Analysis of Two Dimensional Circuits Based onMulti-Conductor
TheoremAkio USHIDA, Yuichi TANJI and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering,
Tokushima University, Tokushima, 770 JAPAN
Tel.: +81-886-56-7469, Fax.: +81-886-56-7471
E-mail: ushidaQee.tokushima-u. ac.jp
ABSTRACT

We discuss an eflicierd algorithm for solving 2-dimensional
circuits based on a multi-conductor theorem. ‘I’here havebeen may papers published about transmission lines, because they are very important for design of high speed
VLSI chips. On the other hand, a device simulation is also
very important to design ICS snd to understand the qualitative behavior. Most of the simulation techniques arebased on the finite-difference time-domsin method, where
the devices are described by many discrete models. This isreally time-consuming because, to get exact solution, thedevice must be divided into many sections. In this paper,
we show an elegant algorithm for solving 2-dimensional
circuits, which can be applied to the device simulations,I.

INTRODUCTION

Most of device simulations are based on the jinite elementtechniques. Recently, new methods have been proposed foranalyzing 2-dimensional circuits, where the linearizationof the discretized semiconductor equations [1-2] permits
the efficient solution for either transient or steady-state.
For three dimensional model, the devices are divided intomuch more pieces so that we need to solve large scaleequations and matrices. The reference [3] has proposed anefficient technique using parallel computation technique.
As one of the applications, it can be used to estimate theresistance between two points on the plate, where two efficient techniques are proposed. One of them is based ona boundary element method [4], and the other is a nodeelimination technique [5].
For conventional semiconductor devices, the circuit equations can be described by partial differential equations,and are usually solved by the finite element techniques. It
is really time-consuming, so that many efficient techniques
are proposed [68]. Mixed-level circuit and device simulations [7-8] are also time-consuming task, because we mustsolve both device and circuit at every time point.
However, for a small input signal, the device is linearizedaround the de operation point so that we can solve it in
the frequency-domain [7].
In thk paper, we propose an elegant method for solving 2-dimensional distributed circuits bssed on the multiconductor theorem. At first, we discretize the plane circuit into two directions such that each transmission line isconnected by the discretized R-L elements in each other.
We found that after some transformations,the circuit
equation is also described by the form of multi-conductortransmission lines. Thus, it cm be efficiently solved in
the time-domain [9-10] and the frequency-domain [11-12]
techniques.
Note that when 2-dimensional circuit is diacretized in
M x N pieces, we must solve the same number of the circuit equations. The discretizing technique is really timeconsuming for larger M and N. Our multi-conductor
method only takes into account N or M lines, so that thecomputationzd efficiency is largely improved in comparedwith the USUSIdlscretized methods. As an illustrated example, we solved a 2-dimensional circuit composed of 2
different plates.
II. ANALYSISOF 2-DIMENSIONALCIRCUITSIn order to anslyze the two-dimensional circuits, we discretize the circuit into the z and y axes as shown in Fig. 1.
Thus, the circuit equation can be described by the following partial equatio-ns:

II~(8Vi~ii-1.~+Latx=–

Copyright 1997 IEEE-‘1~+LmF)(R.~ii_l + Rzii + Rmii+l)- (Gmui-l

~ii+l

(1.1)

+ G=t)i + Gmvi+l)+iu,i— iy,i-l

(1.2)

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i+lline

where each of R,X, G and B matrices has the same form
as follow: i.e. let A be an arbhrmy matrix of R. ,, X. G
and B. Then, we havea=amam......
a=

A=

...

...

am

0

\oam

am
a=a=amwhere the tridlagonal elements of G and B in (4) given byG.+v.
—

1.1 i i.l

jkwBz

2RY

=

( ‘z+R=&‘x&R;+\

C=-R;+(kWLV)2(

Fig.1 Discretized model of 2-dimensional circuitG.

)+ jkwB.

=RyGm -R;+(~ ~iu,i-l
Vi-l—

=y —

&

8iV,ivi+l _ vi = L VF
i=l,2,+(1.3)

Rviy,i-l+jk~cm+(1.4)+ Ruiv,i. . ..n=Em sin(Kt):O<t<=

o:T/K<t~T(kwLV)2 )LV(kuLV)2R;+(

where n denotes the total lines, and (Lz, R=, C=, G=) and(L~,Rm, Cm, Gm) are self-impedances and couplingimpedances between the lines, respectively. The circuit
equation is coupled by (1.3) and (1.4), and different fromthose of multi-conductor transmission lines [11]. Although
there are time-domain and frequency-domain approaches
for solving transmission lines, we apply the frequencydomain technique [12], because it can be essily applied toboth AC and transient analysis of “2-dimensional circuits.
Now, consider the responses of the following impulse waveform:e(t))

2LV+jkUVi(kwLv)zi-1 line)

Thus, A is a tridiagonal and symmetric Toeplitz matrix,
so that the linear transformation is easily achieved. It is
known that the transfomation matrix M is independent of
the element A, and the inverse of M is equal to the transpose of it [11]. Furthermore, the eigenvalues are given
Sa~i = a= + amp(5.1)
The elements of matrix M can be computed as~i,j=(5.2)@i-l(Pi)/~j

where ~i(~) is given in the form

The initial conditions are defined ss: do(p) = 1 and@l(P) = P, and the normtization factor ~i is given b

T/h’and we expand it in the Fourier series ss follows:
Me(t)

=

E. + ~(E2k-1 cos )wt + &sin bt)

(2)

k=l

where w = 2~/T.

Assumevi = Viejkut, ii = Iiejkut,.

and substitutingdVz––iui = Jyiejkut~=-(G+jkUB)V

V = MV’,(3)

I = MI’(6)

Thus, we havethese relations into (1), we have–-(R+jkwL)I,It can be shown that the transformation matrix of thetridiagonsl symmetric matrices has its inverse matrix
equal to its transpose matrix.
Using M, let us transform relations (4) sa follows:(4)

Copyright 1997 IEEE

dV’
x=-MTRMI’– jkwMTXikfI’

(7.1)

1757

The current2.

ary of PI andmust be continuous
Pz: 1(2)(12)= fi3) (0)at the bound--’ag(%K’2)+dK’)*+ _ 1 &3)
Zj:),)~K~3)(72[:)= -diag

3. The voltage must be continuous
ary of PI and P2: V(2)(12) = V(3)(0)+Fig.2 2-dimensional circuitdI’z–--MTGMV’- jkwMTBMV’diag()Y(l)lz
e *.D

4. Current(7.2)

All of the right hand side can be described by the diagonal
matrices as follows:(10.2)

at the bound-(-Y::vf)‘@3)+K~+diag eKf2)

(10.3)

at therightend

is zero:I(3J (13) = O-diag(+?)K’3)
K’)=*(10.4)

5. Voltage

at inputline: V(lJ (0)=V(z)(0)(10.5)(8.1)
6. CurrentdI;
z–– -(gi,i+ .i~~~i,i)~’ = Kl,ieyiiz(K2

impedance

m(o)
w(o)

=E,(10.6)

where ZLie-~i.i.)(9.2)

Zvoo...

and propagation‘ii=ti=i~i,i =()z~ I z~ \ D )(9.1)+ h72,ie-7i’zA(-K1,iey’”+Zi,iwhere the characteristic
stant are given byline:
1(1)(0)Thus, the solution is given byI;=at input(8.2)

u’~(~i,i+jk~~i,i) (gi,i + jk~6i,i)

con-‘“)o,Ek=Zy Zv Zv ... 0Zv Zv Zv ... Zv
1-10...

O1–l(9.4)
D=

where the constants Kl,i and K2,i are decided by theboundary value conditions of the 2-dimensional plate as
shown in Fig.2, which is composed of two different plates
PI (v~~), Z$)), and P2 (-y\~),Z\~)). Assume the input location’is k-th line and 11from the left end, and choose theorigin at (11, k). Then, we have the following 6 boundary
conditions:
1. Current at the left end is zero: l(l) (/1) = O0’1.
zvY8Ek,inkzuysEk,inn1[[)Zv Zv o ... 0=..
....................

.

0...0.....................

000...–1000...1o ... 0 ... 01
0 ... 0 ... 0+O”””A.”” Ok..................
[1 0 ... A . . . On
1...

k...

n-diag($NK’’)+diag(e:’l)’K’l)=Note that the coupling impedance is given by+(10.1)

Copyright 1997 IEEE

Zv= RV + jkuLy,

A = ZYY51758

of discretized basic semiconductor equation,” Solid StateElectnm., VO1.36,no.4, pp.643-648,1993.

2 M.S. Obrecht, M.I. Elmesry and E.L. Heasell, “TRASIM:l’!
Compact and eflicient two-dimensional transient simulator for arbitrary planner semiconductor devices,” IEEEZhzns. on Computer-Aided Desig~ VO1.14, no.4, pp.447458, 1995.

1.00.0[3] S. Odanaka and T. Nogi,
0.20.40.6

1.00.8TrsnsIsINReSPOIISS

Fig. 2-dimensional circuit[4] Z. Wang and Q.

Plate : 48[mm] x 50[mm]
RI = 30[ft/mm] R~l = l[fZ/mm] JZ1= 0.003 ~/mm]~ml = &t)OOl~/mm] G1 = 30[S/mm] G~l = –l[S/mm]
R2 = Zoo[nfmm]
C~l = –0.001~F/mm]
10[$2/mm] Lz = 0.003~/mm] Lmz = O.0001~/mm]
G2 = 200[S/mm] Gm2 = -l[S/mm]
C2 = 0.050~F/mm]
C~z = -0.001 ~F/mm], R,l = 10IQ/mm] L.l = 0.003 ~/mm]
R.z = 200[f2/mm], L.z = 0.003~/mm]
Input position (30[mm], 25[mm]), Time= 0.25[nsec]

Cl = 0.050~F/mm]&2“Massively parallel Computation using a splitting-up operator method forthree-dimensional device simulation,” IEEE Zkans. on
Computer-Aided Desig~ VO1.14,no.7, pp.824-832, 1995.

=~U,‘A two-dimensional resistancesimulator using the boundary element method,” IEEETmns. on Computer-Aided Desig~ vol.11, no.4, pp.497504, 1992.

[5] T. Mitsuhaahi and K. Yoshida, “A resistance calculation
algorithm and its application to circuit extraction,” IEEEZkans. on Computer-Aided Desig~ VOL6, no.3, pp.337345, 1987.

[6] S.P. Chin and C.Y. Wu, ‘A new grid-generation methodfor 2-D simulation of devices with nonplaner semiconductor surface,” IEEE Tmns. on Computer-Aided Desig~
VO1.12,no.9, pp.1337-1344, 1993.

and Ys, Ek,in are input admittance and the Fourier coefficients of the input impulse.
[7] K. Mayaram and D.O. Pederson,”Coupling algorithm for
Solving relations (10), we can estimate the coefficientsmixed-level circuit and device simulation,” IEEE Tmns.
(&),-# ~)),(@2);&))and (K~3), K$)).
Thus, theon Computer-Aided Desig~ vol.11, no.8, pp.1003-1012,
steady-state waveform at any point can be calculated by1992.
the phasor technique.
[8] N. Iwamuro and S. Tagami, “Two-dimensional power
As an illustrative example, we calculated the impulse redevice simulator considering an integral external cirsponse on the two different plates as shown in Fig. 2. We
cuit equation,” IEEE ZYans. on Computer-Aided Desig~
consider here 50 lines in Y-directions located 1[mm] eachVO1.12,no.6, pp.909-912, 1993.
other, and connected by RL circuit, where 11 = 30[7nln],[9] L.Lu, N. Nakhla and Q.J. Zhang, “A resetting algorithm
12= 4[7n7n]
and 13 = 14[mm]. The response waveform at
for transient analysis of coupled transmission linee~ IEEE
0.25[nsec] has two reflections from the boundary of two
Theory and Tech., VO1.42, no.3,
2Yans. on Microwave
plates and the right end as shown in Fig.3.
pp.494500,
III.

CONCLUSIONS AND REMARKS

We proposed here an elegant algorithm for solving 2dimensional circuits based on a multi-conductor transmission line theorem. Since the transfer matrix obtained by
the use of Toeplitz’s theorem is very simple, it can be easily applied to the large scale circuits. As an illustrative
example, we solved 2-dimensional circuit composed of two
different plates. The computatiomd efficiency is largelyimproved compared to the usual finite element technique.
Furthermore, the method can be also applied to solve 3dimensional circuits and devices. Thus, the technique will
be used for wide classes of device simulations.
REFERENCES[1] M.S. Obrecht,”A

new stable method

1994.

[10] F.Y. Chang, “Transient simulation of nonuniform coupledlossy transmission lines characterized with frequencydependent parameters Part II: Discrete -time analysis,”,IEEE Tmns. on Circuits and Syst: Fundamental Theory
and Application, VO1.39,no.11, pp.907-927, 1992.

[11] D.S. Gao, A.T. Yang and S.M. Kang, ‘hlodeling and simulation of interconnection delays and crosstalk in highspeed integrated circuits,” IEEE Trans. on Cinmits andSyst., vol.37, no.1, pp.1-9, 1990.

[12] Y. Tanji, L. Jiang and A. Ushida, “Analysis of pulse responses of multi-conductor transmission lines by a partitioning technique,” IEICE Trans. on Fundamentals,
VO1.E77-A, no.12, pp.2017-2027, 1994.

for linearization

Copyright 1997 IEEE1759