2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Analysis of Voltage Distribution of Interconnects
Hideo Sakaguchi , Yoshifumi Nishio , Yoshihiro Yamagami , Atsumi Hattori , Akio UshidaTokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Phone: +81-88-656-7470
FAX: +81-88-656-7471
Email:  sakaguti, nishio, yamagami, hattori  @ee.tokushima-u.ac.jp

1. Introduction

Tokushima Bunri University
1314-1 Shido, Sanuki, Kagawa, 769-2193, Japan
Phone: +81-87-894-5111
FAX: +81-87-894-4201
Email: ushida@fe.bunri-u.ac.jp

The Telegraph equation is described by

The analysis and design of high speed LSI chips are becoming
more and more important, because PCBs connecting LSI chips may
cause the signal delays and crosstalks, which sometimes happen tofaulty switching operations. In the last decade, many papers havebeen published on the transient analysis of lossy interconnects [1][3],[7]. The recursive convolution methods using moment-matching
technique [2],[3] can be efficiently applied to the lossy interconnects
terminated by nonlinear elements. However, one of serious problems in the moment-matching method is that the poles calculatedby the Maclaurin expansion and Pad  approximation may happen tocause the large errors when the poles are located far from the origin.
To overcome the problem, Nakhla et al. propose CFH (complex frequency hopping) [4] which can calculate the exact poles by properlychanging the origins of the Taylor expansions on imaginary axis.
On the other hand, an equivalent circuit technique is shown in [5],where the interconnects are replaced by the discrete  -type and/or
T-type models. It can be efficiently applied to the relatively shortinterconnect.
In this paper, we propose an asymptotic equivalent circuit technique based on a complex frequency domain. Firstly, we calculate
the exact poles and the residues of the admittance matrix, and after then , the admittance matrix is described by the partial fractions. Secondly, the equivalent circuit is synthesized with the essential poles and residues of the partial fractions. Lastly, replacing
the interconnect by the equivalent circuit, we calculate the transient
response with Spice. In section 2, we show how to calculate the exact poles and residues. Our asymptotic equivalent circuit shown insection 3.
Although most papers have only discussed the transient responses
at the near and far ends of interconnects, knowing the voltage distribution on the interconnect is also important for designing ICs because it may give the large crosstalks to the other equipments. In
this paper, two methods are shown for calculating the distribution
which are based on the numerical Laplace transformation and theasymptotic equivalent circuit. We show the algorithms in section 4,
and some interesting illustrative examples in section 5.

2. Calculation of the poles and the residues of interconnect
Now, consider a uniformly coupled

conductors’ interconnect.


!"#$$#$$&'(!)*
%
Let us introduce the transfer matrix .0/
eigenvalues as follows:

2436587:9 ;=<>11?A@

1

.JI



EF!"BDC[6], and calculate the1BDC1EF$G,&'!)* . /We further introduce the transfer matrix .JI

1

,+-

. /

1>

1K"11LK"1

. /

1

,H-

:

2436587M9 ;=<1N@O,P-

Then, the input and output relation is described by the admittancematrix as follows:#$NQ:
#M$C,C M!
R ?8C M!
R

C? M$
R ?,? M$NQ:
M!

R

NST

where

C,C M$ ?,? M!>  . I 1N243517:9 U$V-W,XY; < 1Z@ .0/ 1 BDC
1N2=36587M9 [,3]\=XY;=<$1Z@ B^C 1> B^C
R C? M!>
R8? C M$
% .JI
. /
,_-
R
R

we have the following theorem for calculations of the poles [7].
Theorem 1: The locations of poles satisfying relations (5) are foundby solving the following equation:`(!",&'($)a^

b

? #


QT

Q:!+-$HZ$c$cc

b 

,d-

On the other hand, the poles for the zero eigenvalue are obtained by
the following relation:

+

e]3gf
1N243517T9 [,3]\=Xp;:q]N[rN2=@ BDC 1ms. I
h!i,jgkml nao . I`($G BDC ,t-vw!

Q:cu Thus, the poles for zero eigenvalues
are calculated by uQThus, all the poles for x 
are located on negative real axis.
Since
the relations (6) and (7) are given by the algebraic equation of, the poles are numerically calculated by the use of the algorithm
such as Bairstow. Now, let us calculate the residues of admittancematrices (4).
M$
M$
Theorem 2: The residues of C?
and ?8C
for the pole

- 439 -

RR

y z are given by{

C?8| z 

8? C,| z}1}8}~}1}1}+- z BDC
§ o 1ÉÆÊ-Ë 11Ì8H¿Ç z| C?  ®¿

Ç o | C?

{Ç o | ?,?

Ç o | C,C ¶Ç z | ?,?

Ç z | C,C ¿Ç o | ?8Ckm $ib +Z$H-$c$c$c,+1S:cH-Ç z | C?

Ç o | C? ¿Ç z | ?8C





where

}8}1}   / g- B^Cz ]Z
1 ] z ]Z]41°F(>²­(m®
Ë 1>
®
²km=i ÉÆÊ-Ë 11Ì,+ª_-


H

­
(

,

®
H
°F(>²
`

Í

]




M!
M$
Corollary: The residue{ of C,C {and C,C
are given by
ÍR
C,C,| R z  ?,?8| z}1}1} ~
}8}8}
Since the poles and residues are composed of the complex conju% I$]ZZ

 }1}1}

% I g-]Z

 }1}8} z ]Z}}1}  / g- B^C

gates, the admittance matrices can be described as follows:
km=$i

Observe that the residues given by (8) and (9) are same except
for the sings, because the denominator of (8) has the coefficient
U$V-[,X;1:
U$V-[,X;1>: km=$iuon+Z$HZtaking
!$$ “1” or +Z“-1”$HZ!$depended
$.
x: b x Using these poles and the residues, the elements of admittancematrix (4) are described by the partial fractions as follows:

<
z<
z<
z<
 C,C,| z < g-   ?,?8| z < ]Z  o | z
 C,|
 ?8|
£ |
}1}}
"¡ oJ¢ " ¡ C ¢ G ¡ ? ¢ G a £ ¢<<z<z <   4¤   C?8| z < g-   ?8C,| z < ]Z  o | z   ,C | z
 8? |   £ |}1}}"¡ o"¡ C ¢ G¡ ?
Ga
 £ ¢
  4¤ ¥Note that since the terms corresponding to poles far from the origin in s-plane don’t give large effect in the transient analysis, thelossy interconnect can be well-modeled with relatively few terms.
In this section, we consider the modeling the admittance matrix (4)described by the partial fractions (10). For simplicity, we consider an
uncoupled uniform interconnect. The admittance equation is given
by ¦



+
§ o 1¨

where The poles b¬ «QT

U$V-W,XY;1
9 [,3]\=XY;1Z@ BDC©

9 [,3]\=XY;1Z@ BDC
 !U V-W,XY;1

NQ:$
© M$,+-+ª

satisfy the following relation:

­Fm®¯,1°±>²³




 b´

M

y o ¶µ o  y z ¶µ zT· ¸ z ­ ®
´ , and
where µ o +Z!H-c$c$cb 

Thus, we have

¹ z »º^¼ *¢ ½¿¾ Á z ¥ º ¾ÀÎ C? M$ Î ?8C M$?,Ð H ¨Ç oÏ  ¿y oz C,+ªdZcH-

b 

Ç

Ç

Ç

Each term of (17) is realized by the circuit by RLC circuit shown by
Fig.1 whose parameters are given as follows:Ùzz Ùzz  Ùz Ú C Ù C ?  o
z
z
½
o º
C À ½
Ù z? Ù C z oÀ Ù z£
C
C
z 
z
Ùz
zÙ ?
z Ù z Ù z ¼  o z Ù z? C  Ù z o  z C Ù z C  Ù z o¾
Ú o zC ¢  o  Ú C C oÚÚ¢  
 Ù
º o
o

o

o Ú Ùthat the admittance ofÎ CObserve? 11 Î ?8C 1,Î C,C 11 Î ?,? 1,

andare composed of the same terms except of
1
Î 
,
their signs. Now, put the admittance having the same sign to C
Î ? 1
and that having the opposite sign to
as follows:ÛÎ C 1


,+-PªzÄÃ ?
?
Â º ¾  ½ ¼ ¢  Å  4Æ ºD¼ ¢¡½¿¾ 
¥ º ¾

Ç

z C ¿ z o+Z!H-c$c$c$!H8Ó ? ÖØ× z C ÈÖ Æ× z o b ,+ªt-o  Î z 1
Ø
Ö × o



,+-Hª

+-!HZc!c$c+- z BDC ÑJÊªÇ z Ìª¶ÑJÊ-Ç zNy=z Ìª1¿ z  ?  ¸ z ?
µz | C,C 
o | C,C Ò o | C? , z For simplicity, we set here o 
Ç z | C?
H8ÓÔÕ+means the order of approximation. Set eachuu , andterm of (16) as follows:Î o 1

3. Asymptotic equivalent circuit

,+ªdZc+-ÇThus, the input-output relations described by (4) is realized with theequivalent asymptotic circuit as shown in the next section.

¦ NQ:$M!Î C,C M! Î ?,? M$?,Ð HÑJÊªÇ z Ìª¿FHÑJÊ-Ç zÄy z Ì
Ç o

Ï ¿
1È z  ?  ¸ z ?
y o
µ
z C

Î ? 1


HÑJÊªÇ ? z B^C Ìª¿FHÑJÊ-Ç ? z BDC y ? z BDC Ì
1È ? z BDC  ?  ¸ ? z
µ
? BDC
z C
Ç o

È y oÛHÑJÊ-Ç z Ì-¶±H8ÑJÊ-Ç zNy z Ì
?
? ?
1È
  ¸ ?z
µ ?z ?
z C?

,+ªÜZc+-,+ªÜZcH-Then, the corresponding residues are given byÇ o | C,CH
ÆÇ z | C,C
 ®È


+
§ o 1>ÉÆÊªË 11Ì8k $ib +Z$H-$c$c$c,+8STc+-

as¦Then, we can realize the interconnect by the equivalent circuit¦
shown by Fig.1. Note that the current controlled current sourcesÝ M
Ý NQ:$in the figure shows the reflection at the far end, andat the near end, respectively.

- 440 -

where we< ê applied
the trapezoid numerical integration formula at
Ã
 ×¯

ò ù
´ for the interval

.
The
step
size is chosen by 
9 Q:ò ù @ s. To carry out the algorithm, we first calculate the transient
responses with Spice given in section 3. Using these data, the transient waveform of the voltage distribution is calculated by Fortranand/or C-program.
4.2 Asymptotic equivalent circuit technique
The next one uses the asymptotic equivalent circuit satisfying therelation (20). To understand the method, we consider the case of
single line. The voltage at “ ” from the near end is given by

Y2
I (0,s)I (d,s)Y1

V (0,s)Y1
I (d,s)r

Near end

V (d,s)I r (0,s)

Far end

(a)

Y2

Y1

R2

R 2N

R1

R3

L2

L 2N

L1

L3

R0

R 2N-1
L 2N-1

L0
C2

G2

C 2N

G 2N

C1

G1 C 3

G3

Y1

Y1
R3

R 2N-1

R 2N-1R3

L1

L3

L 2N-1

L 2N-1L3

G1 C 3G3

C 2N-1

G2N-1

G2N-1

2N-1 Cë ]ì Z  ë  -    :]ZA]³aì: ë g Z    :]Z]ì   :
¢
]

Z

]


À
À
À
+
­ ¥® 
y
b

´ ,
The poles 
are the same as those of (13), and o C  
y o ?  ° ´ ² for b  Q .
However, the corresponding residues for y o C and y o ? are zeros,

I (d,s)
r

R1

C1

G2N-1

(c)

(b)
I r (0,s)V(0,s)

C 2N-1

R1
L1

G3

3C

V(d,s)

G1

C1

because (23) is nonsingular at the poles. On the other hand, the¸ zresidues corresponding to y z ±µ z ·are given by

(e)

(d)

Fig.1: Equivalent circuit of an interconnect.

Ç z
4. Calculations of the voltage distribution on interconnect
It is important to know the voltage distribution on the interconnect, because the potential may give large crosstalk to the other lines
and transistors located close to the lines. These values are estimatedby using the voltage waveforms at the near and far ends of the interconnect as follows:Þ$

1N2=36587[,3g\4Xß; z 1>,

[,3]\=XY; z 1

1N2=36517[,3]\=Xß; z 1
[,3]\4Xß; z 1

 . /
. /

. /

. /

B^C 1>ÞNQ:>

B^C 1>ÞM$,H8Q 

Á gàmáT â-ãç äæå
,Hò

è

ê To

ë ]ì

Ú ¢

í

çZî  â1À
Q:$+ZHZc$c$c!

<?ê

á¯ð ï

  ë g ì Ú ¥À+]¥  ´1ó
ó  . The formula comesfor ñ á  xfrom the inverse Fourier transformation [8], so that it gives the goodQõôô'ò
results in the region. It is known from many numer× ñ
×öò
P±÷ø_
ical experiments that “ ” should be chosen aroundHrù¸ NQ: ¸ M and ó . In this case, the waveforms
ñ andñ at thenear and far ends are not given by the analytical forms, because theinterconnects are usually terminated by the nonlinear sub-circuits.
Therefore, we also need to apply the numerical Laplace transformaNQ:!>
M$
tion to get ©
and ©
as follows:
 Ç
 
ò


û

B
 ï
¸ NQ: 
ã!þGÿ äñ ño üý÷  ¸ NQ:6Q ^ ¸ NQ:6ò  B
 ï s
ùãþ"ÿ

ü
Û BDC

ê
 ï 

¸ NQT$Ç 4 BHãþ"ÿ
ê C©

ú×¡,HªH-

NQT$Ye]3]f 1 y z kn  i ©

NQ:

[,3]\=XY;1,[,3]\=Xß;1

[,3g\4Xß;1, hZjgkml ÈU$V-[,XY;1 kkm $i,H1ST

The residues for the first term of (23) is given by Ç zC

H




?

b ©
S-®³²

NQ:$[,3g\­*°±jÅ B  l  ¨
+- z
b Å

zÄÃ ?
®¯°(­*²³ ?
Å


,Hª_Zc+-The residues for the second term of (23) has the same form as follow:

In this section, we show two methods depending on the numerical
Laplace transformation and the asymptotic equivalent circuit.
4.1 Numerical Laplace transformation method
The transient responses can be obtained by the use of the inverse
Laplace! transformations [8]. The formula for the Laplace function
is given by
©

é BDC

 ©

 Ç z?
M$>[,3]\ j Å l  ¨
+- z
b ©
b

®¯°(­*²³ ?
S-®³² *
­ °± zÄÅ Ã ?

H


?

,Hª_ZcH-

Thus, the first term of (23) is described by the asymptotic partial
fractions as follows:Ðë  - ë C gì Z  ¥4¤  ?
?
zzz  C  C ¢ 6D¥ ¹  zÀ C ¢  ¹ z ¢ Á z  z CÀH-Ç z C
where ó z C  . In the same way, the second term of (23) isgiven byÐë ] -
ë ? gì Z  
?
?
z
z
¹
z  C  ? ¢ 6D¥  zÀ ? ¢  ¹ z ¢ Á z   z ?
ÀH-Ç z ?
where ó z ? ¥4¤ ¥

Thus, the asymptotic equivalent circuit realizing (24) is shown by®
+­
Fig.2, where the parameters are given by z ê  u ´1ó z ê u , z ê  u 
H z
Ç ² zê
 ¸ z?  Ê+Z$cc$c!6Ó`Ç+-$HZÌ?
zzêµ ´1ób u , uó ´ µu , NQb :  ¸ NQT    M . The
ñ 
ñ and
ñ 
M  controlled-current sources¸voltagenear and far
ñ are obtained by the transient responses at j theÅÇÇ[,3]\BÅ  l  ¨
+- z
b

ends in Fig.2. Since C z and z ? have term,
the signs depend on “b ” and “ ”. The parameters denoted by “”’in
the figure correspond to the terms with the negative signs. Thus, thetransient distributed voltage waveform can be given by the voltage+-Y
difference atresistor. Since the netlists will be automaticallyobtained by Spice, the method will be also useful.

- 441 -

i11(t)v1 (t)

i21(t)
R21
i2 (t)
L22 L21(b) Transient response of the fulladder without interconnects.

R 12

R1M

R 2M

L11

L 12

L1M

L 2M

C 11

C 12

C1M

C 2M

+
C 22 C 21
i22(t) v2 (t)

’
R11

R’12

’
R1M

R’2M

’
R22

’
R21

-

L’11
C ’11

L’12
C ’12

L’1M
C’

L’2M
C ’2M

L’22

L’21

v2 (t)+
v1 (t)
1(F) -

R22

R11

i12 (t)v1 (t)
j(0,t)=v(0,t)

1Mv2 (t)[v]
3.0
VS’ C 21
’
C 22

2.0
1.0

1(F)

0.0
[v]
3.0

j(d,t)=v(d,t)

VC

2.0
1.0

+
v(x,t)
-

1(Ω)

0.0
0

i11(t) i (t) i (t) i (t)
12
21
22

Fig.2: Asymptotic equivalent circuit for getting Á ]ì

À

à1 .

¼

5. Illustrative examples
5.1. Interconnect terminated by linear resistors
To investigate the accuracy of our asymptotic method, consider an+8Q 9D@at near and far ends, respectively.
interconnect terminated by
5.0

30

40

50

[n sec] 60 4¤ $ À *!+!&#  À¾$ À  !"!"#

ÀIn this example, we replaced the interconnects by the asymptotic
equivalent circuits, and get the results with transient analysis of
Spice. We found from these results that it has large time delay. Our
simulator is easily applied to any kind of circuits.

Near end
2.5

2.5

Far endFar end0.0

0.0

T=1.2ns

T=1.2ns
-2.5
2T

4T

6T

8T

10T

-2.5

0

2T

4T

6T

8T

10T

(a) Solution by the numerical inverse Laplace transformation.
(b) Solution by our asymptotic method with our 15th order.
Fig. 3: Comparison our asymptotic method to the numerical inverse
Laplace transformation.

º20

6. Conclusions and remarks

5.0

Near end010(b) Transient response of the fulladder coupled with interconnects with 37thorder.
 4¤  ')!"!"#
Fig.4: ,  $  % !"!&# º     - !"!"# 4¤ $ !"!&#
 (')!"!"#
   !"!"# ½
%Â  4À ¤ $ !"* !"!"#   À $ ¾ !+!&#
À
¼


À

In this paper, we have proposed an asymptotic equivalent circuit
for interconnects, and two methods for getting the response of distribution voltage on the interconnect. At first step, we calculate theexact poles and the residues of admittance matrix, and describe itin the form of partial fractions. Secondly, we realize the admittancematrix by the asymptotic equivalent circuit. We found that we canget good result even with our low order approximation. Although
we have not yet shown the example for the voltage distribution inthis paper, we want show some examples in the conference.
As the future problem, we need to extend our algorithm to multicoupled interconnects.

From these results, we found that our method can get the good results with 15th order of the approximation. Furthermore, the timedelay is exactly estimated even with the low order model.
5.2. A LSI circuit coupled with interconnects
Consider a fulladder circuit coupled with interconnects.
Half Adder 1

Half Adder 2

T3
v1
vs
v2

T2
vc

v3

T4

T1

(a) Fulladder circuit consisted of nand circuits.
[v]
3.0
VS

2.0
1.0
0.0
[v]
3.0
VC

References[1] M.Celik, L.Pileggi and A.Odabasioglu,it IC Interconnect Analysis,
Kluer Academic Pub., 2002.
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1994.
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