A NEW CURVE FITTINGFREQUENCY-DEPENDENT
Yuichi TANJI,
Dept.

TECHNIQUE
FOR ANALYSIS
OF
LOSSY TRANSMISSION
LINES

Yoshifumiof Electricaland Electronic
2-1 Minami-Josanjima,
E-mail:tanji@ee.

andEngineering,
Tokushima
tokushima-u.

Analysis of frequency-dependent
lossy transmission
lines isvery importantfor designing
the high-speed
VLSI,
MCM
and PCB. The frequency-dependent
paramet ers are always
obtained
as tabulated
data. In this paper, a new curve fitting techniqueof the tabulated
data for the moment matching technique
in interconnect
analysis are presented.
This
method based on Chebyshev
interpolation
enhances the efficiency of the moment matching
technique.

INTRODUCTION

The high-speed
performanceof microwave
or digital
circuit systems is limitedby the interconnect
effects rather
than the switching
speed of semiconductor
devices. Whenthe operating
frequency
is increase, the current
density of
conductor
tends to be great around the surface of the conductor.
Due to the high packing density, the interconnects
such as VLSI,
MCMand PCB are closely placed on eachother, and the current density is also great at the near side
betweenconductors.
They are known as the skin effectand proximity
effect [1], respectively,
thus the interconnects
of high-speed
integrated
circuits have frequency-dependent
characteristics.
The frequency-dependent
parameters are always obtainedby any numerical
procedure and as tabulated
data in real frequency.
Therefore,
the analysis of frequencydependent
lossy transmission
lines with tabulated
data isvery important
for accurate analysis VLSI’s circuits,
MCM
and PCB.
For such analysis,
FFT based algorithm
is very accurate.
However,
this method
is not useful from computational
point
of view, because the system to be analyzed
contain very large number of transmissionslines, and FFT
based algorithm
needs large numberof data points.
The
moment mat thing technique[3], [4] are efficientand accurate for the interconnect
analysis.
Recently,these methods
are extended to the frequency-dependent
case [5], [6]. Since
the moment
matching
techniques
are essentiallyPad4 approximationof any Laplace functions,
if any transfer function are described in power series of complexs,
the moment
matching
technique can be applied to the analysis.
Thus thekey techniquein reference [5], [6] is how the tabulated
data
in real frequency
is described in power series of complex s,and the piecewise polynomial
approximationin [6] and theleast square approximationin [5] are used.
In this paper, we provide
a new curve fitting
techniquefor the moment
matching
scheme in interconnect
analysis.
The proposed method is based on Chebyshev
interpolation

Akio

USHIDA

Tokushima
77o, JAPANac.jp

technique.

ABSTRACT1.

NISHIO

University

Chebyshev

polynomial

is considered

as an al-most minimax
approximate
polynomial.
Hence the proposed method
based on Chebyshev
interpolation
gives agood approximation
than one by means of the least square
fitting[5]. Moreover,since the proposecl method
does notrequire for any matrix operation,
this method does not suffer from the singularity
problemin the least square fitting[7]. The polynomialmust be constructed
as having realcoefficients
due to realistic
impedanceor admittancefunctions. The discrete orthogonal
propertyof Chebyshev polynomial
allows us to construct
the continuous
polynomialwith real coefficients,
different
from the piecewise
one in[6].
In the numericalexamples,reliable results to the tabulated2.

the proposed
methodgives
data in real frequency.

FREQUENCY-DEPENDENTTRANSMISSION

LOSSY
LINES

The frequency-dependenttransmission
knes are describedby the Telegrapher’sequationin the Laplace-dornain:
d

&

[1

v(s, x)1(s, X)=

V(s, z)1(s, ~)

D(s)

(1)

[1

where
D(s)

=
[

z(s)

=

o

–z(s)

–Y(s)

o

R(s)+

sL(s),1y(s)= G(s)+sC(S).

The parameters
R(s), L(s), C(s), G(s) are per unit length
resistance,
inductance,
capacitance,
conductancematrices!
respectively,
and these matrices
are arbitrary
functionsofcomplex s. Actually,
these matrices are not given as a funtion of complex s, but also tabulated
data to some points,
jw.’s on the imaginaryaxis.
In this paper, our aim is how to apply the moment matching scheme [3], [4] for solving (1). If any transfer functions
are describedin power series of complexs, we can apply
the moment
matching
scheme to the analysis, because themomentmatching
scheme is essentiallyPad6 approximation.
Hence, the input (x = O)-output (z = 1) relationsoftransmission
lines are described in power series of complex
s is the key technique.
Assumingthat the parameter
matrices R(s), L(s), C(s), G(s) are power series of complexs, the matrix
exponential
method[5] [6] is very powerful
techniqueto describe the input-output
relation.
Here webriefly
modifythe matrix
exponential
methodto increase
the efficiency.

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

Applying
put relation[1

the matrix
exponentialof Eq. (1) is given by

V(s, 1)1(s, 1)=

method,the inputout-

V(s, o)1(s, O)

exp(F(s)Z)~+(F(s)O”
?ESo

~:’;)
[1

.

(2)
~T”sn
n=o

[ 7$?

=

exp(F(s)~) . exp(F(s)

~).

(3)

However, dividing
the transmission
lines requires for morecomputationalcost, because this means that some equations are added to the circuit
equation.
Alternatively,exp(–J?(s)~) ismultiplied
from left side of Eq. (2) insteadof dividing,

and we can get the following

relations:(4)
The relation(4) represents
continuityof the voltages
and currents
at the center point of the transmission
lines,
whereas the relation(2) gives a relationof the output
variables to the input.
This means that the relation(4) is more
effective than (2), because the complexity
depends on thelength of the transmission
lines.
Assuming
F(s) as M degree matrix
polynomialof complex s, the coefficientsof the matrix
exponential(2) is obtained in recursive mannar[5], [6]:
‘

.

E

T.

(n= O)

TO,j

j=–1

=

(5)

cm

x

T~,j

(n # O)

j=int(~)

CHEBYSHEVINTERPOLATIONOF FREQUENCY-DEPIENDENTPARAMETERS

In the previous

FkTi_k,j–l

~m’Y)
j+l

&=o

(i=o,...,

j#o)

jM,

section,the matrix

SCHEME

exponential

methodisappliedtothe momentgenerationof lossy transmission

lineswith frequency-dependent
parameters.
Here, it is assumed
that the parameters
given as tabulated
clata to some pointon imaginary
axis are able to write in power series of complex s. So, the procedure
for making the power series fromthe tabulated
data is provided
in this section.
3.1.

Curve

Fitting

Algorithm

Let r(s), l(s), c(s), g(s) be (i,j) element of R(s), L(s), C(s),G(s), respectively,
where these values are given as tabulated
data to some points, jui’son the imaginary
axis. The (i,j)
element z(s) = r(s) + jsl (s) of the series impedance
matrixand y(s) = g(s) + jsc(s) of the parallel
admittancematrixare determinedso that they satisfies
N

N

J2=o

k=o

(i

=

0,1,...,

N)

where N is the numberof the data.
Moreoverthe coefficients zk and yk are assumed as real numbers,
which is areasonable assumption
due to realktic
impedanceor admittance functions.
Let us consider z(s) only, and y(s) can be obtainedbythe same procedure.
In reference [5], eliminating
the losslesspart of l(s) is introduced
in order to approximate
accurately,
namely, the lossless part l(m) is separated from l(jw) such
as
=

l(jw)

– t(cm).

(8)

Then, z’ (jw,)= ~(jw;) + jw~l’ (jw;) is interpolatedby theChebyshev series. A transformz = w/w,. is used to convert
w G [0, Wm] into z c [0, 1]. Assumingz’(—jw)
is complex
conjugate to z’ (jw), the interpolated
polynomial
is obtained
by

.,
=

3.

l’(ju)

where

Ti, j

(6)

] “

In reference [3], the convergenceof infinite
series (2) is illustrated
gratelydependingon the length
1 of transmission lines. If the convergenceis smooth,the transmission
lines must be divided
in some regin.
(This implies
that
exp(F(s)ll)
rapidly
converges than exp(F(s)lz),
if 11 < lz.)
In this case, the following
relation
is very useful in order to
get the whole characteristicsof transmission
lines:
exp(F(s)l)

can be calculated
by multiplying
Ti,~ by
Althoughthe matrix
exponential(2) congerves
after 40-50 terms,
exp(F (s) 1/2) rapidly
converges
thanexp(F(s)l),because the length of transmission
lines is half.
Interchanging
the elements of (4), we can get the ports
relationof the transmission
lines as follows:[1=

exp(-F(s)l)(-l)J+l.

N–1

Ti, j

=

FkT&_k,j–1‘5
j+l
lc=i-j(i=jikf+l,
Ti, O

=

Fil

Ti, O

=

I.

(i=

z’ (jwmz)

=

M

~’

akTk(x)(9)k=O. . ..(j+l)M.
O,...,

M)

j#o)where the symbol~’
denotes the summation
with theFromfirst component
dividedby 2 and Tk (z) = cos M.
the discrete orthogonal
propertyof Chebyshev
polynomial,, N – 1) are given as follows:ak(k=o,l...

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

if N is odd,

= +~}
1

Table 1. Coefficients
met hod.

T(O)COS(k: e“en)

N–3

.27
~~ ~

Zwm

COS 6’.Z(jWm

COS 0,)

series given

value

;=0

ak

of power

COS M’;

co

3.448

c1

4.670

(k: odd)c?

-1.684

X 10-2c~–1.601

C3

5.196 X 10-4

Cg

3.295

–3.196

,

X

10-7

c]--o

I

1

–2.917a

X
X

10-8

10–10

X 10–11
10-14

X

X 10–15

4.447 x 10-7

C5

if N is even,–3.292

C2

C4(lo.a)

C6

1.956

i=o

by the proposed~2r(~wmcos8t).os~ei
where(k: even)

+

$=0

ak

(-1)$

:1

=

.2-j ~ ~2W7?lCOSOi l(jW7n.zL
COS Oi)COS k70i

ico(10.b)
where 0, = =7.
(i = O, I ,. ... IV – 1) are the Chebyshev
points.
Note that if N is even number,
the informationats = O does not reflect the fitting
curve. Hence, N is preferto be odd number.
The coefficientsof a~ are real part or imaginary
part only,
thus we can derive the power series of jw~z
having realcoefficients.
First, z’ (jw~z)of (9) is converted into a power
series with respect to z. Using the recurrence
formulaof
Chebyshev
polynomial,
= 1,

To(z)
~k+l

{

Chebyshev

(z)

=

Tl(z)
%zTk

polynomial

(~)

=

1,

–

Tk-1

Th (i = 2,3,.

T2(x)

=

2X2 – 1,T,(x)=4X3 – 3X,..) is obtained

Tq(z)

=

8X4 – 8Z2 + 1,

T5(z)

=

1635-

T6(z)

=

32z6 – 48x4 + 18ZZ – 1,

20%3+

(11)

(z),

(12)

into

x

=

From

(12),

Tz~ (z)tions,
tivelyrespectively.
Thus,real and imaginary

bz~

part

(z)

are evenand oddfunc-

and b2~+1 in (13) are respeconly as az~ and az~+lin

is expressedin power

series of

N–1Z’(jwmz)

=

~
k=o

=

’20 +

S(21

+

1(00))

+

~

(15)

Zjsk.

kz2

where

all coefficients

3.2.

Shiftedof Sk are real numbers.

CoefRcientsof Power

SeriesWhen the multi-point
Pad& approximation[3], [4] is used
to getting
dominantpoles, the shifted moments,
the coefficients of Taylor expansion at an arbitrary
point sk is needed.
Thus, in section II F(s) in (2) and the matrix
exponential
muSt be a matrix
polynomialof complex o = s — Sk.

NUMERICAL

EXAMPLES

To show the efficiencyof our method(, the 3-conductors
transmission
lines provided by M. Celik alre considered.
Thefrequency-dependent
parameters
are listed as the tables II,
III in reference [5]. Using the proposed method in Sect. 3,(2,2) element of the series impedance
matrix10-degree power series of complex s:

is given by the

=

c~ s’
x

(13)

bhxk.

k=o

(9). Consequently,z’(jw~z)jw~zwith real coefficients:

N-1
z(s)

222 (s)and TZ~+l4(@JAz)k,matrix10

N–1Z’(jwmz)

From (8), (14), a element of the series impedanceof transmission“fines is described by

4.

series (9) is converted(k: odd)as (2), then F(s)Let be F(s) M degree matrix polynomial
is convered into a matrix poiynomail
of complex u = s — .$k:

by

5X,

As a result, the finite Chebyshev
a power series with x:(k: even)(-1)**{(k: odd)

[

$

=

(14)

(17)

i=o
where each coefficient
is listed in Table 1.
For comparison,
the values of 222 in s = ju and the tabulat ed data are shown in Fig. 1. In thk figure, the frequencies range is from O to 7 GHz. The propcwed method gives areliable result.
The time- and frequency-domtin
responses
in the example [5] can be calculatedby the matrix exponential method in Sect. 2 and complex frequency
hopping[3].
The circuit includes two identical
3-conductortransmission
lines, 4 resisters,
7 capacitorsand 1 inductors.
responses to a pulse input (0.8 [ns] pulse width,

0-7803-4455-3/98/$10.00 (c) 1998 IEEE

Transient
0.1 [ns] rise

[volt]
1.500

200 -

1.200

160 -

0.900

160 -

Series Impedance

z22

140 -

0.600

120 -

0.300

100 -

0.000

60

-

-0.30060--0.600

40

-

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

[nsse]

20

Figure

0

h

1

1

1

1

I

1

3.00

4.00

5.00

6.00

7.00

8.00

9.00

2. TransientIelement of the series impedancelines provided
byM.
Celik [3].

matrixofand fall time) and frequency
response to a impulse
input
are shown in Fig. 1, 2, respectively.
These result are compared with the result by the frequency-domain
method[8]
and single Pad4 approximation[2]. Here, in complex frequency hopping,the maximum
frequency
is selected by 5[GHz] and 9 expansion
points is considered.

—

il
‘1

I

............... FFTI............... AWE

1.5

1.0

0.5

0.0I
o

CONCLUSIONS

A new curve fitting
techniquefor analysis
of frequencydependent
lossy transmission
lines have been presented.
This method
is efficiently
incorporated
with the moment
matching
technique[3], [4]. Although
the objectof thispaper is turnedto the moment
mathingtechnique,thismethodis easyly applied to the methodof characteristics
by means of a techniquein reference [9]. This is our future
work.
REFERENCES“ A review of the skin effect[1] L. T. Hwang and I. Turlik,
as applied
to t hh film interconnects,“ IEEE
Trans.
Manuj.
Technol.,
vol. 15, vol. 15, no. 1,
Camp. Hybrids
pp. 43-54, Feb. 1992.
[2] L. T. Pillage
and R. A. Rohrer,“Asymptotic
waveIEEE
Trans.
form evaluation
for timing
analysis,”
Computer-Aided
Design, vol. 9, no. 4, April 1990.
[3] E. Chiproutand M. S. Nakhla,“Analysisof intercon(cfh),”
nect networks using complex frequency hopping
IEEE
Trans. Computer-Aided
Design, vol. 14, no. 2,
Feb. 1995.
“Im[4] M. Celik, O. Ocah, M. A. Tan, and A. Atalar,proving
awe accuracy using multipoint
padb, approxiProc.
ISCAS’94, vol. 1, pp. 379-382,
1994.
mation”

CFH~%.
..........._

1

5.

to a pulse input.

2“5 ~

2.0

Figure 1. (2,2)
thetransmission

response1

2

L% ...-...
\

3

4

5
Frequency

Figure3. Frequency

response

to a impulse-------.........
6

7

[GHz]

input.

[5] M. Celik and A. C. Cangellaris“Efficienttransientsimulation of lossy packaging interconnects
using momentIEEE
Trans.
Comps., Pack., @matching
techniques,”
Manuf.
Technol.,Part-B,
vol. 19, no. 1, pp. 64-73, Feb.
1996.
J. Poltz, M. Nakhla,
Q. J. Zhang, “A fast
[6] R. Khazaka,
methodfor the simulationof lossy interconnects
with
Proc.
IEEE
Multifrequency
dependent
parameters,”
Chip ModuleCont., pp. 95-98, Feb. 1996.
Computation,[7] G. H. Golub and C. F. Van Loan, Matrix
the Johns Hopkins
University
Press, 1983.
“Analysisof pulse[8] Y. Tanji, L. Jiang and A. Ushida,responses of multi-conductortransmission
lines by a
IEICE
Trans. on Fundamenpartitioning
technique,”tals, vol. E77-A,
No. 12, pp. 2017-2!027, Dec. 1994.
A. Kamo and H. Asai, ‘{Time-domain[9] T. Watanabe,simulationof lossy coupledtransmission
lines based
Proc.
Euro. Conf. on
on delay evaluation
technique,”
Theoryand Design, vol. 2, pp. 517-520, Sept.
Circuits
1997.

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