Electromagnetic Field Computation by Network

Electromagnetic Field Computation by Network Methods

Peter Russer Institute for High Frequency Engineering Technische Universit¨ at M¨ unchen Arcisstrasse 21, 80333 Munich, Germany [email protected]

Abstract: Network-oriented methods applied to electromagnetic field problems can contribute significantly to the problem formulation and solution methodology and to a reduction of computational effort and allow a systematic introduction of hybrid methods. In network theory systematic approaches for circuit analysis are based on the separation of the circuit into the connection network and the circuit elements. The connection network represents the topological structure of the circuit and contains only interconnects, including ideal transformers. Applying a network description, electromagnetic structures can be partitioned into subdomains, defining the circuit elements. The boundary surfaces between the subdomains define the interconnection network. Lumped element circuit models of electromagnetic structures can be obtained by analytic or numerical techniques. Keywords: Electromagnetic Fields, Computational Electromagnetics, Networks 1. Introduction With increasing bandwidths and data rates of modern electronic circuits and systems, electromagnetic wave phenomena that in the past were in the domain of the microwave engineer, are now becoming pivotal in the design of analog and digital systems. Design, modeling and optimization of high-speed analog and digital electronic circuits and systems, photonic devices and systems, of antenna, radar, imaging and communications systems, among other applications, require the application of advanced tools in computational electromagnetics. Compared with a network-oriented design a field-oriented design of circuits and systems requires a tremendously higher computational effort. The availability of steadily increasing computing facilities has not reduced the demand for efficient methods of electromagnetic field computation. This is readily understandable especially in the highly competitive design of broadband and high-speed electronic components. The demands for volume, weight and cost reduction foster a compact and complex design of electromagnetic structures yielding a high computational effort in electromagnetic modeling. Whereas in field theory the three-dimensional geometric structure of the electromagnetic field has to be considered [1–3], a network model exhibits a plain topological structure [4–7]. Networkoriented methods applied to electromagnetic field problems may contribute significantly to the problem formulation and solution methodology. In a three-part sequence of papers [8–10] and a book [11], together with L.B. Felsen and M. Mongiardo the author has outlined an architecture for a systematic and rigorous treatment of electromagnetic field representations in complex structures.

Concepts Field

Network Structure

Geometric Structure

Topological Structure Fundamental Laws

Maxwell’s Equations

Kirchhoff Laws Physical Quantities

Electric and Magnetic Field E(x, t), H(x, t)

Voltage and Current v(t), i(t) Mathematical Relations

Integral Equations

Algebraic Equations

Table 1: Field and Network Concepts.

An architecture constitutes a structure that addresses complexity systematically and can provide a systematic framework for proper problem formulation and can contribute considerably to an efficient problem solution. In network theory systematic approaches for circuit analysis are based on the separation of the circuit into the connection circuit and the circuit elements. The connection circuit represents the topological structure of the circuit and contains only interconnects, including ideal transformers. Applying a network description electromagnetic structures can be segmented into substructures. These substructures define the circuit elements and the set of boundary surfaces between the substructures define the interconnection network. Canonical Foster equivalent circuits can represent lossless structures in sub domains. Canonical Cauer networks can describe radiation modes. The lumped element models can be obtained by analytic methods, i.e. via Green’s function or mode matching approaches or by numerical methods techniques (Transmission Line Matrix Method or Transverse Wave Formulation) in connection with system identification techniques. The network approach allows a systematic introduction of hybrid methods. Furthermore, network formulations are well suited for the application of model order reduction methods. Analytic and numerical methods and examples of their application are discussed. Network methods are applicable in connection with the main analytic and numerical methods for electromagnetic field modeling and provide a large variety of tools for efficient modeling of complex electromagnetic structures. 2. Circuit Models of Electromagnetic Structures Whereas the electromagnetic field concept provides the fundamental and complete description of electromagnetic phenomea, network models relate integral quantities like voltage and current. Table 1 contrasts field and network concepts. The network description represents a higher model hierarchical level than the field description and yields a considerable model simplification in all cases where it is applicable. At lower frequencies voltage and current can be defined uniquely by line integrals over electric and magnetic field quantities. An essential point is that network models can also apply to electromagnetic structures at higher frequencies when generalized voltages and currents are introduced by a proper definition of integral field quantities. One particular example of this procedure realized in Method of Moments (MoM) where the coefficients of the expansions of electric and magnetic

(b)

(a) R0

B10

∂R1

B01 ∂R0

∂R4

∂R1 B41 B14 R4 B04

B40

∂R4

B20

B21 B B02 12 ∂R0 ∂R R1 1 R2 ∂R2 ∂R 2 B31 B32 B13 B23 ∂R4 B34

R3

B43

∂R0

I1α

I2α

α

I3

∂R3

B03

B30

β

V1

β

V2

β

V3

Figure 1: The connection circuit: (a) Segmentation of an electromagnetic structure, (b) Canonical form of the connection network. fields can be considered as generalized voltages and currents and the linear equations relating these quantities as network equations. Figure 1a shows schematically the segmentation of the problem space of an electromagnetic structures into subregions. The regions are denominated with Rl and their boundary surfaces are ∂Rl . The part of the boundary separating two regions Rl and Rj is named Blj and Bjl , respectively. If a region is filled by source-free linear media, the relation between the tangential electric field and the tangential magnetic field on the boundary surface ∂Rl may be expressed by either of the integral equations [3, 10–12]

Etl (x, ω) Htl (x, ω)

=

Z

˜ l (x, x′ , ω) H l (x, ω) dA , Z t

(1a)

Z

Y˜ l (x, x′ , ω) Etl (x, ω) dA ,

(1b)

∂Rl

=

∂Rl

˜ l (x, x′ , ω) and Y˜ l (x, x′ , ω) are the dyadic Green’s functions relating the tangential electric where Z l field Et with the the tangential magnetic field Htl on the boundary surface ∂Rl in the impedance representation or admittance representation, respectively. We can discretize (1a) and (1b) by expanding the tangential fields on ∂Rl into a complete set of bi-orthonormal vector basis functions eln (x) and hln (x) by Etl (x, ω) =

X

Vnl (ω)eln (x) ,

(2a)

X

Inl (ω)hln (x) ,

(2b)

n

Htl (x, ω) =

n

where the superscript l refers to the lth subdomain and n numbers the basis functions. We will see that the complex expansion coefficient Vnl (ω) can be considered as a generalized voltage and Inl (ω)

represents a generalized current. We note that these expansions need only to be defined on ∂Rl . On ∂Rl the biorthogonal vector basis functions eln (x) and hln (x) fulfill the orthonormality relations Z

∂Rl

Z

∂Rl

l el∗ m (x) · en (x) dA = δmn ,

(3a)

l hl∗ m (x) · hn (x) dA = δmn .

(3b)

Furthermore the two sets of vector basis functions un (x) and vn (x) are related via hln (x) = nl (x) × eln (x) ,

(4a)

eln (x) = −nl (x) × hln (x) .

(4b)

Inserting (4b) into (3a) yields the biorthonormality relation Z

∂Rl

l l el∗ m (x) × hn (x) · n (x)dA = δmn ,

(5)

where nl (x) is the normal vector on ∂Rl . From (2a) and (2b) and the orthogonality relation (5) we obtain Vnl (ω)

=

Z

hl∗ n (x) × Et (x, ω) dA ,

(6a)

Z

el∗ n (x) × Ht (x, ω) dA .

(6b)

∂Rl

Inl (ω) =

∂Rl

If the domain Rl is partially bounded by an ideal electric or magnetic wall Et or Ht respectively vanish on these walls. If the independent field variable vanishes on the boundary, this part of the boundary does not need to be represented by the basis functions. If only electric walls are involved, the admittance representation of the Green’s function will be appropriate, and if only magnetic walls are involved, the impedance representation will be appropriate.

l Zm,n (ω)

l Ym,n (ω)

=

=

Z

∂Rl

Z

∂Rl

′ l ˜l el∗ n (x) · Z (x, x , ω) · hn (x) dA ,

(7a)

′ l ˜l hl∗ n (x) · Y (x, x , ω) · en (x) dA .

(7b)

Applying the method of moments [13,14], we obtain from the integral equations the linear systems

of equations Vml (ω) =

X

l Zm,n (ω)Inl (ω) ,

(8a)

X

l Ym,n (ω)Vnl (ω) .

(8b)

n

l Im (ω) =

n

l (ω) in the vectors Summarizing the amplitudes Vml (ω) and Im

V l (ω) = [V1l (ω)...VNl (ω)]T ,

l (ω)]T I l (ω) = [I1l (ω)...IN

(9)

l l (ω) in the matrices and the matrix elements Zm,n (ω) and Ym,n

l (ω) Z11  Z l (ω) =  ...



··· .. . l (ω) · · · ZN 1

 l (ω) Z1N  .. , . l ZN N (ω)

l (ω) Y11  Y l (ω) =  ...



··· .. . YNl 1 (ω) · · ·

 l (ω) Y1N  ..  .

(10)

YNl N (ω)

we obtain the linear system of equations in matrix notation V l (ω) = Z l (ω)I l (ω) ,

I l (ω) = Y l (ω)V l (ω) .

(11)

These equations represent a circuit description of the EM structure in domain Rl . The impedance and admittance matrices Z l (ω) and Y l (ω), respectively relate the generalized voltages and currents summarized in the vectors V l (ω) and I l (ω), respectively. 3. Tellegen’s Theorem and the Connection Network A. Field Theoretic Formulation of Tellegen’s Theorem Complex electromagnetic structures may be subdivided into substructures confined to spatial subdomains. Comparing a distributed circuit represented by an electromagnetic structure with a lumped element circuit represented by a network, the spatial subdomains may be considered as the circuit elements whereas the complete set of boundary surfaces separating the subdomains corresponds to the connection circuit [10, 12]. Figure 1a shows the segmentation of an electromagnetic structure into different regions Rl separated by boundaries Blk . The regions Rl may contain any electromagnetic substructure. In our network analogy the two-dimensional manifold of all boundary surfaces Blk represents the connection circuit whereas the subdomains Rl are representing the circuit elements. The tangential electric and magnetic fields on the boundary surface of a subdomain are related via Green’s functions [3, 10, 12, 15]. These Green’s functions can be seen in analogy to the Foster representation of the corresponding reactive network. We can establish a field representation of the Tellegen’s theorem relating the tangental electric and magnetic fields on the two-dimensional manifolds of boundaries Blk [16]. Expanding the tangential electric and magnetic fields on the boundaries again into basis functions allows to give an equivalent circuit representation for the boundary surfaces. The equivalent circuit of the boundary surfaces is a connection circuit exhibiting only connections and ideal transformers.

Tellegen’s theorem states fundamental relations between voltages and currents in a network and is of considerable versatility and generality in network theory [16–18]. A noticeable property of this theorem is that it is only based on Kirchhoff’s current and voltage laws, i.e. on topological relationships, and that it is independent from the constitutive laws of the network. The same reasoning that yields from Kirchhoff’s laws to Tellegen’s theorem allows to directly derive a field form of Tellegen’s theorem from Maxwell’s equations [16]. In order to derive Tellegen’s theorem for partitioned electromagnetic structures let us consider two electromagnetic structures based on the same partition by equal boundary surfaces. The subdomains of either electromagnetic structure however may be filled with different materials. The connection network is established via the relations of the tangential field components on both sides of the boundaries. Since the connection network exhibits zero volume no field energy is stored therein and no power loss occurs therein. Starting directly from Maxwell’s equations we may derive for a closed volume R with boundary surface ∂R and relative normal vector n the following relation: Z

Z

E ′ (x, t′ ) × H ′′ (x, t′′ ) · n dA = −

E′ (r, t′ ) · J′′ (r, t′′ ) dr −

R

∂R

−

Z

E′ (r, t′ ) ·

R

∂D′′ (r, t′′ ) dr − ∂t′′

Z

H′ (r, t′ ) ·

R

∂B′′ (r, t′′ ) dr . ∂t′′

(12)

The prime ′ and double prime ′′ denote the case of a different choice of sources and a different choice materials filling the subdomains. Furthermore also the time argument may be different in both cases. For volumes R of zero measure or free of field the right side of this equation vanishes. Considering an electromagnetic structure as shown in Figure 1a, we perform the integration over the boundaries of all subregions not filled with ideal electric or magnetic conductors respectively. The integration over both sides of a boundary yields zero contribution to the integrals on the right side of (12). Also the integration over finite volumes filled with ideal electric or magnetic conductors gives no contribution to these integrals. We obtain the field form of Tellegen’s theorem Z

E ′ (x, t′ ) × H ′′ (x, t′′ ) · ndA = 0 .

(13)

∂R

We note, that Tellegen’s theorem also holds in frequency domain, Z

E ′ (x, ω ′ ) × H ′′ (x, ω ′′ ) · ndA = 0 ,

(14)

∂R

and also when the frequencies ω ′ and ω ′′ differ from each other. B. The discretized connection network We now consider the fields as expanded on finite orthonormal basis function sets; the assumption of orthonormal basis is not necessary, and is introduced to simplify notation. We consider sets of electric and magnetic field vectorial basis functions eξn (x) and hξn (x), respectively, of dimension Nξ

on the boundary surface ξ with ξ = α, β. Expanding the tangential electric and magnetic Fields on both sides α and β of the boundary as described in Section 2 into

Etξ =

Nξ X

Htξ =

Vnξ eξn (x) ,

n

Nξ X

Vnξ hξn (x) ,

(15)

n

we can summarize the field expansion coefficients for the electric and magnetic field in the vectors V α , I α , V β and I β , leading compactly to the vectors

V =



Vα Vβ



,

I=



Iα Iβ



,

(16)

summarizing the voltage and current amplitudes on both sides of the boundary layer. Since the tangential electric and magnetic field are continuous at the boundary surface the total tangential electric and magnetic fields are the same on both sides of the boundary. However, since the vectorial basis functions eξn (x) and hξn (x) in general will be chosen differently on both sides of the boundary the components of V α , I α may be coupled in a general way to the components of V β and I β , respectively. Inserting the field expansions (15) into the field form of Tellegen’s theorem (13) yields

Z

′

′

′′

′′

E (x, t ) × H (x, t ) · ndA = ∂R

Nξ Nξ X X X

ξ=α,β n

m

′ ′′ Vmξ (t′ )Inξ (t′′ )

Z

∂R

eξm × hξn · ndA = 0 .

(17)

If the vectorial basis fuctions fulfill the biorthonormality relations (5), Tellegen’s theorem takes the standard form V ′T (t′ ) I ′′ (t′′ ) = 0 . (18) where V (t) and I(t) denote the voltage and current vectors of the connection circuit. The prime ′ and double prime ′′ again denote different circuit elements and different times in both cases. It is only required that the topological structure of the connection circuit remains unchanged. C. Canonical Forms of the Connection Network Consistent choices of independent and dependent fields do not violate Tellegen’s theorem and allow to draw canonical networks, which are based only on connections and ideal transformers. Figure 1b shows the canonical form of the connection network when using as independent fields the vectors V β (dimension Nβ ) and I α (dimension Nα ). In this case the dependent fields are V α (dimension Nα ) and I β (dimension Nβ ). In all cases we have Nβ +Nα independent quantities and the same number of dependent quantities. Note that scattering representations are also allowed and that the connection network is frequency independent. It is apparent from the canonical network representations that the scattering matrix is symmetric, S T = S, orthogonal, S T S = I and unitary, i.e. SS † = I, where the † denotes the Hermitian conjugate matrix.

4. Foster Representation of Reactance Multiports Consider again a subregion Rl of an electromagnetic structure as depicted schematically in Figure 1a. Let us assume this subregion to be filled with an arbitrary structure consisting from free space, perfect electric and magnetic conductors and reciprocal lossless electric and magnetic materials. Such a subregion may be characterized by the relation between tangential electric and magnetic fields on the boundary surface ∂Rl as specified by either (1a) or (1b). Covering the closed boundary surface ∂Rl to consist either by a perfect electric conductor or a perfect magnetic conductor makes the complete structure a lossless resonator. The electromagnetic field inside such a lossless closed cavity can be expanded into orthogonal modal functions [19–23]. In the spectral representation the dyadic impedance and admittance Green’s functions introduced in (1a) and (1b) are given by [3, 10, 12, 15] X 1 ω2 1 0 ′ ˜ ˜λ z˜ (x, x′ ) + Z(x, x′ , ω) = 2 z (x, x ) , jω jω ω 2 − ωp,λ

(19a)

X 1 ω2 1 0 ′ ˜λ y˜ (x, x′ ) + Y˜ (x, x′ , ω) = 2 y (x, x ) . jω jω ω 2 − ωs,λ

(19b)

λ

λ

The dyadics z˜0 (x, x′ ) and y˜0 (x, x′ ) represent the static parts of the Green’s functions, whereas each term z˜λ (x, x′ ) and y˜λ (x, x′ ), respectively, corresponds to a pole at the frequency ωλ . Expanding the electric and magnetic field into basis functions eln (x) and hln (x) as described in (2a) and (2b) the impedance and admittance matrices relating the generalized voltages V l (ω) and currents I l (ω) will be obtained as

Z(ω) =

N X 1 ω2 1 Ap,0 + 2 Ap,λ , jωCp,0 jωCp,λ ω 2 − ωp,λ

(20a)

N X 1 1 ω2 As,0 + 2 As,λ , jωLs,0 jωLs,λ ω 2 − ωs,λ

(20b)

p,λ=1

Y (ω) =

s,λ=1

where Ap,λ and As,λ , respectively are real frequency-independent rank 1 matrices given in either case by   nλ1 nλ2 . . . nλ1 nλN n2λ1  nλ2 nλ1 . . . nλ2 nλN  n2λ2   Aλ =  (21)  . .. .. .. ..   . . . . nλN nλ1 nλN nλ2 . . . n2λN We can interpret the respective impedance matrix Z l (ω) and admittance matrix Y l (ω) as describing lumped element equivalent circuits. This allows us to draw directly the equivalent circuits

(a) Cp,N

(b) L p,N nN1 :1

nNM :1

nN3 :1

nN2 :1

Cp,3

L p,3 n31 :1

n32 :1

n 33 :1

n

Cp,2

L p,2 n 21 :1

n 2 2:1

n23 :1

n2M :1

Cp,1

Cp,0

Lp,1 n 11 :1

n 12:1

n 01 :1

3M

L s,0

:1

1: n 3M

Cs,1

Cs,2

Cs,3

Cs,N

L s,1

L s,2

L s,3

L s,N

1: n03

1: n 13

1: n 23

1: n 33

1: nN3

port 2

1: n 02

1: n

12

1: n 22

1: n 32

1: nN2

1: n11

1: n 21

1: n 31

1: nN1

n 0M :1

n03 :1

port 2

1: nNM

1: n 2M

port 3

port 1

port 1

1: n 1M

n 1M :1

n13 :1

n 02:1

port M 1: n 0M

port 3

1: n 01

port M

Figure 2: Canonical Foster representations of reactance multipors: (a) Foster impedance representation, (b) Foster admittance representation. corresponding to the above impedance and admittance matrices. These circuits are the canonical Foster representations of linear reactance multiports [4, 24]. A canonical representation is a unique realization of an impedance or admittance function with a minimum number of circuit elements. The impedance matrix Z l (ω) given in (20a) can be realized by the lumped element circuit shown in Figure 2a. This circuit consists of one Capacitor Cp,0 and a number of parallel resonant circuits connected via an ideal transformer network with the ports. The capacitor realizes the zero frequency pole whereas the parallel resonant circuits realize the frequency poles at ωp,λ . The 2 C values of the inductors are given by Lp,λ = 1/ωp,λ p,λ . Every row of ideal transformers, connected to a resonant circuit with resonant frequency ωp,λ is described by a frequency-independent rank-1 matrix Aλ as given in (21). The determination of the equivalent circuit parameters follows the procidure in circuit theory, where Z parameters are determined under open circuit conditions and separate excitation of every port by a current source. The corresponding procedure to calculate ˜ the Foster impedance representation analytically from the Green’s function Z(x, x′ , ω) is to realize open circuit conditions by choosing a perfect magnetic conductor (PMC) as the boundary ∂RL and exciting every mode independently by impressing modal electric surface polarization on the inner surface of the PMC on ∂RL as described for example in [1, 12]. The equivalent Foster admittance representation realizing Y l (ω) given in (20b) is shown in Figure 2b. The first column in that figure contains the inductor Ls,0 realizing the zero frequency pole and the other colums contain series resonant circuits consisting of the inductors Ls,λ and the 2 L . capacitors Cs,λ = 1/ωs,λ s,λ Every column of transformers again is described by a frequency-independent rank-1 matrix Aλ according to (21). The admittance matrix Y l (ω) can be computed from the Green’s function Y˜ (x, x′ , ω) describing the structure in Rl enclosed in e perfect electric conductor (PEC) on ∂RL . The modal excitation in that case is performed by impressing modal magnetic surface polarizations on the inner surface of the PEC on ∂RL . The the analytic computation of the circuit parameters of waveguide junctions and other distributed microwave circuits already has been treated in numerous papers [25–33]. It is also possible

(a)

(b)

z

R1

S R3

Source 1

Source 1

θ ϕ

r=ro

Source 2

TE 11 REACTANCE MULTIPORT

CONNECTION NETWORK

y

Source 2

x

TM 11

TMm’n’ TEm’’n’’

Source k

R4

Figure 3: (a) Embedding of a radiating electromagnetic structure into a sphere, (b) Equivalent circuit block diagram of the complete radiating electromagnetic structure. to find an equivalent Foster representation from admittance parameters calculated by numerical field analysis. This has been done by numerical Laplace transformation [34, 35], and in a more efficient way by applying system identification and spectral analysis methods [36–48]. So far we have considered Foster equivalent circuit realizations of linear reciprocal lossless structures. Lossy structures exhibit complex poles. Difficulties arise from the circumstance that lossy electromagnetic structures are described by partial differential equations exhibiting non-self adjoint operators. This usually does not allow to find orthogonal modal eigenfunctions. In case of weak losses we can seek the modal eigenfunctions of the lossless structure obtained by neglecting the losses and than compute the complex poles applying the power loss method [49, 50]. A further difficulty arises to find an equivalent lumped element circuit realization for lossy structures. The canonical Foster realizations are only defined for reactance multiports. In the lossy case equivalent lumped element circuits can be found by including also resistors in the equivalent circuits. This already has been demonstrated in [34,35], however, special care has to be taken to maintain stability of the equivalent circuits. The problems of synthesis of RLC impedance functions is discussed in chapter 9 of [51]. 5. The Characterisation of Radiation Modes It is already known that spherical radiation modes can be exactly represented by a lumped element equivalent network model with a finite number of network elements. In 1948 L.J. Chu has investigated the orthogonal mode expansion of the radiated field [52]. Using the recurrence formula for spherical bessel functions he gave the Cauer representation [4, 24] of the equivalent circuits of the T Mmn and the T Emn spherical waves. The equivalent circuit expansion of spherical waves also is treated in [1, 11, 12, 15]. To establish the equivalent circuit of a reciprocal linear lossless radiating electromagnetic structure, we embed the structure in a sphere as shown schematically in Figure 3a. Region R1 is the the infinite free space region outside the sphere S. Region R2 contains the reciprocal passive electromagnetic structure. The internal sources 1 and 2 are enclosed in regions R3 and R4 . The electromagnetic field outside the sphere may be expanded into T Mmn and T Emn spherical waves propagating in outward direction. The T Mmn modes are given by
T M ij Hmn = curl Aij mn er ,

T M ij Emn =

1 T Mi curl Hmn , jωε

(22)

where n = 1, 2, 3, 4, . . . , m = 1, 2, 3, 4, . . . , n, i = e, o, and j = 1, 2. The radial component Aij mn of the vector potential is given by ej m (j) Aej mn = amn Pn (cos θ) cos mϕ hn (kr) ,

oj m (j) Aoj mn = amn Pn (cos θ) sin mϕ hn (kr) ,

(23)

(j)

where the Pnm (cos θ) are the associated Legendre polynomials and hn (kr) are the spherical Hankel oj functions and aej mn and amn are coefficients [1, 11, 12, 15]. The superscript j = 1 denotes an inward propagating wave and the superscript j = 2 indicates an outward propagating wave. Outside the sphere, for r > r0 only outward propagating waves occur and we have only to consider j = 2. The T Emn modes are dual with respect to the T Mmn modes and are given by
1 ij T Eij T Mi T Eij Emn = − curl Fmn er , Hmn curl Emn , =− jωε

(24)

ij where n = 1, 2, 3, 4, . . . , m = 1, 2, 3, 4, . . . , n, i = e, o, and j = 1, 2. The radial component Fmn of the dual vector potential is given by ej ej Fmn = fmn Pnm (cos θ) cos mϕ h(j) n (kr) ,

oj oj Fmn = fmn Pnm (cos θ) sin mϕ h(j) n (kr) .

(25)

(j)

where the Pnm (cos θ) are the associated Legendre polynomials and hn (kr) are the spherical Hankel ej oj functions. The fmn and fmn are coefficients. The wave impedances for the outward propagating T Mmn and T Emn modes are

+ = Zmn

The superscript TE modes are +tm Zmn

+

+ + Emnϕ Emnθ . = − + + Hmnϕ Hmnθ

(26)

denotes the outward propagating wave. The wave impedances for the TM and

= jZF 0

d dr

  (2) rhn (kr) (2) rhn (kr)

(2)

,

+te Zmn = −jZF 0

(27a)

rh (kr)  n , (2) d rh (kr) n dr

(27b)

p where ZF 0 = µ/ε is the wave impedance of the plane wave. We note that the characteristic wave impedances only depend on the index n and the radius r of the sphere. A continued fraction expansions of the wave impedances of the TM modes

+tm Zmn



  = ZF 0   

n jkr

+

1 2n−1 1 + 2n−3 jkr + jkr

..

 . +

1 3 + jkr

1 1 +1 jkr

    

(28)

rε 2n - 3

rε n

(a) +TM

Z mn

rµ 2n - 1

rε 2n - 1

(b)

rµ 2n - 5

ZF0

+TE

Z mn

rµ n

rε 2n - 5

rµ 2n - 3

ZF0

Figure 4: Equivalent circuit of (a) T Mmn , (b) T Emn spherical wave. and the TE modes 

+te Zmn

   = ZF 0    

1

1 n + 2n−1 jkr + 2n−3 1 jkr 1 + 2n−5 jkr + jkr

 ..

. +

1 3 + jkr

1 1 +1 jkr

   .   

(29)

These continued fraction expansions represent the Cauer canonic realizations of the outward propagating TM modes shown in Figure 4a and TE modes shown Figure 4b. We note that the equivalent circuit representing the T Emn mode is dual to the equivalent circuit representing the T Mmn mode. The equivalent circuits for the radiation modes exhibit highpass character. For very low frequencies the wave impedance of the T Mmn mode is represented by a capacitor C0n = εr/n and the wave impedance of the T Emn mode is represented by an inductor L0n = µr/n. Figure 3b shows the block diagram of a complete equivalent circuit of a radiating electromagnetic structure. The linear passive reciprocal electromagnetic structure with exception of the sources can be modeled by a Foster canonical multiport in the lossless case or by lumped element equivalent circuits including RLC elements in the lossy case. Instead of modeling the complete region R2 by a network model it may be useful to partition R2 into subregions and to find a network model for every subregion. These network models of the subregions can be connected via connection networks shown in Figure 1a. This would be an interesting option if it is advantageous, for example to apply an analytic and numerical methods in different subregions. The sources which are represented in a field model by impressed electric and magnetic polarizations [1,11,12,15] after modal expansion may be represented by current and voltage sources. The spherical modes in the free space region R1 are represented by Cauer eqivalent circuits. The connection between regions R1 and R2 is accomplished by connection networks. 6. Network methods based on field discretization A. The Transmission Line Matrix (TLM) Method So far we have considered the introduction of network models by modal or eigenfunction expansion of the electromagnetic field. We are now considering a method based on the spatial discretization of the electromagnetic field. Such a spatial discretization is performed in the finite difference (FD), finite elements (FE) and the transmission line matrix (TLM) methods. In the following we will focus on the TLM method since this method also introduces a network model of the electromagnetic field. The transmission line matrix (TLM) method, developed by Johns and Beurle [53] is a powerful

8

8

6

6

Figure 5: (a) TheTLM cell, (b) the wave amplitudes and (c) the condensed symmetric tlm node. method for computer modeling of electromagnetic fields [54–57]. The TLM method exhibits an excellent numerical stability and is also suitable for modeling of lossy, dispersive and nonlinear media [12,55–60]. The Transmission Line Matrix (TLM) method is based upon the mapping of the electromagnetic field problem into a network problem. This makes the TLM method excellently suited for applying network oriented concepts for problem solution. In the Transmission Line Matrix (TLM) Method the electromagnetic field is modeled by wave pulses propagating in a mesh of transmission lines and being scattered in the mesh nodes. The main advantage of the TLM simulation resides in the capability to model circuits of arbitrary geometry, and to compute and to display the time evolution of the fields. The TLM scheme has been derived from Maxwell’s equations using method of moments [61] and finite integration [62–64]. Figure 5 demonstrates the principle of the TLM method. In a first step the space is discretized into cubical TLM cells. As shown in Figure 5a on every surface of the TLM cell samples of tangential electric and magnetic fields are taken. Then, as shown in Figure 5b the electric and magnetic field amplitudes are represented by incident and scattered waves. The TLM cell is represented by a 12-port node, whch is connection network with instantaneous scattering and dispersion-free transmission lines connected to each port. B. Hybrid Methods Hybrid methods provide a computationally efficient approach to solve complex EM modeling problems. In hybrid methods two or more techniques are combined in such a way that each part of the problem is treated with the most suitable technique. In [65–67] a hybrid method is introduced to model radiating electromagnetic structures. The method combines a discretization based network method inside the sphere with modal expansion based network method in the space outside the sphere. Within a spherical region complex electromagnetic structures are modeled with the TLM method. Outside the spherical region the field is expanded into spherical waves. At the boundaries of the spherical region the TLM solution is matched to the multipole expansion of the field in the outer region. This yields a potentially exact modeling of the radiating boundary conditions. C. System Identification By using system identification (SI) algorithms in parallel to the time-domain TLM simulation it is possible to reduce the total simulation time considerably by estimation of the poles of the impedance function [36–48]. Since system identification yields the poles of the impedance function it allows the generation of Foster type equivalent networks. D. Model Order Reduction Model Order Reduction (MOR) by usage of Krylov Space Methods applied to TLM EM field Simulation are presented in [59, 68–71]. A two-step reduction approach based on the scatteringsymmetric Lanczos algorithm for TLM-MOR [72], oblique-oblique projection in TLM-MOR for

high-Q structures [73], system identification and model order reduction for TLM analysis [74] yield a considerable reduction of the computational effort in modeling complex electromagnetic structures. E. Discrete Electrodynamics and Metamaterials Metamaterials are artificial electromagnetic structures exhibiting special properties like negative permeability, negative permittivity and negative refractive index [75–80]. The discrete nature of metamaterials makes a discrete electrodynamical approach the most natural theoretical framework for the treatment of 3D metamaterials. Beyond being a powerful tool for numerical modeling of metamaterial structures, the TLM scheme provides a fundamental theoretical framework for the finding and exploration of three-dimensional metamaterial structures [81–83]. The numerical modeling of metamaterial structures exhibiting a large number of cells swamps the computational resources of numerical CAD tools. Network oriented modeling with adapted TLM scheme can solve this problem [84]. 7. The Mode Matching Method The generalized network formulation of the electromagnetic field problem, in connection with mode matching (MM) has already been applied to analyze waveguide N -furcation [85–90], to flangemounted radiating waveguides [91] and to model antennas realized with a central circular waveguide [92]. The MM method is excellently suited for the modeling of planar transmission line structures with sub-micrometer cross sections [93–96]. A hybrid combination of the TLM method and the mode matching method yields an efficient tool for full-wave analysis of transmission lines and discontinuities in RF-MMICs [97, 98]. 8. Conclusion Network methods applied to electromagnetic field simulation can contribute substanially to problem formulation, solution methodology and computational efficiency and allow to generate compact models of electromagnetic systems. Various methods for introducing network models of electromagnetic structures are based either on eigenmode expansion or on spatial discretization of the electromagnetic field. Network-oriented modeling can be combined with complexity reduction and system identification techniques and have the potential to reduce the computation time for the modeling of electromagnetic systems bsubstantially. This work is based on research projects supported by the Deutsche Forschungsgemeinschaft.

References [1] R. F. Harrington, Time Harmonic Electromagnetic Fields.

New York: McGraw-Hill, 1961.

[2] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics. John Wiley & Sons, 1965. [3] R. Collin, Field Theory of Guided Waves, 2nd ed. [4] V. Belevitch, Classical Network Theory.

New York:

New York: IEEE Press, Inc., 1991.

San Francisco: Holden-Day, 1968.

[5] H. J. Carlin and A. B. Giordano, Network Theory.

Englewood Cliffs, NJ: Prentice Hall, 1964.

[6] L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits. [7] W. Mathis, Theorie nichtlinearer Netzwerke.

New York: Mc Graw Hill, 1987.

Berlin: Springer, 1987.

[8] L. B. Felsen, M. Mongiardo, and P. Russer, “Electromagnetic field representations and computations in complex structures I: Complexity architecture and generalized network formulation,” International Journal of Numerical Modelling, Electronic Networks, Devices and Fields, vol. 15, pp. 93–107, 2002.

[9] ——, “Electromagnetic field representations and computations in complex structures II: Alternative Green’s functions,” International Journal of Numerical Modelling, Electronic Networks, Devices and Fields, vol. 15, pp. 109–125, 2002. [10] P. Russer, M. Mongiardo, and L. B. Felsen, “Electromagnetic field representations and computations in complex structures III: Network representations of the connection and subdomain circuits,” International Journal of Numerical Modelling, Electronic Networks, Devices and Fields, vol. 15, pp. 127–145, 2002. [11] L. B. Felsen, M. Mongiardo, and P. Russer, Electromagnetic Field Computation by Network Methods, 1st ed. Berlin Heidelberg New York: Springer, 2009. [12] P. Russer, Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, 2nd ed. Boston: Artech House, 2006. [13] R. F. Harrington, “Matrix methods for field problems,” Proceedings of the IEEE, vol. 55, no. 2, pp. 136–149, Feb. 1967. [14] ——, Field Computation by Moment Methods,.

San Francisco: IEEE Press, 1968.

[15] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice Hall, 1972. [16] P. Penfield, R. Spence, and S. Duinker, Tellegen’s theorem and electrical networks. Cambridge, Massachusetts: MIT Press, 1970. [17] B. D. H. Tellegen, “A general network theorem with applications,” Philips Research Reports, vol. 7, pp. 259–269, 1952. [18] ——, “A general network theorem with applications,” Proc. Inst. Radio Engineers, vol. 14, pp. 265–270, 1953. [19] J. C. Slater, “Microwave electronics,” Rev. Mod. Phys., vol. 18, no. 4, pp. 441–512, Oct 1946. [20] T. Teichmann, “Completeness relations for Loss-Free microwave junctions,” J. Appl. Physics, vol. 23, no. 7, pp. 701–710, Jul. 1952. [Online]. Available: http://link.aip.org/link/?JAP/23/701/1 [21] T. Teichmann and E. P. Wigner, “Electromagnetic field expansions in loss-free cavities excited through holes,” J. Appl. Physics, vol. 24, no. 3, pp. 262–267, Mar. 1953. [22] S. A. Schelkunoff, “On representation of electromagnetic fields in cavities in terms of natural modes of oscillation,” J. Appl. Physics, vol. 26, no. 10, pp. 1231–1234, Oct. 1955. [23] K. Kurokawa, “The expansions of electromagnetic fields in cavities,” IEEE Trans. Microwave Theory Techn., vol. 6, no. 2, pp. 178– 187, Apr. 1958. [24] W. Cauer, Theorie der linearen Wechselstromschaltungen.

Berlin: Akademie-Verlag, 1954.

[25] T. Rozzi and W. Mecklenbrauker, “Field and network analysis of waveguide discontinuities,” in European Microwave Conference, 1973. 3rd, vol. 1, 1973, pp. 1–4. [26] T. E. Rozzi and W. F. G. Mecklenbrauker, “Wide-Band network modeling of interacting inductive irises and steps,” IEEE Trans. Microwave Theory Techn., vol. 23, no. 2, pp. 235–245, 1975. [27] P. Arcioni, M. Bressan, G. Conciauro, and L. Perregrini, “Wideband modeling of arbitrarily shaped e-plane waveguide components by the boundary integral-resonant mode expansion method,” IEEE Trans. Microwave Theory Techn., vol. 44, no. 11, pp. 2083–2092, 1996. [28] G. Conciauro, P. Arcioni, M. Bressan, and L. Perregrini, “Wideband modeling of arbitrarily shaped h-plane waveguide components by the boundary integral-resonant mode expansion method,” IEEE Trans. Microwave Theory Techn., vol. 44, no. 7, pp. 1057–1066, 1996. [29] P. Arcioni and G. Conciauro, “Combination of generalized admittance matrices in the form of pole expansions,” IEEE Trans. Microwave Theory Techn., vol. 47, no. 10, pp. 1990–1996, 1999. [30] J. Dittloff, J. Bornemann, and F. Arndt, “Computer aided design of optimum e- or H-Plane N-Furcated waveguide power dividers,” in European Microwave Conference, 1987. 17th, 1987, pp. 181–186. [31] J. Dittloff, F. Arndt, and D. Grauerholz, “Optimum design of waveguide e-plane stub-loaded phase shifters,” IEEE Trans. Microwave Theory Techn., vol. 36, no. 3, pp. 582–587, 1988. [32] M. Guglielmi, F. Montauti, L. Pellegrini, and P. Arcioni, “Implementing transmission zeros in inductive-window bandpass filters,” IEEE Trans. Microwave Theory Techn., vol. 43, no. 8, pp. 1911–1915, 1995. [33] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis. 2000.

New York: John Wiley & Sons,

[34] T. Mangold and P. Russer, “Generation of lumped element equivalent circuits for distributed multiport structures via TLM simulation,” Second International Workshop on Transmission Line Matrix (TLM) Modeling – Theory and Applications, Munich, Germany, 29.-31.10.1997, pp. 236–245, Oct. 1997., pp. 236–245, Oct. 1997. [35] T. Mangold, P. Gulde, G. Neumann, and P. Russer, “A multichip module integration technology on silicon substrate for high frequency applications,” in 1998 Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems Digest, Ann Arbor, 26-28 Sept 1998, 1998, pp. 181–184. [36] V. Chtchekatourov, L. Vietzorreck, W. Fisch, and P. Russer, “Time-domain system identification modeling for microwave structures,” in MMET 2000. International Conference on Mathematical Methods in Electromagnetic Theory, vol. 1, 2000, pp. 137–139 vol.1. [37] V. Chtchekatourov, F. Coccetti, and P. Russer, “Direct Y–parameters estimation of microwave structures using TLM simulation and prony’s method,” in Proc. 17th Annual Review of Progress in Applied Computational Electromagnetics ACES, Monterey, May 2001, pp. 580–586. [38] V. Chtchekatourov, W. Fisch, F. Coccetti, and P. Russer, “Full–wave analysis and model–based parameter estimation approaches for s- and y- matrix computation of microwave distributed circuits,” in 2001 Int. Microwave Symposium Digest, Phoenix, 2001, pp. 1037–1040. [39] F. Coccetti, V. Chtchekatourov, and P. Russer, “Time-domain analysis of RF structures by means of TLM and system identification methods,” in European Microwave Conference, 2001. 31st, 2001, pp. 1–4. [40] Y. Kuznetsov, A. Baev, F. Coccetti, and P. Russer, “The ultra wideband transfer function representation of complex three-dimensional electromagnetic structures,” in 34th European Microwave Conference, Amsterdam, The Netherlands, 11.-15.10.2004, Oct. 2004, pp. 455–458. [41] U. Siart, K. Fichtner, Y. Kuznetsov, A. Baev, and P. Russer, “TLM modeling and system identification of distributed microwave circuits and antennas,” in ICEAA2007, Int. Conf. on Electromagnetics in Advanced Applications, 2007, Torino, Torino, Italy, sept 2007, pp. 352–355. [42] Y. Kuznetsov, A. Baev, P. Lorenz, and P. Russer, “Network oriented modeling of passive microwave structures,” in EUROCON 2007, sept 2007, pp. 10–14. [43] N. Fichtner, U. Siart, Y. Kuznetsov, A. Baev, and P. Russer, “TLM modeling and system identification of optimized antenna structures,” in Kleinheubacher Tagung, Miltenberg, Germany, sept 2007. [44] F. Coccetti and P. Russer, “A prony’s model based signal prediction (psp) algorithm for systematic extraction of y parameters from td transient responses of electromagnetic structures,” in 15th International Conference on Microwaves, Radar and Wireless Communications, MIKON-2004, 17-19 May, vol. 3, May 2004, pp. 791 – 794. [45] P. Russer and F. Coccetti, “Network methods in electromagnetic field computation,” in Proc. Turkish URSI Congress 2004, 8 - 10 August 2004, Ankara, Aug. 2004. [46] Y. Kuznetsov, A. Baev, F. Coccetti, and P. Russer, “The ultra wideband transfer function representation of complex three-dimensional electromagnetic structures,” in 2004 International Symposium on Signals, Systems, and Electronics ISSSE’04, August 10-13, Linz, Austria, Aug. 2004. [47] D. Lukashevich, F. Coccetti, and P. Russer, “System identification and model order reduction for TLM analysis of microwave components,” in Computational Electromagnetics in Time-Domain, 2005. CEM-TD 2005. Workshop on, 2005, pp. 64–67. [48] K. Fichtner, U. Siart, Y. Kuznetsov, A. Baev, and P. Russer, “Bandwidth optimization using transmission line matrix modeling,” in Time-Domain Methods in Modern Engineering Electromagnetics, A Tribute to Wolfgang J. R. Hoefer, P. Russer and U. Siart, Eds. Technische Universit¨ at M¨ unchen: Springer, may 16–17 2007. [49] J. Gustincic and R. Collin, “A general power loss method for attenuation of cavities and waveguides,” in PGMTT National Symposium Digest, vol. 62, 1962, pp. 20–21. [50] J. Gustincic, “A general power loss method for attenuation of cavities and waveguides,” IEEE Trans. Microwave Theory Techn., vol. 11, no. 1, pp. 83–87, 1963. [51] E. A. Guillemin, Synthesis of Passive Networks.

New York: Wiley, 1957.

[52] L. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Physics, vol. 19, no. 12, pp. 1163–1175, Dec. 1948. [53] P. Johns and R. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. IEE, vol. 118, no. 9, pp. 1203–1208, Sep. 1971. [54] C. Christopoulos, The Transmission-Line Modeling Method TLM.

New York: IEEE Press, 1995.

[55] P. Russer, “The transmission line matrix method,” in Applied Computational Electromagnetics, ser. NATO ASI Series. Berlin Heidelberg New York: Springer, 2000, pp. 243–269. [56] W. Hoefer, “The transmission line matrix method-theory and applications,” IEEE Trans. Microwave Theory Techn., vol. 33, pp. 882–893, Oct. 1985. [57] ——, “The transmission line matrix (TLM) method,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, Ed. New York: J. Wiley., 1989, pp. 496–591. [58] C. Christopoulos and P. Russer, “Application of TLM to microwave circuits,” in Applied Computational Electromagnetics, ser. NATO ASI Series. Cambridge, Massachusetts, London, England: Springer, 2000, pp. 300–323. [59] P. Russer and A. C. Cangellaris, “Network–oriented modeling, complexity reduction and system identification techniques for electromagnetic systems,” Proc. 4th Int. Workshop on Computational Electromagnetics in the Time–Domain: TLM/FDTD and Related Techniques, 17–19 September 2001 Nottingham, pp. 105–122, Sep. 2001. [60] P. Russer, “The transmission line matrix method,” in New Trends and Concepts in Microwave Theory and Technics, H. Baudrand, Ed. Trivandrum, India: Research Signpost, 2003, pp. 41–82. [61] M. Krumpholz and P. Russer, “A field theoretical derivation TLM,” IEEE Trans. Microwave Theory Techn., vol. 42, no. 9, pp. 1660–1668, Sep. 1994. [62] N. Pe na and M. M. Ney, “A general and complete two-dimensional TLM hybrid node formulation based on Maxwell’s integral equations,” in Proc. 12th Annual Review of Progress in Applied Computational Electromagnetics ACES, Monterey, Monterey, CA, Mar. 1996, pp. 254–261. [63] N. Pe na and M. Ney, “A general formulation of a three-dimensional TLM condensed node with the modeling of electric and magnetic losses and current sources,” in Proc. 12th Annual Review of Progress in Applied Computational Electromagnetics ACES, Monterey, Monterey, CA, Mar. 1996, pp. 262–269. ¨ Int. J. Electron. Commun., [64] M. Aidam and P. Russer, “Derivation of the TLM method by finite integration,” AEU vol. 51, pp. 35–39, 1997. [65] P. Lorenz and P. Russer, “Hybrid transmission line matrix - multipole expansion TLMME method,” in Fields, Networks, Methods, and Systems in Modern Electrodynamics, P. Russer and M. Mongiardo, Eds. Berlin: Springer, 2004, pp. 157 –168. [66] ——, “Hybrid transmission line matrix (TLM) and multipole expansion method for time-domain modeling of radiating structures,” in IEEE MTT-S International Microwave Symposium, Jun. 2004, pp. 1037 – 1040. [67] ——, “Discrete and modal source modeling with connection networks for the transmission line matrix (TLM) method,” in 2007 IEEE MTT-S Int. Microwave Symp. Dig. June 4–8, Honolulu, USA, jun 2007, pp. 1975–1978. [68] A. Cangellaris, M. Celik, S. Pasha, and L. Zhao, “Electromagnetic model order reduction for system level modeling,” IEEE Trans. Microwave Theory Techn., vol. 47, pp. 840–850, Jun. 1999. [69] A. Cangellaris, D. Lukashevich, and P. Russer, “Model order reduction in electromagnetic field computation,” in 35th European Microwave Conference, Workshop on Advanced Modelling Methods in Microwaves, Paris, France, 3.-7.10.2005, Oct. 2005. [70] D. Lukashevich, A. Cangellaris, and P. Russer, “Model order reduction by krylov space methods applied to TLM electromagnetic field simulation,” in IEEE MTT-S International Microwave Symposium, Jun. 2004, pp. 200–205. [71] D. Lukashevich and P. Russer, “TLM model order reduction,” in Fields, Networks, Methods, and Systems in Modern Electrodynamics, P. Russer and M. Mongiardo, Eds. Berlin: Springer, 2004, pp. 205 –217. [72] D. Lukashevich, A. Cangellaris, and P. Russer, “Two-step reduction approach based on the scattering-symmetric lanczos algorithm for TLM-ROM,” in Proceedings of the 21th Annual Review of Progress in Applied Computational Electromagnetics ACES, 3.-7.04.2005, Honolulu, Hawaii, Apr. 2005, pp. 698–705. [73] D. Lukashevich and P. Russer, “Oblique-oblique projection in tlm-mor for high-q structures,” in 35th European Microwave Conference, Paris, France, 3.-7.10.2005, Oct. 2005, pp. 849–852. [74] D. Lukashevich, F. Coccetti, and P. Russer, “System identification and model order reduction for tlm analysis,” International Journal of Numerical Modelling, Electronic Networks, Devices and Fields, vol. 20, no. 1–2, pp. 75–92, jan 2007. [75] D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett., vol. 85, pp. 2933–2936, Oct. 2000.

[76] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L - C loaded transmission lines,” IEEE Trans. Microwave Theory Techn., vol. 50, no. 12, pp. 2702 – 2712, 12 2002. [77] A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Magazine, vol. 5, no. 3, pp. 34–50, Sep. 2004. [78] A. Sanada, C. Caloz, and C. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microwave Theory Techn., vol. 52, no. 4, pp. 1252 – 1263, Apr. 2004. [79] C. Caloz and T. Itoh, Electromagnetic Metamaterials.

Hoboken, New Jersey: John Wiley & Sons, 2006.

[80] G. Eleftheriades and K. Balmain, Negative-Refraction Metamaterials.

New York: John Wiley & Sons, 2005.

[81] M. Zedler, C. Caloz, and P. Russer, “A 3-d isotropic left-handed metamaterial based on the rotated transmissionline matrix (TLM) scheme,” IEEE Trans. Microwave Theory Techn., vol. 55, no. 12, pp. 2930–2941, dec 2007. [82] M. Zedler and P. Russer, “Three-dimensional CRLH metamaterials for microwave applications,” Proceedings of the European Microwave Association, vol. 3, pp. 151–162, jun 2007. [83] M. Zedler, C. Caloz, and P. Russer, “Analysis of a planarized 3d isotropic lh metamaterial based on the rotated TLM scheme,” in Proc. 37th European Microwave Conference, Munich, Munich, Germany, okt 2007, pp. 624–627. [84] P. Poman, H. Du, and W. Hoefer, “Modeling of metamaterials with negative refractive index using 2 - D shunt and 3 - D SCN TLM networks,” IEEE Trans. Microwave Theory Techn., vol. 53, no. 4, pp. 1496 – 1505, Apr. 2005. [85] M. Mongiardo, P. Russer, M. Dionigi, and L. Felsen, “Generalized networks for waveguide step discontinuities,” Proc. 14th Annual Review of Progress in Applied Computational Electromagnetics ACES, Monterey, pp. 952–956, Mar. 1998. [86] ——, “Waveguide step discontinuities revisited by the generalized network formulation,” 1998 Int. Microwave Symposium Digest, Baltimore, pp. 917–920, Jun. 1998. [87] M. Mongiardo, P. Russer, C. Tomassoni, and L. Felsen, “Analysis of N–furcation in elliptical waveguides via the generalized network formulation,” 1999 Int. Microwave Symposium Digest, Anaheim, pp. 27–30, Jun. 1999. [88] ——, “Analysis of N–furcation in elliptical waveguides via the generalized network formulation,” IEEE Trans. Microwave Theory Techn., vol. 47, pp. 2473–2478, Dec. 1999. [89] L. B. Felsen, M. Mongiardo, P. Russer, G. Conti, and C. Tomassoni, “Waveguide component analysis by a generalized network approach,” Proc. 27th European Microwave Conference, Jerusalem, pp. 949–954, Sep. 1997. [90] M. Mongiardo, P. Russer, C. Tomassoni, and L. Felsen, “Generalized network formulation analysis of the N– furcations application to elliptical waveguide,” Proc. 10th Int. Symp. on Theoretical Electrical Engineering, Magdeburg, Germany, (ISTET), pp. 129–134, Sep. 1999. [91] M. Mongiardo and P. Tomassoni, C.and Russer, “Generalized network formulation: Application to flange– mounted radiating waveguides,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 6, pp. 1667–1678, jun 2007. [92] C. Tomassoni, M. Mongiardo, P. Russer, and R. Sorrentino, “Rigorous computer-aided design of coaxial/circular antennas with semi-spherical dielectric layers,” in Microwave Symposium Digest, 2008 IEEE MTT-S International, 2008, pp. 975–978. [93] J. Kessler, R. Dill, P. Russer, and A. A. Valenzuela, “Property calculations of a superconducting coplanar waveguide resonator,” Proc. 20th European Microwave Conference, Budapest, pp. 798–903, Sep. 1990. [94] J. Kessler, R. Dill, and P. Russer, “Field theory investigation of high-tc superconducting coplanar wave guide transmission lines and resonators,” IEEE Trans. Microwave Theory Techn., vol. 39, no. 9, pp. 1566–1574, Sep. 1991. [95] R. Schmidt and P. Russer, “Modelling of cascaded coplanar waveguide discontinuities by the mode-matching approach,” 1995 Int. Microwave Symposium Digest, Orlando, pp. 281–284, May 1995. [96] ——, “Modelling of cascaded coplanar waveguide discontinuities by the mode-matching approach,” IEEE Trans. Microwave Theory Techn., vol. 43, pp. 2909–2916, Dec. 1995. [97] B. Broido, D. Lukashevich, and P. Russer, “Hybrid method for simulation of passive structures in rf-mmics,” in Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, April 9–11, 2003, Garmisch, Germany, Apr. 2003, pp. 182–185. [98] D. Lukashevich, B. Broido, M. Pfost, and P. Russer, “The Hybrid TLM-MM approach for simulation of MMICs ,” in Proc. 33th European Microwave Conference, Munich, October 2003, pp. 339–342.