Prwedings 4 EEE Asia Pacific Cmfmence on Circuits and Systems '96
N~vember18-21, 1996, Sml, Uaea

Characteristic Curve of Nonlinear Resistive Circuits
and its Stability
Akio USHIDA, Lingge JIANG and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering, Faculty ofEngineering, Tokushima University, Tokushima, 770, Japan
Phone: +81-886-56-7469
F~x.0886-56-7471
Email: ushidaQee.tokushima-u.ac.jp

Abstract
Distinguishing the stability of characteristic curves
for nonlinear resistive circuits is requirement and importance to design various electronic circuit exactly.
Since every resistive element has a parasitic component, solutions on the characteristic curves are stable
or unstable. In this paper, we show that the stability will be mainly changed at the bifurcation points
such as limit point and branch point. Applying thecurve tracing method, we can decide the unstable regions on characteristic curves by the locationsof bifurcation points.

1 Introduction

lar transistors, FETs and so on. The solution curve
can be calculated by solving a resistive circuit composed of n equations in ( n 1) variables -+f(x) = 0,(1)Assume f (z) is C2continuous in z E Rn+'. Let us
describe the variable by x = x(s) as a function ofarc-length s from the starting point 20. Then, thesolution of (1) satisfies the following set of algebraicdifferential equations [7]:f (x)= 0($$+(%) +-*+(+)
2(2.1)
2=1(2.2)

Since the solution curve is a continuous function of

In this paper, we discuss the stability of the solution curves for nonlinear resistive circuits including parasitic elements. Although the DC solution isdetermined by analyzing the nonlinear resistive circuit, its equilibrium point will be the one of stable
or unstable because every resistive element has smallparasitic elements in practice. There are some papers
discussing the stability of nonlinear networks. In references [1,2], a globally asymptotically stable condition is discussed for nonlinear dynamic networks ina qualitative manner. In reference [3], a simple technique is proposed to identify unstable DC operating
points. Of course, the stability can be decided bysolving the variational equation at each equilibrium
point obtained by the DC analysis, however, that isvery time-consuming. We show here that the stability is mainly changed at the bifurcation points suchas turning and branch points [4-51 on the DC characteristic curves, so that the stability of the solution
curve is easily found by the locations of bifurcation
points.

2 Unstable regions of solution curves

s even at the limit point [5], we have from (2)D+)=I\ %......................
2 EI...
&gL/
$f$

(4)

* * *

ds+

Observe that the first n x (n 1) submatrix corresponds to the Jacobian matrix of f(x), and the lastrow shows the derivatives of the curve. Our curve
tracing algorithm 161 efficiently traces the solution
curve satisfying (2) . In this case, it is proved thatwhenever the rank of Jacobian matrix of f(z) is n,
the coefficient matrix DI'(x) is nonsingular, so that
we can trace even for the limit points [5]. Thus, wehave the following relation by the Cramer's formulato ( n 1)th variable

Now, consider electronic circuits containing bipo0-7803-3702-6/96/$5.00@1996 IEEEf : R"+' -+ R"

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where D n f ( x ) is the Jacobian matrix for the variable{ ~ 1 , ~ 2 ., .,. xn)
Now, assume that we have the following dynamic
equation by considering the parasilic elements:dx
P- = f(x),atthe follovvlng theorem around the point.
Let T(x) be a smooth solutaon curme
passang through the branch hafurcataon point. T h e n ,the stahalaty of solutaon 2s changed at the poant.

for P =Then, the variational equation at an equilibrium
point x is given by(9)sion to dl(x,+l) at two
Ax,+l before andpoints x:+~ - Axn+l and x:+lafter the bifurcation point x*, we have+Thus, the stability condition of the resistive circuits
is decided by the eigenvalues of the Jacobian matrix
DnJ(x). We have the following stability property
around the lamat point.

nt of solution curve
Theorem 1 If the startae, w h e n the solution
point, the stahilaty as changed at the point.
Proof: The lamat poant is a turning point such thatthe direction of the solution curve is chdx,+l/ds = 0 at the point. This means t hD n f ( x ) l is changed after passing throint because of the nonsingularity of D r ( x ) in(5) [ 5 ] . Here, we transform (7) as follows:dAx- - P-’Dn f ( ~ ) A X
dt
The eigenvalues of variational equation satisfy thefollowing relation[7]:= d l (x:+l)+d1’(5~+1)nxn+l+. ’ .
(10.2)
where ’ indicates the derivative with respect to zn+1.
At the branching point x*, the following relations
hold [4]dl(x:+1+Axn+l)rank(Df(z*)) = n-1, dl(x:+,)= 0, dl’(xz+l) # 0(11)Multiplying the two equations in (lo), we obtainThus, the sign of the denominator of (4) is changed
whenever it passes through the point. We have thesame result as detlDl?(x)l ford2(xn+1) E detID,f(z)l(12)

ndetlP‘lD,f(x)l= detlP-lI(8)i=l

We assume that detl P-’ I # 0
bility depends on the eigenval&(i = 1,2,. . . ,n) are the eig
real and/or complex conjugates. Thus, the changeof sign ( 5 ) means that the sign of one of the real
eigenvalues is changed after passing through the limit
point, so that the stability isNext, we consider the stabilit
around the branch bafurcataon poant [4], where twoross at a point.
It is known
the rank of the Jacobian matrix to (1)8x1aax,+1n... &ax,

8x1ax,2.L.

430

As a special case, there are many ssuch as Flip-Flop circuit. In this case, they sometimes have an interesting property such that one of
the solution curves is s y m m e t m c with rother one. This type of bifurcation
patchforkpoint 141.

the s a m e stability passang throughthe poant. Thas as hecause that the prtchfork bifurcat i o n has symmetrac solutaon curves.

......................
... &alnuced to less than n. Hence,
becomes singular at the bifurcati

because the rank of Of(.) is less than n at the bifurcation point. Thus, the sign of (4) is not changed
at the point, so that the direction of solution curve
dx,+l/ds is never changed at tlie branchtion point. But the stability of tis changed. The instability of the eafter the bifurcation point will be aax,,,We haveR e m a r k In Theorem 1 and 2 , the instability regions are determined by investigating whether theT5-PC8.2

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where- 10.5~;+ 11.8t~1
g2(~2) 0.43~: - 2.69~; + 4.56v2
gl(W1) E 2.5~:

The driving point characteristic for R = 1.5,
L, = 1 and Cpl = Cp2 = 1 is shown in Fig.l(b),"inwhere the dotted lines show the unstable regions.
Observe that there are small regions of the Hopf bifurcations before and after the dvin/ds < 0 regionsin the dwi,/ds > 0. On the other hand, there is a
closed loop (EaFb), where the region ( E a F ) is stable and (FbE) unstable. Note that once the stability
is checked at a point on the closed loop, the whole ofthe stability can be known by Theorem 1.

r(a)Two-tunnel-diode circuiti
4. -0.

3.2 Hopfield networkHopfield neural networks are sometimes applied tosolve combinatorial problems such z1s the travelingsalesman problem, and the layout of VLSI circuits.
Now, consider the circuits containing 6 synapses
whose equation is given by
3.

6.

9.

12.

V&

15.

(b)Driving-point characteristic curves

dui
=6dt

j=1+axi
w;jzj - -log - Ii
21-xi

i = 1,2, ...,6Figure 1: Two-tunnel-diode circuit

where-2-20-2-2

10-2-2-2-2-2-2-2-2-2-20-2-2

( 3.5

3.5

5.05.0 3.5'variational equation has the positive real eigenvalues
or not. However, it may sometimes happen that it
has the complex conjugate eigenvalues having positive real parts. For this kind of instability, the equilibrium point behaves as an unstable focal point, andthe sign of dx,+l/ds is not changed at the bifurcation
point. This bifurcation is called Hopf bifurcation.

3 Illustrative examples
3.1 Two-tunnel-diode circuit [6]Consider the two-tunnel-diodes circuit shown inFig.l(a). Let us choose such that the normal tree
contains vin and two tunnel diodes. Put the parasitic
element L, into the co-tree R and Cpl,Cp2 between
the tree diodes and ground.
The circuit equation is given byw =I=0
1-2-2-2-2-2-20
1-2-2-2-2
103.5 )T

Setting dui/dt = 0, the stationary solutions are obtained. Choosing a as an additional variable, we havea set of 6 algebraic equations with 7 variables. The
solution curves are obtained starting from a = 0.1[15]. The curves in the (xl,z3,q)-plane are shownin Fig.8, where we choose a = 0.29x~+0.1. We found
9 pitchfork points and 4 limit points.
Note that since the coefficient matrix W is symmetric, all of the eigenvalues are real, and the equilibrium
points belong to the nodal or saddle points. We show
the unstable curves by dotted lines. Their stabilities
are determined by the application of Theorem 2 aqd
Corollary 1.

4 Conclusions and remarks
In this paper, the stability of DC solution curves is
examined by introducing parasitic elements, such as'The example is given by Prof. A.Sakamoto at Tokushima
university.

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[3] M.M.Green and A.N.Willson,Jr, “How to identify unstable dc operating points”, IEEE Trans.
Circuits Syst., vol.CAS-39, pp.821-832, Oct.
1992.
[4] M.Kubicek and M.Marek, Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, 1983.
[5] L.G.Jiang and A.Ushida, “Bifurcation analysisof nonlinear resistive circuits by curve tracing
method”, IEICE Trans., vol.E78-A, pp.12251232, Sept. 1995.

xs[6] A.Ushida and L.O.Chua, “Tracing solution
curve of nonlinear equations with sharp turningpoints”, Int. J. of Circuit Theory Appl., v01.12,
pp.1-21, 1984.
L1[7] L.O.Chua and P.M.Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms andComputational Techniques, Prentice-Hall Englewood Cliff, N.J., 1975.
10.0

X,

1.0

(b) Stability of the solution curve for the Hopfield netw ork

Figure 2: Hopfield networka small capacitor between every resistor and ground,and inductor in series to every co-tree resistor.
We have proved two theorems and one corollary
which are very useful t o check t hsolution curves. Since thechanged at the bifurcation
point and branch bifurcatio
stability of solution curves without investigating thevariational equation.

References
L.O.Chua and G.R.Alexande
parasitic reactances on non
IEEE Trans. Circuits Syst., vol.CT-18, pp.520532, May 1971.
[2] L.O.Chua and D.N.Green, “A qualitative analysis of the behavior of dynamic nonlinear
networks: stability of autonomous networks”,IEEE Trans.
Circuits Syst., vol.CAS-23,
pp.355-530, June 1976.

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