kabir khazaka

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Macromodeling of Interconnect Networks from Frequency Domain Data

using the Loewner Matrix Approach

Muhammad Kabir and Roni KhazakaDepartment of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada

Abstract Recently, Loewner Matrix (LM) based methodswere introduced for generating time-domain macromodels basedon frequency domain measured parameters. These methods wereshown to be very efficient and accurate for systems with avery large number of ports, however they were not suitable fordistributed transmission line networks. In this paper, an LMbased approach is proposed for modeling distributed networks.The new method was shown to be efficient and accurate forlarge-scale distributed networks.

Index Terms Distributed Networks, tangential interpolation,Loewner Matrix, Time-domain macromodel.

I. INTRODUCTION

In microwave and high frequency applications, we are often

faced with linear structures for which it is very difficult to

derive a physics based time-domain lumped equivalent circuit.

In such cases, it is often possible to obtain the frequency

domain Y or S-parameters of the structures through direct

measurement or through full-wave simulation. An algorithm is

then applied to generate a lumped time-domain macromodel

compatible with SPICE. The vector fitting (VF) algorithm

[13] is an effective method to achieve this result. However,

vector fitting can become very inefficient for systems with

a very large number of poles and a large number of ports.

More recently, a new approach based on the Loewner Matrix

(LM) pencil has been proposed [5,6]. This method was shownto be very efficient and accurate compared to vector fitting

[6]. However, the LM approach requires measured/simulated

frequency response data spanning the entire bandwidth of the

system in order to produce a state macromodel. This is not

possible for distributed networks such as transmission lines,

which have infinite bandwidth. In this paper we propose a

new method based on the LM method which is applicable to

distributed transmission line networks.

I I . REVIEW OF LOEWNER MATRIX METHOD

A brief review of the LM method [6] is provided in

this section. Consider a p-port network for which we have

measured/simulated Y-parameter at n frequency points over

a band of 0 to fB. The objective of the LM method is to

obtain a time-domain macromodel based on the frequency

samples. In Sec. II-A the form of the macromodel is defined,

the macromodel in terms of LM is presented in Sec. II-B.

A. Time-Domain Macromodel

A time-domain macromodel of a p-port system can be

expressed as an LTI system with p inputs and outputs:

Ex(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t) + Yu(t) (1)

where, u(t) Rp contains the port voltages, y(t) Rp

contains the port currents and x(t) Rm contains the internalvariables of the system, E,A Rmm, B Rmp, C

Rpm, D Rpp, and Y Rpp.

The poles of the system are the generalized eigenvalues of

(A,E) and the Y-parameters are given by:

Y(s) = C(sE A)1B + D + sY (2)

B. Time-Domain Macromodel using Loewner Matrices

The time-domain macromodel shown in Eq. (1) can be

derived using the LM method [6] as follows:

E = L, A = L, B = F, C = W, D = 0 (3)

where, L and L are the Loewner and the shifted Loewner

matrices respectively. The matrices are defined using tangential

interpolation (TI) [6] which can be classified into Vector

Format Tangential Interpolation (VFTI) [6] and Matrix Format

Tangential Interpolation (MFTI) [7].

1) VFTI : L, L, and the corresponding constraints anddata are defined as follows:

Lj,i =jri lji

j i, Lj,i =

jjri ilji

j i(4)

Y(i)ri = i; i C, ri Cp1,i C

p1,

ljY(j) = j

; j C, lj C1p,

j

C1p

where, i = j = 1, . . . , , is the size of the matrices, i, jare the frequency points and ri, lj are the tangential direction

vectors (i.e. columns and rows of the identity matrix) to derive

the tangential vector data, i and j . F and W are formed by

taking js and is as rows and columns respectively. The

number of available Y-parameter data, n is doubled using

the complex conjugate pair. The selection of (i, ri,i) and(j , lj,j) are performed using the real approach [6].

2) MFTI : Using the same size , the matrices L, L, andthe corresponding constraints and data are defined as follows:

Lj,i =jRi Lji

j i, Lj,i =

jjRi iLjij i

(5)

Y(i)Ri = i; i C,Ri Cpti ,i C

pti ,

LjY(j) = j; j C,Lj Ctip,j C

tip

where, Ri, Lj are the tangential direction matrices and tiis the number of columns and rows to be sampled in the

tangential matrix data i and j respectively. F and W are

formed by adding the block matrices j and i row-wise and

column-wise respectively. The choice of interpolation data is

as follows:

{; R;} {si, si; I, I; Y(i)

I, Y(i)

I}

{; L;} {si+1, si+1; I, I; IY(i+1), IY

(i+1)}

978-1-4673-1088-8/12/$31.00 2012 IEEE


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0 1 2 3 4 50.01

0.015

0.02

0.025

0.03

Y11

Frequency (GHz)

MeasuredParameter

(a)

230 240 250 260 270 280 29015

10

5

0

log10

(i/

1

)

VFTI

MFTI

Singular ValueDrops

Order, m

(b)

Fig. 1. (a) Y11, (b) En for 18-port low bandwidth system

where, i = 1, 3, . . . , n1, si = j2fi, I is the identity matrix

of size pp and() is the complex conjugate.

The equivalent real matrices (Lr , Lr, Fr and Wr) are

formed using congruence transformation with the regular part

extracted using the SVD approach [6].

The LM method determines the order m of the model based

on the prominent drops on the plot of the normalized singular

values En of matrix xLr Lr, x = 2f1 [6] as shownin Fig. 1(b). The location of the drops can also indicate the

existence of a D matrix. Once D is extracted the macromodel

becomes stable [6].

III. PROPOSED APPROACH FOR MODELING DISTRIBUTED

NETWORKS

One of the key features of the LM method [6,7] is the

determination of the order of the system based on the drops in

singular values as shown in Fig. 1(b) and the extraction of the

D matrix in order to ensure stability [6]. However, in order to

obtain such prominent drops in En as shown in Fig. 1(b), the

original frequency domain data must span all the bandwidth

of the underlying system beyond the highest frequency pole as

shown in Fig. 1(a). This requirement is not realistic when the

data describes distributed networks which typically have a very

wide band response that goes well beyond our frequency range

of interest as shown in Fig. 2(a). In such cases the singular

values do not exhibit a large drop. Instead only a change in the

slope can be observed as shown in Fig. 2(b). In this paper we

propose an algorithm for modeling such distributed systems

using the LM approach.

A. Determining the Order of the System

As can be seen in Fig. 2, when the bandwidth of the

measured parameters does not extend beyond the highest

frequency pole, the singular values do not exhibit large drops

as observed in Fig. 1(b). The higher slope region represents

the regular part of xLr Lr, and the lower slope region

represents the singular part. In this case the order of the

system can be heuristically chosen at the point of the largest

0 1 2 3 4 50.015

0.02

0.025

0.03

Y11

Frequency (GHz)

MeasuredParameter

(a)

250 300 350 40020

15

10

5

0

log10

(i/

1

)

MFTI

VFTI

Slope Change

Regular Part

No Prominent Drop

(b)

Fig. 2. (a) Y11, (b) En for 18-port high bandwidth system

drop in the singular value in the regular part nearest to the

point of change in the slope. Alternatively, one can choosem = rank(xLr Lr) in the case of VFTI [6] approach.

B. Extraction of D and Y

The macromodel (A,E,B,C) with D = 0 and Y = 0,extracted using the LM method is typically unstable. In the

case of low bandwidth systems such as the one in Fig. 1,

the reason for this instability is the fact that the D matrix is

embedded in the A, E and C matrices and results in unstable

real poles very far from the origin. Once the matrix D is

extracted, the system becomes stable [6]. In the case of high

bandwidth systems such as the one in Fig. 2, one must also

take into account Y

which can no longer be taken as zero.In this case, the presence of both D and Y embedded in the

(A,E,B,C) system matrices results in both real and complex

unstable poles far from the origin as shown in Fig. 3(a). In

this paper we present a method for extracting both D and Y,

which results in a stable macromodel (A, E, B, C, D, Y

).

The first step of the process is to choose the poles that we

would like to keep in the system. There would be all poles up

to the bandwidth fB of the measured parameters in addition

to some poles up to a frequency fB + fa which are conservedto improve accuracy near fB. The rest of the poles including

the large unstable poles will be extracted. The real ones will

be captured in D and the complex ones will contribute to Y

.

Let Vl and Vr be the matrices containing the left and righteigenvectors of the corresponding poles we want to conserve.

The matrices Ql and Qr are the orthonormal bases of Vl and

Vr respectively. The macromodel (A, E, B, C, D, Y

) can be

formed as follows:

A = QTl AQr, E = QTl EQr, B = Q

Tl B, C = CQr,

D = avg

real

C(sE A)1B C(sE A)1B

, (6)

Y

= avg

imag

C(sE A)1B C(sE A)1B

./s

where, the average in D and Y

are taken over na values of

s equally spaced and spanning the bandwidth fB .

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