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Modeling of Transmission Lines with EM Wave Coupling by

Finite Difference Quadrature Methods

Qinwei Xu and Pinaki Mazumder

Abstract This paper proposes an efficient numerical approxi-

mation technique, called the Finite Difference Quadrature (FDQ)

Method, which has been adapted to model transmission lines

(TLs) with external EM wave coupling. The finite difference

quadrature method can quickly compute finite differences be-

tween adjacent grid points by estimating a weighted linear sum

of derivatives at a set of points belonging to the domain. Similarly

to the Gaussian Quadrature method to compute the numerical in-

tegrals, FDQ method uses global quadrature method to construct

the approximation framework, however, to compute the finite dif-

ference. A discrete modeling approach, FDQ needs much sparser

grid points than the Finite Difference (FD) methods to achieve

comparable accuracy. The FDQ-based equivalent models can be

integrated into simulators like SPICE to efficiently simulate cir-

cuits involving EM interference. The FDQ procedure can be ap-

plied to solve other classes of partial differential equations. This

paper thus demonstrates a general numerical approach, though

the focus of the paper is transmission line modeling.

1 INTRODUCTION

Fast operation and large integration scale have made

the interconnect effect an important issue in high-speed

systems. In addition to the on-chip and on-board ef-

fects of delay, crosstalk, and reflections, electrically

long interconnects pose the antenna effect, as they

receive considerable dose of incident electromagnetic(EM) waves emitted by external electronic devices [1].

Fast clocking rate and short rise time result in sig-

nals with wavelengths comparable to interconnect sizes

which increases the radiation efficiency of the conduct-

ing traces, while small feature sizes lead to high electro-

magnetic interference (EMI) susceptibility among the

circuit parts. The antenna effect becomes a more seri-

ous challenge to signal integrity with progressive

scaling.

Equivalent source modeling, based on the quasi-TEM

assumption, is one of the mainstream approaches to

solve the problem of external field coupling to TLs [2],

in which incident EM waves illuminating the TLs aremodeled as equivalent sources. These effects of the in-

cident field appear as forcing functions on the TL equa-

tions, which are incorporated into the circuit simulators

altogether with other devices. Compared to the field

solvers, the equivalent circuit approach is computation-

ally efficient and numerically accurate as long as the

quasi-TEM assumption is valid. The equivalent source

approach in the literature [3] is mathematically based on

This work was supported by MURI grant.

EECS Dept., University of Michigan, Ann Arbor, MI 48109-

2122, Email:

qwxu,mazum

@eecs.umich.edu

0

i

j

zj

zi

yj

yi

- xA0(x,0,0)

Aj(x,yj,zj)

Ai(x,yi,zi)z

yEH

d

E

H

E

Figure 1: MTL illuminated by EM wave.

the finite difference (FD) methods. As the FD methods

have low order accuracy, the electrical length of each

section has to be a considerably small fraction (1/12-

1/20) of the minimum wave length of the signal; there-

fore, the equivalent models consist a excessive number

of lumped elements. On the other hand, numerical in-

tegration (quadrature) is more stable and reliable than

differentiation because it globally integrates the local

information. Integral approaches like Gaussian Quadra-

ture generally give more accurate solutions [4]. In thispaper, the Finite Difference Quadrature (FDQ) method

is proposed to model TLs. The idea of the FDQ method

is to quickly compute the finite difference of two neigh-

boring grid points by estimating a weighted linear sum

of derivatives at a small set of points belonging to the

domain. The weighted linear sum is like the numeri-

cal integral in Gaussian Quadrature method, yet it is to

compute finite differences rather than the integrals. As

a result, FDQ needs much sparser grid points than the

Finite Difference (FD) methods to achieve comparable

accuracy.

2 QUASI-TEM FORMULATIONS OF TLSWITH EM WAVE COUPLING

The quasi-TEM assumption of TLs is equivalent to the

condition that the dimensions of TLs cross-sectional

sizes are much smaller than a wavelength of the ex-

ternal EM wave. Under such a condition, the principal

propagation mode of the TLs is TEM, and can be ac-

curately described by the Telegraphers equations. The-

oretical analysis and experimental results have shown

that the external EM wave illuminating can be modeled

as forcing terms which are added in the Telegraphers

equations [1]. Let the illuminating EM wave be repre-


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sented in the Cartesian coordinate system by

!

"

#

%

!

"

% #

'

!

"

' ) (1)

and the TLs stretch along0

direction. Assume that

the reference of conductor (defined as line 0) is colin-ear with the

0-axis (see Fig.1). By assuming a quasi-

TEM mode of propagation along an MTL consisting of 1

# 3

conductors, the voltages5

0

)

and currents6

0

)

along the conductors can be represented by the

-domain Telegraphers equations:

8

8

0

5

0

)

#

9

0

6

0

)

# B

0

6

0

)

5 E

0

)

(2)8

8

0

6

0

)

#

G

0

5

0

)

# I

0

5

0

)

6

E

0

)

(3)

where9

0

,G

0

, B

0

and I

0

denote the per-unit-

length (PUL) inductance, capacitance, resistance, andconductance

1 R 1

matrices at point 0 , respectively.

The1

-dimensional vector forcing functions5 E

0

) V

and6

E

0

) V

are derived as in [1].

3 FINITE DIFFERENCE QUADRATURE

METHODS

There are two major approaches to numerically solve

partial differential equations (PDEs): one is the numer-

ical finite differencing and the other is the numerical

integration (quadrature). Finite difference (FD) meth-

ods have been fully developed in the literature and are

widely used to numerically solve differential equationsbecause of its simplicity and applicability. Consider a

smooth function X

0

and its derivative function X Y

0

both defined on the domain a b ) 3 d , which is uniformly

divided by grid points e 0 ) f

3 g g h i , then the central

FD framework is given by

X

0

q r

t

X

0

u r

y

X

Y

0

(4)

where

0

q r

t

0

is the distance between two ad-

jacent grid points. This framework is a local approxi-

mation in that the finite difference is only represented

by the immediately neighboring grid points. Despite

its wide popularity and uses, it requires very dense gridpoints and therefore takes computationally prohibitive

time to solve large problems. One the other hand,

the numerical integration is more stable and converges

faster as it globally integrates the local information. The

approximation framework of Gaussion Quadrature can

be adapted as

X

t

X

"

X

Y

0

8

0

r

X

Y

0

) (5)

where

s are the coefficients which are determined by

the orthogonal polynomials in the particular Gaussian

1/N

V0 V1 VN-1 VN

I0 I1/2 I1+1/2 IN-1/2 IN

0 1

...

...

V2

Figure 2: FDQ framework.

rules. Compared to the FD framework in Eqn. 4, the

Gaussian Quadrature in Eqn. 5 is a global approxima-

tion in that the difference between X

and X

"

is rep-

resented by all the grid points over the entire domain.

We propose a new technique called the finite dif-

ference quadrature (FDQ) method to integrate the fi-

nite difference and quadrature methodsin one approach.

The general framework of FDQ is shown by

X

0

q r

t

X

0

r

0

X

Y

0

g (6)

where

s are the FDQ coefficients to be determined,

0

is the weight at point 0 . The RHS representa-

tion of the FDQ Eqn. 6 is apparently a global quadra-

ture approximation, which is the same as the RHS in

the Gaussian Quadrature Eqn. 5; however, the LHS of

Eqn. 6 is the finite difference, compared to the numeri-

cal integral over the entire domain in Eqn. 5. The FDQ

method is as simple and applicable as the finite differ-

ence method, and is as stable and efficient as the quadra-

ture method. It is expected to be able to solve all kinds

of PDEs with higher efficiency. In this paper, we use

the FDQ method to solve the Telegraphers equations

for transmission line modeling.

For simplicity, the FDQ modeling of TLs is first de-

veloped on a uniform two-wire TL with one of the wires

being the reference. In this case, the vectors and matri-

ces in Eqns. 2 and 3 become scalars:

0

)

,

0

)

,

,

,

and

. A direct numerical technique, FDQ

methods do not need to decouple the MTL, therefore its

application to non-uniform and/or multi-conductor TL

is straightforward extended. Assume that a TL has been

normalized to one unit of length along0

direction.

As shown in Fig. 2, the TL is equally discretized to

small sections. There are two sets of grid points: one

set are at integer-spatial positions 0

f ) f

b

g g ,

the other are at half-spatial positions 0 q r k

f #

3

y

) f

b

g g

t

3

. The grid points are num-

bered in such a way that the voltages are evaluated at

integer-spatial positions as

0

)

) f

b

g g

and the currents are evaluated at half-spatial positions

as

q r k

0

q r k

)

) f

b

g g

t

3

. In addition,

the currents o

b

)

and

3 )

are input and

output currents at the port, respectively. For clarity of

the intermediate use in the context, we define the inter-


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mediate voltages q r k

0

q r k

)

) f

b

g g

t

3

and the intermediate currents

0

)

) f

3 g g

t

3

.

For the finite difference of voltage, the FDQ approx-

imation framework is

q r

t

u r

o

"

8

8

0

q r k

) f

b

g g

t

3

(7)

For the finite difference of current, the FDQ approxi-

mation framework is

q r k

t

u r k

o

8

8

0

) f

3 g g

t

3

(8)

for internal grid points and

r k

t

o

o

o

8

8

0

)

t

u r k

o

8

8

0

(9)

for the left and right boundary, respectively.

Substituting Eqns. 2 and 3 into Eqns. 7-9, we obtain:

q r

t

u r

o

"

a

#

q r k

#

E

0

q r k

)

d

f

b

g g

t

3

) (10)

q r k

t

u r k

o

a

#

#

E

0

)

d )

f

3 g g

t

3

)

(11)

for the internal grid points and

r k

t

o

o

o

a

#

#

E

0

)

d (12)

t

u r k

o

a

#

#

E

0

)

d (13)

for the boundary points.

Eqns. 10-13 are the FDQ approximation framework

of the two-wire TL in

-domain, the coefficients of

which have yet to be determined.Following the concept of Weighting Residual

Method, a testing-function approach of the Galerkins

method is employed by using the following function set:

3 )

0

)

0

k

) g g g )

0

) g g g (14)

The function set constitute a linear vector space

of polynomials with respect to the operations of addi-

tion and multiplication. In order for Eqn. 7 to be exact

in the -dimensional subspace, every item in the base

0

) h

b

g g serves as a test function to fit Eqn. 7. As

a result, an R

matrix of coefficients is obtained as

a

"

d and, similarly, the

# 3

R

# 3

matrix

a

d .

The FDQ approximations in Eqns. 7 and 8-9 re-

spectively have approximation orders of

0

and

0

q r

, compared to the low approximation of FD

methods.

Once the positions of the grid points are given, the

above testing-function approach gives constant FDQ

coefficient matrices, no matter in what applications the

TL equations appear. For the equally spaced grid points,

the coefficients can be calculated and saved for future

use, which reduce the computational cost in practice.

With the pre-calculated coefficient matrices,

Eqns. 10-13 are rewritten as

5

6

!

!

5E

6

E

#

#

!

o

(15)

where

#

)

#

5

a o

rg g g

d

6

a

r k

r q r k

g g g

u r k

d

5 E

a E

0

r k

E

0

r q r k

g g g

E

0

u r k

d

6

E

a E

0o

E

0

r

g g g

E

0

d

is an R

# 3

matrix, and#

is an

# 3

R y

con-

necting matrix of the external exiting current sources:

%'

'

'(

t

3 3

t

3 3

. . .. . .

t

3 3

) 0

0

0

2

)

#

%'

'

'(

3

b

b b

......

b

t

3

) 0

0

0

2

Eqn. 15 can be transformed into and solved in the

time domain. In this paper we will not introduce the

time domain algorithm; instead, we derive the SPICE-

compatible equivalent model from the above equations

for circuit simulation.

4 NUMERICAL RESULTS

The heuristic for the order of FDQ modeling is to takey

voltage differences per wavelength, i.e., y

in

Eqn. 15 per wavelenght (see Fig. 2). It is well estab-

lished that for the finite difference method, a resolu-

tion of more than dozen points per wavelength (PPW) is

needed for required accuracy [4]. Therefore, the resolu-

tions of 4 PPW is a significant improvement, which ap-

proaches the Nyquist limit ofy

PPW. The intrinsic rea-

son for the improvement is the global quadrature with

the Weighting Residual method. The more global a

quadrature framework is, the closer its resolution could


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B

A

C2

D3

D4D2

D1

C1

0 2 4 6 8 10 120.5

0

0.5

1

1.5

2

Time(ns)

Voltage

(V)

Output w/o EM wave

D1 with EM waveD2 with EM waveD3 with EM waveD4 with EM wave

Figure 3: H clock tree with EM wave interference.

approach the Nyquist limit to achieve the required ac-

curacy. The globality of FDQ approximation frame-

work heuristically makes it one of the approaches which

closely approach the Nyquist limit.

The example is about an H clock tree in an MCM cir-

cuit of g 3 m technique to offer the clock signal for

four modules located at points r , k , and as

shown in Fig. 3. The H tree is etched on a silicon sub-

strate which is assumed to be the ideal ground. The cir-

cuit is illuminated by a plane EM wave. There are seven

TLs , r , k , r r r k , k and k ,each of them having the same length of

y

cm. The first

three TLs have the PUL parameters y

b b nH/m,

3

b pF/m,

3 3

b /m and

b S/m, and the

other TLs

b bnH/m,

3

y

bpF/m,

b b /m

and

bS/m. At points ,

r and

k , there are three

identical inverters used as buffers to drive the clock sig-

nal.

The applied input is periodic clock signal whose

rise/fall time is b

ps. The illuminating EM wave is

a trapezoidal pulse having rise/fall time of b

ps and

width ofy

ns. The phase of the EM wave is referred

by point at which the delay of EM wave is ns.

As we focus on the effect of the illuminating EM wave

on the integrated circuits, for simplicity we assume that

the EM wave is still the uniform wave in free space at

the top of the circuit. For more accurate modeling the

nonuniformity should be considered due to nonuniform

distribution of media and metals. Using the heuristic,

each of the TLs is modeled by using FDQ with y

(4

PPW) as in Eqn. 15, whose coefficients have been

pre-calculated.

Let the EM wave have the propagation configuration

as!

4 #,

! %

4 #, and

'

b #(refer to Fig. 1).

The responses are shown in Fig. 3.

Fig. 3 shows that, without EM wave interference, the

waveforms at the leaf points

r

t

are identical. With

the EM wave coupling in this configuration, the wave-

forms at

r and

are still identical, so are the wave-

forms at k and , but the two sets of clock wave-

forms are severely skewed, due to the effect of the illu-

minating EM waves.

5 CONCLUSIONS

An efficient numerical approximation technique, call

the Finite Difference Quadrature (FDQ) Method, is first

proposed to solve partial differential equations (PDEs),

and has been adapted to model transmission lines (TLs)

with external field coupling. A discrete modeling ap-

proach, the FDQ adapts grid points along the transmis-

sion lines to compute the finite difference between ad-

jacent grid points by estimating a weighted linear sum

of derivatives at a set of points belonging to the do-main. Similar to the Gaussian Quadrature method to

compute the numerical integrals, the FDQ method uses

global quadrature method to construct the approxima-

tion framework, however, to numerically compute fi-

nite differences. As a global approximation technique,

the FDQ needs much sparser grid points than the fi-

nite difference (FD) method does to achieve required

accuracy. Equivalent voltage and current sources are

derived, which excite the TLs at the grid points. Equiv-

alent circuit models are therefore derived to represent

the TLs illuminated by external electromagnetic waves.

A heuristic of FDQ modeling resolution of two differ-

ences per wavelength is given to determine the order in

the application of FDQ modeling. The FDQ method

supplies a general numerical technique that can solve

partial differential equations (PDEs) accurately and ef-

ficiently.

References

[1] C. R. Paul, Analysis of multiconductor transmission

lines. New York, NY: Wiley, 1994.

[2] I. Erdin, M. S. Nakhla, and R. Achar, Circuit anal-

ysis of electromagnetic radiation and field coupling

effects for networks with embedded full-wave mod-ules, IEEE Trans. Electromagnetic Compatibility,

vol. 42, no. 4, pp. 449460, 2000.

[3] M. Omid, Y. Kami, and M. Hayakawa, Field

coupling to nonuniform and uniform transmission

lines, IEEE Trans. Electromagnetic Compatibility,

vol. 39, no. 3, pp. 201211, 1997.

[4] M. N. O. Sadiku, Numerical techniques in electro-

magnetics. Boca Raton, FL: CRC, 2001.

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