+(53760A
10
2−
9139
2A8 2+52
640A
6 2−
117
60A
4 2+810A
2 2−5)A
2 1+8A
2 2(512A
10
2−
1280A
8 2+11
84A
6 2−
496A
4 2+90A
2 2−
5))
Z4,11=88A
2 1A
2
( 756A
12
1+126
(188A
2 2−
19)A
10
1+84(1848A
4 2−
650A
2 2+35)A
8 1+2(1632
00A
6 2−1216
32A
4 2+2296
0A
2 2−
875)A
6 1+
2(124
160A
8 2−
15782
4A
6 2+63
168A
4 2−854
0A2 2+255
)A4 1+(64512A
10
2−
12467
2A
8 2+85
248A
6 2−
245
28A
4 2+266
0A
2 2−
63)A
2 1+
2(204
8A12
2−56
32A
10
2+588
8A
8 2−
2912A
6 2+680A
4 2−
62A
2 2+1))
77
Z4,12=12A
2 1
( 693A
14
1+2688
(33A
2 2−
1)A
12
1+168(809
6A
4 2−
156
8A2 2+25)A
10
1+336
0(1760A
6 2−
872A
4 2+90A
2 2−1)A
8 1+5(186
700
8A
8 2−
1731
072A
6 2+4583
04A
4 2−
33600A
2 2+28
7)A
6 1+16
(35481
6A
10
2−526
080A
8 2+260
736A
6 2−
4928
0A
4 2+28
70A
2 2−
19)A
4 1+8(152
576A
12
2−
330
752A
10
2+2626
56A
8 2−93
184A
6 2+14
140A
4 2−684A
2 2+3)A
2 1+64A
2 2(1−
2A2 2)2(256A
8 2−
512A
6 2+32
0A
4 2−64A
2 2+3)
)
Z4,13=104A
2 1A
2
( 2772A
14
1+92
4(124A
2 2−11)A
12
1+42(248
16A
4 2−
7612A
2 2+357)A
10
1+60
(53856A
6 2−351
12A
4 2+572
6A
2 2−
189)A
8 1+
60(6617
6A
8 2−
7392
0A
6 2+2564
8A
4 2−
2968A
2 2+77
)A6 1+2(986
112A
10
2−
1675
520A
8 2+9964
80A
6 2−
24595
2A
4 2+22
680A
2 2−48
3)A
4 1+
(3563
52A
12
2−
8570
88A
10
2+77
696
0A
8 2−
329
472A
6 2+651
84A
4 2−
5068A
2 2+87)A
2 1+2(8192A
14
2−266
24A
12
2+343
04A
10
2−
222
72A
8 2+
7616A
6 2−128
8A
4 2+86A
2 2−
1)
)
Z4,14=56A
2 1
( 572A
16
1+13
2(702A
2 2−19)A
14
1+66
(284
96A
4 2−
4864A
2 2+69)A
12
1+42
(26998
4A
6 2−
117
744A
4 2+106
26A
2 2−105
)A10
1+
12(21964
80A
8 2−17
98720A
6 2+41
580
0A
4 2−264
60A
2 2+203
)A8 1+12(215
987
2A
10
2−
2837
120A
8 2+1232
000A
6 2−
20160
0A
4 2+
101
50A
2 2−
63)A
6 1+(10702
848A
12
2−
20545
536A
10
2+14
319
360A
8 2−
440
832
0A6 2+574
560A
4 2−
2419
2A
2 2+119)A
4 1+(164
659
2A
14
2−
43540
48A
12
2+444
364
8A
10
2−
2200
320A
8 2+5416
32A
6 2−
59808A
4 2+21
42A
2 2−
7)A
2 1+8A
2 2(8192A
14
2−
28672A
12
2+40
448A
10
2−29
440A
8 2+
11712A
6 2−
2464A
4 2+23
8A
2 2−
7))
Z4,15=120A
2 1A
2
( 1029
6A
16
1+1716
(316A
2 2−
25)A
14
1+3696
(1768A
4 2−
481A
2 2+20)A
12
1+462
(607
36A
6 2−
35152A
4 2+50
48A
2 2−14
7)A
10
1+
12(422
822
4A
8 2−420
5344A
6 2+128
620
8A
4 2−
1301
30A
2 2+2975
)A8 1+4(10203
648A
10
2−
15466
880A
8 2+81
312
00A
6 2−175
560
0A
4 2+
140
700A
2 2−
2667)A
6 1+16(888
832A
12
2−
19052
80A
10
2+152
7680A
8 2−567
600A
6 2+974
40A
4 2−
6573A
2 2+10
5)A
4 1+(1884
160A
14
2−
543
539
2A
12
2+6174
720A
10
2−
35059
20A
8 2+10
387
20A
6 2−151
200A
4 2+884
8A
2 2−115)A
2 1+65
536A
16
2−245
760A
14
2+37
683
2A
12
2−
304
128A
10
2+1382
40A
8 2−
34944A
6 2+44
80A
4 2−
228A
2 2+2
)
78
Z55=5( 36
A10
1+90
(16A
2 2−
1)A
8 1+80
(84A
4 2−27A
2 2+1)A
6 1+10(640A
6 2−
528A
4 2+96A
2 2−3)A
4 1+5(16A
4 2−12A
2 2+1)2A
2 1
)
Z56=60A
2 1A
2
( 60A
8 1+44(20A
2 2−
3)A
6 1+4(608A
4 2−
298A
2 2+25)A
4 1+2(832A
6 2−86
4A4 2+23
6A
2 2−
15)A
2 1+25
6A
8 2−
448A
6 2+25
6A
4 2−
52A
2 2+3
)
Z57=35A
2 1
( 20A
10
1+60(20A
2 2−
1)A
8 1+4(2240A
4 2−60
0A
2 2+17
)A6 1+4(4096A
6 2−
276
0A4 2+40
8A
2 2−
9)A
4 1+(844
8A
8 2−
106
24A
6 2+
3936A
4 2−
432A
2 2+9)A
2 1+102
4A
10
2−
2048A
8 2+14
08A
6 2−
384A
4 2+36A
2 2−1
)
Z58=80A
2 1A
2
( 210A
10
1+10
(448A
2 2−
57)A
8 1+(20048A
4 2−
8220A
2 2+57
5)A
6 1+8(326
4A
6 2−
2832A
4 2+635A
2 2−
33)A
4 1+2(52
48A
8 2−
7776A
6 2+36
72A
4 2−
604A
2 2+27)A
2 1+4(256A
10
2−
576A
8 2+46
4A
6 2−160A
4 2+22A
2 2−
1)
)
Z59=45A
2 1
( 60A
12
1+210
(24A
2 2−
1)A
10
1+10(548
8A
4 2−
1260A
2 2+29
)A8 1+8(203
84A
6 2−11
620A
4 2+14
50A
2 2−
25)A
6 1+8(19
904A
8 2−
21280A
6 2+65
80A
4 2−600A
2 2+9)A
4 1+(512
00A
10
2−
875
52A
8 2+505
60A
6 2−
1136
0A4 2+864A
2 2−
13)A
2 1+409
6A
12
2−10
240A
10
2+
9472A
8 2−
396
8A6 2+736A
4 2−
52A
2 2+1
)
79
Z5,10=20A
2 1A
2
( 3780A
12
1+42
0(264A
2 2−
29)A
10
1+14
(51984A
4 2−183
80A
2 2+1095
)A8 1+20(769
92A
6 2−
57232A
4 2+10
920A
2 2−
475)A
6 1+
40(292
48A
8 2−373
44A
6 2+149
52A
4 2−2065A
2 2+75)A
4 1+10(30720A
10
2−596
48A
8 2+4089
6A
6 2−11
808A
4 2+13
40A
2 2−
45)A
2 1+
5(16A
4 2−
12A
2 2+1)2(16A
4 2−20A
2 2+5)
)
Z5,11=55A
2 1
( 189A
14
1+75
6(28A
2 2−1)A
12
1+11
2(2826A
4 2−
567A
2 2+11)A
10
1+210(659
2A
6 2−326
4A
4 2+352A
2 2−5)A
8 1+4(54
8160A
8 2−
5080
32A
6 2+1349
60A
4 2−
10500A
2 2+12
5)A
6 1+4(333
312A
10
2−
4965
12A
8 2+2464
00A
6 2−
47040A
4 2+30
00A
2 2−33)A
4 1+
2(14540
8A
12
2−
3164
16A
10
2+2519
04A
8 2−
89600A
6 2+13
920A
4 2−
792A
2 2+9)A
2 1+1638
4A
14
2−
49152A
12
2+573
44A
10
2−32
768A
8 2+
9472A
6 2−
129
6A4 2+72A
2 2−
1)
Z5,12=120A
2 1A
2
( 2772A
14
1+84
(1276A
2 2−
123)A
12
1+42
(23056A
4 2−
7172A
2 2+371)A
10
1+60(5033
6A
6 2−
327
60A
4 2+544
6A
2 2−
203)A
8 1+
2(18585
60A
8 2−
207
840
0A6 2+722
064A
4 2−
8596
0A2 2+26
25)A
6 1+2(92467
2A
10
2−
157
824
0A8 2+940
416A
6 2−
2334
08A
4 2+22
540A
2 2−
613)A
4 1+(3399
68A
12
2−8202
24A
10
2+74
5216A
8 2−31
667
2A
6 2+635
04A
4 2−
537
2A2 2+141)A
2 1+2(8192A
14
2−26
624A
12
2+34
304A
10
2−
2227
2A8 2+7648A
6 2−13
36A
4 2+10
6A
2 2−
3))
80
Z5,13=65A
2 1
( 616A
16
1+2772
(32A
2 2−
1)A
14
1+16
8(1038
4A
4 2−
184
8A2 2+31)A
12
1+84(1253
12A
6 2−
54912A
4 2+52
08A
2 2−
63)A
10
1+30
(819
456A
8 2−
6702
08A
6 2+1562
40A
4 2−
10584A
2 2+10
5)A
8 1+3(806
502
4A
10
2−
1061
632
0A8 2+4614
400A
6 2−76
608
0A
4 2+42
000A
2 2−
371
)A6 1+
2(501
3504A
12
2−9664
512A
10
2+67
507
20A
8 2−
208
5888A
6 2+278
880A
4 2−
133
56A
2 2+113)A
4 1+8(19660
8A
14
2−52
121
6A
12
2+53
2992A
10
2−
2643
84A
8 2+6563
2A
6 2−76
02A
4 2+33
9A
2 2−3)A
2 1+65
536A
16
2−
229
376A
14
2+323
584A
12
2−
23552
0A
10
2+93
952A
8 2−
2016
0A6 2+21
44A
4 2−
96A
2 2+1
)
Z5,14=140A
2 1A
2
( 10296A
16
1+264
(1924A
2 2−
165)A
14
1+528
(11440A
4 2−
3177A
2 2+145)A
12
1+12(21690
24A
6 2−
125
664
0A4 2+18
5108A
2 2−
6027
)A10
1+30(157
414
4A
8 2−
1564
992A
6 2+4804
80A
4 2−
50260A
2 2+13
23)A
8 1+48
(79206
4A
10
2−1203
840A
8 2+6336
00A
6 2−
13790
0A
4 2+11
585A
2 2−266
)A6 1+2(66600
96A
12
2−
143
308
80A
10
2+115
174
40A
8 2−
4291
200A
6 2+7479
36A
4 2−
54264A
2 2+115
5)A
4 1+
2(9011
20A
14
2−
26050
56A
12
2+296
448
0A
10
2−
1685
760A
8 2+5022
72A
6 2−
75264A
4 2+49
84A
2 2−105
)A2 1+6553
6A
16
2−24
576
0A
14
2+
3768
32A
12
2−
3041
28A
10
2+13
8496A
8 2−
35456A
6 2+481
6A
4 2−
308A
2 2+7
)
Z5,15=15A
2 1
( 10296A
18
1+514
80(36A
2 2−
1)A
16
1+13
20(3515
2A
4 2−
5616A
2 2+83)A
14
1+18
480(1996
8A
6 2−
785
2A4 2+664A
2 2−
7)A
12
1+
210(5674
240A
8 2−41
550
08A
6 2+86
345
6A
4 2−
517
44A
2 2+441
)A10
1+6(2880
317
44A
10
2−33
935
6160A
8 2+1312
326
40A
6 2−
192
808
00A
4 2+
92610
0A
2 2−
6881
)A8 1+20
(579
60448A
12
2−1001
594
88A
10
2+6226
176
0A
8 2−
17001
600A
6 2+19
950
00A
4 2−
82572A
2 2+56
7)A
6 1+
10(34766
848A
14
2−
82694
144A
12
2+75
333
632A
10
2−3303
1680A
8 2+7190
400A
6 2−
7230
72A
4 2+27
216A
2 2−18
3)A
4 1+5(819
200
0A
16
2−
256
57344A
14
2+321
863
68A
12
2−
2068
070
4A10
2+72
23040A
8 2−13
440
00A
6 2+12
185
6A
4 2−
439
2A2 2+31)A
2 1+
5(16A
4 2−12A
2 2+1)2(102
4A
10
2−
2560A
8 2+2240A
6 2−
800A
4 2+100A
2 2−
1))
81
Z66=6( 1
00A
12
1+288(25A
2 2−1)A
10
1+6(880
0A
4 2−
240
0A2 2+51)A
8 1+16(6080A
6 2−40
80A
4 2+61
2A
2 2−9)A
6 1+3(166
40A
8 2−
2099
2A
6 2+
7776A
4 2−864A
2 2+9)A
4 1+24A
2 2(16A
4 2−
16A
2 2+3)
2A
2 1
)
Z67=84A
2 1A
2
( 200A
10
1+20
(220A
2 2−27)A
8 1+20(976A
4 2−
404A
2 2+27)A
6 1+4(6400A
6 2−
5536A
4 2+12
50A
2 2−
61)A
4 1+8(128
0A
8 2−
190
4A
6 2+900A
4 2−14
9A
2 2+6)A
2 1+10
24A
10
2−
230
4A8 2+1856A
6 2−64
0A
4 2+88A
2 2−
3)
Z68=48A
2 1
( 50A
12
1+10
(490A
2 2−
17)A
10
1+5(106
40A
4 2−
244
8A2 2+45)A
8 1+5(31584A
6 2−18
064A
4 2+225
0A
2 2−
29)A
6 1+
2(774
40A
8 2−8268
8A
6 2+25
640A
4 2−232
0A
2 2+23)A
4 1+2(2483
2A
10
2−
42624A
8 2+24
656A
6 2−
556
0A4 2+414A
2 2−
3)A
2 1+
16A
2 2(256A
10
2−
640A
8 2+592A
6 2−24
8A
4 2+46A
2 2−
3))
Z69=36A
2 1A
2
( 2100A
12
1+560(115A
2 2−
12)A
10
1+30
(14000A
4 2−
498
4A2 2+279
)A8 1+12(742
40A
6 2−
5521
6A4 2+10
580A
2 2−
425)A
6 1+
8(8480
0A
8 2−
108
288A
6 2+434
16A
4 2−
6005A
2 2+195)A
4 1+24(7424A
10
2−
144
64A
8 2+993
6A
6 2−2876A
4 2+325A
2 2−9)A
2 1+
3(1−
4A2 2)2(256A
8 2−57
6A
6 2+43
2A
4 2−112A
2 2+3))
Z6,10=30A
2 1
( 315A
14
1+112(360A
2 2−11)A
12
1+14
(43440A
4 2−
862
4A2 2+139
)A10
1+16(1656
48A
6 2−82
432A
4 2+87
57A
2 2−
99)A
8 1+
4(10531
20A
8 2−
975
744A
6 2+2601
20A
4 2−
1980
0A2 2+17
5)A
6 1+32
(80128A
10
2−119
488A
8 2+593
60A
6 2−
11360A
4 2+70
0A
2 2−
5)A
4 1+
(5611
52A
12
2−
1224
704A
10
2+97
689
6A
8 2−
348
160A
6 2+540
80A
4 2−2880A
2 2+15)A
2 1+8A
2 2(409
6A
12
2−
1228
8A10
2+143
36A
8 2−
8192A
6 2+
2368A
4 2−320A
2 2+15)
)
82
Z6,11=13
2A
2 1A
2
( 2520A
14
1+84
(1220A
2 2−
111)A
12
1+28
(33024A
4 2−
102
84A
2 2+497)A
10
1+2(14409
60A
6 2−
94046
4A
4 2+15
610
0A
2 2−
5355
)A8 1+
4(8896
00A
8 2−
99456
0A
6 2+34
630
4A
4 2−4102
0A
2 2+11
25)A
6 1+8(22144
0A
10
2−
378
560A
8 2+22
582
4A
6 2−
561
68A
4 2+53
65A
2 2−12
5)A
4 1+
2(1638
40A
12
2−
396
288A
10
2+36
0704A
8 2−
15353
6A
6 2+308
16A
4 2−
254
0A2 2+51)A
2 1+1638
4A
14
2−
5324
8A
12
2+68
608A
10
2−44
544A
8 2+
152
96A
6 2−265
6A
4 2+19
6A
2 2−
3)
Z6,12=24A
2 1
( 154
0A16
1+226
8(110A
2 2−
3)A
14
1+1134
(4400A
4 2−768A
2 2+11)A
12
1+14(214
473
6A
6 2−94
3056A
4 2+87
318A
2 2−
875)A
10
1+
90(780
032A
8 2−
63884
8A
6 2+14
918
4A
4 2−
9800A
2 2+77)A
8 1+30(2309
120A
10
2−303
974
4A8 2+13
231
68A
6 2−
21963
2A
4 2+11
550A
2 2−
75)A
6 1+
(2873
344
0A12
2−554
926
08A
10
2+388
108
80A
8 2−1201
356
8A
6 2+1601
040A
4 2−72
000A
2 2+38
7)A
4 1+3(15
155
20A
14
2−
40263
68A
12
2+41
24160A
10
2−
20487
68A
8 2+50
915
2A
6 2−
583
20A
4 2+2322A
2 2−
9)A
2 1+24A
2 2(16A
4 2−16A
2 2+3)2(32A
6 2−
48A
4 2+18A
2 2−
1)
)
Z6,13=15
6A
2 1A
2
( 9240A
16
1+18
48(260A
2 2−21)A
14
1+33
6(1707
2A
4 2−471
9A
2 2+201
)A12
1+12(2056
384A
6 2−11
95040A
4 2+17
4748A
2 2−524
3)A
10
1+
30(149
5296A
8 2−14
875
52A
6 2+457
632A
4 2−4734
8A
2 2+1127
)A8 1+20(1809
408A
10
2−
27512
32A
8 2+14
496
00A
6 2−
31584
0A
4 2+26
096A
2 2−
525
)A6 1+2(63
488
00A
12
2−13
687
808A
10
2+110
156
80A
8 2−
4110
720A
6 2+7161
28A
4 2−
50540A
2 2+88
7)A
4 1+2(86
835
2A
14
2−25
14944A
12
2+
286
6176A
10
2−16
320
00A
8 2+48
6784A
6 2−725
76A
4 2+455
6A
2 2−69)A
2 1+65
536A
16
2−
2457
60A
14
2+37
6832A
12
2−
30412
8A
10
2+13
8496A
8 2−
35392A
6 2+47
20A
4 2−
268A
2 2+3
)
83
Z6,14=42A
2 1
( 343
2A
18
1+2112
(325A
2 2−8)A
16
1+132(133
120A
4 2−20
736A
2 2+26
7)A
14
1+192
(72758
4A
6 2−2862
64A
4 2+2349
6A
2 2−
211
)A12
1+
6(7515
622
4A8 2−
55154
176A
6 2+11
447
744A
4 2−
661
696A
2 2+465
5)A
10
1+48(1365
4784A
10
2−16
093
440A
8 2+62
339
20A
6 2−
9128
00A
4 2+
418
95A
2 2−
245)A
8 1+(44036
096
0A
12
2−
7615
365
12A
10
2+47
384
8320A
8 2−
12956
160
0A
6 2+15
103
200A
4 2−
588
000A
2 2+293
3)A
6 1+
8(1654
784
0A14
2−
394
36288A
12
2+359
772
16A
10
2−
1579
776
0A8 2+3440
640A
6 2−34
160
0A
4 2+11
732A
2 2−
49)A
4 1+(157
941
76A
16
2−
495
452
16A
14
2+622
346
24A
12
2−
4003
430
4A
10
2+13
996
800A
8 2−259
840
0A
6 2+229
264A
4 2−7056A
2 2+21
)A2 1+8A
2 2(6553
6A16
2−
262
144A
14
2+4341
76A
12
2−
38502
4A
10
2+197
376A
8 2−
588
80A
6 2+9744A
4 2−
784A
2 2+21
))
Z6,15=60A
2 1A
2
( 10296
0A
18
1+34
32(194
0A
2 2−
141)A
16
1+79
2(1283
36A
4 2−
318
76A
2 2+121
5)A
14
1+1584
(3644
16A
6 2−19
073
6A
4 2+249
13A
2 2−
665)A
12
1+16
(90586
496A
8 2−
811
942
56A
6 2+223
617
24A
4 2−
2052
897A
2 2+4321
8)A
10
1+18(9569
996
8A
10
2−
13102
003
2A
8 2+61
8203
52A
6 2−
119
775
04A
4 2+871
640A
2 2−1543
5)A
8 1+24
(40785
920A
12
2−
7921
305
6A
10
2+5708
736
0A
8 2−18
955
200A
6 2+29
148
00A
4 2−
1797
60A
2 2+
277
9)A
6 1+2(127
43475
2A
14
2−332
931
072A
12
2+340
116
480A
10
2−1724
236
80A
8 2+4544
640
0A
6 2−
59330
88A
4 2+32
328
8A
2 2−
442
5)A
4 1+
12(222
822
4A16
2−752
844
8A
14
2+1034
0352A
12
2−
74265
60A
10
2+29
875
20A
8 2−668
736A
6 2+773
92A
4 2−
3818A
2 2+45)A
2 1+
3(1−4A
2 2)2(1638
4A14
2−
61440A
12
2+92
160A
10
2−
6988
8A8 2+2764
8A
6 2−
5184A
4 2+32
8A
2 2−
3))
84
Z77=7( 4
00A
14
1+1400
(24A
2 2−
1)A
12
1+84(4400A
4 2−
100
0A2 2+23)A
10
1+70(156
16A
6 2−
8960A
4 2+11
04A
2 2−
19)A
8 1+28(3840
0A
8 2−
4096
0A6 2+1272
0A
4 2−
1140A
2 2+17
)A6 1+28
(122
88A
10
2−2112
0A
8 2+12
224A
6 2−276
0A
4 2+204A
2 2−
3)A
4 1+7(−64A
6 2+80A
4 2−
24A
2 2+1)
2A
2 1
)
Z78=11
2A
2 1A
2
( 700A
12
1+750
(28A
2 2−
3)A
10
1+20(6888A
4 2−
2440A
2 2+14
1)A
8 1+5(583
04A
6 2−
4347
2A4 2+8296A
2 2−34
7)A
6 1+
12(18560A
8 2−23
680A
6 2+95
04A
4 2−
131
0A2 2+45)A
4 1+2(2918
4A
10
2−56
960A
8 2+391
68A
6 2−1135
2A
4 2+1280A
2 2−
39)A
2 1+
4(1024A
12
2−
281
6A10
2+2944A
8 2−14
56A
6 2+34
4A
4 2−34A
2 2+1))
Z79=63A
2 1
( 175A
14
1+700(28A
2 2−1)A
12
1+20(1484
0A
4 2−
294
0A2 2+57)A
10
1+10(129
472A
6 2−
64400A
4 2+68
40A
2 2−97)A
8 1+
20(1028
48A
8 2−954
24A
6 2+254
24A
4 2−1940A
2 2+23
)A6 1+8(156
928A
10
2−2339
20A
8 2+1162
88A
6 2−
22240A
4 2+13
80A
2 2−
15)A
4 1+
16(1715
2A12
2−3750
4A
10
2+2995
2A
8 2−
10688A
6 2+166
0A
4 2−90A
2 2+1)A
2 1+1638
4A
14
2−49
152A
12
2+573
44A
10
2−
3276
8A8 2+
9472A
6 2−
128
0A4 2+64A
2 2−1
)
Z7,10=140A
2 1A
2
( 2520A
14
1+14
0(708A
2 2−
67)A
12
1+672
(1338A
4 2−415A
2 2+21)A
10
1+2(140
025
6A
6 2−914
928A
4 2+151
424A
2 2−
5493
)A8 1+
4(86528
0A8 2−968
000A
6 2+337
232A
4 2−
3988
0A
2 2+11
75)A
6 1+8(215
808A
10
2−
368
960A
8 2+2202
24A
6 2−5479
2A
4 2+523
5A
2 2−135
)A4 1+
8(399
36A
12
2−
9676
8A
10
2+88
192A
8 2−375
84A
6 2+754
8A
4 2−
625A
2 2+15
)A2 1+16
384A
14
2−532
48A
12
2+6860
8A
10
2−
44544A
8 2+152
96A
6 2−
2656A
4 2+20
0A
2 2−
5)
85
Z7,11=77A
2 1
( 560A
16
1+25
20(32A
2 2−1)A
14
1+28
(57360A
4 2−
100
80A
2 2+169
)A12
1+14(6923
52A
6 2−
30384
0A
4 2+28
392A
2 2−
343
)A10
1+30(75
494
4A
8 2−
619
136A
6 2+144
368A
4 2−
9604A
2 2+95
)A8 1+40
(55961
6A
10
2−
7368
96A
8 2+3209
92A
6 2−
53228A
4 2+28
50A
2 2−25
)A6 1+8(116
224
0A
12
2−
22456
32A
10
2+15
71520A
8 2−
48675
2A
6 2+64
860A
4 2−300
0A
2 2+25)A
4 1+(14745
60A
14
2−
3923
968A
12
2+40
243
20A
10
2−
2001
408A
8 2+
497
792A
6 2−
5712
0A
4 2+24
00A
2 2−
21)A
2 1+6553
6A
16
2−229
376A
14
2+323
584A
12
2−
23552
0A
10
2+93
952A
8 2−20
096A
6 2+20
80A
4 2−
84A
2 2+1
)
Z7,12=168A
2 1A
2
( 9240A
16
1+42
0(110
0A
2 2−
93)A
14
1+33
6(1645
6A
4 2−
454
5A2 2+204
)A12
1+2(11905
344A
6 2−
69165
60A
4 2+10
1186
4A
2 2−
322
91)A
10
1+
8(541
024
0A8 2−53
882
40A
6 2+16
571
52A
4 2−
171
815A
2 2+441
0)A
8 1+10(349
798
4A
10
2−53
196
80A
8 2+28
0473
6A
6 2−
61096
0A
4 2+50
708A
2 2
−11
25)A
6 1+8(15
360
00A
12
2−33
13920A
10
2+2668
800A
8 2−
99664
0A
6 2+17
365
6A
4 2−
123
65A
2 2+25
0)A
4 1+(1687
552A
14
2−48
947
20A
12
2+
558
4896A
10
2−
3183
360A
8 2+95
0336A
6 2−
141
840A
4 2+908
0A
2 2−
177)A
2 1+65
536A
16
2−
2457
60A
14
2+376
832A
12
2−
3041
28A
10
2+138
496A
8 2−
35392A
6 2+47
36A
4 2−
284A
2 2+6
)
Z7,13=91A
2 1
( 184
8A18
1+924
0(36A
2 2−
1)A
16
1+504(167
20A
4 2−26
40A
2 2+39
)A14
1+336
(19993
6A
6 2−
7854
0A
4 2+655
2A
2 2−
69)A
12
1+2(10
844
697
6A
8 2−
796
2662
4A
6 2+16
513
056A
4 2−97
3728A
2 2+828
1)A
10
1+6(52569
088A
10
2−62
008
320A
8 2+24
026
240A
6 2−
3517
920A
4 2+16
5620A
2 2−
1225)A
8 1+
80(265
369
6A
12
2−
459
008
0A10
2+285
772
8A
8 2−7815
36A
6 2+9121
0A
4 2−
3675A
2 2+25)A
6 1+2(3194
880
0A
14
2−76
20608
0A
12
2+69
574
656A
10
2−
305
740
80A
8 2+66
617
60A
6 2−
6636
00A
4 2+2400
0A
2 2−15
9)A
4 1+(766
7712A
16
2−24
084
480A
14
2+302
8582
4A
12
2−
19501
056A
10
2+68
236
80A
8 2−
12678
40A
6 2+112
800A
4 2−38
16A
2 2+27)A
2 1+2621
44A
18
2−
104
857
6A16
2+173
670
4A
14
2−15
40096A
12
2+78
9504A
10
2−
2355
20A
8 2+39
040A
6 2−
321
6A
4 2+10
8A
2 2−
1)
86
Z7,14=28A
2 1A
2
( 24024
0A
18
1+92
40(161
2A
2 2−
123)A
16
1+18
48(122
928A
4 2−
306
20A
2 2+123
3)A
14
1+48(2692
518
4A
6 2−
1408
052
8A
4 2+184
699
9A
2 2−
5252
1)A
12
1+11
2(2897
523
2A
8 2−
259
918
56A
6 2+715
453
2A
4 2−
6608
49A
2 2+1499
4)A
10
1+210(183
838
72A
10
2−
251
83488A
8 2+11
88844
8A
6 2−
2302
944A
4 2+1690
64A
2 2−
3283)A
8 1+140
(15695
872A
12
2−
3049
267
2A10
2+21
987
328A
8 2−
730
3680A
6 2+11
235
84A
4 2−701
96A
2 2+12
25)A
6 1+
14(4092
723
2A
14
2−
1070
284
80A
12
2+1094
277
12A
10
2−
55518
720A
8 2+14
641
920A
6 2−
191
452
8A4 2+106
540A
2 2−
1743
)A4 1+28(21
626
88A
16
2−
7315
456A
14
2+1005
772
8A
12
2−722
9952A
10
2+291
072
0A
8 2−
6520
32A
6 2+7576
8A
4 2−38
92A
2 2+63
)A2 1+
7(64A
6 2−112A
4 2+56A
2 2−
7)(−
64A
6 2+80A
4 2−24A
2 2+1)2)
Z7,15=10
5A
2 1
( 6240A
20
1+34
320
(40A
2 2−
1)A
18
1+26
4(1632
80A
4 2−
234
00A
2 2+309
)A16
1+132(332
134
4A
6 2−
11814
40A
4 2+88
992A
2 2−831
)A14
1+
336(5564
416A
8 2−
3697
408A
6 2+6907
12A
4 2−
36564A
2 2+27
3)A
12
1+16
(23473
049
6A
10
2−
2507
00736A
8 2+874
547
52A
6 2−11
468
688A
4 2+
4815
72A
2 2−
308
7)A
10
1+2(186
107
084
8A
12
2−
291
428
966
4A10
2+1634
00371
2A
8 2−
3999
959
04A
6 2+4154
304
0A
4 2−
1481
760A
2 2+85
75)A
8 1+
(181
501
952
0A
14
2−
3919
052
800A
12
2+3222
269
952A
10
2−
12677
913
60A
8 2+24
568
320
0A
6 2−
216
249
60A
4 2+686
000A
2 2−3743
)A6 1+
2(20696
268
8A
16
2−
5891
686
40A
14
2+6677
463
04A
12
2−38
520
627
2A
10
2+119
953
920A
8 2−
196
869
12A
6 2+153
552
0A
4 2−
4491
6A2 2+24
3)A
4 1+
2(19398
656A
18
2−
7032
012
8A16
2+1049
886
72A
14
2−83
431
424A
12
2+380
743
68A
10
2−
1003
622
4A
8 2+1457
792A
6 2−
1041
68A
4 2+29
16A
2 2−
17)A
2 1+
104
857
6A
20
2−
471
859
2A18
2+897
843
2A
16
2−
9404
416A
14
2+59
146
24A
12
2−228
556
8A
10
2+5332
48A
8 2−
70848A
6 2+47
84A
4 2−13
6A2 2+1
)
87
Z88=8( 1
225A
16
1+1600
(98A
2 2−
3)A
14
1+400(588
0A
4 2−
117
6A2 2+19)A
12
1+64(160
720A
6 2−79
800A
4 2+85
50A
2 2−
97)A
10
1+
40(40812
8A8 2−
3790
08A
6 2+10
0880A
4 2−
7760A
2 2+69)A
8 1+128
(77952A
10
2−
1161
60A
8 2+57
760A
6 2−110
40A
4 2+690A
2 2−
5)A
6 1+64(3404
8A
12
2−74
496A
10
2+595
20A
8 2−
2124
8A6 2+3300A
4 2−18
0A
2 2+1)A
4 1+51
2A
2 2(−
16A
6 2+24A
4 2−10A
2 2+1)2A
2 1
)
Z89=14
4A
2 1A
2
( 245
0A
14
1+700(140A
2 2−
13)A
12
1+70(1265
6A
4 2−
394
0A2 2+195
)A10
1+20(138
208A
6 2−9021
6A
4 2+1498
0A
2 2−529
)A8 1+
5(682
752A
8 2−
76448
0A
6 2+26
622
4A
4 2−315
76A
2 2+899
)A6 1+68(250
88A
10
2−
42880A
8 2+25
600A
6 2−
6368A
4 2+610A
2 2−
15)A
4 1+
2(1576
96A
12
2−
38246
4A
10
2+348
800A
8 2−
1487
36A
6 2+2988
0A
4 2−
2480A
2 2+55
)A2 1+4(409
6A
14
2−1331
2A
12
2+1715
2A
10
2−
111
36A
8 2+
3824A
6 2−664A
4 2+50A
2 2−1))
Z8,10=80A
2 1
( 490A
16
1+70(113
4A
2 2−
31)A
14
1+70(225
12A
4 2−
3968A
2 2+57
)A12
1+14
(67939
2A
6 2−
2983
20A
4 2+2793
0A
2 2−281
)A10
1+
(22240
512A
8 2−
18237
184A
6 2+42
554
40A
4 2−
283
248A
2 2+223
9)A
8 1+(21980
672A
10
2−28
962
560A
8 2+12
616
800A
6 2−20
9384
0A
4 2+
111
950A
2 2−
735)A
6 1+2(4573
184A
12
2−88
35072A
10
2+61
843
20A
8 2−
191
564
8A6 2+255
480A
4 2−
1176
0A2 2+65)A
4 1+2(724
992A
14
2−
193
126
4A12
2+198
220
8A
10
2−
9864
96A
8 2+2454
88A
6 2−
28200A
4 2+11
70A
2 2−5)A
2 1+16A
2 2(4096A
14
2−
14336A
12
2+20
224A
10
2−
14720A
8 2+
5872A
6 2−
1256A
4 2+13
0A
2 2−
5))
Z8,11=17
6A
2 1A
2
( 882
0A
16
1+210(215
6A
2 2−17
7)A
14
1+112(482
16A
4 2−
13365A
2 2+58
0)A
12
1+42(554
304A
6 2−
3219
36A
4 2+4724
0A
2 2−
1449
)A10
1+
10(423
270
4A8 2−42
168
96A
6 2+12
969
60A
4 2−
134
792A
2 2+329
7)A
8 1+5(6845
440A
10
2−10
414
848A
8 2+54
924
80A
6 2−
1196
720A
4 2+99
484A
2 2−
207
5)A
6 1+8(15052
80A
12
2−
324
787
2A10
2+261
632
0A
8 2−9772
80A
6 2+1703
52A
4 2−
12135A
2 2+22
5)A
4 1+2(827
392A
14
2−
2402
304A
12
2+
274
329
6A10
2−15
648
00A
8 2+46
742
4A
6 2−698
16A
4 2+446
0A
2 2−75)A
2 1+4(1638
4A
16
2−61
440A
14
2+94
208A
12
2−
760
32A
10
2+34
624A
8 2−
884
8A6 2+118
4A
4 2−70A
2 2+1)
)
88
Z8,12=48A
2 1
( 3234A
18
1+84
0(770A
2 2−19)A
16
1+210
(78232A
4 2−1231
2A
2 2+159)A
14
1+896(145
992A
6 2−574
50A
4 2+477
0A
2 2−43
)A12
1+
7(603
947
52A
8 2−
4433
356
8A
6 2+9208
320A
4 2−
53939
2A
2 2+38
15)A
10
1+4(153
714
176A
10
2−1814
054
40A
8 2+702
912
00A
6 2−
10306
240A
4 2+48
069
0A
2 2−283
5)A
8 1+5(82847
744A
12
2−14
334
156
8A
10
2+892
70016A
8 2−
244
177
92A
6 2+285
292
0A
4 2−
11340
0A
2 2+
575)A
6 1+16(780
288
0A
14
2−
1861
632
0A12
2+17
001
984A
10
2−
747
392
0A8 2+162
904
0A
6 2−1624
00A
4 2+57
50A
2 2−25
)A4 1+8(18
75968A
16
2−
5898
240A
14
2+74
229
76A
12
2−
478
310
4A10
2+167
472
0A
8 2−3113
60A
6 2+2770
0A
4 2−
900A
2 2+3)A
2 1+64A
2 2(1−2A
2 2)2(204
8A
12
2−
6144A
10
2+
691
2A8 2−35
84A
6 2+856A
4 2−88A
2 2+3))
Z8,13=208A
2 1A
2
( 3234
0A
18
1+46
20(448A
2 2−
33)A
16
1+42
(75028
8A
4 2−
1872
20A
2 2+7245
)A14
1+336(533
632A
6 2−
27922
4A
4 2+366
65A
2 2−
996
)A12
1+16
(28158
592A
8 2−
252
592
56A
6 2+695
734
2A
4 2−
6427
47A
2 2+1381
8)A
10
1+(536
094
720A
10
2−734
75072
0A
8 2+34
69075
20A
6 2−
672
456
96A
4 2+493
192
0A
2 2−
8967
0)A
8 1+5(611
000
32A
12
2−
118
725
632A
10
2+8563
5840A
8 2−
2845
209
6A6 2+4379
872A
4 2−272
916A
2 2+
437
5)A
6 1+8(99696
64A
14
2−
260
812
80A
12
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764
80A
10
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984
0A
8 2+3572
160A
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46737
6A
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865A
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)A4 1+2(42
270
72A
16
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143
114
24A
14
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152
0A
12
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14164
992A
10
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062
40A
8 2−
127
900
8A6 2+148
680A
4 2−
7540A
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)A2 1+4(65536A
18
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528A
16
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495
616A
14
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232A
12
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408A
10
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504A
8 2+18
608A
6 2−198
4A
4 2+94A
2 2−
1)
)
89
Z8,14=112A
2 1
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20
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0(200
2A
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18
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672A
4 2−
180
00A
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)A16
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868
8A
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2521
464
0A
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001
0A
2 2−
1533
3)A
14
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142
4A
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410
56A
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613
5A
4 2−
1226
64A
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12
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77766
4A
10
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543
488A
8 2+424
111
38A
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556
749
9A
4 2+23
0202A
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10
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896
32A
12
2−
11310
858
24A
10
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433
497
6A
8 2−155
339
520A
6 2+16
142
112A
4 2−
5644
80A
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259
7)A
8 1+5(352
509
952A
14
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761
303
040A
12
2+6261
37600A
10
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24641
356
8A
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772
480A
6 2−420
425
6A
4 2+129
850A
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525)A
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2(20
122
828
8A16
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112
320A
14
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406
40A
12
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37504
204
8A
10
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83584
0A
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19185
152A
6 2+14
943
60A
4 2−42
000A
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2(1893
990
4A18
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6871
449
6A
16
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26662
40A
14
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816
394
24A
12
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787
20A
10
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936A
8 2+142
878
4A
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1016
40A
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46A
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7)A
2 1+
16A
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36A
18
2−
29491
2A
16
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152A
14
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5877
76A
12
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966
4A
10
2−142
848A
8 2+333
28A
6 2−
4424A
4 2+29
4A
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7)
)
Z8,15=24
0A
2 1A
2
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120A
20
1+85
80(10
92A
2 2−
73)A
18
1+264
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4A
4 2−15
2360A
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25)A
16
1+66
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790
4A
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928
699
2A4 2+
1102
760A
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2691
3)A
14
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875
2A
8 2−
217
835
904A
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238
08A
4 2−4513
971A
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688)A
12
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989
824A
10
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54723
468
8A
8 2+23
38691
52A
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26478A
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7101A
2 2−432
18)A
10
1+16
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1664A
12
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901
888A
10
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499
7760A
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12714
4776A
6 2+17
52256
8A
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967
5A
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720
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055
552A
14
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5929
226
240A
12
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981
760
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10
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25090
7008
0A8 2+
5925
984
00A
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6895
996
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920A
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5)A
6 1+4(125
468
672A
16
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3866
460
16A
14
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18534
40A
12
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227
2384A
10
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1126
604
80A
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22466
688A
6 2+230
630
4A
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70A
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592A
18
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690
624A
16
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176
012
8A
14
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114
866
176A
12
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7335
68A
10
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178
836
48A
8 2+314
521
6A
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832A
4 2+1210
4A
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127)A
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2144A
20
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1245
184A
18
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2523
136A
16
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672
0A14
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788
8A
12
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0A
10
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681
6A
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363
04A
6 2+316
0A
4 2−
122A
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)
90
Z99=9
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18
1+22
050
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16
1+18
00(784
0A
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137
2A2 2+23)A
14
1+1680
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6 2−22
260A
4 2+20
70A
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25)A
12
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144(1382
080A
8 2−
1132
880A
6 2+2646
00A
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17500A
2 2+17
3)A
10
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24032A
10
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872
0A
8 2+6272
000A
6 2−
10416
00A
4 2+55
360A
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485
)A8 1+12(6823
936A
12
2−13
181
952A
10
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752
0A
8 2−2858
240A
6 2+3813
60A
4 2−1746
0A
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145)A
6 1+36(360
448A
14
2−9605
12A
12
2+98
611
2A
10
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490
880A
8 2+122
176A
6 2−1404
0A
4 2+58
0A
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5)A
4 1+9(256A
8 2−
448A
6 2+240A
4 2−
40A
2 2+1)2A
2 1
)
Z9,10=180A
2 1A
2
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0A
16
1+3724
0(12A
2 2−1)A
14
1+56(9542
4A
4 2−
26390A
2 2+116
5)A
12
1+56(411
072A
6 2−
2389
52A
4 2+34990A
2 2−
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10
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976A
8 2−
18623
6A
6 2+57
312A
4 2−594
5A
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)A8 1+4(8465
408A
10
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12884
480A
8 2+67
952
00A
6 2−
148
120
0A4 2+122
920A
2 2−
2645
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188
8A
12
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643
328
0A10
2+5182
720A
8 2−19
360
00A
6 2+33
756
8A
4 2−24
010A
2 2+46
5)A
4 1+
8(2048
00A
14
2−
594
944A
12
2+679
680A
10
2−
38784
0A
8 2+11
588
8A
6 2−
173
16A
4 2+110
5A
2 2−20)A
2 1+65
536A
16
2−
245
760A
14
2+37
6832A
12
2−
30412
8A
10
2+138
496A
8 2−
3539
2A
6 2+4736A
4 2−
280A
2 2+5
)
Z9,11=99A
2 1
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18
1+882
0(36A
2 2−1)A
16
1+280
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4 2−
4536A
2 2+67
)A14
1+56
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64A
6 2−
452
760A
4 2+375
20A
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395)A
12
1+
112(1859
136A
8 2−
1365
168A
6 2+2835
40A
4 2−
16590A
2 2+14
1)A
10
1+56(541
030
4A
10
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6384
960A
8 2+247
4720A
6 2−36
280
0A
4 2+
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2 2−
125)A
8 1+4(5103
6160A
12
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126
4A
10
2+55
01824
0A
8 2−
15052
800A
6 2+17
584
00A
4 2−700
00A
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009
60A
14
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743
168A
12
2+33
560
576A
10
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1475
4560A
8 2+3216
640A
6 2−
3206
00A
4 2+11
400A
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75)A
4 1+
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568A
16
2−
23298
048A
14
2+29
335
552A
12
2−1891
1232A
10
2+6624
000A
8 2−12
320
00A
6 2+10
960
0A
4 2−360
0A
2 2+25)A
2 1+
26214
4A
18
2−1048
576A
16
2+17
36704A
14
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154
009
6A12
2+789
504A
10
2−23
5520A
8 2+39
040A
6 2−
320
0A4 2+100A
2 2−
1)
91
Z9,12=72A
2 1A
2
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0A
18
1+11
760
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2 2−
39)A
16
1+25
2(3683
68A
4 2−
917
00A
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5)A
14
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1328A
6 2−
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96A
4 2+
107
805A
2 2−
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12
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891
2A
8 2−42
522
336A
6 2+11
716
920A
4 2−108
0580A
2 2+240
87)A
10
1+60(263
280
64A
10
2−
360
899
84A
8 2+170
424
00A
6 2−
3304
448A
4 2+2420
32A
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4599)A
8 1+30(300
195
84A
12
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752
0A
10
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093
056A
8 2−
139
878
40A
6 2+215
364
8A
4 2−
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92A
2 2+2275
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664A
14
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769
84320A
12
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530
24A
10
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824
0A
8 2+
105
494
40A
6 2−
13804
56A
4 2+76
405A
2 2−120
0)A
4 1+3(83230
72A
16
2−
281
968
64A
14
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177
92A
12
2−2793
676
8A
10
2+
112
588
80A
8 2−
25246
08A
6 2+29
355
2A
4 2−
149
20A
2 2+225
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4A2 2)2(8192A
14
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20A
12
2+4608
0A
10
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3494
4A8 2+
138
24A
6 2−
2592A
4 2+16
6A
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3))
Z9,13=117A
2 1
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0A20
1+3234
0(40A
2 2−
1)A
18
1+420(973
28A
4 2−
13860A
2 2+18
3)A
16
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56A
6 2−8808
80A
4 2+65
880A
2 2−
615
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718
4A
8 2−
35087
36A
6 2+65
594
0A
4 2−
344
40A
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12
1+84(4244
275
2A
10
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453
41824A
8 2+158
272
64A
6 2−
207
640
0A4 2+863
52A
2 2−
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10
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2A
12
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324
48A
10
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886
72A
8 2−38
063
424A
6 2+39
532
64A
4 2−
139
356A
2 2+805
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764
48A
14
2−
18658
3040A
12
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348
531
2A
10
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604
139
52A
8 2+117
15200A
6 2−1030
624A
4 2+
322
00A
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6 1+2(1974
599
68A
16
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56246
2720A
14
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789
056
0A
12
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368
222
208A
10
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36640A
8 2−
1884
467
2A6 2+
146
776
0A4 2−42
000A
2 2+22
5)A
4 1+(3722
444
8A
18
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1351
352
32A
16
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147
20A
14
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071
884
8A
12
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420
800A
10
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193
720
32A
8 2+281
612
8A
6 2−
2004
80A
4 2+5400A
2 2−
31)A
2 1+104
857
6A
20
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592A
18
2+89
784
32A
16
2−
940
441
6A14
2+
591
462
4A12
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855
68A
10
2+533
248A
8 2−
707
84A
6 2+4720A
4 2−12
4A
2 2+1
)
92
Z9,14=252A
2 1A
2
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120A
20
1+3696
0(247A
2 2−
17)A
18
1+132(131
185
6A
4 2−
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0A
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725
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1+24
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896A
6 2−249
369
12A
4 2+
295
8560A
2 2−
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5)A
14
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073
92A
8 2−
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236
8A6 2+1768
676
8A
4 2−
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35A
2 2+29
427
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1+48
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442
56A
10
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178
20633
6A
8 2+761
79664A
6 2−132
981
80A
4 2+872
221A
2 2−
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8)A
10
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33440
0A
12
2−
2544
837
120A
10
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8976
00A
8 2−
49730
448
0A
6 2+68
532
912A
4 2−379
666
0A
2 2+570
85)A
8 1+4(6093
045
76A
14
2−
1449
943
040A
12
2+13
42745
600A
10
2−
613798
400A
8 2+
14500
032
0A
6 2−
1687
302
4A4 2+8250
20A
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94464A
16
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7573
012
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14
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40051
20A
12
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7040A
10
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3040A
8 2−
440
476
16A
6 2+452
211
2A
4 2−2014
60A
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688A
18
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080
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16
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249
92A
14
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120A
12
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878
336A
10
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672A
8 2+1547
648A
6 2−
1446
48A
4 2+60
20A
2 2−
77)A
2 1+1048
576A
20
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807
36A
18
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925
44A
16
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880A
14
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1552A
12
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200A
10
2+9072
64A
8 2−14
521
6A
6 2+12
656A
4 2−
504A
2 2+7
)
Z9,15=45A
2 1
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22
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360
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2 2−1)A
20
1+396
0(1536
08A
4 2−
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18
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341
76A
6 2−
18935
280A
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59852
80A
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4 2−
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00A
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6(3306
434
56A
10
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629
12A
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200
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116
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44A
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12
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12
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641
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10
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254
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8 2−
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728A
4 2−6509
16A
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118
528A
14
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198
848A
12
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256A
10
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954
112
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79678
72A
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680A
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384
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5)A
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37516
8A
16
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013
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14
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270
080
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12
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516
800
00A
10
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17271
040A
8 2−
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349
12A
6 2+5414
976
0A
4 2−13
720
00A
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25)A
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148
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18
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71552
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16
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14
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12
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43232A
10
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350
528A
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59680A
6 2−9327
360A
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400A
20
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774
950
4A
18
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704
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16
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14
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247
308
8A
12
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075
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10
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833
6A
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20A
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4 2−
393
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2 1+
3(1−4A
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144A
18
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648A
16
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840A
14
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8224A
12
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10
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44A
8 2+74
560A
6 2−
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2 2−
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93
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6A
20
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200
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2)A
18
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50(328
32A
4 2−
518
4A2 2+67)A
16
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780
8A
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40A
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132
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14
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964
8A
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060A
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12
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113
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10
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9160A
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451
200A
4 2+211
50A
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10
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169
792A
12
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907
52A
10
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630
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400A
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000
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4 2−
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142
5)A
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712A
14
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292
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12
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086
08A
10
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920A
8 2+1601
600A
6 2−
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00A
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994
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14
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335
36A
12
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3767
500
8A
10
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198
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550
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400A
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2 2(256A
8 2−512A
6 2+336A
4 2−
80A
2 2+5)2A
2 1
)
Z10,11=220A
2 1A
2
( 3175
2A
18
1+88
20(228A
2 2−
17)A
16
1+11
76(260
64A
4 2−650
0A2 2+255
)A14
1+56(3115
008A
6 2−
16302
72A
4 2+21
399
0A
2 2−
5905
)A12
1+16
8(2609
664A
8 2−234
182
4A
6 2+645
120A
4 2−
5959
0A2 2+13
05)A
10
1+15
68(332
928A
10
2−
456
352A
8 2+2155
32A
6 2−
4178
4A
4 2+30
65A
2 2−
57)A
8 1+4(74403
840A
12
2−144
64307
2A
10
2+104
357
120A
8 2−346
832
00A
6 2+533
960
0A
4 2−
3329
20A
2 2+54
95)A
6 1+
12(64880
64A
14
2−
16973
824A
12
2+17
364
480A
10
2−881
536
0A8 2+232
640
0A
6 2−
3044
16A
4 2+1687
0A
2 2−
255)A
4 1+14(589
824A
16
2−
19988
48A
14
2+27
525
12A
12
2−
1981
440A
10
2+7987
20A
8 2−17
913
6A
6 2+20
832A
4 2−
106
0A2 2+15)A
2 1+2621
44A
18
2−
11141
12A
16
2+
19824
64A
14
2−
1916
928A
12
2+10
93632A
10
2−374
016A
8 2+744
32A
6 2−
7936A
4 2+380A
2 2−
5)
Z10,12=12
0A
2 1
( 5292A
20
1+58
8(2178A
2 2−
49)A
18
1+29
40(137
28A
4 2−1960A
2 2+23)A
16
1+84(4886
112A
6 2−
173
9920A
4 2+130
410A
2 2−
1065
)A14
1+
28(62548
992A
8 2−
4158
873
6A6 2+7775
880A
4 2−40
896
0A
2 2+26
15)A
12
1+28(125
773
824A
10
2−
1343
877
12A
8 2+4691
131
2A
6 2−615
564
0A
4 2+
25627
0A
2 2−
1367
)A10
1+14(249
556
992A
12
2−
3910
696
96A
10
2+21
93799
68A
8 2−
53740
416A
6 2+55
829
60A
4 2−
196
848A
2 2+90
9)A
8 1+
2(8536
47360A
14
2−
18443
264
00A
12
2+15
173
365
76A
10
2−
597
319
680A
8 2+115
836
000A
6 2−
1019
368
0A4 2+3181
50A
2 2−
1295)A
6 1+
(39066
009
6A
16
2−
111
28012
80A
14
2+126
211
276
8A
12
2−
7286
128
64A
10
2+2270
566
40A
8 2−
37295
104A
6 2+29
058
40A
4 2−8288
0A
2 2+295
)A4 1+
(36831
232A
18
2−133
758
976A
16
2+200
024
064A
14
2−1591
828
48A
12
2+72
73881
6A
10
2−191
969
28A
8 2+279
1264A
6 2−
198
800A
4 2+53
10A
2 2−
15)A
2 1+
8A2 2(131
072A
18
2−589
824A
16
2+1122
304A
14
2−
11755
52A
12
2+739
328A
10
2−2856
96A
8 2+6665
6A
6 2−
8848A
4 2+59
0A
2 2−
15))
94
Z10,13=260A
2 1A
2
( 1164
24A
20
1+1293
6(696A
2 2−
47)A
18
1+588(289
872A
4 2−
657
80A
2 2+232
5)A
16
1+168
(73687
68A
6 2−
3507
504A
4 2+
416
920A
2 2−
10305
)A14
1+336
(12227
072A
8 2−
9979
904A
6 2+2488
528A
4 2−20
705
5A
2 2+40
35)A
12
1+56
(12069
273
6A
10
2−
150
481
408A
8 2+643
360
32A
6 2−
112
304
16A
4 2+737
670A
2 2−
1211
7)A
10
1+4(1418
231
808A
12
2−2507
774
976A
10
2+1637
8835
20A
8 2−
490
17283
2A
6 2+67
553
136A
4 2−
374
679
2A2 2+541
59)A
8 1+4(6005
882
88A
14
2−1429
442
560A
12
2+13
23929
600A
10
2−60
5272
320A
8 2+
14300
160
0A
6 2−
1664
219
2A
4 2+8143
80A
2 2−10
675
)A6 1+2(2424
176
64A
16
2−
7472
414
72A
14
2+9315
532
80A
12
2−60
392
1408A
10
2+
2179
609
60A
8 2−
434
818
56A
6 2+446
476
8A
4 2−
1989
40A
2 2+2415
)A4 1+2(20447
232A
18
2−
792
33024A
16
2+127
909
888A
14
2−11
159
961
6A
12
2+
5710
540
8A
10
2−
173
998
08A
8 2+306
201
6A
6 2−
2862
72A
4 2+11
900A
2 2−
135)A
2 1+104
857
6A
20
2−
4980
736A
18
2+10
09254
4A
16
2−
1138
688
0A
14
2+
7831
552A
12
2−
337
920
0A10
2+9072
64A
8 2−
14521
6A
6 2+12
656A
4 2−500A
2 2+5
)
Z10,14=70A
2 1
( 3603
6A
22
1+73
92(140
4A
2 2−
29)A
20
1+92
4(4311
84A
4 2−
561
44A
2 2+603)A
18
1+96(5242
036
8A
6 2−17
057
040A
4 2+116
077
5A
2 2−
8650)A
16
1+12(228
389
990
4A
8 2−1386
936
320A
6 2+2359
156
80A
4 2−11
210
400A
2 2+65
235
)A14
1+19
2(378
9660
16A
10
2−369
8928
64A
8 2+
11742
192
0A
6 2−
139
501
20A
4 2+521
880A
2 2−
2529
)A12
1+28(356
229
120
0A
12
2−
5099
077
632A
10
2+2601
599616A
8 2−57
6816
384A
6 2+
53966
880A
4 2−
169
948
8A2 2+713
3)A
10
1+32(2242
215
936A
14
2−44
253
224
96A
12
2+331
12907
52A
10
2−117
9930
240A
8 2+20
601
974
4A
6 2−
16231
824A
4 2+44
937
9A
2 2−
167
3)A
8 1+(265
827
77856A
16
2−69
175
869
440A
14
2+71
363
174
400A
12
2−
372
930
560
00A
10
2+10
46462
208
0A
8 2−
15386
327
04A
6 2+10
663
072
0A
4 2−26
768
00A
2 2+892
5)A
6 1+8(595
853
312A
18
2−
1976
893
440A
16
2+2689
105920A
14
2−
1937
40800
0A12
2+
7972
638
72A
10
2−1883
79648A
8 2+243
622
40A
6 2−
1532
160A
4 2+3570
0A
2 2−10
5)A
4 1+(361
75872
0A
20
2−14
900
264
96A
18
2+258
342
912
0A
16
2−
2453
667
840A
14
2+13
91747
072A
12
2−482
15654
4A
10
2+100
184
064A
8 2−117
555
20A
6 2+687
120A
4 2−
1512
0A2 2+35)A
2 1+8A
2 2(104
857
6A20
2−
5242
880A
18
2+1127
219
2A
16
2−
136
31488A
14
2+101
826
56A
12
2−
4857
856A
10
2+14
78656A
8 2−
27852
8A
6 2+30
352A
4 2−
1680A
2 2+35
))
95
Z10,15=60A
2 1A
2
( 216
216
0A22
1+12
0120(166
8A
2 2−
103)A
20
1+147
84(312
078A
4 2−6493
5A
2 2+20
90)A
18
1+66
0(630
814
08A
6 2−
2753
732
8A4 2+
29892
80A
2 2−66
975)A
16
1+120
(14694
085
12A
8 2−
1099
731
776A
6 2+250
489
008A
4 2−
1895
0360A
2 2+33
3075)A
14
1+56
0(68
39439
36A
10
2−
78202
009
6A
8 2+30
53656
32A
6 2−
484
793
76A
4 2+288
145
5A
2 2−4245
3)A
12
1+56
0(7932
395
52A
12
2−
128
620
544
0A10
2+767
29945
6A
8 2−
20882
558
4A
6 2+26
04951
6A
4 2−130
041
3A
2 2+167
58)A
10
1+4(69227
05305
6A
14
2−15
110
373
376
0A
12
2+12
78326
246
40A
10
2−531
506
976
00A
8 2+
1136
554
320
0A
6 2−
119
094
494
4A4 2+5214
258
0A
2 2−60
539
5)A
8 1+80(113
128
2432A
16
2−
319
793
971
2A14
2+36
4166
656
0A
12
2−
2147
174
400A
10
2+
70143
200
0A
8 2−
1259
892
48A
6 2+1158
040
8A
4 2−45
860
5A
2 2+49
00)A
6 1+10
(14514
913
28A
18
2−51
589
283
84A
16
2+76
08696
832A
14
2−
6038
609
920A
12
2+27
973
376
00A
10
2−
7675
601
92A
8 2+1209
26976A
6 2−
1005
580
8A4 2+3687
60A
2 2−
3675
)A4 1+20(49
807
360A
20
2−
21731
737
6A
18
2+40
284
979
2A
16
2−4139
66336A
14
2+2580
582
40A
12
2−10
038
528
0A
10
2+24
15462
4A
8 2−
344
204
8A6 2+265
104A
4 2−
9170A
2 2+
85)A
2 1+5(16A
4 2−
12A
2 2+1)
2(16384A
14
2−
614
40A
12
2+921
60A
10
2−70
400A
8 2+28
800A
6 2−601
6A
4 2+520A
2 2−5))
Z11,11=11
( 635
04A
22
1+34
9272(40A
2 2−
1)A
20
1+10
780
(41040A
4 2−583
2A
2 2+77)A
18
1+4851
0(926
72A
6 2−
3302
4A
4 2+24
64A
2 2−
23)A
16
1+
88(2180
505
60A
8 2−
144
958
464A
6 2+2711
856
0A
4 2−14
200
20A
2 2+10
595
)A14
1+12
32(313
159
68A
10
2−
3346
444
8A
8 2+116
80704A
6 2−
15334
20A
4 2+63
570A
2 2−
407)A
12
1+36
96(103
577
60A
12
2−
162
309
12A
10
2+9105
600A
8 2−
2230
480A
6 2+23
1800A
4 2−
8140A
2 2+47)A
10
1+
22(850
32960
0A
14
2−18
373
017
60A
12
2+15
116
165
12A
10
2−
595
096
320A
8 2+115
404
800A
6 2−
101
584
00A
4 2+315
840A
2 2−
1715)A
8 1+
88(486
60480A
16
2−
13860
864
0A
14
2+15
720
857
6A
12
2−
9075
7632A
10
2+2828
352
0A
8 2−46
45760A
6 2+36
204
0A
4 2−10
290A
2 2+55)A
6 1+
22(183
50080A
18
2−
66650
112A
16
2+99
680
256A
14
2−7933
542
4A
12
2+3625
574
4A
10
2−
95692
80A
8 2+13
91488A
6 2−991
20A
4 2+26
40A
2 2−
15)A
4 1+
11(−
1024A
10
2+23
04A
8 2−17
92A
6 2+560A
4 2−60A
2 2+1)2A
2 1
)
96
Z11,12=26
4A
2 1A
2
( 1164
24A
20
1+20
286
0(44A
2 2−3)A
18
1+58
80(287
76A
4 2−6522A
2 2+233)A
16
1+84
(1462
560
0A
6 2−
6964
272A
4 2+82
6840A
2 2−
2068
5)A
14
1+16
8(2427
392
0A
8 2−
198
113
28A
6 2+494
144
0A
4 2−
4106
90A
2 2+8115
)A12
1+28
(2396
037
12A
10
2−
29877
657
6A
8 2+
127
736
064A
6 2−
223
025
04A
4 2+146
340
0A
2 2−
2443
3)A
10
1+28
0(2011
545
6A
12
2−
3556
9152A
10
2+232
325
12A
8 2−695
294
4A
6 2+95
839
2A
4 2−
531
04A
2 2+783
)A8 1+23
0(103
710
72A
14
2−
246
865
92A
12
2+228
659
20A
10
2−
1045
452
8A8 2+24
70080A
6 2−28
750
4A
4 2+14
056A
2 2−
189)A
6 1+
2(24
084
480
0A
16
2−
7423
918
08A
14
2+9255
321
60A
12
2−60
004
044
8A
10
2+216
570
880A
8 2−
43206
400A
6 2+44
369
92A
4 2−
197
540A
2 2+24
85)A
4 1+
(406
323
20A
18
2−157
483
008A
16
2+2542
796
80A
14
2−22
189
260
8A
12
2+113
55955
2A
10
2−
346
060
80A
8 2+60
90752A
6 2−569
520A
4 2+236
60A
2 2−
285)A
2 1+2(524
288A
20
2−
249
036
8A18
2+5046
272A
16
2−
56934
40A
14
2+391
5776A
12
2−
168
960
0A10
2+45
3632A
8 2−
72608A
6 2+632
8A
4 2−
250A
2 2+3))
Z11,13=143A
2 1
( 194
04A
22
1+11
6424(44A
2 2−
1)A
20
1+2352
0(8382A
4 2−
1089A
2 2+13
)A18
1+294
0(846
912A
6 2−
2756
16A
4 2+18
720A
2 2−
157)A
16
1+
252
(538
13760A
8 2−
32682
496A
6 2+55
596
80A
4 2−
263
760A
2 2+175
5)A
14
1+672
(53582
848A
10
2−52
30156
8A
8 2+16
60467
2A
6 2−
197
274
0A4 2+
737
10A
2 2−
417)A
12
1+56(881
516
544A
12
2−12
619
284
48A
10
2+64
387
814
4A
8 2−
142
771
776A
6 2+13
357
440A
4 2−
42033
6A
2 2+21
19)A
10
1+
60(591
986
688A
14
2−116
841
267
2A
12
2+874
338
304A
10
2−
3115
741
44A
8 2+5440
691
2A
6 2−42
863
52A
4 2+11
866
4A
2 2−55
3)A
8 1+
5(2632
384
512A
16
2−685
100
236
8A
14
2+706
822
144
0A
12
2−3694
014
464A
10
2+10
36626
432A
8 2−
1524
311
04A
6 2+10
562
944A
4 2−
265
440A
2 2+118
3)A
6 1+2(1180
958
720A
18
2−
39182
008
32A
16
2+53
300
428
80A
14
2−
384
03276
80A
12
2+158
04395
52A
10
2−
37345
536
0A
8 2+
483
015
68A
6 2−
30374
40A
4 2+70
980A
2 2−
315
)A4 1+4(44826
624A
20
2−184
680
448A
18
2+32
027
443
2A
16
2−304
250
880A
14
2+172
607
488A
12
2−
59808
768A
10
2+12
429
312A
8 2−
1458
688A
6 2+8526
0A
4 2−18
90A
2 2+9)A
2 1+41
943
04A
22
2−209
715
20A
20
2+450
887
68A
18
2−
545
259
52A
16
2+
407
306
24A
14
2−
194
3142
4A
12
2+591
462
4A
10
2−1114
112A
8 2+12
1408A
6 2−67
20A
4 2+144A
2 2−1
)
97
Z11,14=308A
2 1A
2
( 4324
32A
22
1+55
44(712
4A
2 2−447
)A20
1+9240
(98592A
4 2−
204
84A
2 2+671
)A18
1+12(684
889
920A
6 2−
2990
766
24A
4 2+32
42162
0A
2 2−
740
775
)A16
1+24(1450
592
000A
8 2−
10856
672
64A
6 2+24
734
740
8A
4 2−
186
894
40A
2 2+335
835
)A14
1+48(157
558
169
6A
10
2−
180
167
1168A
8 2+
703
555
072A
6 2−
11171
675
2A
4 2+66
327
25A
2 2−
100
239
)A12
1+56(1566
572
544A
12
2−
254
033
5104A
10
2+151
557
683
2A
8 2−412
499
808A
6 2+
514
638
72A
4 2−
25666
86A
2 2+34
083
)A10
1+60
(91167
129
6A
14
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04A
4 2−20
9412A
2 2+58
1)+
120A
4 1(23488
102
40A
26
2−131
407
544
32A
24
2+323
317
923
84A
22
2−
4603
065
139
2A20
2+
4198
170
624
0A
18
2−
25652
920
320A
16
2+106
721
935
36A
14
2−3011
412
992A
12
2+5637
749
76A
10
2−
670
1376
0A
8 2+46
838
40A
6 2−
1678
32A
4 2+23
24A
2 2−
7)+15A
2 1(−
16384A
14
2+532
48A
12
2−
6758
4A
10
2+42
240A
8 2−134
40A
6 2+201
6A
4 2−
112A
2 2+1)
2
)
105
C. Matlab function
function [T0n, Zmn]=function T0nZmn(a,b)%[T0n, Zmn]=function Z15152(a,b)%a=sin(d*pi/(2*b))*sin((d+2*dp)*pi/(2*b));%b=cos(d*pi/(2*b))*cos(((d+2*dp)*pi)/(2*b));%b=Height of the parallel plate guide%dp=Height of the wall%d=Height of the gap%T0n=Transfer coeffcients, which gives a vector 1x15%Zmn=Impedance parameters, which gives a matrix 15x15, which is reciprocal% Zmn=Znm%To get the actual Zmn parameters it should be divided by jweb/pi%j=imaginary unit%w=omega, Angular frequency%e=permittivity of free space%%This function will not give Y00, but Y00=jweb/pi ln(1/a)%%%% - ---------------------% | |% b | -% | | d% | | -% | | | dp% - --------------------- -%T0n=[2.ˆ(1/2).*b,2.ˆ(1/2).*((-1)+a.ˆ2+2.*b.ˆ2),(-1).*2.ˆ(1/2).*b.*(3+(-6).* ...
a.ˆ2+(-4).*b.ˆ2),2.ˆ(1/2).*(1+3.*a.ˆ4+(-8).*b.ˆ2+8.*b.ˆ4+(-1).*a.ˆ2.*(4+ ...(-24).*b.ˆ2)),2.ˆ(1/2).*b.*(5+30.*a.ˆ4+(-20).*b.ˆ2+16.*b.ˆ4+10.*a.ˆ2.*(( ...-3)+8.*b.ˆ2)),2.ˆ(1/2).*((-1)+10.*a.ˆ6+18.*b.ˆ2+(-48).*b.ˆ4+32.*b.ˆ6+( ...-1).*a.ˆ4.*(18+(-180).*b.ˆ2)+3.*a.ˆ2.*(3+(-48).*b.ˆ2+80.*b.ˆ4)),2.ˆ(1/2) ....*b.*((-7)+140.*a.ˆ6+56.*b.ˆ2+(-112).*b.ˆ4+64.*b.ˆ6+210.*a.ˆ4.*((-1)+4.* ...b.ˆ2)+28.*a.ˆ2.*(3+(-20).*b.ˆ2+24.*b.ˆ4)),2.ˆ(1/2).*(1+35.*a.ˆ8+(-32).* ...b.ˆ2+160.*b.ˆ4+(-256).*b.ˆ6+128.*b.ˆ8+80.*a.ˆ6.*((-1)+14.*b.ˆ2)+60.* ...a.ˆ4.*(1+(-24).*b.ˆ2+56.*b.ˆ4)+16.*a.ˆ2.*((-1)+30.*b.ˆ2+(-120).*b.ˆ4+ ...112.*b.ˆ6)),2.ˆ(1/2).*b.*(9+630.*a.ˆ8+(-120).*b.ˆ2+432.*b.ˆ4+(-576).* ...b.ˆ6+256.*b.ˆ8+420.*a.ˆ6.*((-3)+16.*b.ˆ2)+54.*a.ˆ4.*(15+(-140).*b.ˆ2+ ...224.*b.ˆ4)+36.*a.ˆ2.*((-5)+60.*b.ˆ2+(-168).*b.ˆ4+128.*b.ˆ6)),2.ˆ(1/2).*( ...(-1)+126.*a.ˆ10+50.*b.ˆ2+(-400).*b.ˆ4+1120.*b.ˆ6+(-1280).*b.ˆ8+512.* ...b.ˆ10+350.*a.ˆ8.*((-1)+18.*b.ˆ2)+350.*a.ˆ6.*(1+(-32).*b.ˆ2+96.*b.ˆ4)+ ...30.*a.ˆ4.*((-5)+210.*b.ˆ2+(-1120).*b.ˆ4+1344.*b.ˆ6)+5.*a.ˆ2.*(5+(-240).* ...b.ˆ2+1680.*b.ˆ4+(-3584).*b.ˆ6+2304.*b.ˆ8)),2.ˆ(1/2).*b.*((-11)+2772.* ...a.ˆ10+220.*b.ˆ2+(-1232).*b.ˆ4+2816.*b.ˆ6+(-2816).*b.ˆ8+1024.*b.ˆ10+ ...2310.*a.ˆ8.*((-3)+20.*b.ˆ2)+6160.*a.ˆ6.*(1+(-12).*b.ˆ2+24.*b.ˆ4)+66.* ...
a.ˆ4.*((-35)+560.*b.ˆ2+(-2016).*b.ˆ4+1920.*b.ˆ6)+22.*a.ˆ2.*(15+(-280).* ...b.ˆ2+1344.*b.ˆ4+(-2304).*b.ˆ6+1280.*b.ˆ8)),2.ˆ(1/2).*(1+462.*a.ˆ12+(-72) ....*b.ˆ2+840.*b.ˆ4+(-3584).*b.ˆ6+6912.*b.ˆ8+(-6144).*b.ˆ10+2048.*b.ˆ12+ ...1512.*a.ˆ10.*((-1)+22.*b.ˆ2)+630.*a.ˆ8.*(3+(-120).*b.ˆ2+440.*b.ˆ4)+ ...1120.*a.ˆ6.*((-1)+54.*b.ˆ2+(-360).*b.ˆ4+528.*b.ˆ6)+45.*a.ˆ4.*(7+(-448).* ...b.ˆ2+4032.*b.ˆ4+(-10752).*b.ˆ6+8448.*b.ˆ8)+12.*a.ˆ2.*((-3)+210.*b.ˆ2+( ...-2240).*b.ˆ4+8064.*b.ˆ6+(-11520).*b.ˆ8+5632.*b.ˆ10)),2.ˆ(1/2).*b.*(13+ ...12012.*a.ˆ12+(-364).*b.ˆ2+2912.*b.ˆ4+(-9984).*b.ˆ6+16640.*b.ˆ8+(-13312) ....*b.ˆ10+4096.*b.ˆ12+36036.*a.ˆ10.*((-1)+8.*b.ˆ2)+2730.*a.ˆ8.*(15+(-220) ....*b.ˆ2+528.*b.ˆ4)+3120.*a.ˆ6.*((-7)+140.*b.ˆ2+(-616).*b.ˆ4+704.*b.ˆ6)+ ...780.*a.ˆ4.*(7+(-168).*b.ˆ2+1008.*b.ˆ4+(-2112).*b.ˆ6+1408.*b.ˆ8)+26.* ...a.ˆ2.*((-21)+560.*b.ˆ2+(-4032).*b.ˆ4+11520.*b.ˆ6+(-14080).*b.ˆ8+6144.* ...b.ˆ10)),2.ˆ(1/2).*((-1)+1716.*a.ˆ14+98.*b.ˆ2+(-1568).*b.ˆ4+9408.*b.ˆ6+( ...-26880).*b.ˆ8+39424.*b.ˆ10+(-28672).*b.ˆ12+8192.*b.ˆ14+6468.*a.ˆ12.*(( ...-1)+26.*b.ˆ2)+9702.*a.ˆ10.*(1+(-48).*b.ˆ2+208.*b.ˆ4)+1470.*a.ˆ8.*((-5)+ ...330.*b.ˆ2+(-2640).*b.ˆ4+4576.*b.ˆ6)+420.*a.ˆ6.*(7+(-560).*b.ˆ2+6160.* ...b.ˆ4+(-19712).*b.ˆ6+18304.*b.ˆ8)+84.*a.ˆ4.*((-7)+630.*b.ˆ2+(-8400).* ...b.ˆ4+36960.*b.ˆ6+(-63360).*b.ˆ8+36608.*b.ˆ10)+7.*a.ˆ2.*(7+(-672).*b.ˆ2+ ...10080.*b.ˆ4+(-53760).*b.ˆ6+126720.*b.ˆ8+(-135168).*b.ˆ10+53248.*b.ˆ12)), ...2.ˆ(1/2).*b.*((-15)+51480.*a.ˆ14+560.*b.ˆ2+(-6048).*b.ˆ4+28800.*b.ˆ6+( ...-70400).*b.ˆ8+92160.*b.ˆ10+(-61440).*b.ˆ12+16384.*b.ˆ14+60060.*a.ˆ12.*(( ...-3)+28.*b.ˆ2)+16632.*a.ˆ10.*(15+(-260).*b.ˆ2+728.*b.ˆ4)+34650.*a.ˆ8.*(( ...-5)+120.*b.ˆ2+(-624).*b.ˆ4+832.*b.ˆ6)+200.*a.ˆ6.*(315+(-9240).*b.ˆ2+ ...66528.*b.ˆ4+(-164736).*b.ˆ6+128128.*b.ˆ8)+180.*a.ˆ4.*((-63)+2100.*b.ˆ2+( ...-18480).*b.ˆ4+63360.*b.ˆ6+(-91520).*b.ˆ8+46592.*b.ˆ10)+120.*a.ˆ2.*(7+( ...-252).*b.ˆ2+2520.*b.ˆ4+(-10560).*b.ˆ6+21120.*b.ˆ8+(-19968).*b.ˆ10+7168.* ...b.ˆ12))];
Zmn=[a.ˆ2,4.*a.ˆ2.*b,3.*a.ˆ2.*((-1)+a.ˆ2+4.*b.ˆ2),8.*a.ˆ2.*b.*((-2)+3.*a.ˆ2+ ...4.*b.ˆ2),5.*a.ˆ2.*(1+2.*a.ˆ4+(-12).*b.ˆ2+16.*b.ˆ4+3.*a.ˆ2.*((-1)+8.* ...b.ˆ2)),12.*a.ˆ2.*b.*(3+10.*a.ˆ4+(-16).*b.ˆ2+16.*b.ˆ4+4.*a.ˆ2.*((-3)+10.* ...b.ˆ2)),7.*a.ˆ2.*((-1)+5.*a.ˆ6+24.*b.ˆ2+(-80).*b.ˆ4+64.*b.ˆ6+10.*a.ˆ4.*(( ...-1)+12.*b.ˆ2)+6.*a.ˆ2.*(1+(-20).*b.ˆ2+40.*b.ˆ4)),16.*a.ˆ2.*b.*((-4)+35.* ...a.ˆ6+40.*b.ˆ2+(-96).*b.ˆ4+64.*b.ˆ6+20.*a.ˆ4.*((-3)+14.*b.ˆ2)+6.*a.ˆ2.*( ...5+(-40).*b.ˆ2+56.*b.ˆ4)),9.*a.ˆ2.*(1+14.*a.ˆ8+(-40).*b.ˆ2+240.*b.ˆ4+( ...-448).*b.ˆ6+256.*b.ˆ8+35.*a.ˆ6.*((-1)+16.*b.ˆ2)+10.*a.ˆ4.*(3+(-84).* ...b.ˆ2+224.*b.ˆ4)+2.*a.ˆ2.*((-5)+180.*b.ˆ2+(-840).*b.ˆ4+896.*b.ˆ6)),20.* ...a.ˆ2.*b.*(5+126.*a.ˆ8+(-80).*b.ˆ2+336.*b.ˆ4+(-512).*b.ˆ6+256.*b.ˆ8+280.* ...a.ˆ6.*((-1)+6.*b.ˆ2)+14.*a.ˆ4.*(15+(-160).*b.ˆ2+288.*b.ˆ4)+12.*a.ˆ2.*(( ...-5)+70.*b.ˆ2+(-224).*b.ˆ4+192.*b.ˆ6)),11.*a.ˆ2.*((-1)+42.*a.ˆ10+60.* ...b.ˆ2+(-560).*b.ˆ4+1792.*b.ˆ6+(-2304).*b.ˆ8+1024.*b.ˆ10+126.*a.ˆ8.*((-1)+ ...20.*b.ˆ2)+140.*a.ˆ6.*(1+(-36).*b.ˆ2+120.*b.ˆ4)+70.*a.ˆ4.*((-1)+48.*b.ˆ2+ ...(-288).*b.ˆ4+384.*b.ˆ6)+3.*a.ˆ2.*(5+(-280).*b.ˆ2+2240.*b.ˆ4+(-5376).* ...b.ˆ6+3840.*b.ˆ8)),24.*a.ˆ2.*b.*(462.*a.ˆ10+420.*a.ˆ8.*((-3)+22.*b.ˆ2)+ ...420.*a.ˆ6.*(3+(-40).*b.ˆ2+88.*b.ˆ4)+80.*a.ˆ4.*((-7)+126.*b.ˆ2+(-504).* ...b.ˆ4+528.*b.ˆ6)+a.ˆ2.*(105+(-2240).*b.ˆ2+12096.*b.ˆ4+(-23040).*b.ˆ6+ ...14080.*b.ˆ8)+2.*((-3)+70.*b.ˆ2+(-448).*b.ˆ4+1152.*b.ˆ6+(-1280).*b.ˆ8+ ...512.*b.ˆ10)),13.*a.ˆ2.*(1+132.*a.ˆ12+(-84).*b.ˆ2+1120.*b.ˆ4+(-5376).* ...b.ˆ6+11520.*b.ˆ8+(-11264).*b.ˆ10+4096.*b.ˆ12+462.*a.ˆ10.*((-1)+24.*b.ˆ2) ...
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