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

i

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

ii

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.

iii

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

iv

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.

1

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)

3

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)

4

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].

5

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

6

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)

7

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).

8

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.

9

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

2+266

764

80A

10

2−1353

984

0A

8 2+3572

160A

6 2−

46737

6A

4 2+25

865A

2 2−375

)A4 1+2(42

270

72A

16

2−

143

114

24A

14

2+1969

152

0A

12

2−

14164

992A

10

2+57

062

40A

8 2−

127

900

8A6 2+148

680A

4 2−

7540A

2 2+99

)A2 1+4(65536A

18

2−278

528A

16

2+

495

616A

14

2−479

232A

12

2+273

408A

10

2−93

504A

8 2+18

608A

6 2−198

4A

4 2+94A

2 2−

1)

)

89

Z8,14=112A

2 1

( 5460A

20

1+66

0(200

2A

2 2−45)A

18

1+330(126

672A

4 2−

180

00A

2 2+211

)A16

1+6(7075

868

8A

6 2−

2521

464

0A

4 2+188

001

0A

2 2−

1533

3)A

14

1+96

(1887

142

4A

8 2−125

410

56A

6 2+234

613

5A

4 2−

1226

64A

2 2+783)A

12

1+32

(113

77766

4A

10

2−121

543

488A

8 2+424

111

38A

6 2−

556

749

9A

4 2+23

0202A

2 2−

1225)A

10

1+5(7219

896

32A

12

2−

11310

858

24A

10

2+63

433

497

6A

8 2−155

339

520A

6 2+16

142

112A

4 2−

5644

80A

2 2+

259

7)A

8 1+5(352

509

952A

14

2−

761

303

040A

12

2+6261

37600A

10

2−

24641

356

8A

8 2+47

772

480A

6 2−420

425

6A

4 2+129

850A

2 2−

525)A

6 1+

2(20

122

828

8A16

2−573

112

320A

14

2+6498

406

40A

12

2−

37504

204

8A

10

2+116

83584

0A

8 2−

19185

152A

6 2+14

943

60A

4 2−42

000A

2 2+147

)A4 1+

2(1893

990

4A18

2−

6871

449

6A

16

2+10

26662

40A

14

2−

816

394

24A

12

2+372

787

20A

10

2−9831

936A

8 2+142

878

4A

6 2−

1016

40A

4 2+26

46A

2 2−

7)A

2 1+

16A

2 2(655

36A

18

2−

29491

2A

16

2+561

152A

14

2−

5877

76A

12

2+36

966

4A

10

2−142

848A

8 2+333

28A

6 2−

4424A

4 2+29

4A

2 2−

7)

)

Z8,15=24

0A

2 1A

2

( 120

120A

20

1+85

80(10

92A

2 2−

73)A

18

1+264

(67194

4A

4 2−15

2360A

2 2+53

25)A

16

1+66

(1950

790

4A

6 2−

928

699

2A4 2+

1102

760A

2 2−

2691

3)A

14

1+16

(26701

875

2A

8 2−

217

835

904A

6 2+543

238

08A

4 2−4513

971A

2 2+86

688)A

12

1+16

(438

989

824A

10

2−

54723

468

8A

8 2+23

38691

52A

6 2−408

26478A

4 2+267

7101A

2 2−432

18)A

10

1+16

(36824

1664A

12

2−650

901

888A

10

2+42

499

7760A

8 2−

12714

4776A

6 2+17

52256

8A

4 2−96

967

5A

2 2+13

720

)A8 1+(2492

055

552A

14

2−

5929

226

240A

12

2+548

981

760

0A

10

2−

25090

7008

0A8 2+

5925

984

00A

6 2−

6895

996

8A4 2+3363

920A

2 2−4287

5)A

6 1+4(125

468

672A

16

2−

3866

460

16A

14

2+48

18534

40A

12

2−31

227

2384A

10

2+

1126

604

80A

8 2−

22466

688A

6 2+230

630

4A

4 2−1022

70A

2 2+1191

)A4 1+2(21102

592A

18

2−81

690

624A

16

2+13

176

012

8A

14

2−

114

866

176A

12

2+58

7335

68A

10

2−

178

836

48A

8 2+314

521

6A

6 2−293

832A

4 2+1210

4A

2 2−

127)A

2 1+4(26

2144A

20

2−

1245

184A

18

2+

2523

136A

16

2−284

672

0A14

2+195

788

8A

12

2−

84480

0A

10

2+22

681

6A

8 2−

363

04A

6 2+316

0A

4 2−

122A

2 2+1)

)

90

Z99=9

( 4900A

18

1+22

050

(32A

2 2−1)A

16

1+18

00(784

0A

4 2−

137

2A2 2+23)A

14

1+1680

(50624A

6 2−22

260A

4 2+20

70A

2 2−

25)A

12

1+

144(1382

080A

8 2−

1132

880A

6 2+2646

00A

4 2−

17500A

2 2+17

3)A

10

1+18

(109

24032A

10

2−1439

872

0A

8 2+6272

000A

6 2−

10416

00A

4 2+55

360A

2 2−

485

)A8 1+12(6823

936A

12

2−13

181

952A

10

2+922

752

0A

8 2−2858

240A

6 2+3813

60A

4 2−1746

0A

2 2+

145)A

6 1+36(360

448A

14

2−9605

12A

12

2+98

611

2A

10

2−

490

880A

8 2+122

176A

6 2−1404

0A

4 2+58

0A

2 2−

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

( 882

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−

1095)A

10

1+224(186

976A

8 2−

18623

6A

6 2+57

312A

4 2−594

5A

2 2+149

)A8 1+4(8465

408A

10

2−

12884

480A

8 2+67

952

00A

6 2−

148

120

0A4 2+122

920A

2 2−

2645

)A6 1+4(298

188

8A

12

2−

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

( 1764A

18

1+882

0(36A

2 2−1)A

16

1+280

(28896A

4 2−

4536A

2 2+67

)A14

1+56

(11504

64A

6 2−

452

760A

4 2+375

20A

2 2−

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

2−

6384

960A

8 2+247

4720A

6 2−36

280

0A

4 2+

16920A

2 2−

125)A

8 1+4(5103

6160A

12

2−8833

126

4A

10

2+55

01824

0A

8 2−

15052

800A

6 2+17

584

00A

4 2−700

00A

2 2+475

)A6 1+

4(154

009

60A

14

2−36

743

168A

12

2+33

560

576A

10

2−

1475

4560A

8 2+3216

640A

6 2−

3206

00A

4 2+11

400A

2 2−

75)A

4 1+

(7405

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

2−

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

( 9702

0A

18

1+11

760

(517A

2 2−

39)A

16

1+25

2(3683

68A

4 2−

917

00A

2 2+364

5)A

14

1+336

(157

1328A

6 2−

8226

96A

4 2+

107

805A

2 2−

3020)A

12

1+28(4739

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

2 2−

4599)A

8 1+30(300

195

84A

12

2−5834

752

0A

10

2+42

093

056A

8 2−

139

878

40A

6 2+215

364

8A

4 2−

1340

92A

2 2+2275

)A6 1+8(29425

664A

14

2−

769

84320A

12

2+787

530

24A

10

2−

3997

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

2+388

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

)A2 1+6(1−

4A2 2)2(8192A

14

2−307

20A

12

2+4608

0A

10

2−

3494

4A8 2+

138

24A

6 2−

2592A

4 2+16

6A

2 2−

3))

Z9,13=117A

2 1

( 588

0A20

1+3234

0(40A

2 2−

1)A

18

1+420(973

28A

4 2−

13860A

2 2+18

3)A

16

1+168

(24738

56A

6 2−8808

80A

4 2+65

880A

2 2−

615

)A14

1+336(527

718

4A

8 2−

35087

36A

6 2+65

594

0A

4 2−

344

40A

2 2+257)A

12

1+84(4244

275

2A

10

2−

453

41824A

8 2+158

272

64A

6 2−

207

640

0A4 2+863

52A

2 2−

553)A

10

1+20

(176

79155

2A

12

2−2770

324

48A

10

2+1553

886

72A

8 2−38

063

424A

6 2+39

532

64A

4 2−

139

356A

2 2+805

)A8 1+20(863

764

48A

14

2−

18658

3040A

12

2+15

348

531

2A

10

2−

604

139

52A

8 2+117

15200A

6 2−1030

624A

4 2+

322

00A

2 2−

175)A

6 1+2(1974

599

68A

16

2−

56246

2720A

14

2+63

789

056

0A

12

2−

368

222

208A

10

2+1147

36640A

8 2−

1884

467

2A6 2+

146

776

0A4 2−42

000A

2 2+22

5)A

4 1+(3722

444

8A

18

2−

1351

352

32A

16

2+2020

147

20A

14

2−16

071

884

8A

12

2+73

420

800A

10

2−

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

2−

4718

592A

18

2+89

784

32A

16

2−

940

441

6A14

2+

591

462

4A12

2−22

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

( 120

120A

20

1+3696

0(247A

2 2−

17)A

18

1+132(131

185

6A

4 2−

29708

0A

2 2+10

725

)A16

1+24

(52376

896A

6 2−249

369

12A

4 2+

295

8560A

2 2−

7489

5)A

14

1+48(869

073

92A

8 2−

7092

236

8A6 2+1768

676

8A

4 2−

14691

35A

2 2+29

427

)A12

1+48

(1429

442

56A

10

2−

178

20633

6A

8 2+761

79664A

6 2−132

981

80A

4 2+872

221A

2 2−

1479

8)A

10

1+4(1439

33440

0A

12

2−

2544

837

120A

10

2+1661

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

2 2−

11375

)A6 1+2(2456

94464A

16

2−

7573

012

48A

14

2+94

40051

20A

12

2−

61192

7040A

10

2+

22082

3040A

8 2−

440

476

16A

6 2+452

211

2A

4 2−2014

60A

2 2+2625

)A4 1+4(10354

688A

18

2−401

080

32A

16

2+647

249

92A

14

2−

56453

120A

12

2+28

878

336A

10

2−8796

672A

8 2+1547

648A

6 2−

1446

48A

4 2+60

20A

2 2−

77)A

2 1+1048

576A

20

2−49

807

36A

18

2+100

925

44A

16

2−

11386

880A

14

2+783

1552A

12

2−

3379

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

( 60060A

22

1+360

360

(44A

2 2−1)A

20

1+396

0(1536

08A

4 2−

2002

0A2 2+239)A

18

1+132(582

341

76A

6 2−

18935

280A

4 2+12

90600A

2 2−

10825

)A16

1+36(1162

45043

2A

8 2−70

59852

80A

6 2+12

00038

40A

4 2−

57156

00A

2 2+38

037

)A14

1+33

6(3306

434

56A

10

2−

3226

629

12A

8 2+

10243

200

0A

6 2−

1216

116

0A4 2+4564

44A

2 2−25

83)A

12

1+112

(13593

681

92A

12

2−

1945

641

984A

10

2+99

254

073

6A

8 2−

220058

784A

6 2+

20575

728A

4 2−6509

16A

2 2+3283

)A10

1+12(9124

118

528A

14

2−

18005

198

848A

12

2+134

70944

256A

10

2−

479

954

112

0A8 2+83

79678

72A

6 2−

65983

680A

4 2+18

384

80A

2 2−

857

5)A

8 1+3(13514

37516

8A

16

2−

35163

013

120A

14

2+36

270

080

000A

12

2−189

516

800

00A

10

2+53

17271

040A

8 2−

7817

349

12A

6 2+5414

976

0A

4 2−13

720

00A

2 2+61

25)A

6 1+2(363

148

083

2A

18

2−1204

71552

00A

16

2+1638

531072

0A

14

2−

11803

443

200A

12

2+

48565

43232A

10

2−1147

350

528A

8 2+1483

59680A

6 2−9327

360A

4 2+22

0500A

2 2−

983)A

4 1+6(91750

400A

20

2−37

774

950

4A

18

2+654

704

640A

16

2−

62160

896

0A

14

2+35

247

308

8A

12

2−122

075

136A

10

2+2535

833

6A

8 2−

29747

20A

6 2+17

3880A

4 2−

393

2A2 2+19)A

2 1+

3(1−4A

2 2)2(262

144A

18

2−1179

648A

16

2+2211

840A

14

2−222

8224A

12

2+129

331

2A

10

2−4285

44A

8 2+74

560A

6 2−

5568A

4 2+144A

2 2−

1)

)

93

Z10,10=10

( 1587

6A

20

1+39

200

(81A

2 2−

2)A

18

1+24

50(328

32A

4 2−

518

4A2 2+67)A

16

1+160(400

780

8A

6 2−

15758

40A

4 2+13

132

0A

2 2−

118

5)A

14

1+560(369

964

8A

8 2−271

7568A

6 2+564

060A

4 2−

3318

0A2 2+235)A

12

1+448(673

113

6A

10

2−794

3040A

8 2+307

9160A

6 2−

451

200A

4 2+211

50A

2 2−

125)A

10

1+10

(203

169

792A

12

2−3516

907

52A

10

2+2190

630

40A

8 2−

59942

400A

6 2+70

000

00A

4 2−

2800

00A

2 2+

142

5)A

8 1+80(7667

712A

14

2−18

292

736A

12

2+167

086

08A

10

2−

7345

920A

8 2+1601

600A

6 2−

15960

0A

4 2+57

00A

2 2−

25)A

6 1+

5(1474

560

0A16

2−

463

994

88A

14

2+584

335

36A

12

2−

3767

500

8A

10

2+13

198

080A

8 2−24

550

40A

6 2+218

400A

4 2−720

0A

2 2+25

)A4 1+

40A

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

2−

1990

066

176A

12

2+1683

713

024A

10

2−

70011

200

0A

8 2+14

9720

160A

6 2−

156

901

92A

4 2+686

448A

2 2−8267

)A8 1+16

(11177

164

80A

16

2−315

994

521

6A

14

2+3598

73024

0A

12

2−21

220

41600A

10

2+69

3276

800A

8 2−

124

534

560A

6 2+114

476

04A

4 2−4531

45A

2 2+5075

)A6 1+2(14352

38400A

18

2−

51013

877

76A

16

2+75

243

192

32A

14

2−

597

209

088

0A12

2+

276

675

0720A

10

2−

75923

200

0A

8 2+11

962

598

4A

6 2−99

482

88A

4 2+364

840A

2 2−3885

)A4 1+2(985

661

44A

20

2−430

178

304A

18

2+79

76386

56A

16

2−

819

838

976A

14

2+511

180

800A

12

2−1988

904

96A

10

2+47

86585

6A

8 2−

68221

44A

6 2+52

550

4A

4 2−

182

00A

2 2+189

)A2 1+419

4304A

22

2−

220

200

96A

20

2+500

695

04A

18

2−

6461

849

6A16

2+52

117

504A

14

2−272

629

76A

12

2+929

382

4A

10

2−20

21376A

8 2+26

6624A

6 2−193

76A

4 2+64

4A

2 2−

7)

Z11,15=165A

2 1

( 665

28A

24

1+432

432

(48A

2 2−

1)A

22

1+184

8(5160

48A

4 2−

6177

6A

2 2+67

3)A

20

1+92

4(1585

958

4A

6 2−

474

864

0A4 2+29

6120A

2 2−

225

5)A

18

1+12

(82637

06880A

8 2−

461

91425

28A

6 2+72

034

424

0A

4 2−

312

543

00A

2 2+187

725

)A16

1+24(140

009

492

48A

10

2−

125

76605

184A

8 2+

366

145

472

0A6 2−39

721

5280A

4 2+1351

620

0A

2 2−

68547

)A14

1+336

(181

03255

04A

12

2−

23850

844

16A

10

2+11

158

850

56A

8 2−22

59980

80A

6 2+

192

239

80A

4 2−

54837

6A

2 2+24

57)A

12

1+28

(21598

175

232A

14

2−

392

318

484

48A

12

2+269

205

094

40A

10

2−

8761

855

872A

8 2+139

138

5408A

6 2−

991

920

48A

4 2+247

665

6A

2 2−

1014

3)A

10

1+4(824

762

695

68A

16

2−1975

54470

912A

14

2+1869

014

71232A

12

2−

89216

945

664A

10

2+227

680

588

80A

8 2−

303

015

955

2A6 2+189

015

120A

4 2−4260

060A

2 2+1641

5)A

8 1+8(1198

260

224

0A

18

2−

36591

796

224A

16

2+45

648117

760A

14

2−30

04198

400

0A

12

2+

112

43904

000A

10

2−

24047

96160A

8 2+28

00344

96A

6 2−

15760

080A

4 2+32

830

0A

2 2−122

5)A

6 1+2(69216

501

76A

20

2−

262

46905

856A

18

2+

417

441

382

40A

16

2−

36224

860

160A

14

2+18

691

993

600A

12

2−586

288

332

8A

10

2+109

711

718

4A

8 2−

1152

614

40A

6 2+5990

880A

4 2−

117

600A

2 2+

441

)A4 1+(864

026

624A

22

2−

39824

916

48A

20

2+786

37957

12A

18

2−

869

859

328

0A16

2+5917

409

280A

14

2−25

584517

12A

12

2+70

200

934

4A

10

2−

11849

932

8A

8 2+11

495

680A

6 2−562

240A

4 2+105

84A

2 2−

43)A

2 1+16

777

216A

24

2−922

746

88A

22

2+221

249

536A

20

2−3035

6275

2A

18

2+

26299

5968A

16

2−1497

825

28A

14

2+5660

672

0A

12

2−

14000

128A

10

2+218

0608A

8 2−

198

912A

6 2+929

6A

4 2−

172A

2 2+1

)

98

Z12,12=12

( 213

444A

24

1+25

401

6(242A

2 2−

5)A

22

1+846

72(278

30A

4 2−

3630A

2 2+39

)A20

1+31

360

(949

608A

6 2−

30888

0A

4 2+21

060A

2 2−

157)A

18

1+189

0(85

8035

20A

8 2−

5211

8528A

6 2+8862

336A

4 2−4220

16A

2 2+24

57)A

16

1+23

04(186

909

184A

10

2−

182

434

560A

8 2+

5792

505

6A

6 2−687

960

0A

4 2+257

985A

2 2−1251

)A14

1+11

2(52712

81664A

12

2−

7546

429

440A

10

2+38

50447

104A

8 2−

8538

48576A

6 2+

7986

384

0A

4 2−252

201

6A

2 2+105

95)A

12

1+576

(73756

672

0A

14

2−14

557

491

20A

12

2+10

89381

888A

10

2−388

209

920A

8 2+677

924

80A

6 2−

533

988

0A4 2+148

330A

2 2−

553)A

10

1+15(1049

552

4864A

16

2−27

316

715

520A

14

2+28

183

658

496A

12

2−

147

29838

592A

10

2+41

336

1715

2A

8 2−

6078

535

68A

6 2+421

164

80A

4 2−10

617

60A

2 2+35

49)A

8 1+16

(17661

95200A

18

2−

585

990

1440A

16

2+797

14713

60A

14

2−

574

35750

40A

12

2+

2363

741

184A

10

2−558

558

720A

8 2+722

444

80A

6 2−

4542

720A

4 2+1064

70A

2 2−

315)A

6 1+24(893

911

04A

20

2−

36831

2320A

18

2+63

877

939

2A

16

2−

6068

633

60A

14

2+34

43077

12A

12

2−11

931

033

6A

10

2+24

796

160A

8 2−

291

020

8A6 2+170

100A

4 2−

378

0A2 2+9)A

4 1+192A

2 2(−

512A

10

2+

128

0A8 2−11

52A

6 2+448A

4 2−70A

2 2+3)2A

2 1

)

Z12,13=312A

2 1A

2

( 426

888A

22

1+38

808(1012A

2 2−

63)A

20

1+97

020

(9328A

4 2−

194

0A2 2+63)A

18

1+84

0(972

259

2A

6 2−

42448

56A

4 2+460

614A

2 2−

10423

)A16

1+84

(411

82592

0A

8 2−

308

256

960A

6 2+702

20304A

4 2−531

076

0A

2 2+943

95)A

14

1+168

(447

36102

4A

10

2−51

155

456

0A

8 2+

199

778

304A

6 2−

3171

929

6A4 2+18

848

90A

2 2−

281

31)A

12

1+84(1037

910

016A

12

2−168

316

5184A

10

2+100

42081

28A

8 2−

2733

355

52A

6 2+

340

991

28A

4 2−

17021

04A

2 2+22

271

)A10

1+12

0(4530

831

36A

14

2−9890

406

40A

12

2+8368

174

08A

10

2−

3479

705

60A

8 2+74

4178

40A

6 2−

779

839

2A4 2+341

460A

2 2−

4039

)A8 1+80

(22221

619

2A

16

2−

6282

79296A

14

2+7155

563

52A

12

2−

42195

686

4A

10

2+13

78596

80A

8 2−

247

651

44A

6 2+22

764

42A

4 2−

9018

1A2 2+987)A

6 1+2(14273

740

80A

18

2−

507

346

9440A

16

2+748

329

369

6A

14

2−

593

969

152

0A12

2+

275

183

104

0A10

2−

75516

416

0A

8 2+11

899

008

0A

6 2−98

954

24A

4 2+363

160A

2 2−3745

)A4 1+(1960

837

12A

20

2−85

590

016

0A

18

2+

158

7216

384A

16

2−

16315

8425

6A14

2+10

174

259

20A

12

2−395

900

928A

10

2+952

883

20A

8 2−13

58233

6A

6 2+10

46304A

4 2−

362

60A

2 2+35

7)A

2 1+

419

4304A

22

2−

2202

009

6A20

2+50

069

504A

18

2−6461

8496A

16

2+5211

750

4A

14

2−

27262

976A

12

2+92

938

24A

10

2−

202

137

6A8 2+26

662

4A

6 2−

19376A

4 2+64

4A

2 2−

6)

99

Z12,14=16

8A

2 1

( 6098

4A24

1+55

44(371

8A

2 2−71)A

22

1+27

72(340

912A

4 2−

408

96A

2 2+405

)A20

1+462

0(31437

12A

6 2−

94123

2A

4 2+58806A

2 2−

403)A

18

1+60(163

813

478

4A

8 2−9156

913

92A

6 2+1427

986

56A

4 2−

6206

200A

2 2+3313

1)A

16

1+12

(27755

67308

8A

10

2−

249

33553

920A

8 2+

7259

143

584A

6 2−

787

537

520A

4 2+268

36110A

2 2−119

067

)A14

1+12

(50248

43366

4A

12

2−662

038

446

08A

10

2+309

758

085

12A

8 2−

6273

619

968A

6 2+

5336

858

80A

4 2−152

405

76A

2 2+585

17)A

12

1+12(499

620

577

28A

14

2−

9075

869

286

4A

12

2+62

280

300

544A

10

2−202

715

05408A

8 2+321

920

043

2A

6 2−

2295

197

52A

4 2+5734

666A

2 2−

19565

)A10

1+240

(13630

341

12A

16

2−

3264

970

752A

14

2+30

890

531

84A

12

2−147

461

529

6A

10

2+3763

388

16A

8 2−

5008

763

2A6 2+3124

758A

4 2−70

434A

2 2+21

7)A

8 1+80

(11883

970

56A

18

2−

3629

33452

8A16

2+4527

854

592A

14

2−29

800

422

40A

12

2+111

540

576

0A

10

2−

2385

704

64A

8 2+2778

207

6A

6 2−

15637

86A

4 2+32

550A

2 2−

91)A

6 1+(137

373

941

76A

20

2−

52093

255

680A

18

2+82

853

953

536A

16

2−71

90216

704

0A

14

2+

3710

31449

60A

12

2−116

38243

328A

10

2+217

797

120

0A

8 2−228

824

064A

6 2+118

955

20A

4 2−

2329

60A

2 2+581)A

4 1+(857

735

168A

22

2−39

54180

096A

20

2+

7809

138

688A

18

2−

863

941

427

2A16

2+587

796

480

0A

14

2−

2541

727

744A

12

2+69

750

5792A

10

2−117

752

832A

8 2+114

24448A

6 2−

558

880A

4 2+10

458A

2 2−

21)A

2 1+8A

2 2(2097

152A

22

2−

11534

336A

20

2+27

656

192A

18

2−

379

453

44A

16

2+3287

449

6A

14

2−18

722

816A

12

2+70

758

40A

10

2−17

50016A

8 2+27

2576A

6 2−

2486

4A4 2+1162A

2 2−21

))

Z12,15=120A

2 1A

2

( 47567

52A

24

1+72

072

(717

2A

2 2−

411)A

22

1+66

528

(21392

8A

4 2−410

93A

2 2+1224

)A20

1+396(394

515

264A

6 2−

15903

4512A

4 2+158

852

40A

2 2−

3284

05)A

18

1+52

8(1557

350

080A

8 2−

107

635

320

0A6 2+225

632

736A

4 2−

1565

0915A

2 2+2530

50)A

16

1+36(6332

730

060

8A

10

2−

66864

328

960A

8 2+

2403

074

995

2A

6 2−349

836

6256A

4 2+189

90818

0A

2 2−256

583

7)A

14

1+1008

(34644

459

52A

12

2−518

765

9776A

10

2+284

830

515

2A

8 2−

7108

823

04A

6 2+

8099

352

8A

4 2−367

720

3A

2 2+433

24)A

12

1+28(107

470

454

784A

14

2−

2166

242

181

12A

12

2+16

867

38124

80A

10

2−

643

13632

768A

8 2+125

619

721

92A

6 2−

1197

048

888A

4 2+4744

510

8A

2 2−

50229

9)A

10

1+72(201

194

864

64A

16

2−

5252

608

819

2A

14

2+55

055

05689

6A

12

2−297

69610

752A

10

2+888

323

744

0A

8 2−

1451

164

176A

6 2+1207

241

28A

4 2−

4306

260A

2 2+4189

5)A

8 1+48(7846

952

960A

18

2−257

568

76800A

16

2+349

657

989

12A

14

2−

2545

028

864

0A

12

2+

1076

967

360

0A

10

2−

268

772

784

0A8 2+383

301

600A

6 2−

2869

813

8A4 2+9427

95A

2 2−

8575)A

6 1+16(307

678

412

8A

20

2−

1240

154

1120A

18

2+21

16642

406

4A

16

2−

1995

313

971

2A

14

2+1136

529

408

0A

12

2−40

221

699

84A

10

2+87

627

328

0A

8 2−

112

463

136A

6 2+775

504

8A

4 2−

2390

15A

2 2+2058

)A4 1+3(93

532

979

2A

22

2−

4541

382

656A

20

2+9517

400

064A

18

2−

11278

942

208A

16

2+831

97788

16A

14

2−

3962

84313

6A12

2+1224

153

088A

10

2−

23998

361

6A

8 2+28

363

392A

6 2−

1834

784A

4 2+5387

2A

2 2−

441)A

2 1+6(1−4A

2 2)2(524

288A

20

2−275

251

2A

18

2+6193

152A

16

2−77

824

00A

14

2+595

968

0A

12

2−

2843

136A

10

2+82

6880A

8 2−

1368

00A

6 2+1123

2A

4 2−38

2A

2 2+3)

)

100

Z13,13=13( 85

377

6A

26

1+554

9544(48A

2 2−1)A

24

1+55

036

8(2226

4A

4 2−

266

2A2 2+29)A

22

1+5045

04(373

120A

6 2−

11176

0A

4 2+69

60A

2 2−

53)A

20

1+

182

0(7000

266

24A

8 2−39

127

334

4A

6 2+61

039

440A

4 2−

264

448

8A2 2+158

83)A

18

1+163

8(2635

685

888A

10

2−2367

805

440A

8 2+68

93286

40A

6 2−

748

070

40A

4 2+254

128

0A

2 2−1288

7)A

16

1+31

2(2505

221

734

4A

12

2−

3300

70343

68A

10

2+1544

39577

60A

8 2−

3127

821

312A

6 2+26

61444

80A

4 2−

757

755

6A2 2+339

47)A

14

1+124

8(6227

460

096A

14

2−11

312

795

648A

12

2+77

631

375

36A

10

2−252

686

649

6A

8 2+401

271

584A

6 2−28

615

314A

4 2+

712

887A

2 2−

2919)A

12

1+156(271

849

88160A

16

2−

65118

535

680A

14

2+61

610

823

680A

12

2−

294

114

078

72A

10

2+750

626

688

0A

8 2−

99901

984

0A

6 2+

623

347

20A

4 2−14

011

20A

2 2+53

97)A

10

1+78

0(1580

204

032A

18

2−

482

603

827

2A16

2+6020

95616

0A

14

2−

3962

8144

64A

12

2+14

832

742

40A

10

2−

317

258

368A

8 2+369

45664A

6 2−207

984

0A4 2+431

76A

2 2−

161)A

8 1+52

(34256

977

92A

20

2−1299

054

5920A

18

2+2066

153

472

0A

16

2−

179

306

496

00A

14

2+925

274

112

0A

12

2−

2902

373

376A

10

2+54

31564

80A

8 2−57

066

240A

6 2+29

668

80A

4 2−

579

60A

2 2+217

)A6 1+26(42

78190

08A

22

2−

197

237

145

6A20

2+389

545

984

0A

18

2−

4309

843

968A

16

2+29

324

08320A

14

2−

126

807

6544A

12

2+348

002

304A

10

2−

5875

200

0A

8 2+57

003

52A

6 2−

278

880A

4 2+520

8A

2 2−

21)A

4 1+13

(4096A

12

2−1126

4A

10

2+11

520A

8 2−

537

6A6 2+112

0A

4 2−

84A

2 2+1)

2A

2 1

)

Z13,14=364A

2 1A

2

( 15855

84A

24

1+32

931

36(52A

2 2−

3)A

22

1+11

088

(42556

8A

4 2−

816

86A

2 2+245

1)A

20

1+3960

(13078

208A

6 2−52

729

60A

4 2+52

631

6A

2 2−

1096

9)A

18

1+19

80(137

682

688A

8 2−

951

561

60A

6 2+199

503

36A

4 2−

1382

892A

2 2+2256

1)A

16

1+24(3149

269

094

4A

10

2−

33253

200

640A

8 2+

1195

096

320

0A

6 2−174

00287

52A

4 2+94

394

300A

2 2−128

837

1)A

14

1+24(4824

380

211

2A

12

2−

722

411

233

28A

10

2+396

656

128

00A

8 2−98

99805

696A

6 2+

1128

043

840A

4 2−511

817

18A

2 2+610

097

)A12

1+72(138

577

838

08A

14

2−

2793

363

865

6A

12

2+21

750

95193

6A

10

2−

82936

663

04A

8 2+161

996

665

6A

6 2−

1543

829

28A

4 2+6115

214A

2 2−

65639

)A10

1+60(800

627

097

6A

16

2−

20902

379

520A

14

2+21

909

307

392A

12

2−118

471

77216A

10

2+353

528

806

4A

8 2−

5775

336

00A

6 2+4804

934

4A

4 2−17

12928A

2 2+16

947

)A8 1+240(520

486

912A

18

2−

170

852

352

0A16

2+231

945

830

4A

14

2−

16883

005

44A

12

2+

7144

506

88A

10

2−178

306

688A

8 2+254

292

00A

6 2−

1904

028A

4 2+6251

7A

2 2−58

1)A

6 1+24

(680

52582

4A

20

2−

2743

009

280A

18

2+46

81728

000A

16

2−

441

345

638

4A14

2+2513

96096

0A

12

2−

88971

1872A

10

2+19

383

872

0A

8 2−

248

784

00A

6 2+171

5616A

4 2−

528

50A

2 2+469)A

4 1+2(46

556

774

4A

22

2−

226

072

985

6A

20

2+473

825

280

0A

18

2−

5615

714

304A

16

2+41

426

94400A

14

2−

197

337

9072A

12

2+609

635

328A

10

2−1195

21280A

8 2+14

126

976A

6 2−

913

920A

4 2+268

24A

2 2−

231)A

2 1+16

777

216A

24

2−

964

689

92A

22

2+243

269

632A

20

2−3536

322

56A

18

2+32

761446

4A

16

2−

2019

000

32A

14

2+

83869

696A

12

2−

23293

952A

10

2+42

019

84A

8 2−

465

536A

6 2+286

72A

4 2−812A

2 2+7

)

101

Z13,15=19

5A2 1

( 226

512A

26

1+15

855

84(52A

2 2−

1)A

24

1+110

88(402

688A

4 2−4461

6A

2 2+447)A

22

1+11

088

(739

4816A

6 2−

20511

92A

4 2+11

8008A

2 2−

823)A

20

1+39

6(170

450

508

8A

8 2−88

241

753

6A

6 2+12

708

976

0A

4 2−

506

9680A

2 2+277

55)A

18

1+132

(215

3238

630

4A

10

2−17

915

589

120A

8 2+

48158

880

00A

6 2−480

896

640A

4 2+1498

770

0A

2 2−

68901

)A16

1+24(273

620

492

288A

12

2−

333

896

003

584A

10

2+14

42494

3104

0A

8 2−

2688

488

140

8A

6 2+20

973

444

80A

4 2−

545

69592A

2 2+220

181)A

14

1+24(359

268

614

144A

14

2−

604

468

953

088A

12

2+38

3001

95225

6A

10

2−

11472

242

073

6A

8 2+16

70461

900

8A

6 2−

10878

442

96A

4 2+24

660

272A

2 2−902

09)A

12

1+12

(5417

778

544

64A

16

2−

12020

090

1427

2A14

2+

1050

084

016

128A

12

2−

461

311

262

720A

10

2+1079

521

963

52A

8 2−

1312

182

995

2A6 2+7444

60528A

4 2−

1515

511

2A

2 2+515

97)A

10

1+

12(2318

459

207

68A

18

2−65

58178

344

96A

16

2+7554

878

668

80A

14

2−

45759

697

715

2A

12

2+15

705

078

681

6A

10

2−

306

794

315

52A

8 2+

32488

6150

4A

6 2−165

510

240A

4 2+309

582

0A

2 2−

1004

5)A

8 1+8(813

862

748

16A

20

2−

285

87851

776

0A

18

2+419

873

587

200A

16

2−

3353

483

673

60A

14

2+15

86837

504

00A

12

2−

454

624

55808A

10

2+773

701

824

0A

8 2−735

705

600A

6 2+34

433

280A

4 2−

602

700A

2 2+19

11)A

6 1+

8(967

206

502

4A22

2−413

015

7363

2A

20

2+75

32314

624

0A

18

2−

76697

468

928A

16

2+478

545

51040A

14

2−

18902

008

832A

12

2+47

17669

376A

10

2−

72090

8160A

8 2+62

9771

52A

6 2−

27577

20A

4 2+45

864A

2 2−147

)A4 1+(4043

309

056A

24

2−20

627

587

072A

22

2+457

325

936

64A

20

2−

5782

162

636

8A

18

2+45

99244

3904A

16

2−

239

524

577

28A

14

2+8241

422

336A

12

2−

18467

921

92A

10

2+25

924

147

2A

8 2−21

186

816A

6 2+88

1216A

4 2−

1411

2A

2 2+49

)A2 1+67

108

864A

26

2−4026

53184A

24

2+1065

353

216A

22

2−

16357

785

60A

20

2+16

140

206

08A

18

2−

10705

960

96A

16

2+

48526

131

2A

14

2−14

97825

28A

12

2+30

80806

4A

10

2−404

275

2A

8 2+313

600A

6 2−1254

4A

4 2+19

6A

2 2−

1)

102

Z14,14=14( 2

9446

56A

28

1+683

020

8(169A

2 2−

3)A

26

1+426

888(1460

16A

4 2−

1622

4A

2 2+149)A

24

1+2822

4(4057

081

6A

6 2−11

250

096A

4 2+64

904

4A

2 2−

4115

)A22

1+27

72(340

033

408

0A

8 2−176

04787

20A

6 2+25

348

400

0A

4 2−101

393

60A

2 2+499

59)A

20

1+246

4(161

0887

449

6A

10

2−134

029

2096

0A

8 2+

360

304

560

0A6 2−

35970

480

0A

4 2+11

240

775A

2 2−459

34)A

18

1+42(2183

493

238

784A

12

2−

26645

446

164

48A

10

2+11

511

325

9008

0A

8 2−

214

554

869

760A

6 2+167

346

748

80A

4 2−

4365

56736A

2 2+154

126

7)A

16

1+96(1254

338

854

912A

14

2−21

104

342

138

88A

12

2+13

372

254

945

28A

10

2−

4005

47051

520A

8 2+583

252

158

72A

6 2−

3797

681

888A

4 2+863

109

52A

2 2−

2706

27)A

14

1+84

(108

09071

370

24A

16

2−

239

817

8770

944A

14

2+

209

508

850

073

6A12

2−

92040

216

576

0A

10

2+21

538

626

278

4A

8 2−

261

813

601

28A

6 2+148

520

0976A

4 2−

303

102

24A

2 2+859

95)A

12

1+

672

(57823

068

160A

18

2−

163

564

09344

0A

16

2+188

424

192

000A

14

2−

1141

291

82720A

12

2+3917

044

454

4A

10

2−

76519

027

20A

8 2+810

330

640A

6 2−

412

776

00A

4 2+773

955A

2 2−

2009

)A10

1+84(108

261

277

696A

20

2−

3802

870

579

20A

18

2+5585

412

096

00A

16

2−44

610

682

880

0A

14

2+21

109

630

976

0A

12

2−

604

792

545

28A

10

2+102

927

596

80A

8 2−

9787

45600A

6 2+458

052

00A

4 2−8036

00A

2 2+1911

)A8 1+224(482

554

675

2A

22

2−20

6060

912

64A

20

2+

375

80308

480A

18

2−38

26630

656

0A

16

2+23

876

075

520A

14

2−

943

088

025

6A12

2+235

383

859

2A

10

2−

3596

947

20A

8 2+3142

272

0A

6 2−13

759

20A

4 2+

229

32A

2 2−

49)A

6 1+7(8069

840

896A

24

2−

4117

128

806

4A

22

2+91

28273

510

4A

20

2−

115

41676

032

0A

18

2+918

078750

72A

16

2−47

814

2791

68A

14

2+

16452

165

632A

12

2−3686

817

792A

10

2+5175

475

20A

8 2−

42298

368A

6 2+17

592

96A

4 2−282

24A

2 2+49)A

4 1+56A

2 2(4096A

12

2−

1228

8A10

2+14

080A

8 2−

7680A

6 2+20

16A

4 2−

224A

2 2+7)2A

2 1

)

103

Z14,15=420A

2 1A

2

( 588

931

2A

26

1+15

8558

4(468A

2 2−25)A

24

1+1585

584

(15184A

4 2−

2708A

2 2+75)A

22

1+158

4(199

403

776A

6 2−

746

665

92A

4 2+69

048

98A

2 2−

13237

7)A

20

1+105

6(191

5037

696A

8 2−12

293

355

36A

6 2+23

869

045

2A

4 2−152

815

95A

2 2+228

480

)A18

1+132

(527

83637

504A

10

2−51

765

725

440A

8 2+

17230

699

200A

6 2−23

16182

400A

4 2+11

56654

80A

2 2−

14408

31)A

16

1+264

(515

95661

312A

12

2−71

760

79872

0A

10

2+36

492

135

680A

8 2−

840

933

120

0A6 2+

881

719

776A

4 2−

36692

908A

2 2+39

7355)A

14

1+24

(645

85649

356

8A

14

2−

1209

196

34739

2A12

2+8720

406

3232

0A

10

2−

30700

8542

720A

8 2+55

183

383

552A

6 2−

482

168

825

6A4 2+174

483

078A

2 2−

169

260

5)A

12

1+48

(21484

503

040

0A

16

2−

520

991

178

752A

14

2+5057

74596

096A

12

2−252

520

167

424A

10

2+

69343

307

264A

8 2−

1038

676

284

8A6 2+78

917577

2A

4 2−

255

90859A

2 2+227

451)A

10

1+12(329

057

304

576A

18

2−

1003

275

288

576A

16

2+126

1493

747

712A

14

2−

847

827

599

360A

12

2+330

169

964

544A

10

2−75

554

657

024A

8 2+98

408

981

44A

6 2−

67001

356

8A

4 2+19

915

980A

2 2−165

095

)A8 1+16

(522

56309

248A

20

2−

195

658

383

360A

18

2+309

318

451

200A

16

2−26

925

404

160

0A

14

2+14

114

666

496

0A

12

2−458

046

973

44A

10

2+91

143

6064

0A

8 2−106

37700

00A

6 2+66

389

400A

4 2−

184

180

5A2 2+144

48)A

6 1+8(1134

139

801

6A

22

2−511

54780

160A

20

2+993

070

28480A

18

2−

1086

82444

800A

16

2+73

78786

7136A

14

2−

322

320

3020

8A

12

2+

909

505

408

0A10

2−16

216

467

20A

8 2+17

348

6400A

6 2−1010

486

4A

4 2+2655

66A

2 2−

1995

)A4 1+16(272

62976

0A

24

2−

145

804

492

8A22

2+34

09838

080A

20

2−

458

244

096

0A18

2+39

11417

856A

16

2−

2212

7697

92A

14

2+84

041

574

4A

12

2−212

480

640A

10

2+3472

384

0A

8 2−

3466

704A

6 2+19

1268A

4 2−

482

3A2 2+35)A

2 1+

671

088

64A

26

2−

419

430

400A

24

2+11

6182

220

8A

22

2−18

790

48192A

20

2+19

676

528

64A

18

2−139

821

056

0A16

2+687

161

344A

14

2−

2336

522

24A

12

2+

5410

201

6A

10

2−

8244

736A

8 2+77

9136A

6 2−

41216A

4 2+10

08A

2 2−

7)

104

Z15,15=15

( 1177

862

4A

30

1+883

396

80A

28

1(56A

2 2−1)

+1829

520A

26

1(17035

2A

4 2−

17576A

2 2+16

3)+

396

3960A

24

1(170

060

8A

6 2−

439

296A

4 2+23

472A

2 2−

151)+

151

20A

22

1(438

68830

72A

8 2−

2114

9173

76A

6 2+282

913

488A

4 2−

1045

101

2A

2 2+5257

3)+

1584A

20

1(214

48422

1952A

10

2−16

61892

46080A

8 2+

414

913

699

20A

6 2−

38372

334

00A

4 2+11

040

3300A

2 2−

46428

9)+22

0A

18

1(4433

825

550

336A

12

2−

5038

578

395

136A

10

2+20

21750

56512

0A

8 2−

34897

262

400

0A

6 2+251

362

440

00A

4 2−6017

18544A

2 2+220

952

9)+

990A

16

1(1651

061

161

984A

14

2−

2586

957

381

632A

12

2+15

2241

367

449

6A

10

2−

4223

3478

144

0A

8 2+567

775

488

00A

6 2−

3402

556

416A

4 2+7070

492

8A

2 2−23

3939)

+360A

14

1(4520

995

4549

76A

16

2−

934

098

396

7744A

14

2+

757

930

626

252

8A12

2−

308

373

960

294

4A10

2+666

275

850

240A

8 2−

745

246

064

64A

6 2+387

678

121

6A

4 2−

720

532

12A

2 2+22

0181)+

40A

12

1(240

62643

4048

00A

18

2−

6338

800

8972

288A

16

2+67

822

938

095

616A

14

2−

380

466

11226

624A

12

2+120

563

41118

976A

10

2−

2167

274

672

640A

8 2+

210

429

808

512A

6 2−

97905

986

64A

4 2+16

645

683

6A

2 2−

479

829

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10

1(92136

045

281

28A

20

2−

301

39969

699

840A

18

2+41

116

09430

0160A

16

2−

304

144

941

056

00A

14

2+13

28824

9139

200A

12

2−

3503

376

543

744A

10

2+5466

739

801

60A

8 2−

47473

912

640A

6 2+20

20510

800A

4 2−

319

886

00A

2 2+

8875

3)+60A

8 1(106

4128

479

232A

22

2−423

208

629

0432A

20

2+716

922

421

2480A

18

2−6761

40072

960

0A

16

2+38

9536

972

800

0A

14

2−14

159

572

008

96A

12

2+

324

046

514

176A

10

2−45

22469

952

0A

8 2+35

925

792

00A

6 2−

142

380

000A

4 2+213

007

2A

2 2−

5817

)+40A

6 1(1587

795

722

24A

24

2−

7544

210

718

72A

22

2+

1553

6577

576

96A

20

2−18

194

910

412

80A

18

2+133

64158

464

00A

16

2−640

550

633

472A

14

2+202

106

146

816A

12

2−

4136

66964

48A

10

2+528

092

352

0A

8 2−

390

633

600A

6 2+146

291

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) ...

+630.*a.ˆ8.*(1+(-44).*b.ˆ2+176.*b.ˆ4)+420.*a.ˆ6.*((-1)+60.*b.ˆ2+(-440).* ...b.ˆ4+704.*b.ˆ6)+20.*a.ˆ4.*(7+(-504).*b.ˆ2+5040.*b.ˆ4+(-14784).*b.ˆ6+ ...12672.*b.ˆ8)+3.*a.ˆ2.*((-7)+560.*b.ˆ2+(-6720).*b.ˆ4+26880.*b.ˆ6+(-42240) ....*b.ˆ8+22528.*b.ˆ10)),28.*a.ˆ2.*b.*(7+1716.*a.ˆ12+(-224).*b.ˆ2+2016.* ...b.ˆ4+(-7680).*b.ˆ6+14080.*b.ˆ8+(-12288).*b.ˆ10+4096.*b.ˆ12+1848.*a.ˆ10.* ...((-3)+26.*b.ˆ2)+1386.*a.ˆ8.*(5+(-80).*b.ˆ2+208.*b.ˆ4)+120.*a.ˆ6.*((-35)+ ...770.*b.ˆ2+(-3696).*b.ˆ4+4576.*b.ˆ6)+20.*a.ˆ4.*(63+(-1680).*b.ˆ2+11088.* ...b.ˆ4+(-25344).*b.ˆ6+18304.*b.ˆ8)+24.*a.ˆ2.*((-7)+210.*b.ˆ2+(-1680).* ...b.ˆ4+5280.*b.ˆ6+(-7040).*b.ˆ8+3328.*b.ˆ10)),15.*a.ˆ2.*((-1)+429.*a.ˆ14+ ...112.*b.ˆ2+(-2016).*b.ˆ4+13440.*b.ˆ6+(-42240).*b.ˆ8+67584.*b.ˆ10+(-53248) ....*b.ˆ12+16384.*b.ˆ14+1716.*a.ˆ12.*((-1)+28.*b.ˆ2)+924.*a.ˆ10.*(3+(-156) ....*b.ˆ2+728.*b.ˆ4)+462.*a.ˆ8.*((-5)+360.*b.ˆ2+(-3120).*b.ˆ4+5824.*b.ˆ6)+ ...210.*a.ˆ6.*(5+(-440).*b.ˆ2+5280.*b.ˆ4+(-18304).*b.ˆ6+18304.*b.ˆ8)+4.* ...a.ˆ4.*((-63)+6300.*b.ˆ2+(-92400).*b.ˆ4+443520.*b.ˆ6+(-823680).*b.ˆ8+ ...512512.*b.ˆ10)+4.*a.ˆ2.*(7+(-756).*b.ˆ2+12600.*b.ˆ4+(-73920).*b.ˆ6+ ...190080.*b.ˆ8+(-219648).*b.ˆ10+93184.*b.ˆ12));4.*a.ˆ2.*b,2.*(a.ˆ4+8.* ...a.ˆ2.*b.ˆ2),12.*a.ˆ2.*b.*((-1)+2.*a.ˆ2+4.*b.ˆ2),8.*a.ˆ2.*(a.ˆ4+8.*b.ˆ2.* ...((-1)+2.*b.ˆ2)+a.ˆ2.*((-1)+18.*b.ˆ2)),20.*a.ˆ2.*b.*(1+6.*a.ˆ4+(-12).* ...b.ˆ2+16.*b.ˆ4+a.ˆ2.*((-6)+32.*b.ˆ2)),6.*a.ˆ2.*(5.*a.ˆ6+8.*a.ˆ4.*((-1)+ ...20.*b.ˆ2)+8.*b.ˆ2.*(3+(-16).*b.ˆ2+16.*b.ˆ4)+a.ˆ2.*(3+(-144).*b.ˆ2+400.* ...b.ˆ4)),28.*a.ˆ2.*b.*((-1)+20.*a.ˆ6+24.*b.ˆ2+(-80).*b.ˆ4+64.*b.ˆ6+10.* ...a.ˆ4.*((-3)+20.*b.ˆ2)+4.*a.ˆ2.*(3+(-40).*b.ˆ2+72.*b.ˆ4)),16.*a.ˆ2.*(7.* ...a.ˆ8+5.*a.ˆ6.*((-3)+70.*b.ˆ2)+10.*a.ˆ4.*(1+(-48).*b.ˆ2+168.*b.ˆ4)+16.* ...b.ˆ2.*((-1)+10.*b.ˆ2+(-24).*b.ˆ4+16.*b.ˆ6)+2.*a.ˆ2.*((-1)+90.*b.ˆ2+( ...-600).*b.ˆ4+784.*b.ˆ6)),36.*a.ˆ2.*b.*(1+70.*a.ˆ8+(-40).*b.ˆ2+240.*b.ˆ4+( ...-448).*b.ˆ6+256.*b.ˆ8+140.*a.ˆ6.*((-1)+8.*b.ˆ2)+2.*a.ˆ4.*(45+(-700).* ...b.ˆ2+1568.*b.ˆ4)+4.*a.ˆ2.*((-5)+120.*b.ˆ2+(-504).*b.ˆ4+512.*b.ˆ6)),10.* ...a.ˆ2.*(42.*a.ˆ10+112.*a.ˆ8.*((-1)+27.*b.ˆ2)+35.*a.ˆ6.*(3+(-160).*b.ˆ2+ ...672.*b.ˆ4)+8.*a.ˆ4.*((-5)+420.*b.ˆ2+(-3360).*b.ˆ4+5376.*b.ˆ6)+8.*b.ˆ2.*( ...5+(-80).*b.ˆ2+336.*b.ˆ4+(-512).*b.ˆ6+256.*b.ˆ8)+a.ˆ2.*(5+(-720).*b.ˆ2+ ...8400.*b.ˆ4+(-25088).*b.ˆ6+20736.*b.ˆ8)),44.*a.ˆ2.*b.*((-1)+252.*a.ˆ10+ ...60.*b.ˆ2+(-560).*b.ˆ4+1792.*b.ˆ6+(-2304).*b.ˆ8+1024.*b.ˆ10+210.*a.ˆ8.*(( ...-3)+28.*b.ˆ2)+560.*a.ˆ6.*(1+(-18).*b.ˆ2+48.*b.ˆ4)+a.ˆ4.*((-210)+5600.* ...b.ˆ2+(-28224).*b.ˆ4+34560.*b.ˆ6)+2.*a.ˆ2.*(15+(-560).*b.ˆ2+4032.*b.ˆ4+( ...-9216).*b.ˆ6+6400.*b.ˆ8)),24.*a.ˆ2.*(66.*a.ˆ12+42.*a.ˆ10.*((-5)+154.* ...b.ˆ2)+84.*a.ˆ8.*(3+(-180).*b.ˆ2+880.*b.ˆ4)+140.*a.ˆ6.*((-1)+90.*b.ˆ2+( ...-840).*b.ˆ4+1584.*b.ˆ6)+5.*a.ˆ4.*(7+(-896).*b.ˆ2+12096.*b.ˆ4+(-43008).* ...b.ˆ6+42240.*b.ˆ8)+8.*b.ˆ2.*((-3)+70.*b.ˆ2+(-448).*b.ˆ4+1152.*b.ˆ6+( ...-1280).*b.ˆ8+512.*b.ˆ10)+a.ˆ2.*((-3)+630.*b.ˆ2+(-11200).*b.ˆ4+56448.* ...b.ˆ6+(-103680).*b.ˆ8+61952.*b.ˆ10)),52.*a.ˆ2.*b.*(1+924.*a.ˆ12+(-84).* ...b.ˆ2+1120.*b.ˆ4+(-5376).*b.ˆ6+11520.*b.ˆ8+(-11264).*b.ˆ10+4096.*b.ˆ12+ ...924.*a.ˆ10.*((-3)+32.*b.ˆ2)+42.*a.ˆ8.*(75+(-1540).*b.ˆ2+4752.*b.ˆ4)+ ...240.*a.ˆ6.*((-7)+210.*b.ˆ2+(-1232).*b.ˆ4+1760.*b.ˆ6)+20.*a.ˆ4.*(21+( ...-840).*b.ˆ2+7056.*b.ˆ4+(-19008).*b.ˆ6+15488.*b.ˆ8)+2.*a.ˆ2.*((-21)+ ...1120.*b.ˆ2+(-12096).*b.ˆ4+46080.*b.ˆ6+(-70400).*b.ˆ8+36864.*b.ˆ10)),14.* ...a.ˆ2.*(429.*a.ˆ14+528.*a.ˆ12.*((-3)+104.*b.ˆ2)+462.*a.ˆ10.*(5+(-336).* ...b.ˆ2+1872.*b.ˆ4)+1680.*a.ˆ8.*((-1)+99.*b.ˆ2+(-1056).*b.ˆ4+2288.*b.ˆ6)+ ...30.*a.ˆ6.*(21+(-2800).*b.ˆ2+43120.*b.ˆ4+(-177408).*b.ˆ6+201344.*b.ˆ8)+ ...16.*a.ˆ4.*((-7)+1260.*b.ˆ2+(-25200).*b.ˆ4+147840.*b.ˆ6+(-316800).*b.ˆ8+ ...

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end

TRITA EE 2017:178

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Circuit analysis of a parallel plate waveguide1182100/FULLTEXT01.pdf · goal and that was...

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