Krylov Model Order Reduction of Finite Element Models of

Electromagnetic Structures with Frequency-Dependent Material Properties

Hong Wu and Andreas Cangellaris

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Urbana, Illnois 61801, Email: hongwu, cangella@uiuc.edu

Abstract— Krylov subspace-based model order reduction(MOR) of finite element models of electromagnetic structuresis not readily applicable when the electromagnetic properties ofthe materials exhibit arbitrary frequency dependence. This paperpresents a methodology for overcoming this hurdle. The proposedKrylov MOR process is demonstrated through its application tothe expedient broadband analysis of the impact of skin-effectloss on the transmission properties of a microstrip bandpassfilter and the extraction of the propagation characteristics ofa microstrip line on a dielectric substrate with frequency-dependent permittivity described by a Debye model.

Index Terms— Model order reduction, finite element method,dispersive media, frequency dependent rational function.

I. INTRODUCTION

In the context of state-space representations of linear sys-

tems of high dimension the objective of model order reduction

(MOR) is the development of a new model of the linear

system of much smaller dimension than the original one, yet

capable of describing the behavior of the original model with

acceptable accuracy over a broad frequency bandwidth. It is

this attribute of MOR that has prompted its application for

expediting solution efficiency and reducing modeling com-

plexity of the large linear circuits associated with high-speed

interconnect modeling in state-of-the art integrated electronic

systems [1]–[5].

The discretization of the spatial derivatives in Maxwell’s

time-dependent equations using finite methods results in state-

space semi-discrete approximations of the electromagnetic

system. Hence, the same techniques that have been used

successfully for model order reduction of large linear cir-

cuits can be adopted with only slight modifications for the

reduced-order modeling of electromagnetic systems. Among

the various techniques currently available for the reduced-order

modeling of passive electromagnetic devices and structures the

Asymptotic Waveform Evaluation (AWE) method [6] was the

first one to be used. This was followed by the application of

the computationally more robust class of Krylov subspace-

based methods and in particular the Pade via Lanczos (PVL)

process [1], [7]. For applications that called for the utilization

of the reduced-order model as a multiple-input multiple-output

(MIMO) matrix transfer function representation of the electro-

magnetic device, the requirement for preserving the passivity

attributes of the original system in its reduced macromodel

led to the development of passivity-preserving model order

reduction techniques. For the case of model order reduction

of finite-difference/finite-element models of electromagnetic

systems, the block Arnoldi process-based, passive reduced-

order macro-modeling algorithm (PRIMA) [2] was shown to

be particularly useful [3], [8].

The aforementioned Krylov-based model order reduction

techniques are most suitable for the expedient broadband

analysis and macromodeling of electromagnetic devices and

systems with constant material properties. When the elec-

tromagnetic properties of the materials are frequency de-

pendent, characterized by Krylov subspace methods are not

as readily applicable and alternative model order reduction

techniques are sought (see, for example, [9]). In addition to

media that naturally exhibit frequency dependence in their

electromagnetic properties, there are numerous application

of practical interest where the ability to incorporate in a

convenient manner in the electromagnetic model frequency-

dependent relationships between the fields and associated

flux quantities provides for enhanced modeling versatility and

increased computational efficiency. Most typical examples are,

a) the utilization of frequency-dependent surface impedance

boundary conditions to account for skin effect loss in metallic

volumes [10], [11]; b) the utilization of frequency-dependent

multi-port, matrix transfer functions for the macromodeling of

sub-domains of arbitrary (passive) material composition [3]; c)

the macroscopic description of the electromagnetic attributes

of artificially constructed materials (metamaterials) [12].

In this paper a systematic methodology is proposed for

extending the application of Krylov subspace techniques to the

model order reduction of passive electromagnetic structures

with material and or boundary conditions exhibiting general

frequency dependence. The proposed methodology is based

on the extension of the technique introduced in [13]–[15] for

the reduction of second-order dynamical systems. The paper

is organized as follows. First, the development of the pertinent

finite element model of the electromagnetic system is briefly

reviewed. This is followed by the mathematical formulation

of the proposed model order reduction process. The paper

concludes with the validation of the proposed methodology

through its application to the analysis of a bandpass microstrip

filter with skin effect loss taken into account and the extraction

of the propagation characteristics of a microstrip line on a

dispersive dielectric substrate.

II. FINITE ELEMENT APPROXIMATION

The methodology described in [16] is followed for the

development of the finite element approximation of the vector

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Helmholtz equation for the electric field using edge ele-

ments. For the purposes of reduced-order macromodeling it

is assumed that the device under modeling interacts with

its exterior through Np ports. Hence, the Laplace-domain

representation of the finite element model of the Np-port

system is described by the following system of equations [16]

(Y + sZ + s2Pe)x = sBI

y = LHx

(1)

where N is the number of degrees of freedom in the finite

element approximation, the vector x contains the coefficients

in the finite element approximation of the electric field, the

matrices Y, Z, Pe are in RN×N , the matrices B, L are in

CN×Np , the vectors I , y are in C

Np×1 and s is the complex

frequency. The vector I indicates unit excitation at each port.

The matrix B is dependent on the port characteristics and

is used to map the unit excitation to the excitation on state-

space variables. Similarly, the matrix L is used to sample

the calculated electric field to generate the desirable output

quantity output.

As far as the matrices Y, Z and Pe are concerned, under

the assumption of perfect electric conductors and materials

of constant permittivity and permeability, these matrices are

frequency independent. This is not the case anymore when the

electromagnetic properties of the media are frequency depen-

dent. For example, assuming media of constant permeability

but frequency-dependent permittivity ǫ(s), the elements of Pe

are given by

Peij =

∫∫∫

Ω

~wti· ǫ(s) ~wtj

dv, (2)

where Ω is computation domain and ~wt indicates the tangen-

tially continuous edge element. Furthermore, when conductor

loss is taken into account and a surface impedance, η(s), is

used for its modeling, the Z matrix is augmented by an extra

term Zsuf , with elements given by

Zsuf ij = s

∫∫

SSIBC

n × ~wti· 1

η(s)n × ~wtj

ds. (3)

Thus, equation (1) becomes

(Y + sZ + s2Pe(s) + Zsuf (s))x = sBI,

y = LHx.

(4)

III. MODEL ORDER REDUCTION WITH EQUATION

STRUCTURE PRESERVED

For simplicity and for the sake of brevity we will present

the proposed model order reduction methodology for the case

where Pe is constant and the general frequency dependence in

(4) is only due to the surface impedance of lossy conductors.

Under this assumption we can write

Zsuf (s) = H(s)Zs, (5)

where it is Zs is defined as

Zsij =

∫∫

SSIBC

n × ~wti· n × ~wtj

ds, (6)

and

H(s) =s

η(s)= s

(

(1 + j)

√

sµ

2σ

)−1

. (7)

In the above equation σ and µ are, respectively, conductivity

and permeability of the conductors. Thus, equation (4) may

be cast in the following form

(Y + sZ + s2Pe + H(s)Zs)x = sBI,

y = LHx.

(8)

Next, use is made of the vector fitting technique of [17]

to express H(s) in terms of a rational function approximation

H(s). Thus, we have,

H(s) ≈ H(s) = h0 + h1s +

K∑

i=1

ri

s − pi

. (9)

Such a rational function fit is used for all terms in the

finite element approximation that exhibit general frequency

dependence. It is worth noting that rational function represen-

tations are commonly used for the representation of dispersive

media [18].

We proceed to define the impedance matrix for the Np-port

MIMO system as follows

ZG(s) = sLH(Y + sZ + s2Pe + H(s)Zs)

−1B. (10)

The Taylor expansion of ZG(s) around an expansion complex

frequency point s0 has the form

ZG(s) = sLH

∞∑

i=0

ri(s − s0)i, (11)

where LHri is often referred to as the block moment of

ZG(s). By matching the coefficients (or moments) in the Tay-

lor’s series expansion of (10) with those in the above equation,

the following expressions are obtained for the calculation of

ri:

r0 = R,

r1 = A1r0,

r2 = A1r1 + A2r0,

rn =

n∑

i=1

Airn−i, n ≥ 3,

(12)

where R, A1,A2, ... are defined as

R = (Y + s0Z + s20Pe + H(s0)Zs)

−1B,

A1 = −(Y + s0Z + s20Pe + H(s0)Zs)

−1

(

Z + 2s0Pe +

(

h1 −K∑

i=1

ri

(s0 − pi)2

)

Zs

)

,

A2 = −(Y + s0Z + s20Pe + H(s0)Zs)

−1

(

Pe +

(

K∑

i=1

ri

(s0 − pi)3

)

Zs

)

,

An = −(Y + s0Z + s20Pe + H(s0)Zs)

−1

(

(−1)n

(

K∑

i=1

ri

(s0 − pi)n+1

)

Zs

)

, n ≥ 3.(13)

53

Hence, a Krylov subspace is constructed

Kq(A1,A2, ...;R) = colsp[r0, r1, ..., rn−1, r0, ..., rl], (14)

where q is the order of the subspace, ri is ith column of the

matrix rn, n = ⌊q/Np⌋, and l = q −nNp. For the purpose of

model-order reduction, it is desired that q << N.An orthogonalization process is used to develop the or-

thonormal basis for Kq . Let

F = [F1; F2; ...; Fq ]. (15)

be the matrix with columns the generated orthonormal basis.

This matrix is used as the congruence transformation matrix

in the reduction process of the original MIMO system. The

reduced-order impedance matrix is obtained as follows

ZG(s) = sLH(Y + sZ + s2Pe + H(s)Zs)

−1B, (16)

where

Y = FHYF, Zs = F

HZsF, Z = F

HZF,

Pe = FHPeF, L = F

HL, B = F

HB.

(17)

It is evident from (13) that the number of moments that

should be kept in the Taylor series expansion is dependent on

the distance between the expansion frequency and the poles

in the rational function fit of H(s). Thus, if the distance is

sufficiently large, the Taylor series expansion of H(s) can be

safely truncated after only a few terms. In any case, since Zs is

a very sparse matrix, its multiplication with the vectors ri does

not impact adversely the computational efficiency of the model

order reduction process. Thus, higher order derivative terms in

the Taylor series expansion of H(s) can be accommodated, if

needed, at negligible additional computational cost.

IV. VALIDATION STUDIES

A. A microstrip bandpass filter

The first validation study considers the microstrip bandpass

filter depicted (in top and side view)in Fig. 1. The center

strip is, approximately, one-half wavelength long at the center

frequency of the pass band. The resonator is capacitively

coupled to the two strips at its ends. At the ends of the

input and output strips wave ports are used to drive and

terminate the structure. The computational domain used for

the finite element analysis is truncated at its top and side

(parallel to the resonator) boundaries by means of a first order

absorbing boundary condition. The observable quantities are

the scattering parameters over a frequency band centered at

the first resonant frequency.

The objective of this study is to examine the impact of

skin-effect loss on the transmission properties of the filter

for different values of the conductivity of the metallization.

The proposed model order reduction process was used for the

expedient calculation of the response (fast frequency sweep)

over the desired bandwidth. The expansion frequency was set

at 15 GHz.

The magnitude of the calculated S-parameters are depicted

in Fig. 2. For the lossless case, the first resonant frequency

is, approximately, the frequency at which the length of the

Fig. 1. Top view and side view of a single-resonator microstrip bandpassfilter.

resonator is one-half wavelength [19]. For the structure of Fig.

1 this first frequency is 17.22 GHz, as correctly predicted by

the finite element solution. The inclusion of the metal loss

causes the resonant frequency to shift to the left [20]. The

higher the loss the larger the shift. This trend is correctly

captured by the finite element solution.

1.6 1.65 1.7 1.75 1.8

x 1010

−25

−20

−15

−10

−5

0

Frequency (Hz)

|S11| and |S

12| (d

B)

Lossless

σ = 5.8e7 S/m

σ = 3.5e7 S/m

σ = 1.7e7 S/m

Fig. 2. |S11| and |S12| of the microstrip bandpass filter.

B. Microstrip Line on a Debye Substrate

The purpose of the second example study is to demonstrate

the ability of the proposed model order reduction scheme to

handle electromagnetic structures involving dispersive media.

The structure under consideration is a microstrip line of the

same cross-sectional geometry as the one depicted in Fig.

1. However, for this case the substrate is taken to be a

Debye medium with frequency-dependent relative complex

permittivity as follows:

ǫsub = ǫ∞ +A

1 + jωτ, (18)

where ǫ∞ = 2, A = 10, and τ = 5 × 109. The calculated

quantity is the input impedance of a 6 cm-long section of

the microstrip line terminated at a 60-Ohm resistive load. In

addition a point-by-point finite element solution was carried

out at 21 equidistant frequency points over the frequency

bandwidth of 2 GHz. The way the resistive load is introduced

in the finite element model is discussed in [21].

To provide for a reference solution, the input impedance of

the short-circuited microstrip line was calculated using trans-

mission line theory. For this purpose, the effective permittivity

for the microstrip at different frequencies was calculated using

54

the formula in [22] with values for ǫsub obtained from (18).

In the above equation H is the substrate thickness and W is

the width of the microstrip. The calculated relative effective

permittivity, ǫeff (f), at each frequency was combined with

the characteristic impedance, Z0, of the microstrip cross

sectional geometry in free space, to calculate the frequency-

dependent characteristic impedance of the microstrip on the

Debye substrate through the formula Z0/√

ǫeff .

The frequency dependence of the real and imaginary parts

of the calculated input impedance is depicted in Fig. 3. The

results from the MOR-assisted fast frequency sweep are in

excellent agreement with the point-by-point finite element so-

lution, validating the accuracy of the proposed MOR method-

ology. The finite element solution results are also in very good

agreement with the transmission line theory results, especially

over the lower frequency range. At higher frequencies a

discrepancy is observed between the finite element solution

result and the transmission line result. This is attributed to the

fringing capacitance at the terminated end of the microstrip

which, even though present in the finite element model, is not

taken into account in the transmission line model.

0 0.5 1 1.5 2

x 109

20

30

40

50

60

Frequency (Hz)

Real(Z

in)

(Ohm

)

(a) resistance

MOR with rational function

Point by point simulation

Transmission line analysis

0 0.5 1 1.5 2

x 109

−20

−10

0

10

20

30

Frequency (Hz)

imag(Z

in)

(Ohm

)

(b) reactance

MOR with rational function

Point by point simulation

Transmission line analysis

Fig. 3. Input impedance of a microstrip line on a Debye substrate.

V. CONCLUSION

In summary, we have proposed and validated a Krylov

subspace-based model order reduction methodology for use

in conjunction with finite element approximation of electro-

magnetic systems containing dispersive materials and sub-

structures characterized in terms of frequency-dependent trans-

fer impedance matrices. The versatility of the proposed method

in handling dispersive materials and transfer functions of

arbitrary frequency dependence stems from its utilization of

a rational function fit of the frequency dependent parameters

in the structure.

VI. ACKNOWLEDGEMENT

This research was supported in part by the Air Force Office

for Scientific Research through a MURI on the “Analysis and

Design of Ultrawide-band and High-power Microwave Pulse

Interactions with Electronic Circuits and Systems.”

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