university of pavia · 2014. 10. 29. · university of pavia faculty of engineering department of...

UNIVERSITY OF PAVIA

FACULTY OF ENGINEERING

DEPARTMENT OF ELECTRONICS

State-Space / Integral-Equation Method for theS-Domain Modeling of Rectangular Waveguides with

Dielectric Insets

Advisor:Prof. Giuseppe Conciauro

Co-Advisor:Prof. Marco Bressan

Doctoral Thesis

of Wissam Yussef Sabri Eyssa

Italy, Pavia 2006

I dedicate this thesis to my mother Nabila, for her great sacrifice for me. I dedicate

it also to my father Yussef, and to my brother Samer .

I would like to express a feeling of respect and affection to Professor Marisa Grieco.

I would like to express my best thanks and gratitude to Professor Giuseppe Conci-

auro for his patience, his gentleness and his kindness in treating me.

Special thanks are dedicated to Professor Marco Bressan: he was always patient and

available to help me to overcome all of the problems and difficulties found in realizing

this work. I learned from him many things.

Professor Paolo Arcioni was always available and helped me, together with Profes-

sor Giuseppe Conciauro and Professor Luca Perregrini, to solve some serious problems

regarding my permanence in Italy.

Professor Luca Perregrini was really very kind and friendly. He has never got bored

or tired all the times I needed to discuss with him different types of problems concerning

my work. His advices were always precious and valuable.

Dott. Ing. Maurizio Bozzi was very gentle and welcoming: I appreciate his generos-

ity.

Special thanks are dedicated to my colleague Dott. Ing. Gaia Cevini for her helpful-

ness.

Finally, I would like to thank all my colleagues in the Microwave Laboratory for their

sympathy and generosity: Marco Formaggi, Simone Germani, Marco Pasian and Matteo

Repossi.

Contents

1 Introduction 5

1.1 Overview on existing methods of analysis . . . . . . . . . . . . . . . . 6

1.2 The State-Space / Integral-Equation method . . . . . . . . . . . . . . . 9

2 Formulation of the problem and the general lines of the new method 12

2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 BI-RME representation of the scattered field in the air region . . . . . . 16

2.3 BI-RME representation of the field in the dielectric region . . . . . . . 21

2.4 Representation of the incident field . . . . . . . . . . . . . . . . . . . . 22

2.5 Representation of the total field in the air region . . . . . . . . . . . . . 24

2.6 Determination of the unknowns and State-Space formulation . . . . . . 24

2.7 Pole expansion of the Y-matrix and resonant modes of the structure . . . 30

2.8 Other equivalent formulations . . . . . . . . . . . . . . . . . . . . . . 34

3 Application of the method in a simple two-dimensional case 39

3.1 Definition of the structure and State-Space formulation of the problem . 39

3.2 Validation of the method . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Application of the method in a three-dimensional case 55

4.1 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3

4

4.2 Separation of the singular terms of Green’s functions . . . . . . . . . . 60

4.3 Example of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Graphics of the basis functions . . . . . . . . . . . . . . . . . . . . . . 65

A 71

B 74

C Expressions of the resonant fields of the rectangular box 76

D Expressions of the integrals of the singular matrices (3D case) 78

Bibliography 84

Chapter 1

Introduction

Recently, waveguide structures loaded with dielectric insets have received growing im-

portance in microwave systems applications. This is due to the fact that temperature-

stable dielectric materials, with high relative dielectric constants and with very small

losses are available today. A typical application of such structures is in the design of

microwave filters for some applications where a very high quality factor is required (for

example, in high power transmission systems and in base station receivers for cellular

radio).

This thesis proposes a new method for rapid and accurate analysis of a rectangular

waveguide loaded with a dielectric inset, which constitutes the basic building block of

dielectric filters. We implemented this method in the case of one inset of cylindrical

shape which is typically used. However, the theory proposed in this thesis is applicable

to insets of arbitrary shape and could be easily extended to multi-dielectric insets. This

method permits the determination of a reliable wideband model of the structure, yielding

the analytical expression of the General Admittance Matrix (GAM) in pole expansion

form.

5

1.Introduction 6

1.1 Overview on existing methods of analysis

In the last decades, the development of new full-wave methods for the electromagnetic

modeling of passive microwave structures has been at the focus of attention of many

researchers and still has a noticeable growing interest. Particular importance has been

given to inhomogeneously filled waveguides and cavities [1]-[11]. The aim is to create

software, able to carry out an accurate and reliable analysis of such structures in very

short times, and suitable for the inclusion in CAD tools.

Many of the approaches adopted to analyze microwave linear passive structures solve

directly Maxwell’s equations in the time-domain or in the frequency-domain. A method

often used in the time-domain is the Finite-Difference Time-Domain method (FDTD),

because of its flexible applicability to structures with complicated geometry. For its sim-

plicity and versatility, this method has been applied for solving eigenvalue and scatter-

ing problems in waveguiding structures including dielectrics (see [4], [8], for instance).

In its standard formulation, this method is based on the discretization of the unknown

field in both space and time domain. For three-dimensional structures, the field is sam-

pled over a volumetric mesh and, in order to obtain a reasonable accuracy, the number

of variables may be enormous. This implies that the memory requirements and com-

puting times could be prohibitive, in particular, for the full-wave analysis of hybrid

mode waveguiding problems in inhomogeneous waveguiding structures. Furthermore,

the electromagnetic problem has to be solved for each time step in order to obtain the

temporal evolution of the field in the region of interest and, hence, the transient response

of the circuit.

The same drawbacks hold for the Finite Element Method (FEM), usually used in

the frequency-domain. For three-dimensional structures, even not very complicated,

the number of variables (discretized field on a volumetric mesh) may be very large,

and single-frequency analysis may require a long time. Field solutions are generated at

1.Introduction 7

many frequency points in the band of interest, then interpolation is used to determine

the field over the whole frequency band. It is obvious that, with this technique, for very

sharp narrow-band responses some resonances could be ignored which may compromise

the accuracy of the results. To avoid this, the number of frequency points considered

should be sufficiently large, implying further increment of the analysis time and memory

occupance.

New full-wave methods for the simultaneous time-domain and frequency-domain

modeling of linear passive microwave structures have been developed by some researchers

in the last few years [12] [13], with the aim of improving the efficiency of wideband

modeling of electromagnetic structures. These methods lead to the determination of a

reduced-order mathematical model of the structure of interest in the form of pole ex-

pansion of some representative matrix (scattering, admittance, etc.), in the complex fre-

quency plane (s-domain) . It is well known that the transfer function of a closed electro-

magnetic linear device may be represented by means of rational polynomials involving

infinite number of poles and zeros. For excitations that can occur in a certain bandwidth,

a reduced-order model of the structure, representing a good approximation of the sys-

tem in a certain region of the s-plane, can be calculated by the so-called model-order

reduction techniques (MOR). Thanks to these techniques the system is represented with

a finite number of poles [12]. The s-domain analytical expression of this reduced-order

model represents an approximation of the transfer function of the circuit, in the band

of interest. The time-domain response can be simply computed as the inverse Laplace

transform of the transfer function.

One of the known procedures used in the determination of reduced-order models is

the so-called Asymptotic Waveform Evaluation procedure (AWE) [14]-[17]. Originally,

this procedure was used to analyze electronic circuits. The extension to electromagnetic

analysis is reported in [12], and it can be applied to differential- and integral-equation

methods. Solving Maxwell’s equations in the s-domain, either in its differential or inte-

1.Introduction 8

gral form leads to the following system

A(s)x(s) = b(s) (1.1)

where x is the vector containing the desired solution (field samples for differential meth-

ods, current densities and charges for integral methods), b is the excitation vector and

A is the coefficients matrix. The transfer function is defined as H(s) = A(s)−1u(s)

where u(s) is the Laplace transform of the unit impulse. The idea is to approximate

H(s) by a rational function whose numerator and denominator coefficients are obtained

by matching the coefficients of the Taylor-series expansion of it to those of the Taylor-

series expansion of A(s)−1u(s), around a center frequency s0. Numerical simulations

reveal that increasing the order of the Taylor expansion beyond 10 or 12, does not neces-

sarily improve the accuracy of the approximation at frequencies that are far away from

the center frequency [12]. In order to have a wider bandwidth, an improvement may

be obtained by using a procedure that combine the results computed at multiple fre-

quencies. This procedure is called complex frequency hopping (CFH) [17], and has

been employed in the FEM-based electromagnetic code HFSS, to perform the so-called

“fast-sweep”.

MOR is often necessary for differential-equation methods, since the number of the

unknowns may be very large. These model-order reduction techniques, however, repre-

sent an approximation of the transfer function of the circuit of interest.

A different approach is given by Boundary Integral (BI) methods, based on the nu-

merical solution of integral equations (EFIE, MFIE or their combination), where the

number of variables (i.e., the order of the matrices to be calculated) is much smaller.

The application of these methods requires a knowledge of the Green’s functions (GFs)

appropriate to the structure of interest, in a closed form or under the form of a rapidly

converging series. These methods, however, do not lead to state-variable equations, and

therefore do not permit to obtain a mathematical model of the structure under the form

1.Introduction 9

of pole expansion in the s-domain. These methods give rise to a non-linear eigenvalue

problem where the matrices are dependent on the frequency s. The determination of the

resonances of the circuit in a certain band requires to solve the problem in a sufficiently

large number of frequency points in the band of interest, which increases the time of

calculation necessary to obtain a reasonable accuracy. Furthermore, their exist the risk

to skip some resonances of the circuit.

1.2 The State-Space / Integral-Equation method

The method we propose is based on a State-Space/Integral Equation (SS-IE) approach

and is powerful for the s-domain modeling of passive waveguide and quasi-planar inte-

grated structures [18] [19]. The philosophy of the method is based on an appropriate rep-

resentation of the electric and magnetic field in the cavity obtained by closing the ports

of the structure of interest by conducting planes. This type of representation consists

in expressing the field as the sum of its low-frequency limit (in the form of Boundary

Integrals: BI) plus a high frequency correction in the form of rapidly converging series

involving the resonant modes of the cavity (Resonant Mode Expansion: RME). Using

this representation, all matrices are independent of s and a linear eigenvalue problem is

obtained.

The SS-IE approach yields the model of the structure in the standard state-space

form (M + s2N)x = Lv and i = Hx, where the state-variable vector x contains the

disecretized equivalent currents densities, the charges and the mode amplitudes of the

truncated RMEs. The model relates the input, represented by modal voltage excitations

v applied on the waveguide ports, to the corresponding modal currents i, considered as

the output of the system1. As it will be shown, this model yields the currents/voltages

relationship in the from of pole expansion, where the poles are related to the resonances

1Since we consider a lossless structure, the model depends on s2 and not on s.

1.Introduction 10

of the cavity obtained by closing the ports of the structure by perfect conductors. This

means that we obtain the formula of the admittance matrix depending on few parameters

that are calculated once and for all; then the evaluation of the frequency response at a

particular value s is obtained by simple operations of multiplication and summation.

The order of the model obtained is much smaller than that obtained by differential-

equation algorithms due to the small number of state variables involved in the inte-

gral formulation. For this reason MOR is not mandatory and, in any case, less time-

consuming. In the case in which the response of the circuit has to be evaluated in many

frequency points, the SS-IE method is very efficient because, once the coefficients of the

pole expansion are calculated, the GAM can be evaluated at any frequency by simple

multiplication and summation of small matrices. Furthermore, we do not have the risk

of skipping some resonances since the model yields the analytic expression of the Y

parameters 2.

The advantages of this approach goes further. A waveguide section loaded with a

dielectric inset may be a building block of a larger microwave system. For example,

waveguide filters are typically obtained as the cascade of resonators separated by metal

irises coupling the resonators. The whole structure can be subdivided into simple sub-

structures. Each substructure is delimited by two sections which constitutes its physical

ports; the waveguide modes considered on these sections are considered as electrical

ports and a pole expansion of the GAM is calculated for the substructure. Then, using

the efficient algorithm described in [22], the pole expansion of the GAM of the whole

structure is obtained, starting from the pole expansions of the single blocks. Analyzing

simple structures to obtain the model of the overall one, may be more advantageous.

Furthermore, analysis of repeated substructures is avoided.

On the other hand the implementation of the SS-IE method is complicated because

2However, this analytic expression is valid within a certain upper frequency, which depends on the

number of modes retained in the RME.

1.Introduction 11

the computation of the matrices involved is laborious. However, the implementation

of such a complicated method is justified by obtaining a very efficient and reliable code

suitable for inclusion in optimization procedures used in the design of microwave filters.

Chapter 2

Formulation of the problem and the

general lines of the new method

2.1 Formulation of the problem

Fig. 2.1 represents a rectangular waveguide section loaded with a dielectric obstacle

of relative permittivity ε. The dielectric and the waveguide walls are considered loss-

less. The excitation at the physical ports S(1) and S(2) is represented by a set of modal

voltages {v(ν)n } (ν = 1, 2;n = 1, 2, ..., N ), corresponding to the first N waveguide

modes, including all evanescent modes that interact significantly with the dielectric, in

the band of interest. The index n corresponds to a well-determined mode of the waveg-

uide (n↔ TEαβ or TMαβ).

According to the equivalence theorem, the electromagnetic field ( ~E a, ~H a) in the air

region (Fig. 2.2.1) can be determined as the combination of the effects of the applied

12

2.Formulation of the problem and the general lines of the new method 13

S (1)

y

0

x

z

S (2)

a b

c

n

Figure 2.1: A waveguide section loaded with a dielectric obstacle.

�z1

( 2 )nv

�x

J rM r

Sn u l l f i e l d( 1 )

nv

aE raH r

�z2

�x

J rM r

Sn u l l f i e l d

aE raH r

( 1 )nv ( 2 )

nv�z

11

( 2 )nv

�x

J rM r

Sn u l l f i e l d

J rM r

Sn u l l f i e l d( 1 )

nv

aE raH raE raH r

�z22

�x

J rM r

Sn u l l f i e l d

aE raH raE raH r

( 1 )nv ( 2 )

nv

Figure 2.2: Determination of the field in air.

voltages {v(ν)n } and of appropriate electric and magnetic surface currents of density

~M = −~n× ~ET (2.1)defined on S

~J = ~n× ~HT (2.2)

where S indicates the air/dielectric interface, ~ET and ~HT are the components of the field

tangent to S, ~n is the unit vector normal to S, oriented as in Fig. 2.1, and the subscript

T denotes the component tangent to S. The effect of these currents, combined with the

effect of the voltages, creates a null field in the dielectric region; therefore, the field in

the air region is not modified if we remove the dielectric and consider the sources {v (ν)n },

~J and ~M operating in a homogeneous structure totally filled with air (Fig. 2.2.2). The


�z1

�x

J- r

M- r

S

n u l l f i e l ddE rdH r

�z2

�x

J- r

M- r

S

n u l l f i e l ddE rdH r

�z11

�x

J- r

M- r

S

n u l l f i e l ddE rdH rdE rdH r

�z22

�x

J- r

M- r

S

n u l l f i e l ddE rdH rdE rdH r

Figure 2.3: Determination of the field in the dielectric.

field in the air region can be written as

~E a =2∑

ν=1

N∑

n=1

v(ν)

n~E(ν)

n + ~E sc (2.3)

~H a =2∑

ν=1

N∑

n=1

v(ν)

n~H (ν)

n + ~H sc (2.4)

where the summations represent the direct effect of the voltages (incident field) and ~E sc,~H sc (scattered field) are the effect of the equivalent currents in absence of the applied

voltages, i.e., in the air-filled cavity obtained by removing the dielectric inset and by

closing S(1) and S(2) by perfectly conducting planes. Vectors ~E(ν)n , ~H (ν)

n are the fields

generated in the air-filled waveguide when we excite the n-th mode on S (ν) with a unit

voltage and all the other voltages are zero; their expressions are known from waveguide

theory and will be given in a subsequent section of this chapter.

The same currents ( ~J, ~M), with the sign reversed and in the absence of excitation

on ports (ports closed by perfect conductors), generate the correct field1 ( ~Ed, ~Hd) in the

dielectric and a null field in air (Fig. 2.3.1). Also in this case, for the determination

of the field in the dielectric, we can substitute the air with dielectric and consider the1The correct field in the dielectric is such that the continuity condition of the tangential components

of the electric and magnetic fields is fulfilled across the air/dielectric interface.


sources (− ~J,− ~M) as operating in a homogeneous dielectric-filled rectangular cavity

(Fig. 2.3.2).

The current of the m-th mode on the port S (µ) of the waveguide is defined as

i(µ)

m = (−1)µ−1

∫

S(µ)

~hm · ~H a dxdy (2.5)

where ~hm is the corresponding magnetic modal vector, with the normalization condition∫

S(µ) |~hm|2 dxdy = 1. The expression of ~hm can be found in many textbooks (e.g., [26]).

Substituting (2.4) in (2.5) and using the reciprocity theorem to transform the contribution

of the scattered field (see appendix A), we obtain

i(µ)

m = (−1)µ−1

2∑

ν=1

v(ν)

m

∫

S(µ)

~hm · ~H (ν)

m dxdy + 〈 ~E(µ)

m , ~J〉 − 〈 ~H (µ)

m , ~M〉 (2.6)

where 〈~f,~g〉 :=∫

S~f · ~g dS. The first term on the r.h.s. of (2.6) is the direct effect of the

incident field and represents the modal currents/voltages relationship for the m-th mode

in a homogeneous waveguide, which is known in closed form (see later). The other two

terms represent the effect of the field scattered by the dielectric.

Equation (2.6) shows that the crucial point for the calculation of the Generalized

Admittance Matrix (GAM), relating the modal currents {i(µ)m } to the voltages {v(ν)

n }, is

the determination of the currents ~J and ~M as functions of the applied voltages. To de-

termine the unknowns ~J and ~M , we use integral representation to express the fields ~Ea,~Ha, ~Ed and ~Hd as functions of their sources and resolve the pair of equations obtained

by enforcing one of the conditions (2.1) and (2.2) in each region. In this way we can

obtain the following systems of integral equations

A :

~n× ~M = E scT( ~J, ~M) + ~E inc

T

~n× ~M = E dT(− ~J,− ~M)

B :

~J × ~n = H scT( ~J, ~M) + ~H inc

T

~J × ~n = H dT(− ~J,− ~M)

C :

~n× ~M = E scT( ~J, ~M) + ~E inc

T

~J × ~n = H dT(− ~J,− ~M)

D :

~J × ~n = H scT( ~J, ~M) + ~H inc

T

~n× ~M = E dT(− ~J,− ~M)

(2.7)


where E aT, H a

T, E dT, H d

T are integral operators representing the field generated by the

surface currents in the air-filled and in the dielectric-filled rectangular cavity, and ~E incT ,

~H incT represent the incident field tangent to S. Each system of (2.7) could be used to

determine the equivalent sources. Alternatively, we can enforce directly the continuity of

the tangent components of the fields across S and obtain the following pair of equations

E scT( ~J, ~M) + ~E inc

T = E dT(− ~J,− ~M)

H scT( ~J, ~M) + ~H inc

T = H dT(− ~J,− ~M)

(2.8)

Apparently, using this pair of equations is less convenient with respect to the use of

the pairs in (2.7), because they require the introduction of integral representations of

both the electric and the magnetic field in each region. As we shall see later, however,

solving these equations results in a better accuracy of the numerical results, using the

same number of variables.

2.2 BI-RME representation of the scattered field in the

air region

As observed previously, the scattered field in the air region ( ~E sc, ~H sc) can be calculated

as the effect of the currents ~J and ~M in the air-filled rectangular cavity of the same

dimensions of the waveguide section (Fig. 2.4). Observe that the currents create a field

opposite to the incident field in the region occupied by the obstacle.

The BI-RME representation of a cavity field is based on the use of the cavity po-

tentials, in the Coulomb gauge [20] [21] [25]. Furthermore, as it will be shown later,

the vector potentials are expressed as the sum of their low frequency limit plus a high

frequency correction. In the air-filled rectangular cavity, the field is expressed in terms


�z

�x

J rM r

S

s cE rs cH r i n cE- r

i n cH- r

�z

�x

J rM r

S

s cE rs cH rs cE rs cH r i n cE- r

i n cH- r

Figure 2.4: Effect of the currents ~J and ~M in the air-filled rectangular cavity.

of potentials as

~E = −∇φ (e) − jω ~A− 1

ε0∇× ~F (2.9)

~H = −∇φ (m) − jω ~F +1

µ0

∇× ~A (2.10)

where ε0 and µ0 are the electric and the magnetic permittivity in air, ω is the operating

frequency, φ (e) and φ (m) are the electric and the magnetic scalar potentials , ~A and ~F

are the electric and the magnetic vector potentials. In the Coulomb gauge the vector

potentials are solenoidal (∇ · ~A = 0, ∇ · ~F = 0) and the scalar potentials φ (e), φ (m) have

the same form of the electrostatic and the magnetostatic scalar potentials of the cavity.

In order to relate the fields to their sources, we introduce the scalar Green’s functions

(GFs) ge0 and gm

0 for the electrostatic and magnetostatic potentials, and the dyadic GFs

GA and GF for the electric and magnetic vector potentials; we have

φ (e) =−η0

s

∫

S

ge0(~r, ~r

′)∇′S · ~J(~r ′)dS ′ φ (m) =

−1

s η0

∫

S

gm0(~r, ~r

′)∇′S · ~M(~r ′)dS ′ (2.11)

~A = µ0

∫

S

GA(~r, ~r ′) · ~J(~r ′)dS ′ ~F = ε0

∫

S

GF(~r, ~r ′) · ~M(~r ′)dS ′ (2.12)

where s stands for jω√ε0µ0 and η0 =

√µ0

ε0. By substituting (2.11) and (2.12) in (2.9)

and (2.10), we obtain the following expressions of the scattered field tangent to S at any


observation point ~r ∈ S

E scT(~r) =

η0

s∇S g

e0 • ∇S · ~J − sη0G

A • ~J −∇×GF • ~M +1

2~n× ~M(~r) (2.13)

H scT(~r) =

1

sη0

∇S gm0 • ∇S · ~M − s

η0

GF • ~M + ∇×GA • ~J +1

2~J(~r) × ~n (2.14)

where ∇S is the surface nabla operator and we use the notation

g • ∇S ·~V :=

∫−

S

g(~r, ~r ′)∇′S · ~V (~r ′) dS ′ (2.15)

G • ~V :=

(∫−

S

G(~r, ~r ′) · ~V (~r ′) dS ′

)

T

(2.16)

∇×G • ~V :=

(∫−

S

∇×G(~r, ~r ′) · ~V (~r ′) dS ′

)

T

(2.17)

The terms 12~n× ~M and 1

2~J × ~n arise from the integration of the singularity of the curls

of the dyadic GFs. The symbol∫−

Sdenotes integral of singular functions.

Notice that the GFs satisfy the boundary conditions on the cavity walls, so that the

only sources are the currents ~J and ~M on S.

Eigenfunction expansions of the potentials and, hence, of their GFs, could be deter-

mined by using the eigenfunction expansion technique to solve the differential equations

of which the potentials are solutions, taking into account the correct boundary condi-

tions. However, we observe that in equations (2.13), (2.14) the field is split into its

irrotational and solenoidal parts due to the use of the Coulomb gauge. The same split

is obtained in the theory of cavity resonators where the irrotational part is represented

by an expansion into irrotational eigenvectors, and the solenoidal part is represented by

a resonant mode expansion. Thus, by comparing (2.13) and (2.14) with the representa-

tion obtained using the theory of cavity resonators [27], we directly get the following

eigenvector expansions of the solenoidal dyadic GFs

GA(~r, ~r ′) =∞∑

m=1

~Em(~r)~Em(~r ′)

k2m + s2

GF(~r, ~r ′) =∞∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m + s2

(2.18)


where ~Em, ~Hm and km are the electric and magnetic resonant eigenvectors and the cor-

responding wavenumber for the m-th resonant mode of the rectangular cavity; these

eigenvectors are normalized according to∫

V|~Em|2dV =

∫V| ~Hm|2dV = 1, where V is

the volume of the cavity, and are related together by the relations ∇× ~Hm = km~Em and

∇× ~Em = km~Hm. In order to accelerate the convergence of (2.18) , we extract from the

GFs their quasi-static terms (which contains the singularity of the GFs); we have [18]

[19]:

GA(~r, ~r ′) ' GA0(~r, ~r

′) − s2

M a∑

m=1

~Em(~r)~Em(~r ′)

k2m(k2

m + s2)(2.19)

GF(~r, ~r ′) ' GF0(~r, ~r

′) − s2

M a∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m(k2

m + s2)(2.20)

where GA0 and GF

0 are the quasi-static dyadics, independent of s. The singularities of the

GFs, the correct representation of which is crucial for representing the field discontinu-

ities across the currents sheets, are explicitly taken into account in g e0, gm

0 , GA0 and GF

0,

which can be expressed by rapidly convergent image series. Therefore, the truncation of

the modal series in (2.19) and (2.20), necessary for the numerical implementation, does

not affect the accuracy of the representation of the discontinuities. We retain the first

M a modes such that

km ≤ kM a where kM a ≤ ξωmax

√ε0µ0 ≤ kM a+1 (2.21)

where ωmax is the maximum frequency of interest and ξ is an accuracy parameter. The

convergence of the series is sufficiently rapid (∼ 1/k4m for the dyadic GFs and ∼ 1/k3

m

for the curls of the dyadic GFs) for allowing us to choose reasonably small values of ξ

(typically ξ = 2 ÷ 3). Furthermore, (2.20) can be rewritten as

GF(~r, ~r ′) ' GF0(~r, ~r

′) −M a∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m︸︷︷︸

∑∞

m=M a+1

~Hm(~r) ~Hm(~r ′)

k2m

+M a∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m + s2

(2.22)


Scrutiny of equation (2.22) shows that it is equivalent to the representation given in

(2.20); the infinite terms ignored in the calculation of the quasi-static part of the third

term of (2.22) are considered in the calculation of GF0 because it is calculated exactly.

We shall use (2.22) in the magnetic field equation instead of (2.20), because it permits

to use the same state variables to represent the high frequency correction terms of the

electric and magnetic cavity fields.

By substituting (2.19), (2.20) into (2.13) and by substituting (2.19), (2.22) into (2.14)

we obtain

E scT ' η0

s∇S g

e0 • ∇S · ~J − sη0G

A0 • ~J −∇×GF

0 • ~M + s2

M a∑

m=1

a′m

(~Em

)T

+1

2~n× ~M (2.23)

H scT ' 1

sη0

∇S gm0 • ∇S · ~M − s

η0

GF0

a

• ~M + ∇×GA0 • ~J − s

η0

M a∑

m=1

km a′m

(~Hm

)T

+1

2~J× ~n

(2.24)

where

a′m :=sη0〈~Em, ~J〉 + km〈 ~Hm, ~M〉

k2m(k2

m + s2)(2.25)

GF0

a

• ~M := GF0 • ~M −

M a∑

m=1

〈 ~Hm, ~M〉k2

m

(~Hm

)T

(2.26)

a′m are the mode amplitudes for the field in the air-filled rectangular cavity. In equations

(2.23) and (2.24) the electromagnetic field has an hybrid representation given by a low

frequency part calculated as Boundary Integrals plus a high frequency correction in a

Resonant Mode Expansion form (BI-RME representation).


2.3 BI-RME representation of the field in the dielectric

region

As introduced in section 2.1, the field in the dielectric region can be calculated as the

effect of the currents − ~J , − ~M in the dielectric-filled rectangular cavity (Fig. 2.3.2).

Therefore, the BI-RME representation of the field in the inner region of the dielectric-

filled rectangular cavity is obtained by reversing the sign of ~J , ~M and ~n in equations

(2.23) and (2.24) and taking into account the different type of the medium; this means

that η0 is substituted by η0/√ε and s is substituted by s

√ε, where ε is the relative

dielectric constant of the inset. We have

E dT '− η0

s ε∇S g

e0 • ∇S · ~J + sη0G

A0 • ~J + ∇×GF

0 • ~M − s2

M d∑

m=1

bm

(~Em

)T

+1

2~n× ~M (2.27)

H dT '− 1

sη0

∇S gm0 • ∇S · ~M +

s ε

η0

GF0

d

• ~M −∇×GA0 • ~J +

s

η0

M d∑

m=1

km bm

(~Hm

)T

+1

2~J× ~n

(2.28)

where

bm := εsη0〈~Em, ~J〉 + km〈 ~Hm, ~M〉

k2m(k2

m + ε s2)(2.29)

GF0

d

• ~M := GF0 • ~M −

M d∑

m=1

〈 ~Hm, ~M〉k2

m

(~Hm

)T

(2.30)

Coefficients bm are the mode amplitudes for the field in the dielectric-filled rectangular

cavity. The number of the resonant modes considered M d (greater than M a) is deter-

mined by the same criteria (2.21) after substituting ε0 by ε0 ε.


2.4 Representation of the incident field

The incident field is created in the-air filled waveguide by the voltages {v (ν)n } applied on

the ports; it can be written as

~E inc =2∑

ν=1

N∑

n=1

v(ν)

n~E(ν)

n (2.31)

~H inc =2∑

ν=1

N∑

n=1

v(ν)

n~H (ν)

n (2.32)

where ~E(ν)n and ~H (ν)

n have been introduced in section 2.1. From waveguide theory we

have

~E(ν)

n =~en sinh γnζ

(ν) − ~uz(−1)ν(∇ · ~en)γ−1n cosh γnζ

(ν)

sinh γnc(2.33)

~H (ν)

n =

1sη0

−~hn(−1)ν γn cosh γnζ(ν)+~uz (∇·~hn) sinh γnζ(ν)

sinh γncTE modes

sη0

−~hn(−1)ν cosh γnζ(ν)

γn sinh γncTM modes

(2.34)

where ~en is the n-th modal electric vector of the waveguide (normalization∫

S(ν) | ~en |2

dxdy = 1), γn =√κ2

n + s2, κn and ζ (ν) are defined in Table 2.1. Also the incident

field should be expressed in the same form as the cavity field, that is as the sum of a

quasi-static part plus a high frequency correction in the form of pole expansion. To this

aim, the transcendental functions in (2.33) and (2.34) can be put in the required form by

extracting their Laurent expansions around s = 0 (truncated to the first order) and by

representing the residual functions by pole expansions:

sinh γnζ(ν)

sinh γnc= s(ν)

n − s2

∞∑

r=1

(rπ/c)Sr

√2c[−(−1)r]ν−1

k2nr(k

2nr + s2)

(2.35)

cosh γnζ(ν)

γn cosh γnc= c(ν)

n + s2(−1)ν

∞∑

r=0

Cr

√2−δ0r

c[−(−1)r]ν−1

k2nr(k

2nr + s2)

(2.36)


where s(ν)n , c(ν)

n , Sr and Cr are defined in Table 2.1. By substituting (2.35), (2.36) in

(2.33), (2.34) and after truncating the infinite series, we obtain the required form of ~E(ν)n

and ~H (ν)n

~E(ν)

n ' ~U (ν)

n − s2

Rn∑

r=0

θ(ν)nr

knr(k2nr + s2)

~Enr (2.37)

~H (ν)

n ' 1

η0s~V (ν)

n +s

η0

~W (ν)

n − s3

η0

Rn∑

r=0

θ(ν)nr

k2nr(k

2nr + s2)

~Hnr (2.38)

(all symbols defined in Table 2.1). The series in equations (2.37), (2.38) converge rapidly

(at least as fast as 1/r3) and we use the criteria given in (2.21) to truncate them (note

that the number of terms considered Rn depends on n). As done in equation (2.22), we

can put equation (2.38) in the following form

~H (ν)

n ' 1

η0s~V (ν)

n +s

η0

~Wn(ν) +

s

η0

Rn∑

r=0

θ(ν)nr

k2nr + s2

~Hnr (2.39)

where

~Wn(ν) = ~W (ν)

n −Rn∑

r=0

θ(ν)nr

k2nr

~Hnr (2.40)

The infinite terms ignored due to the truncation of the quasi-static part of the last term

in equation (2.39) are considered in the calculation of ~Wn, since it is calculated exactly.

This means that the representation of ~H (ν)n in (2.39) is equivalent to that given in (2.38).

We express ~H (ν)n as in (2.39) instead of (2.38) because it permits to use the same state

variables (mode amplitudes) to represent the RMEs of the electric and magnetic field in

the air region (incident field plus scattered field).


2.5 Representation of the total field in the air region

By substituting equations (2.23), (2.24), (2.37), (2.39) into equations (2.3), (2.4) we

obtain the following expressions of the field tangent to S in the air region

~E aT '

∑

ν,n

v(ν)

n

(~U (ν)

n

)T+η0

s∇S g

e0 • ∇S · ~J − sη0G

A0 • ~J −∇×GF

0 • ~M

+ s2

M a∑

m=1

am

(~Em

)T+~n× ~M

2

(2.41)

~H aT '

1

η0s

∑

ν,n

v(ν)

n

(~V (ν)

n + s2 ~Wn(ν)

)T+

1

sη0

∇S gm0 • ∇S · ~M − s

η0

GF0

a

• ~M + ∇×GA0 • ~J

− s

η0

M a∑

m=1

km am

(~Hm

)T+~J× ~n

2

where am are the mode amplitudes for the field in the air region, defined by

am =sη0〈~Em, ~J〉 + km〈 ~Hm, ~M〉 − km

∑2ν=1 θ

(ν)m v

(ν)n

k2m(k2

m + s2)(2.42)

Note the correspondence m 7→ (n, r).

2.6 Determination of the unknowns and State-Space for-

mulation

The unknowns are given by the currents ~J , ~M and by the mode amplitudes am, bm. As

discussed in section 2.1, the determination of the unknowns, under a given voltage exci-

tation, is essential for the determination of the modal currents/voltages relationship. The

unknowns are determined as the solution of the equations system obtained by enforcing

the continuity of the tangential components of the electric and magnetic field across the

air/dielectric interface (equations 2.8) and taking into account the relations between the


currents and the mode amplitudes represented by equations (2.29) and (2.42). By using

the expressions of ~E aT, ~H a

T, E dT and H d

T in function of the unknowns (equations (2.27),

(2.28), (2.41)) to enforce the continuity condition on S, we obtain

η0

s

1 + ε

ε∇S g

e0 • ∇S · ~J − sη02G

A0 • ~J − 2∇×GF

0 • ~M + s2

M a∑

m=1

am

(~Em

)T+s2

M d∑

m=1

bm

(~Em

)T

=−∑

ν,n

v(ν)

n

(~E(ν)

n

)T

(2.43)

2∇S gm0 • ∇S · ~M − s2GF

0

a

• ~M − s2εGF0

d

• ~M + 2sη0∇×GA0 • ~J − s2

M a∑

m=1

km am

(~Hm

)T

− s2

M d∑

m=1

km bm

(~Hm

)T= −

∑

ν,n

v(ν)

n

(~V (ν)

n + s2 ~Wn(ν)

)T

(2.44)

Equations (2.43), (2.44) and the definitions of the mode amplitudes given in (2.29),

(2.42) constitute a system of four equations in the unknowns. In order to solve (2.43),

(2.44) by the method of moments (MoM) we discretize the currents as

~J = − s

η0

P e∑

p=1

dp ~up (2.45)

~M = −P m∑

p=1

cp ~wp (2.46)

where {~up}, {~wp} are appropriate vector basis functions defined on S and, if the dielec-

tric is in contact with the waveguide walls, subject to the same boundary conditions of~J and ~M respectively. It is important that for each one of these two sets, should exist an

appropriate linear combination of the basis functions that gives rise to a new set in which

it is possible to distinguish solenoidal and non-solenoidal basis functions; the presence

of solenoidal basis functions is necessary to meet the physical constraint at s = 0 (for


s → 0: ∇S · ~J = 0, ∇S · ~M = 0). The divergences of the non solenoidal basis functions

should constitute a set of independent functions suited to represent the charges on S; we

indicate with Q e and Q m the number of independent functions representing the electric

and magnetic charges, respectively.

We substitute (2.45), (2.46) into equations (2.42), (2.29), (2.44), (2.43) and enforce the

field continuity conditions by using the Galerkin method: equations (2.43) and (2.44)

are tested by functions ~up and ~wp respectively, since ~up and ~wp are subject to the same

boundary conditions of the electric and the magnetic field, respectively. Thus, we obtain

the following equations in matrix form

Ka2(Ka2 + s2I) a + s2Ea d + KaHa c = −KaΘv (2.47)

Kd2(ε−1Kd2 + s2I) b + s2Ed d + Kd Hd c = 0 (2.48)

2Smc − 2s2Td + s2GFc − s2 H a K a a − s2 H d K d b = −Vv − s2 Wv (2.49)1 + ε

εSed + 2Tc + 2s2GAd + s2 E a a + s2 E d b = −Uv (2.50)

where the tilde denotes the transpose, v =[{v(1)

n } {v(2)n }]

is the vector of voltages,

a = {am}, b = {bm}, c = {cp}, d = {dp} and the other matrices are defined in table

2.2. In equations (2.47) to (2.50) matrices E, H and K are marked by the letters “a”

and “d” in order to distinguish the different number of resonant modes considered in the

air-filled and in the dielectric-filled cavity.

In order to obtain the system equations in the standard form, we observe that Sm is a


symmetric semidefinite positive matrix2; it can be put in the form

2Sm = Qmλm−1Qm (2.51)

where λm is a positive definite diagonal matrix of dimensions equal to the rank of Sm

(Qm). By introducing a new set of state variables qm defined by

−s2Qmqm = 2Smc + Vv (2.52)

into equation (2.49), the system equations can be written in the form3



Qmqm + 2Td − GFc + HaKa a + HdKd b = Wv (2.55)1 + ε

εSed + 2 Tc + 2 s2GAd + s2 Ea a + s2 Ed b = −Uv (2.56)

s2λmqm + Qmc = −λmQmVv (2.57)

Since matrix GF in (2.55) is non singular it is possible to express variable c in function

of the other variables and eliminate it. Thus, the system equations can be written in

matrix form as follows

(M + s2N)x = Lv (2.58)

c = GF−1(hx − Wv) (2.59)2Let matrix T represents a linear application that transforms the set of basis functions {~wp} in a new set

{~w′p} in which solenoidal basis functions are explicitly distinguished. According to this transformation,

matrix Sm is transformed in

Sm′ = TSmT

defined by [Sm′]kl = 〈∇S · ~w′k, g

m0 • ∇S · ~w′

l〉. In [21] it is shown that matrix Sm′ is semidefinite positive.

Since matrix T is non singular, it is easy to conclude that also matrix Sm is semidefinite positive.3Equation (2.57) is obtained by substituting (2.51) into (2.52) and using the property for which matrix

Qm is a row orthonormal matrix (QmQm = Im, where Im denotes the identity matrix of dimensions equal

to the rank of Sm).


where

M =

Ka4 0 0 0

0 ε−1Kd4 0 0

0 0 1+εε

Se 0

0 0 0 0

+ hGF−1h N =

Ka2 0 Ea 0

0 Kd2 Ed 0

Ea Ed 2GA 0

0 0 0 λm

x =

a

b

d

qm

L = hGF−1W −

Ka Θ

0

U

λmQmV

h =

Ka Ha

Kd Hd

2T

Qm

(2.60)

The solution of the system (2.58), (2.59) determines the mode amplitudes and the coef-

ficients qm, c,d .

Substituting equations (2.37), (2.39), (2.45) and (2.46) in (2.6) we find the modal cur-

rents in the form4

i =1

sη0

Av +s

η0

Bv − s

η0

ΘKa−2Θv − s

η0

Ud +1

sη0

Vc +s

η0

Wc − s

η0

ΘKa a

(2.61)

where matrices A and B are defined in table 2.1. By introducing (2.59) we have

i =1

sη0

Av +s

η0

Bv − s

η0

ΘKa−2Θv − s

η0

W GF−1Wv +

s

η0

Lx +s

η0

VQmλmqm

+1

sη0

Vc

4The result of the following integral is used

∫S(µ)

~hm · ~Hnr dxdy = −δm,n(−1)µ θ(µ)nr


Then by using (2.57) we obtain

i =1

sη0

Av +s

η0

Bv − s

η0

ΘKa−2Θv − s

η0

W GF−1Wv +

s

η0

Lx

− 1

sη0

VQmλmQmVv +1

sη0

V(I − QmQm)︸︷︷︸= 0

c

(2.62)

The last term is null, as demonstrated in appendix B.

In conclusion, we state that the system is described in the s-domain by the following

state-space equations

(M + s2N)x = Lv (2.63)

i =1

sη0

A′v +s

η0

B′v +s

η0

Lx (2.64)

where

A′ = A − VQmλmQmV

B′ = B − ΘKa−2Θ − W GF−1W

Equations (2.63) and (2.64) represent the state-space model of the structure5, relating

the inputs (excitation voltages) to the outputs (modal currents on the ports). The state-

space variables are the mode amplitudes, the coefficients of the electric currents and the

variables qm related to the coefficients of the magnetic currents. All matrices are real

and independent of s. Matrices M and N are symmetric, sparse and of order Mtot =

M a +M d + P e +Q m.5This model is slightly different with respect to that introduced in section 1.2, due to the extraction of

the low-frequency limit of the field.


2.7 Pole expansion of the Y-matrix and resonant modes

of the structure

From equations (2.63) and (2.64) we obtain the GAM relating the modal currents to the

voltages

Y =1

sη0

A′ +s

η0

B′ +s

η0

L(M + s2N)−1L (2.65)

If we need to evaluate Y in few frequency points, the time required to invert the matrix

M + s2N, frequency by frequency, could be acceptable. In general, it is necessary to

consider a large number of frequency points in the band of interest, so the inversion may

require too long times.

In order to avoid the inversion process, we exploit the well-known fact that the in-

verse of M + s2N can be put in the form of pole expansion, where the poles are related

to the resonant modes of the cavity obtained by closing the ports of the structure S (1)

and S(2) with perfect conductors. The procedure can be found in specific textbooks and

is reported in this section for completeness.

We observe that matrix M− λ2N is singular when λ2 is an eigenvalue of the gener-

alized problem:

(M − λ2N)y = 0 (2.66)

In our case matrix N is a symmetric positive definite matrix, so using the Cholesky

decomposition it can be written as

N = UU (2.67)

where U is an upper triangular non singular matrix. By substituting (2.67) in (2.66) and

after simple algebraic manipulations, the eigenvalue problem (2.66) can be transformed

in the standard one

(A − λ2I)z = 0 (2.68)


where A = U−1MU−1 and the eigenvectors matrix Z is related to that of the original

problem by the relation

Z = UY (2.69)

Since matrix A is symmetric, it is possible to diagonalize it through an orthogonal matrix

Z; we have

ZAZ = D (2.70)

Z Z = I (2.71)

where D = diag{λ2i } is the diagonal form of matrix A and (2.71) is the eigenvectors

normalization condition. The eigenvectors of the original eigenvalue problem Y can be

obtained from relation (2.69). By substituting (2.69) in (2.70) and (2.71) we find that Y

satisfies the following conditions

Y MY = D (2.72)

Y NY = I (2.73)

that is matrix Y diagonalizes simultaneously matrices M and N.

By extracting M and N from equations (2.72), (2.73) and substituting in (M+ s2N)−1,

we obtain the required pole expansion of the inverse matrix after simple algebraic ma-

nipulations

(M + s2N)−1 =Mtot∑

i=1

yi yi

λ2i + s2

= Y diag{ 1

λ2i + s2

}Y (2.74)

Finally, by substituting (2.74) in (2.65) we obtain the pole expansion of the GAM of the

structure

Y =1

sη0

A′ +s

η0

B′ +s

η0

LY diag{ 1

λ2i + s2

}Y L (2.75)


It is interesting to note that equation (2.66) coincide with equation (2.63) when v = 0

( s = jλ). This means that the elements of the set {λi} are resonant wavenumbers cor-

responding to the frequencies at which resonant fields can exist in the rectangular cavity

loaded by the dielectric inset obtained by closing the ports of the structure with perfect

conductors, and the eigenvectors y are the State-Space variables corresponding to these

resonant fields. Hence, as expected, the poles of the GAM occur at the resonance fre-

quencies of the structure.

Beside these resonances, the spectrum of (2.66) contains other spurious resonances cor-

responding to the resonant fields that can exist in the cavity obtained by interchanging

the air with the dielectric, with the ports short-circuited. In fact, in enforcing the con-

tinuity of the tangential components of the field at the air/dielectric interface, the terms

~n × ~M/2 and ~J × ~n/2 were simplified (see equations (2.27), (2.28), (2.41) , (2.43),

(2.44)), which means that we lost the information on the location of the air and the di-

electric inside the structure with the ports short-circuited, and that interchanging air with

dielectric we obtain the same eigenvalue problem (2.66).

The spurious modes, however, do not couple with the ports of the waveguide, there-

fore, do not yield actual poles in the GAM. This is due to the fact that the eigensolutions

of (2.66) determine equivalent sources that generate resonant fields that differ from zero

either outside or inside of S, in the air-filled cavity. The resonant fields of the first type

couple with ports and give rise to actual poles in the GAM. On the contrary, the resonant

fields of the other type do not couple with ports and do not correspond to poles of the

GAM, because they are zero on the ports6. This means that these spurious resonances

do not affect the GAM, and, apart from numerical approximations, the corresponding

eigenvectors shall satisfy

Lyi = 0 (2.76)6Remember that to calculate the modal currents on the ports we involve the scattered field created by

the currents in the air-filled cavity (see equation (2.5)).


For this reason elimination of the spurious modes is not necessary. However, we

report an efficient criteria to separate spurious and non spurious modes. For the i-th

solution of (2.66), the electric field in the air-filled cavity tangent externally to S is

given by the first of (2.41) with v (1)n = v (2)

n = 0:

~E ext,i ' η0

s∇S g

e0 • ∇S · ~Ji − sη0G

A0 • ~Ji −∇×GF

0 • ~Mi +s2

M a∑

m=1

am,i

(~Em

)T+~n× ~Mi

2

(2.77)

whereas the electric field in the same cavity tangent internally to S is given by

~E int,iT ' η0

s∇S g

e0 • ∇S · ~Ji − sη0G

A0 • ~Ji −∇×GF

0 • ~Mi +s2

M a∑

m=1

am,i

(~Em

)T− ~n× ~Mi

2

(2.78)

where ~Ji, ~Mi and am,i are the equivalent currents and the mode amplitudes relative to

the i-th solution of (2.66). By substituting (2.45), (2.46) into equations (2.77), (2.78)

and by testing them with the basis functions {~up} we obtain the column vectors

Eexti = Sedi + Tci + s2GAdi + s2 E a ai −

1

2Fci (2.79)

Einti = Sedi + Tci + s2GAdi + s2 E a ai +

1

2Fci (2.80)

where [F]pq = 〈~up, ~n× ~wq〉. Theoretically, if the i-th solution of (2.66) is a non-spurious

resonant mode, vector Einti is zero, whereas if it is a spurious mode, vector Eext

i is zero. In

practice, for numerical approximations, these vectors are not perfectly zero and we use

the following criteria to decide the type of the i-th solution:

‖ Einti ‖<‖ Eext

i ‖ 7→ non-spurious mode (2.81)

‖ Einti ‖>‖ Eext

i ‖ 7→ spurious mode (2.82)

where ‖ E ‖ denotes the Euclidean norm of the column vector E.


2.8 Other equivalent formulations

As shown in this chapter, the formulation adopted to solve the problem is based on

representing both the electric and magnetic field in each region. As mentioned in section

2.1, the formulation of the problem based on the use of other pairs of equations given in

(2.7) may appear more convenient. For example, the use of (2.7.A) requires to represent

only the electric field in each region. However, accuracy being equal, adopting (2.7.A)

requires to consider more cavity modes (larger M a and M d) in order to achieve a good

representation of the discontinuity of the magnetic field across S, and this may increase

significantly the order of the state-space model. In this section we discuss the reason

why this happens with reference to (2.7.A). The same conclusion is valid for all the

other formulations in (2.7).

According to the theory exposed in the previous sections, we represent the electric

cavity field by potentials of which the low frequency limit is extracted and calculated

exactly, whereas the residual part is represented by truncated pole expansions as follows

~E ' −∇φ (e) − jω

(~A0 − µ0s

2

M∑

i=1

~Ei〈~Ei, ~Je〉k2

i (k2i + s2)

)− 1

ε

(∇× ~F0 − εs2

M∑

i=1

~Ei〈 ~Hi, ~Jm〉ki(k2

i + s2)

)

(2.83)

where φ (e) is the quasi-static electric scalar potential, ~A0 and ~F0 are the quasi-static elec-

tric and magnetic vector potentials, the two series are the high frequency corrections for

the electric vector potential and for the curl of the magnetic vector potential respectively,~Je and ~Jm are the electric and magnetic currents in the cavity. In the Coulomb gauge,~F0 is solution of the following differential equation7

∇2 ~F0 = −ε( ~Jm − jωµ0∇φ (m)) (2.85)

7To obtain this equation we consider the low frequency limits of the fields ~E and ~H created by mag-

netic sources

~E0 = −1

ε∇× ~F0

~H0 = −∇φ (m) (2.84)


In formulation (2.7.A), the electric field has the representation (2.83) whereas the dual

explicit representation of the magnetic field is not considered. The magnetic field, how-

ever, should satisfy Maxwell’s equation

−∇× ~E = jωµ0~H + ~Jm (2.86)

By substituting (2.83) in (2.86), by using (2.85) and the relationship existing between

the solenoidal eigenvectors ∇ × ~Ei = ki~Hi, we obtain the following representation of

the magnetic field

~H ' −∇φ (m) +1

µ0

∇× ~A0 − jωεM∑

i=1

~Hi〈 ~Hi, ~Jm〉k2

i + s2− s2

M∑

i=1

~Hi〈~Ei, ~Je〉ki(k2

i + s2)

(2.87)

Equation (2.87) is the implicit representation of the magnetic field in the formulation

(2.7.A). After extraction of the low frequency limit of the first series in (2.87) and rear-

rangement, we have

~H '−∇φ (m) −jω

ε

M∑

i=1

〈 ~Hi, ~Jm〉k2

i

~Hi

︸︷︷︸'~F0

+εM∑

i=1

〈 ~Hi, ~Jm〉k2

i (k2i + s2)

~Hi

+1

µ0

(∇× ~A0− µ0s

2

M∑

i=1

~Hi〈~Ei, ~Je〉ki(k2

i + s2)

)

As we can see, ~F0 is not determined exactly, but is calculated as a slowly converging se-

ries. This requires to consider a larger number of cavity modes in order to achieve a good

representation of ~F0 and, hence, to obtain an accurate representation of the discontinuity

of the magnetic field across the air/dielectric interface.

and substitute them in Maxwell’s equation −∇× ~E0 = jωµ0~H0 + ~Jm taking into account that ∇· ~F0 = 0

in the Coulomb gauge.


On the contrary, adopting the formulation based on (2.8), as done in this chapter,

the quasi-static vector potentials are evaluated exactly, for both fields, whereas the high

frequency corrections are rapidly converging series.


Table 2.1:

~U (ν)n = ~ens

(ν)n − (−1)ν z (∇ · ~en) c(ν)

n

~V(ν)n =

−(−1)ν~hn κ2n c

(ν)n + z (∇ · ~hn) s

(ν)n

0

~W(ν)n =

−(−1)ν 12~hn

(ζ(ν)s

(ν)n + (1 − κnc cothκnc) c

(ν)n

)

+z (∇ · ~hn) 12

(ζ(ν) c

(ν)n − c coth κnc

κns(ν)n

)

−(−1)ν~hnc(ν)n

θ(ν)nr :=

√2c

πrknrc

[−(−1)r]ν−1

√2−δr0

c[−(−1)r]

ν−1

A =

diag(An) diag(A′′

n)

diag(A′′n) diag(An)

B =

diag(Bn) diag(B′′

n)

diag(B′′n) diag(Bn)

An =

κn cothκnc

0A′′

n =

−κncschκnc

0

Bn =

coth κnc−κnc csch2κnc2κn

coth κncκn

B′′n =

− 1−κnc coth κnc2κn sinh κnc

− cschκncκn

ζ(1) = c− z ζ(2) = z

s(ν)n = sinh κnζ(ν)

sinh κncc(ν)n = cosh knζ(ν)

κn sinh κnc

Sr =√

2csin rπz

cCr =

√2−δr0

ccos rπz

c

κn =√

(πpa

)2 + (πqb

)2 knr =√κ2

n + (πrc

)2 n 7→ (p, q)

a, b, c are the dimensions of the waveguide.

δ denotes the Kronecher delta.

The upper (lower) definition holds when n corresponds to a TE (TM) mode.


Table 2.2: Definitions of matricesK a = diag{k1, k2, ..., kMa} K d = diag{k1, k2, ..., kMd}[E]mk =< ~Em, ~uk > [H]mk =< ~Hm, ~wk >

[Se]kl = 〈∇S ·~uk , ge0 • ∇S ·~ul〉 [Sm]kl = 〈∇S · ~wk , g

m0 • ∇S · ~wl〉

[GA]kl = 〈~uk, GA0 • ~ul〉 [GF]kl = 〈~wk, G

F0 • ~wl〉

[GF]kl = (1 + ε)[GF]kl − [H aKa−2H a]kl − ε[H dKd−2H d]kl

[T]kl = 〈~wk ,∇×GA0 • ~ul〉

[U](ν)

kn = 〈~uk, ~U(ν)n 〉 [V](ν)

kn = 〈~wk, ~V(ν)n 〉 ν = 1, 2

[W ](ν)

kn = 〈~wk, ~W(ν)n 〉 − [H aKa−2[Θ](ν)]kn

[Θ](ν)

i,m = δn,mθ(ν)

i where i 7→ (n, r) ν = 1, 2

Θ = [Θ(1) Θ(2)]

I is the identity matrix.

θ(ν)nr :=

√2/c(πr)/(knrc) [−(−1)r]ν−1

Chapter 3

Application of the method in a simple

two-dimensional case

In this chapter we present the application of the SS-IE method to the simple case of a

waveguide section in the H-plane, with a centered full-hight cylindrical dielectric inset

and with a TE2n−1,0 excitation on the ports. The focus is the validation of the method

and to demonstrate, by plotting the corresponding electric and magnetic fields, the inter-

pretation of the solutions of the eigenvalue problem obtained by this method.

3.1 Definition of the structure and State-Space formula-

tion of the problem

Fig. 3.1.a represents the structure of interest. The TE2n−1,0 excitation at the physical

ports S(1) and S(2) is represented by a set of modal voltages v(ν)n (ν = 1, 2 ;n = 1, ..., N),

corresponding to the first N even modes, including all evanescent modes that interact

significantly with the inset. Since the excitation and the geometry are independent of the

y coordinate, also the field is invariant with y; therefore, it can be studied in the H-plane

39

3.Application of the method in a simple two-dimensional case 40

S (1)

y

0

x

z

S (2)

a b

ca

f

b

{v (2)}E a

H a

M, Jnullfield{v (1)}

n n

n

ts

�M, �Jnullfield

E d

H d

n

ts

Figure 3.1: Rectangular waveguide with a centered full-hight cylindrical dielectric inset

(a). H-plane view of the structure with equivalent sources for the determination of the

field in the air and in the dielectric region (b).

section shown in Fig. 3.1.b., where the air/dielectric interface is represented by the line

σ. The electric field is parallel to the y axis whereas the magnetic field is transverse to

it. Therefore, in accordance with the definition of the equivalent currents introduced in

(2.1) and (2.2), the magnetic current is tangent to σ and the electric current is along the

y axis; that is

~M = t E (3.1)

~J = y H (3.2)

whereE andH , respectively, are the intensities of the electric and magnetic field tangent

to σ and t is the unit vector tangent to σ. Note that no electric charges are induced on

the interface σ1.

We use the same type of representation of the incident field and of the field created

by the equivalent sources as discussed in the previous chapter. In this simple 2D case,

we obtain the following expressions of the tangential field at a generic point ~r ∈ σ:

1It is easy to verify that ∇ · ~J = 0


ON THE AIR-SIDE

E a '∑

ν,n

v(ν)

n U (ν)

n − sη0GAH − TaE + s2

M a∑

m=1

am Em +E

2(3.3)

H aT '

1

η0s

∑

ν,n

v(ν)

n

(~V (ν)

n + s2 ~Wn(ν)

)T+

1

sη0

SE − s

η0

GFaE +TH− s

η0

M a∑

m=1

km am

(~Hm

)T+H

2

(3.4)

ON THE DIELECTRIC-SIDE

E d ' sη0GAH + TaE − s2

M d∑

m=1

bm Em +E

2(3.5)

H dT '− 1

sη0

SE +sε

η0

GFdE − TH +

s

η0

M d∑

m=1

km bm

(~Hm

)T+H

2(3.6)

where Em is the intensity of the m-th resonant electric field of the air-filled rectangular

cavity (m 7→ 2n−1, 0, r), ~Hm and km are the corresponding magnetic field and resonant

wavenumber, am and bm are the resonant-mode amplitudes in the air-filled and in the

dielectric-filled cavity, defined by

am :=sη0 < Em, H > +km〈 ~Hm, tE〉 − km

∑2ν=1 θ

(ν)m v

(ν)n

k2m(k2

m + s2)(3.7)

bm := εsη0 < Em, H > +km〈 ~Hm, tE〉

k2m(k2

m + ε s2)(3.8)

where < f, g >:=∫

σfg dσ, 〈~f,~g〉 :=

∫σ~f ·~g dσ and M a, M d are the numbers of modes


retained in the RMEs. The other symbols represent the integral operators

GAH :=

∫

σ

GA0(~r, ~r

′)H(~r ′) dσ′ TH := −∫

σ

∂

∂nGA

0(~r, ~r′)H(~r ′) dσ′

SE :=∂

∂t

∫

S

gm0(~r, ~r

′)∂

∂t′E(~r ′) dσ′ TaE := −

∫

σ

∂

∂n′GA

0(~r, ~r′)E(~r ′) dσ′

GFaE :=

∫

S

t ·(GF

0(~r, ~r′) −

M a∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m

)· t′E(~r ′) dσ′

GFdE :=

∫

S

t ·

GF

0(~r, ~r′) −

M d∑

m=1

~Hm(~r) ~Hm(~r ′)

k2m

· t′E(~r ′) dσ′

All GFs refer to the 2D rectangular box bounded by electric walls. More specifically:

gm0 denotes the GF for the magnetostatic scalar potential; y GA

0 y and GF0 are the dyadic

GFs for the electric and magnetic vector potentials ( ~A and ~F ) in the coulomb gauge

(∇ · ~A = 0, ∇ · ~F = 0) and in the zero frequency limit. All GFs are known in the

spatial-domain (see Table. 3.1), so that their singularity is represented in closed form, as

required for the correct representation of the field discontinuities across the electric and

magnetic current sheets. The summations are truncated, according to the criteria (2.21).

By enforcing the continuity of the tangential components of the field at the air/dielectric

interface we obtain the following integral equations in the unknowns E, H , am and bm:

2TaE + sη02GAH − s2

M a∑

m=1

amEm − s2

M d∑

m=1

bmEm =∑

ν,n

v(ν)

n U (ν)

n (3.9)

2

sη0

SE + 2TH − s

η0

GFaE − s ε

η0

GFdE − s

η0

M a∑

m=1

km am

(~Hm

)T− s

η0

M d∑

m=1

km bm

(~Hm

)T

= − 1

η0s

∑

ν,n

v(ν)

n

(~V (ν)

n + s2 ~Wn(ν)

)T

(3.10)

Harmonic basis functions are simple and convenient for a good approximation of E


Table 3.1:

GA0 = − 1

4π

∞∑

n=−∞

1∑

p,q=0

(−1)p+q lnT pqn

gm0 =

c

3a+z2 + z′

2

2ac− z + z′ + |z − z′|

2a− 1

4π

∞∑

n=−∞

1∑

p,q=0

lnT pqn

GF0 =

1

4π

∞∑

n=−∞

1∑

p,q=0

{xx (−1)q

[ln T pq

n

2+ |Zp

n|Epn

cosXq − Epn

T pqn

]

− zx(−1)qZpnE

pn

sinXq

T pqn

− xz(−1)pZpnE

pn

sinXq

T pqn

+ zz (−1)p

[ln T pq

n

2− |Zp

n|Epn

cosXq −Epn

T p qn

]}

Xq = πa

(x+ (−1)q x′) Zpn = π

a(z + (−1)p z′ − 2nc)

Epn = e−|Zp

n| T pqn = 1 − 2Ep

n cosXq + (Epn)2

and H over σ; we have

E(φ) ' −P∑

p=1

cp cos(p− 1)φ (3.11)

H(φ) ' − s

η0

P∑

p=1

dp cos(p− 1)φ (3.12)

where φ is defined in Fig. 3.1.a, and the coefficients cp, dp are unknowns. We substitute

(3.11) and (3.12) in (3.9) and (3.10); then, we use the Galerkin method to obtain the

following matrix equations:

2 Tc + 2 s2GAd + s2E aa + s2E db = −Uv (3.13)

2Smc − 2 s2Td + s2GFc − s2H aa − s2H db = −(V+s2W)v (3.14)

Some matrices are marked by the letters “a” and “d” in order to distinguish the differ-

ent number of modes considered in the air-filled and in the dielectric-filled cavity. In

the above equations, the tilde denotes the transpose. The matrices are defined in the


following prospect:

Sp,q := −〈Cp,SCq〉 Tp,q := 〈Cp,TCq〉

Em,p := 〈Em, Cp〉 Hm,p := 〈( ~Hm)T, Cp〉

Up,j := 〈Cp , U(ν)n 〉 Vp,j := 〈Cp, (~V

(ν)n )T〉

W p,j := 〈Cp, ( ~W(ν)n )T〉 K := diag{km}

GAp,q := 〈Cp,GACq〉 GF

p,q := 〈Cp,GFaCq〉 + ε〈Cp,GF

dCq〉

Θ = [Θ(1) Θ(2)] [Θ](ν)

i,m = δn,mθ(ν)

i (i 7→ (n, r), ν = 1, 2)

a := col{am} b := col{bm} c := col{cp} d := col{dp}

where Cp stands for cos(p − 1)φ, j 7→ (ν)n and θ(ν)

nr :=√

2/c (πr)/(knrc) [−(−1)r]ν−1.

All matrices are real and independent of s. Further equations are deduced by substituting

(3.11), (3.12) into (3.7), (3.8):



where Θ is defined in the prospective above. Starting from these equations, we obtain

the model of the structure as the one obtained in the previous chapter, except for the fact

that matrix Se is zero in this 2D case and that the other matrices have simplest definitions.


3.2 Validation of the method

Figure 3.2: Dimensions of the structure in the considered example.

This method has been validated, in this simple case, by comparison with the FEM-

based code HFSS. We considered a waveguide section of the dimensions indicated in

Fig. 3.2 loaded by an inset of relative dielectric constant ε = 9. The frequency band of

analysis is 8 ÷ 12 GHz. Perfect agreement of our results with those of HFSS is demon-

strated in Fig. 3.3. In these plots, the S parameters have been evaluated in 300 frequency

points.

The data relative to this simulation are:

Waveguide modes (TE10,TE30,TE50) : N = 3

Resonant modes in the air : M a = 30

Resonant modes in the dielectric : M d = 77

Basis functions : P = 4

Model order : Mtot = 114

The computing time is ∼ 1 second (3 GHz PC).

A list of the resonance frequencies within 12 GHz, relative to the eigensolutions of

(2.66), is presented in Table. 3.2. The criteria used to distinguish spurious and non

spurious modes is that presented in section 2.7. In Table.3.2, the quantities ‖ Eexti ‖2,


‖ Einti ‖2, defined in section 2.7, and their ratio are indicated for each mode. Only

the frequencies denoted with * correspond to non spurious modes; calculation of the

resonance frequencies of the structure with HFSS yields the same set of frequencies

denoted with *. Interchanging air with dielectric, HFSS yields as resonance frequencies

the remanent frequencies shown in table. This results confirm our interpretation of the

solutions of the eigenvalue problem obtained by this method, and validate the criteria

exposed in section 2.7 used to distinguish spurious and non spurious modes.

Furthermore, Fig. 3.4 to 3.13 show the electric and magnetic field patterns in the

air-filled and in the dielectric-filled cavity and the total field for each of the modes con-

sidered in Table 3.2. For non spurious modes the field is zero in the inner region of

the air-filled cavity and is zero in the external region of the dielectric-filled cavity. The

contrary occurs for spurious modes.

In order to highlight the relation between the resonances of the structure and the

poles of the Y - parameters, in Fig. 3.15 we report a plot of the amplitudes of the normal-

ized Y -parameters for some modes (y(ij)mn relates the current of them-th waveguide mode

on the port S(i) to the voltage of the n-th mode on the ports S(j)). The Y -parameters are

plotted in the band 9.4 ÷ 10.2 GHz: in this band f6 and f9 are non spurious modes and

give rise to actual poles, whereas f7 and f8 are spurious modes and have no effect on the

Y -parameters, because the corresponding magnetic field in the air-filled cavity is zero

on the ports (see fields patterns).


S S - I EH F S S

S S - I EH F S S

| S 1 1 | d B

| S 2 1 | d B

[ d B ]

[ d B ]

F r e q [ G H z ]Figure 3.3: Scattering parameters for the fundamental TE10 mode for a WR-90 rectan-

gular waveguide with a centered alumina dielectric inset (ε = 9).


Table 3.2: Frequencies corresponding to eigenvalues calculated in the band 0 ÷ 12 Ghz.

Freq (GHz) ‖ Eexti ‖2 ‖ Eint

i ‖2 ‖Eexti ‖2

‖Einti ‖2

1∗ 3.72 0.0333 0.0037 9.0

2 5.09 0.008 0.0721 0.11

3∗ 6.63 0.054 0.0054 10.0

4 6.74 0.006 0.0516 0.116

5 8.64 0.006 0.0533 0.113

6∗ 9.49 0.07 0.008 8.75

7 9.54 0.005 0.05 0.1

8 9.95 0.002 0.0206 0.097

9∗ 10.14 0.066 0.0074 8.92

10 11.81 0.001 0.0117 0.085

The * denotes non spurious resonances.

+ =

+ =

a i r - f i l l e d c a v i t y d i e l e c t r i c - f i l l e d c a v i t y c a v i t y w i t h i n s e tM a

g n et i c

f i e ld

Electri

c field

Figure 3.4: Non spurious mode. f1=3.72 GHz.


+ =

+ =

a i r - f i l l e d c a v i t y d i e l e c t r i c - f i l l e d c a v i t y c a v i t y w i t h i n s e t

M ag n e

t i c f i e l

dEle

ctric fi

eld

Figure 3.5: Spurious mode. f2=5.09 GHz.

+ =

+ =


g n et i c

f i e ld

Electri

c field



+ =

+ =


M ag n e

t i c f i e l

dEle

ctric fi

eld


+ =

+ =


g n et i c

f i e ld

Electri

c field



+ =

+ =


M ag n e

t i c f i e l

dEle

ctric fi

eld


+ =

+ =+ =


g n et i c

f i e ld

Electri

c field



+ =

+ =


M ag n e

t i c f i e l

dEle

ctric fi

eld


+ =

+ =


g n et i c

f i e ld

Electri

c field



+ =

+ =


M ag n e

t i c f i e l

dEle

ctric fi

eld


0

M A XM A X

0M a g n e t i c f i e l dE l e c t r i c f i e l d

Figure 3.14: Color bar for electric and magnetic field representation in the previous

figures.


1 1

| h 0 y ( 1 1 ) | d B2 2

y ( 1 1 ) | d B1 1 y ( 1 1 ) | d B1 2

f 6 f 7 f 8 f 9

F r e q ( G H z )

| h 0 y ( 1 2 ) | d B

| h 0

y ( 1 2 ) | d B1 1| h 0

| h 0

y ( 1 2 ) | d B1 2| h 02 2

Figure 3.15: f7 and f8 do not give rise to poles in the GAM, because they are resonance

frequencies for spurious modes. f6 and f9 are relative to real resonant modes and give

rise to actual poles.

Chapter 4

Application of the method in a

three-dimensional case

In the this chapter we remove the simplifying hypothesis on the structure of interest

considered in chapter 3. The cylindrical dielectric inset is not full-hight and is not nec-

essarily centered with respect to the waveguide. We consider all the first N waveguide

modes that interact significantly with the inset.

It is necessary to define carefully an appropriate set of basis functions, well-suited

to approximate the unknown currents on the air/dielectric interface. In section 4.1 we

will discuss this set of basis functions and the physical requirements to be satisfied. In

section 4.2 we will extract the singular terms of the quasi-static GFs for the potentials

of the rectangular cavity. Section 4.3 presents and example in which the results of the

simulation based on this method are compared to those obtained by the commercial code

HFSS. Excellent results are shown.

55

4.Application of the method in a three-dimensional case 56

4.1 Basis functions

Let {~vi} indicate the set of basis functions used to approximate the surface currents

({~vi} stands for {~ui} and {~wi} used to descretize ~J and ~M , respectively). The two sets

of basis functions are selected such that non-solenoidal {~v ′i} and solenoidal {~v ′′

i } basis

functions are explicitly distinguished.

Figure 4.1: Cylindrical coordinate system of the cylindrical inset.

Consider the cylindrical coordinate system shown in Fig. 4.1, centered on the cylin-

drical inset. In the local reference (ρ, ϕ, ξ) we define the i-th basis function as follows:

on the plane surface (ξ = const.):

~v ′i(ρ, ϕ) = ρ th(ρ)

cosmϕ even

sinmϕ odd(4.1)

~v ′′i (ρ, ϕ) = ρ m th(ρ)

cosmϕ

sinmϕ+ ϕ (th(ρ) + ρ th(ρ))

− sinmϕ even

cosmϕ odd(4.2)


on the cylindrical surface (ρ = const.):

~v ′i(ξ, ϕ) = ξ th(ξ)

cosmϕ even

sinmϕ odd(4.3)

~v ′′i (ξ, ϕ) = ξ m th(ξ)

cosmϕ

sinmϕ+ ϕ ρ th(ξ)

− sinmϕ even

cosmϕ odd(4.4)

where ρ, ϕ, ξ are the unit vectors, m = 1, ...,M and {th} is a finite-dimensional set of

basis functions. For m = 0, the expressions are:

on the plane surface (ξ = const.):

~v ′i(ρ, ϕ) = ρ th(ρ) even (4.5)

~v ′′i (ρ, ϕ) = ϕ th(ρ) odd (4.6)

on the cylindrical surface (ρ = const.):

~v ′i(ξ, ϕ) = ξ th(ξ) even (4.7)

~v ′′i (ξ, ϕ) = ϕ th(ξ) odd (4.8)

Thus we use vector basis functions with two components: one of them is azimuthal

and the other one is radial or axial. The variation of all components in the azimuthal

direction is represented by harmonic functions, whereas the variation in the radial/axial

direction is related to the functions {th}. The single index i corresponds to a pair (h,m)

with upper or lower definition (even or odd). As required, (4.2), (4.4), (4.6), (4.8) define

solenoidal vectors, whereas (4.1), (4.3), (4.5), (4.7) define vectors that cannot be com-

bined to obtain nonzero solenoidal field. We got excellent results by using triangular

continuous functions as elements of {th}.

In the following considerations we will focus our attention on some requirements and

constraints to be fulfilled by the basis functions in order to represent correctly physical


currents on the air/dielectric interface. Initially, the discussion refers to the electric basis

functions {~ui}. Then, the differences between electric and magnetic basis functions shall

be taken into account. Graphics of electric and magnetic basis functions, for m = 0, 1,

are grouped in section 4.4.

The divergence of the basis functions should non give rise to concentrated charges,

the presence of such charges being unphysical1. This requires that the azimuthal and

radial/axial components of the basis functions should be continuous with respect to ϕ

and ρ/ξ, respectively. This condition is fulfilled for the azimuthal components since the

dependence on ϕ is represented by harmonic functions. The continuity of the other two

components is obtained because {th} are continuous functions.

In Fig. 4.4 and 4.5 are traced the triangular functions {th} for non-solenoidal and

solenoidal electric basis functions, when m = 0. The continuous lines are relative to

radial/axial components whereas the dashed lines are relative to azimuthal components.

Fig. 4.6 and 4.7 refer to the case m = 1.

Functions traced in red (tα and tδ) can not be associated to all values of m. Function

tα (non solenoidal with one extreme on the axis) can only be considered with m = 0 2.

Function tδ (nonzero on the axis) can be considered only with m = 1; this function is

necessary to represent nonzero currents passing by the center of the plane surface of the

inset.

Functions traced in green (indicated with tβ) are nonzero on both surfaces and are

necessary to insure the continuity of the currents in the radial/axial direction.

Finally, functions traced in blue (indicated with tγ) can belong only to the set of the

electric basis functions, since they have nonzero axial component on the edge of contact

1Furthermore, introducing concentrated charges, represented by Dirac delta distributions, in the irrota-

tional terms of the cavity field tangent to S (see equations (2.27), (2.28), (2.41)) will transform the surface

integrals in linear integrals of the singular part of the scalar GF: these integrals diverge.2Associating function tα to values of m > 0 leads to non well-defined charges on the center of the

plane surface.


with the waveguide. They are necessary to insure the continuity of the axial electric

current on the edge. Furthermore, the azimuthal component of these basis function

should be zero on the edge for a correct representation of the boundary condition of the

magnetic field on the waveguide wall.

The set of basis functions with m > 1 is the same of that shown for m = 1 except

for the fact that the basis function tδ is not considered.

Fig. 4.8, 4.9, 4.10 and 4.11 refer to the magnetic basis functions. The same consid-

erations hold, except for the functions that are nonzero on the waveguide wall (traced in

blue). Simply, the boundary conditions are dual to that of the electric current. Further-

more, we note the presence of the function tσ (indicated in yellow), necessary for the

representation of ξ-independent azimuthal magnetic current on the cylindrical surface,

for m > 0.


4.2 Separation of the singular terms of Green’s func-

tions

As shown in Tab. 2.2, the calculation of matrices Se, Sm, GA, GF and T requires to

evaluate the quasi-static GFs ge0, gm

0 , GA0, GF

0 and ∇×GF0. The study of the singularity of

ge0 and gm

0 shows that they can be written as:

ge0(~r, ~r

′) =1

4πR+ ge

reg(~r, ~r′) gm

0(~r, ~r ′) =1

4πR+ gm

reg(~r, ~r′) (4.9)

where R is the euclidean distance between observation and source points, and g ereg, g

mreg

are non-singular functions.

The study of the dyadics shows that they can be expressed as [23]:

GA0(~r, ~r

′) =1

8πR(I +

~R~R

R2) +GA

reg(~r, ~r′) (4.10)

GF0(~r, ~r

′) =1

8πR(I +

~R~R

R2) +GF

reg(~r, ~r′) (4.11)

∇×GA0(~r, ~r

′) = ∇× 1

8πR(I +

~R~R

R2) + ∇×GA

reg(~r, ~r′) (4.12)

where GAreg and GF

reg are non-singular dyadics. The non singular GFs gereg, g

mreg, G

Areg, G

Freg

and ∇×GAreg are calculated as image series whose convergence is accelerated by Ewald

technique [24]. The explicit expressions of the electric GFs, for instance, are reported in

[18]; the expressions of the magnetic GFs are similar.

For ~r, ~r ′ ∈ S, where S denotes the air/dielectric interface, it is possible to express


(4.10)-(4.12 ) as3

GA0(~r, ~r

′) =I

4πR+

1

8π(∇S + ~n

∂

∂n)(∇′

S + ~n′ ∂

∂n′)R +GA

reg(~r, ~r′) (4.15)

GF0(~r, ~r

′) =I

4πR+

1

8π(∇S + ~n

∂

∂n)(∇′

S + ~n′ ∂

∂n′)R +GF

reg(~r, ~r′) (4.16)

∇×GA0 =

1

4π∇ 1

R× I + ∇×GA

reg(~r, ~r′) (4.17)

We note that the terms containing the derivative with respect to n and n′ don’t contribute,

since they are normal to S and the currents are tangent. By using the expressions of GFs

(4.9), (4.15), (4.16) and (4.17) we can express the matrices involving the GFs as:

Se = Sesing + Se

reg Sm = Smsing + Sm

reg (4.18)

GA = GAsing + IA + GA

reg GF = GFsing + IF + GF

reg (4.19)

T = Tsing + Treg (4.20)

where all matrices are defined in Tab. 4.1.

3The singular term of the dyadics can be transformed in:

1

8πR(I +

~R~R

R2) =

I

4πR+

∇∇′R

8π=

I

4πR+

1

8π(∇S + ~n

∂

∂n)(∇′

S + ~n′ ∂

∂n′)R (4.13)

where ~n is the unit vector normal to S as indicated in Fig. 2.1, whereas the singular term of the curl of the

dyadics can be expressed as:

∇× 1

8πR(I +

~R~R

R2) = ∇× I

4πR=

1

4π∇ 1

R× I. (4.14)


Table 4.1: Matrices definitions

[Sesing]ij =

1

4π

∫

S

∫

S

(∇S · ~ui(~r))(∇′

S · ~uj(~r′))

1

RdSdS ′ [Sm

sing]ij =1

4π

∫

S

∫

S

(∇S · ~wi(~r))(∇′

S · ~wj(~r′))

1

RdSdS ′

[GAsing]ij =

1

4π

∫

S

∫

S

~ui(~r) · ~uj(~r′)

1

RdSdS ′ [GF

sing]ij =1

4π

∫

S

∫

S

~wi(~r) · ~wj(~r′)

1

RdSdS ′

[Tsing]ij = −1

4π

∫

S

∫

S

~wi(~r) × ~uj(~r′) · ∇

1

RdSdS ′

[Sereg]ij =

∫

S

∫

S

(∇S · ~ui(~r))gereg(~r, ~r ′)(∇′

S · ~uj(~r′))dSdS ′ [Sm

reg]ij =

∫

S

∫

S

(∇S · ~wi(~r))gmreg(~r, ~r ′)(∇′

S · ~wj(~r′))dSdS ′

[GAreg]ij =

∫

S

∫

S

~ui(~r) · GAreg(~r, ~r ′) · ~uj(~r

′)dSdS ′ [GFreg]ij =

∫

S

∫

S

~wi(~r) · GFreg(~r, ~r ′) · ~wj(~r

′)dSdS ′

[Treg]ij =

∫

S

∫

S

~wi(~r) · ∇ × GAreg(~r, ~r ′) · ~uj(~r

′)dSdS ′

[IA]ij =1

8π

∫

S

∫

S

~ui(~r) · ∇S∇′

SR · ~uj(~r′)dSdS ′

=1

8π

(∫

S

∫

S

(∇S · ~ui(~r))(∇′

S · ~uj(~r′)) R dS dS ′ +

∫

l

∫

l

(~ui(~r) · m(~r))(~uj(~r′) · m′(~r ′)) R dl dl ′

−

∫

S

∫

l

(∇S · ~ui(~r))(~uj(~r′) · m′(~r ′)) R dS dl ′ −

∫

l

∫

S

(~ui(~r) · m(~r))(∇′

S · ~uj(~r′)) R dl dS ′

)

[IF]ij =1

8π

∫

S

∫

S

~ui(~r) · ∇S∇′

SR · ~wj(~r′)dSdS ′

=1

8π

(∫

S

∫

S

(∇S · ~wi(~r))(∇′

S · ~wj(~r′)) R dS dS ′ +

∫

l

∫

l

(~wi(~r) · m(~r))(~wj(~r′) · m′(~r ′)) R dl dl ′

−

∫

S

∫

l

(∇S · ~wi(~r))(~wj(~r′) · m′(~r ′)) R dS dl ′ −

∫

l

∫

S

(~wi(~r) · m(~r))(∇′

S · ~wj(~r′)) R dl dS ′

)

l denotes the circle of contact of the cylindrical surface with the wall of the waveguide.

m is the unit vector normal to l.

The matrices that depend on the regular part of the quasi-static GFs have been cal-

culated numerically. The integrals in the other matrices have been treated analytically.

Furthermore, the double surface integrals in the singular matrices have been reduced in

triple simple integrals, by exploiting the properties of the trigonometric functions used

to represent the variation of currents in azimuth. The expressions of the integrals to be

computed in order to calculate the singular matrices are reported in appendix D.


4.3 Example of simulation

Figure 4.2: Dimensions of the structure in the 3D example.

The structure shown in Fig. 4.2 has been analyzed by this method in the frequency

band 8÷ 12 GHz, and the results have been compared to those obtained by the commer-

cial code HFSS. The comparison is satisfactory as shown in Fig. 4.3. In these plots, the

S parameters have been evaluated in 300 frequency points.


S S - I EH F S S

[ d B ]

| S 1 1 | d B

| S 2 1 | d B

Figure 4.3: Scattering parameters for the fundamental TE10 mode for the structure in the

considered example.

The data relative to this simulation are:

Waveguide modes : N = 10

Resonant modes in the air : M a = 93

Resonant modes in the dielectric : M d = 193

Electric basis functions : P e = 130

Magnetic basis functions : P m = 130

Model order : Mtot = 480

The computing time is 20 seconds (3 GHz PC).


4.4 Graphics of the basis functions

Figure 4.4: Non-solenoidal electric basis functions (m = 0).

Figure 4.5: Solenoidal electric basis functions (m = 0).


Figure 4.6: Non-solenoidal electric basis functions (m = 1).


Figure 4.7: Solenoidal electric basis functions (m = 1).


Figure 4.8: Non-solenoidal magnetic basis functions (m = 0).

Figure 4.9: Solenoidal magnetic basis functions (m = 0).


Figure 4.10: Non-solenoidal magnetic basis functions (m = 1).


Figure 4.11: Solenoidal magnetic basis functions (m = 1).

Appendix A

We use the reciprocity theorem to demonstrate the relation

−(−1)(µ)

∫

S(µ)

~hm · ~Hscdxdy = 〈 ~E(µ)

m , ~J〉 − 〈 ~H (µ)

m , ~M〉 µ = 1, 2 (A.1)

where ~Hsc is the magnetic field generated in the volume V of the air-filled rectangular

cavity by the surface sources ~J and ~M defined on S, and ~hm is the m-th magnetic modal

vector of the wavegiude (see chapter 2).

From waveguide theory we recall the relation

~hm = z × ~em (A.2)

where ~em is them-th electric modal vector and z is the unit vector of the z-axis indicated

as in Fig. 2.1. It is noted that ~em represents the electric field tangent to S (µ) when the m-

th waveguide mode on this port is excited by a unitary voltage and all the other voltages

are zero. In (A.1), as well as in chapter 2, this field and the corresponding magnetic field

are denoted by ~E(µ)m and ~H (µ)

m , respectively. Thus, in order to demonstrate (A.1), we need

to verify that

−(−1)(µ)

∫

S(µ)

~E(µ)

m × ~Hsc · z dxdy = 〈 ~E(µ)

m , ~J〉 − 〈 ~H (µ)

m , ~M〉 µ = 1, 2

(A.3)

For simplicity, we view the demonstration when µ = 1. The reciprocity theorem is

applied to the volume V where we consider the field ~Esc, ~Hsc created by the sources ~J

71

APPENDIX A. 72

and ~M when the boundary is short-circuited, and the field ~E(1)m , ~H(1)

m (see Fig. A.1.a and

Fig. A.1.b). We have

J rM rs cE r

�n

s cH r

SVS

J rM rs cE r

�n

s cH r

SVS

�n

( 1 )mE r ( 1 )

mH r( 1 )mE r ( 1 )

mH r�z

VS

( 1 )S

aa

bb

unit vo

latge

Figure A.1: The reciprocity theorem is applied to the fields: ~Esc, ~Hsc created by the

sources ~J , ~M in the rectangular cavity (a), and ~E(1)m , ~H(1)

m (b).

∫

SV

~Esc × ~H(1)m · n dSV =

∫

SV

~E(1)m × ~Hsc · n dSV + 〈 ~E(1)

m , ~J〉 − 〈 ~H(1)m , ~M〉

(A.4)

where SV is the boundary of V , 〈 ~f,~g〉 :=∫

S~f · ~g dS and n is the unit vector normal to

SV . The first member of (A.4) is null because the electric field generated by the sources

in the rectangular cavity is normal to SV . The tangential component of the electric field~E

(1)m is zero on all the boundary except on S(1), where n = −z. Therefore, we have

∫

S(1)

~E(1)m × ~Hsc · z dSV = 〈 ~E(1)

m , ~J〉 − 〈 ~H(1)m , ~M〉 (A.5)

APPENDIX A. 73

Analogously, for µ = 2 we obtain

−∫

S(2)

~E(2)m × ~Hsc · z dSV = 〈 ~E(2)

m , ~J〉 − 〈 ~H(2)m , ~M〉 (A.6)

Appendix B

We demonstrate that the non symmetric term in equation (2.62) is null. For simplicity we

suppose that it is possible to classify the basis functions of the set {~wp} into solenoidal

and non-solenoidal ones. The number of the non-solenoidal functions isQ m (see section

2.6) and we suppose that they precede the solenoidal ones in the set. According to this

subdivision of the basis functions, we partition matrix 2Sm as follows

2Sm =

2Sm 0

0 0

Matrix Sm is symmetric positive definite [21] and is related to its diagonal form λm−1,

defined in section 2.6, by the relation

2Sm = Rλm−1 R (B.1)

where R satisfies the orthogonality condition

RR = Im (B.2)

and Im is the identity matrix of dimension Q m. In equation (2.51) we decomposed 2Sm

as 2Sm = Qmλm−1Qm. By comparing with the equations above, we can state that

Qm =[

R 0

](B.3)

74

APPENDIX B. 75

and that

Qm Qm =

Im 0

0 0

(B.4)

On the other hand, the waveguide magnetic field ~H(ν)n tends to 1

η0s~V

(ν)n for s → 0 (see

equation (2.39)). Since the waveguide magnetic field should be irrotational at low fre-

quency, it is clear that ~V (ν)n is irrotational. At this point, it easy to demonstrate that

〈~wk, ~V(ν)n 〉 = 0 when ~wk is a solenoidal basis function1. This means that the subma-

trix of V corresponding to the solenoidal basis functions is null, and that matrix V is

partitioned as

V =

V

0

(B.6)

where V is a non null matrix.

Based on these considerations, it is clear that V(I − QmQm) c is null for any current

vector c.

1Vector ~V (ν)n can be written as the gradient of a scalar function: ~V (ν)

n = ∇ψ(ν)n . The generic element

of matrix V is given by

〈~wk, ~V(ν)

n 〉 =

∫

S

~Wk · ∇ψ(ν)n dS

=

∫

S

∇ · (ψ(ν)n ~wk) dS −

∫

S

ψ(ν)n ∇ · ~wk dS (B.5)

where the Green theorem has been applied. The first term of (B.5) is zero because the flux of all basis

functions multiplied by ψ(ν)n is zero across S. The second term is zero when ~wk is solenoidal.

Appendix C

Expressions of the resonant fields of the

rectangular box

Let a,b,c be the dimensions of the rectangular cavity as indicated in Fig. 2.1.

Magnetic solenoidal eigenvectors:

~H TEmnl =

√2c

kmnl

[~h TEmn (

lπ

c) cos(

lπz

c) − z γmnψmn sin(

lπz

c)]

~H TMmnl =

√ξlc~h TM

mn cos(lπz

c)

~h TEmn(x, y) =

√ξmξna b

π

γmn

[xm

asin(

mπx

a) cos(

nπy

b) + y

n

bcos(

mπx

a) sin(

nπy

b)]

~h TMmn(x, y) =

2√ab

π

γmn

[xn

bsin(

mπx

a) cos(

nπy

b) − y

m

acos(

mπx

a) sin(

nπy

b)]

ψmn(x, y) =

√ξmξna b

cos(mπx

a) cos(

nπy

b)

76

Expressions of the resonant fields of the rectangular box 77

Electric solenoidal eigenvectors:

~E TEmnl =

√2

c~e TE

mn sin(lπz

c)

~E TMmnl =

√ξl

c

kmnl

[~e TMmn (

lπ

c) sin(

lπz

c) + z γmn ϕmn cos(

lπz

c)]

~e TEmn(x, y) =

√ξmξna b

π

γmn

[xn

bcos(

mπx

a) sin(

nπy

b) − y

m

asin(

mπx

a) cos(

nπy

b)]

~e TMmn(x, y) = − 2√

ab

π

γmn

[xm

acos(

mπx

a) sin(

nπy

b) + y

n

bsin(

mπx

a) cos(

nπy

b)]

ϕmn(x, y) =2√a b

sin(mπx

a) sin(

nπy

b)

where:ξn =

2 per n 6= 0

1 per n = 0

γmn =

√(mπ

a)2 + (

nπ

b)2

kmnl =

√γ2

mn + (lπ

c)2

{m,n} 6= {0, 0} e l 6= 0 TE modes

m 6= 0 e n 6= 0 TM modes

We observe that:

∇ · ~h TEmn = γmn ψmn ∇ · ~e TM

mn = γmn ϕmn

Appendix D

Expressions of the integrals of the

singular matrices (3D case)

Expressions of the basis functions and of their divergencesThe plane surface of the cylinder (base) is segmented in the radial direction, whereas thecylindrical surface (side) is segmented in the axial direction. Each triangular basis func-tion is defined on two adjacent segments. On each segment the generic basis functionhas the expression:on the plane surface (ξ = const.):

~u = ρ (a0 + a1 ρ)

{cos(mϕ)

sin(mϕ)+ ϕ (b0 + b1 ρ)

{− sin(mϕ) even

cos(mϕ) odd

∇S · ~u = (a0 − m b0

ρ+ 2a1 − m b1)

{cos(mϕ) even

sin(mϕ) odd

on the cylinderical surface (ρ = const.):

~u = ξ (a0 + a1 ξ)

{cos(mϕ)

sin(mϕ)+ ϕ (b0 + b1 ξ)

{− sin(mϕ) even

cos(mϕ) odd

∇S · ~u = (a1 ρ − m b0

ρ+

−m b1

ρξ)

{cos(mϕ) even

sin(mϕ) odd

78

Expressions of the integrals of the singular matrices 79

In elaborating the expressions of the elements of the singular matrices, the double sur-face integrals have been transformed by using the following relations:

I1 =

∫ 2π

0

∫ 2π

0cos mϕ cos nϕ′ f(ϕ − ϕ′) dϕ dϕ′ = δmn(1 + δ0m) π

∫ 2π

0cos(m ζ) f(ζ) dζ (D.1)

I2 =

∫ 2π

0

∫ 2π

0cos mϕ sin nϕ′ f(ϕ − ϕ′) dϕ dϕ′ = −δmn π

∫ 2π

0sin(m ζ) f(ζ) dζ (D.2)

I3 =

∫ 2π

0

∫ 2π

0sin mϕ cos nϕ′ f(ϕ − ϕ′) dϕ dϕ′ = δmn π

∫ 2π

0sin(m ζ) f(ζ) dζ (D.3)

I4 =

∫ 2π

0

∫ 2π

0sin mϕ sin nϕ′ f(ϕ − ϕ′) dϕ dϕ′ = δmn(1 − δ0m) π

∫ 2π

0cos(m ζ) f(ζ) dζ (D.4)

where δmn denotes the kronecker symbol. We took into account that if function f(ζ)is even and periodic I2 and I3 are zero, whereas I1 and I4 are zero if f(ζ) is odd andperiodic. The elements of all matrices depend on a pair of indexes (i, j) correspondingto the test and the basis functions involved in the integrals. In defining the expressionsof the integrals we will make use of the following symbols:

∆mi =

1 + δ0mi if both basis functions are even

1 − δ0mi if both basis functions are odd

0 otherwise

(D.5)

∆′mi =

1 − δ0mi if both basis functions are even

1 + δ0mi if both basis functions are odd

0 otherwise

(D.6)

∆mi =

1 + δ0mi if the i-th basis functions is even and the other is odd

−1 + δ0mi if the i-th basis functions is odd and the other is even

0 otherwise

(D.7)

∆′

mi =

−1 + δ0mi if the i-th basis functions is even and the other is odd

1 + δ0mi if the i-th basis functions is odd and the other is even

0 otherwise

(D.8)


Expressions of [Sesing]ij = 1

4π

∫S

∫S(∇S · ~ui(~r))(∇

′

S · ~uj(~r′)) 1

RdS dS′

Base-Base:

δmi,mj

4∆mi

[a0 − mib0 2a1 − mib1

]{∫ ρ2

ρ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[1 ρ′

ρ ρρ′

]cos miϕ

Rdρ dρ′ dϕ

}[a′0 − mib

′0

2a′1 − mib

′1

]

Base-Side:

δmi,mj

4∆mi

[a0 − mib0 2a1 − mib1

]{ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ρ ρξ′

]cos miϕ

Rdρ dξ′ dϕ

}

a′

1ρ′−mib′0ρ′

2a′


Side-Base:

δmi,mj

4∆mi

[a1ρ−mib0

ρ2a2ρ−mib1

ρ

]{ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[1 ρ′

ξ ξρ′

]cos miϕ

Rdξ dρ′ dϕ

}[a′0 − mib

′0

2a′1 − mib

′1

]

Side-Side:

δmi,mj

4∆mi

[a1ρ−mib0

ρ2a2ρ−mib1

ρ

]{ρρ′

∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos miϕ

Rdξ dξ′ dϕ

}

a′


2a′



Expressions of [GAsing]ij = 1

4π

∫S

∫S

~ui(~r) · ~uj(~r′) 1

RdS dS′

Base-Base:

δmi,mj

4{ ∆mi

[a0 a1

]IBB cc 1R

[a′0

a′1

]+ ∆′

mi

[b0 b1

]IBB cc 1R

[b′0

b′1

]

+[

a0 a1

]IBB ss 1R

[b′0

b′1

]+

[b0 b1

]IBB ss 1R

[a′0

a′1

]}

Base-Side:

δmi,mj

4{∆′

mi

[b0 b1

]IBL cc 1R

[b′0

b′1

]+[

a0 a1

]IBL ss 1R

[b′0

b′1

]}

Side-Base:

δmi,mj

4{∆′

mi

[b0 b1

]ILB cc 1R

[b′0

b′1

]+[

b0 b1

]ILB ss 1R

[a′0

a′1

]}

Side-Side:

δmi,mj

4{∆mi

[a0 a1

]ILL c 1R

[a′0

a′1

]+ ∆′

mi

[b0 b1

]ILL cc 1R

[b′0

b′1

]}

where:

IBB cc 1R =

∫ ρ2

ρ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρρ′ ρρ′2

ρ2ρ′ ρ2ρ′2

]cos(miϕ) cos(ϕ)

Rdρ dρ′ dϕ

IBB ss 1R =

∫ ρ2

ρ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρρ′ ρρ′2

ρ2ρ′ ρ2ρ′2

]sin(miϕ) sin(ϕ)

Rdρ dρ′ dϕ

IBL cc 1R = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[ρ ρξ′

ρ2 ρ2ξ′

]cos(miϕ) cos(ϕ)

Rdρ dξ′ dϕ

IBL ss 1R = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[ρ ρξ′

ρ2 ρ2ξ′

]sin(miϕ) sin(ϕ)

Rdρ dξ′ dϕ

ILB cc 1R = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]cos(miϕ) cos(ϕ)

Rdξ dρ′ dϕ

ILB ss 1R = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]sin(miϕ) sin(ϕ)

Rdξ dρ′ dϕ

ILL c 1R = ρρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos(miϕ)

Rdξ dξ′ dϕ

ILL cc 1R = ρρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos(miϕ) cos(ϕ)

Rdξ dξ′ dϕ


Expressions of [−Tsing]ij = 14π

∫S

∫S~wi(~r) × ~uj(~r

′) · ∇ 1RdS dS ′

Base-Base:

δmi,mj

4{ ∆mi

[a0 a1

]IBB ss D1R ξ

[a′0

a′1

]+ ∆mi

[a0 a1

]IBB cc D1R ξ

[b′0

b′1

]

− ∆′

mi

[b0 b1

]IBB cc D1R ξ

[a′0

a′1

]− ∆′

mi

[b0 b1

]IBB ss D1R ξ

[b′0

b′1

]}

Base-Side:

δmi,mj

4{ −∆′

mi

[a0 a1

]IBL s D1R ϕ

[a′0

a′1

]+ ∆mi

[a0 a1

]IBL cc D1R ξ

[b′0

b′1

]

+ ∆′

mi

[b0 b1

]IBL c D1R ρ

[a′0

a′1

]+ ∆mi

[b0 b1

]IBL ss D1R ξ

[b′0

b′1

]}

Side-Base:

δmi,mj

4{ −∆mi

[a0 a1

]ILB sc D1R ϕ

[a′0

a′1

]− ∆mi

[a0 a1

]ILB ss D1R ρ

[a′0

a′1

]

+ ∆mi

[a0 a1

]ILB cs D1R ϕ

[b′0

b′1

]− ∆mi

[a0 a1

]ILB cc D1R ρ

[b′0

b′1

]

− ∆′

mi

[b0 b1

]ILB cc D1R ξ

[a′0

a′1

]− ∆′

mi

[b0 b1

]ILB ss D1R ξ

[b′0

b′1

]}

Side-Side:

δmi,mj

4{ ∆mi

[a0 a1

]ILL cs D1R ϕ

[b′0

b′1

]− ∆mi

[a0 a1

]ILL cc D1R ρ

[b′0

b′1

]

+ ∆′

mi

[b0 b1

]ILL c D1R ρ

[a′0

a′1

]− ∆′

mi

[b0 b1

]ILL ss D1R ξ

[b′0

b′1

]}


where:

IBB ss D1R ξ =

∫ ρ2

ρ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρρ′ ρρ′2

ρ2ρ′ ρ2ρ′2

]sin(miϕ) sin(ϕ)

∂

∂ξ

1

Rdρ dρ′ dϕ

IBB cc D1R ξ =

∫ ρ2

ρ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρρ′ ρρ′2

ρ2ρ′ ρ2ρ′2

]cos(miϕ) cos(ϕ)

∂

∂ξ

1

Rdρ dρ′ dϕ

IBL s D1R ϕ = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ρ ρξ′

]sin(miϕ)

∂

∂ϕ

1

Rdρ dξ′ dϕ

IBL cc D1R ξ = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[ρ ρξ′

ρ2 ρ2ξ′

]cos(miϕ) cos(ϕ)

∂

∂ξ

1

Rdρ dξ′ dϕ

IBL c D1R ρ = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[ρ ρξ′

ρ2 ρ2ξ′

]cos(miϕ)

∂

∂ρ

1

Rdρ dξ′ dϕ

IBL ss D1R ξ = ρ′∫ ρ2

ρ1

∫ ξ′2

ξ′1

∫ 2π

0

[ρ ρξ′

ρ2 ρ2ξ′

]sin(miϕ) sin(ϕ)

∂

∂ξ

1

Rdρ dξ′ dϕ

ILB sc D1R ϕ =

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]sin(miϕ) cos(ϕ)

∂

∂ϕ

1

Rdξ dρ′ dϕ

ILB ss D1R ρ = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]sin(miϕ) sin(ϕ)

∂

∂ρ

1

Rdξ dρ′ dϕ

ILB cs D1R ϕ =

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]cos(miϕ) sin(ϕ)

∂

∂ϕ

1

Rdξ dρ′ dϕ

ILB cc D1R ρ = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]cos(miϕ) cos(ϕ)

∂

∂ρ

1

Rdξ dρ′ dϕ

ILB cc D1R ξ = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]cos(miϕ) cos(ϕ)

∂

∂ξ

1

Rdξ dρ′ dϕ

ILB ss D1R ξ = ρ

∫ ξ2

ξ1

∫ ρ′

2

ρ′

1

∫ 2π

0

[ρ′ ρ′2

ξρ′ ξρ′2

]sin(miϕ) sin(ϕ)

∂

∂ξ

1

Rdξ dρ′ dϕ

ILL cs D1R ϕ = ρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos(miϕ) sin(ϕ)

∂

∂ϕ

1

Rdξ dξ′ dϕ

ILL cc D1R ρ = ρρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos(miϕ) cos(ϕ)

∂

∂ρ

1

Rdξ dξ′ dϕ

ILL c D1R ρ = ρρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]cos(miϕ)

∂

∂ρ

1

Rdξ dξ′ dϕ

ILL ss D1R ξ = ρρ′∫ ξ2

ξ1

∫ ξ′2

ξ′1

∫ 2π

0

[1 ξ′

ξ ξξ′

]sin(miϕ) sin(ϕ)

∂

∂ξ

1

Rdξ dξ′ dϕ

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