IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Assessment of Stability for Oscillatory Circuit
by Using SpiceHiroshige Kataoka, Yoko Uwate, Yoshihiro Yamagami and Yoshifumi Nishio
Department of Electrical and Electronic Engineering
Tokushima University
Tokushima, Japan
Email: {hiroshige, uwate, yamagami, nishio} @ee.tokushima-u.ac.jp
Abstract— SPICE is a very convenient tool for circuit simulation and is used by many researchers. Nowadays, various SPICEoriented algorithm are proposed. By using these methods, we can
extend a function of SPICE and can analyze various circuit.
In this study, we propose a SPICE-oriented algorithm forassessment of stability for oscillatory circuit. We combine the harmonic balance method, Newton homotopy method and Floquet
theory. We find out a oscillatory parameter by using harmonicbalance method and Newton homotopy method, and we assess
the stability by applying our method. As an example, we assess
the stability of the periodic solutions for Cauer oscillator. The
result shows our propose method gives the correct results.

I. I NTRODUCTION

II. SPICE-O RIENTED ANALYSIS OF OSCILLATOR
A. Sine-Cosine Circuit
The sine-cosine circuit has been introduced in order to solve
the determining equations of the harmonic balance method byusing SPICE. In this section, we explain how we can derive
the sine-cosine circuit for simple passive element cases.
First, we set a voltage and a current with Fourier series;

n
∑



v
=
V
+
(Vsk sin kωt + Vck cos kωt)

0

k=1
(1)
n
∑



(Isk sin kωt + Ick cos kωt)
 i = I0 +k=1

SPICE is used for various analysis of the electrical circuit.
For example, AC analysis, DC analysis, sensitivity analysis,transient analysis etc. In addition, we can easily perform a
SPICE simulation. We only have to make a netlist or schematicby computer, without writing a complex program. From thisreason, many people use SPICE for circuit simulation. On theother hand, many researchers have proposed SPICE simulation
method, which is the introduction of analytical dynamics. We
propose a SPICE-oriented algorithm by applying the analytical
dynamics and combine to the conventional method.
For designing oscillatory circuit, it is important to theassessment of the stability. We have proposed a SPICEoriented algorithm to the assessment of the stability for periodic solutions which is based on the Floquet theory [1]. In
the conventional method, we assessed the stability of resonant
circuit. In this study, we apply the our method to the oscillatory
circuit by combining the Newton homotopy method [2]-[4].
The article is organized as follows. Section II-A shows
how to use the sine-cosine circuits [5], which is based on
the HB (harmonic balance) [6]-[8] method. We use the sinecosine circuit to obtain the value of the voltages which arerequired in order to solve variational circuits. Section II-B
shows the Newton homotopy method. This method is realized
by using solution-curve tracing circuit(STC) [9]. Section IIC shows the Floquet theory [8][10]. Section III shows anillustrative example and how to solve the variational circuits
by using SPICE. Section IV shows the results and confirms
the effectiveness of the proposed method. Section V concludes
this article.

A current through a capacitor is given bydv.
dt
From Eqs. (1) and (2), we can express the current as
i=C

i=

n
∑

(−kωCVck sin kωt + kωCVsk cos kωt).

(2)

(3)

k=1

From Eq. (3), we can express the relation between the coefficients of sine and cosine components as follows;{
Isk = −kωCVck(4)
Ick = kωCVsk
In the case of an inductor, we can express the voltage acrossan inductor asn∑v=(−kωLIck sin kωt + kωLIsk cos kωt),(5)
k=1where the current through an inductor is given by
di
.
(6)
dt
Equations for coefficient of sin kωt and cos kωt are given by{
Vsk = −kωLIck.
(7)
Vck = kωLIsk
v=L

If we make the circuit model satisfying this method, acapacitor is replaced by coupled voltage-controlled current
sources and an inductor is replaced by coupled currentcontrolled voltage sources in the sine-cosine circuit.

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v&ρ

B. Newton Homotopy Method
Newton homotopy method is one of method for finding
multiple dc solutions. The circuit model of Newton homotopy
method is shown in Fig. 1. We assume equations as follows;g0 (V0 , V1 , V2 , . . . , VM ) = 0g1 (V0 , V1 , V2 , . . . , VM ) = 0g2 (V0 , V1 , V2 , . . . , VM ) = 0.
(8)..................

gM −1 (V0 , V1 , V2 , . . . , VM ) = 0gM (V0 , V1 , V2 , . . . , VM ) = 0
These determining equations are described by a set of algebraic equations, which consists of M -equations and same
number of variables. However, it is not easy to solve theequations, because they may have the multiple solutions.
Applying the Newton homotopy method to solve Eq. (8),we obtain the following relation;
G(V , ρ) = g(V ) − (1 − ρ)g(V(0) ) = 0.

(9)where the initial state is set by a point (V(0) , ρ = 0) and getsthe solutions satisfying g(v) = 0 at ρ = 1 on the path. ρ shows
solutions curves called homotopy paths, and find the multiple
solutions lying on the paths. A solution curve is traced by
ark-length method as follows;G(V , ρ) = 0


 M (
)2 ( )2

∑

dρ
dVi

+
=1
.
(10)
ds
ds



 i=1


i ̸= 2Equation (10) is realized by using ABMs. Figure 2 shows thecircuit diagram of solution-curve tracing circuit (STC).

v0

g 1(V,ρ )gM(V,ρ )( ρ - 1) g 1(0)VCCS

g 1(0)

1[Ω]

・
・
・

・
・
・

vM

( ρ - 1) gM(0)VCCS

gM (0)

1[Ω]

ρ

STC

1[A]

Fig. 2.

R

Solution-curve tracing circuit (STC).

C. Stability of Periodic Solutions
We suppose that there is a circuit equation asf (ẋ, x, y, ωt) = 0,

(11)

and make the variational equation for the regular period
solution of x̂. First, we assume the small change quantity as(∆x, ∆y) as{ x = x̂ + ∆x,
(12)y = ŷ + ∆yand substitute Eq. (12) to Eq. (11). We obtain the equation as


˙
[
]
∆x
∂f
∂f
∂f
˙ x̂, ŷ, ωt) +f (x̂,|x=x̂,y=ŷ  ∆x  = 0. (13)∂ ẋ ∂x ∂y∆y
In Eq. (13), the first term is regular period solution and second
term is variational equation. We change the second term as
˙ = A(t)∆x.
∆x

(14)

In Eq. (14), A(t) is the periodic function. We apply the Floquet
theory for this periodic function. We write the Jacobian matrixof the periodic solution as Φ(t). From this, the solution after
one period from initial value of ∆x(0) is given as follows;

Σ

(15)

III. I LLUSTRATIVE EXAMPLE-1
Fig. 1.

int

Hence, when the eigenvalues (λ1 , λ2 , . . . , λn ) of Φ(T ) satisfy|λk | < 1 (k = 1, 2, . . . , n), the regular periodic solution x̂ isstable.
In this study, we derive the variational circuit which corresponds to the variational equation and perform the transient
analysis of Spice just for one period in order to obtainthe components of Φ(T ). We should repeat the transient
analysis by giving different initial conditions to obtain all thecomponents numerically. However, the number of the repeat
is at most the same as the number of the state variables of thecircuit. Further, it should be mentioned that we do not have tochange the structure of the variational circuit even when thevoltages of the regular periodic solution are changed.

g 0(0)

1[Ω]

v1

vρ

Iτ=v&ρ 2

i= 1∆x(T ) = Φ(T )∆x(0).

( ρ - 1) g 0(0)VCCS

g 0(V,ρ )M

IM=∑v&i 2

Circuit model of Newton homotopy method.

As an illustrative example, we assess the stability with thecircuit in Fig. 3. The circuit parameters in Fig. 3 are set as α =β = 1, R1 = R2 = R3 = 0.1, L1 = 0.4167, L2 = 0.2618,
L3 = 1.058, C1 = 0.1, C2 = 0.3429 and C3 = 0.439.

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IV. S IMULATION RESULTS

iG
+
vG
−

C3

Fig. 3.

Cauer oscillator.
Fig. 5.

The circuit equation can be written as

dv1


+ i1 = αv1 − βv1 3
C1


dt



dv2
.
(16)
i1 = i2 + C2

dt





 i2 = i3 + C3 dv3dt
If we write the variables as periodic solutions with small
variations;{
ik = ik0 + ∆ik(17)
vk = vk0 + ∆vk
We obtain the following variational equations as

dv10


+ ∆i1 = α∆v1 − 3βv10 2 ∆v1
 C1

dt



∆dv2
,
(18)
∆i1 = ∆i2 + C2

dt





 ∆i2 = ∆i3 + C3 ∆dv3
dt
where we neglect higher-order small terms. From these equations, we can make the variational circuit of Fig. 3 as shownin Fig. 4.

∆v1

∆v 2

Figure 5 is the time-response obtained by solving the circuitin Fig. 3 with HB method combining with Newton homotopy
method. We found 5 equilibrium points, where ρ (=V(RO))
satisfying ρ = 1. We show the solutions in Table I.
TABLE IE IGENVALUES OBTAINED BY CONVENTIONAL METHOD
time [sec]V(Omega)
V(VC1)
V(IL1)
V(VC2)
V(IL2)
V(VC3)
V(IL3)

2

C3

In Fig. 4, v10 is a steady periodic solution and are calculatedasv10 = Vc cos ωt + Vs sin ωt,(19)where Vc and Vs are given by the sine-cosine circuit obtained
from the circuit in Fig. 3.

20.26
2.003
9.446
65.94554.84
28.39869.43
32.727

32.72
4.002
1.216
0.499
0.416
1.03
0.677
0.160

34.723
4.999
2.346
3.339
6.410
7.853
3.91600.740(∆v10 , ∆i10 , ∆v20 , ∆i20 , ∆v30 , ∆i30 )(V(VC1), 0, 0, 0, 0, 0)





(0,V(IL1), 0, 0, 0, 0) (0, 0,V(VC2), 0, 0, 0)=.
(0, 0, 0,V(IL2), 0, 0)





(0, 0, 0, 0,V(VC3), 0) (0, 0, 0, 0, 0,V(IL3))∆v3

Variational circuit of Fig. 3.

1.57280.993
1.211
0.1563
1.1520
0.520
1.009
0.956

36.6945.97971.1918
0.716
0.598
0.513
0.208
0.033

We give the obtained value to the circuit of Fig. 4 as initialcondition. The state of initial condition are given as follows;(α − 3β v10 )∆v1

Fig. 4.

Result of transient analysis in Fig.4.

(20)

(21)

We perform the transient analysis with the circuit in Fig. 4
just for one period and obtain the components of Φ(T ) as thevalues of the state variables of the variational circuit. We showΦ(T ) for the 5 cases in Table II. Table III shows eigenvaluesof Φ(T ) for 5 cases, which calculated by MATLAB. We cansee that the point ω = 0.993, ω = 4.0022 and ω = 5.9797 arestable, because all of eigenvalues satisfy |λ| < 1. However, forthe other two points, ω = 2.0028 and ω = 4.999, the solutions
are unstable, because some eigenvalues do not satisfy |λ| < 1.
These results agree with the results obtained in [4]. Namely,
we can say that our algorithm gives the correct results.

- 154 -TABLE II

TABLE IIIT HE VALUES AFTER ONE PERIOD Φ(T )E IGENVALUES OF Φ(T ).
ω
λ1
λ2
λ3
λ4
λ5
λ6

(a) ω=0.993
∆v1
1.74m
-5.06m
2.76m
1.06µ
6.16m
-8.51m∆i1-9.31m27.07m-6.56m368.49µ-34.57m
43.34m∆v2
838.9µ-1.07m-181.00m-13.33m
37.01m
48.33m∆v1
8.97µ-107.69m-1.04m-38.26m
11.41m32.32m∆i1-2.41m28.93
269.11m
10.28
-3.03
-8.68

∆v2
-33.81µ368.68m
10.02
488.71m-42.771.93∆v1-2.11m20.57m
-16.04m-7.00m35.59m-22.70m∆i1
11.89m-115.25m
82.74m58.45m-200.25m
122.12m∆v2-13.55m
120.45m
74.55m-468.93m
216.63m-29.50m∆v1-133.99µ14.93m-61.98m
111.37m-43.34m-41.11m∆i1
2.42m-269.94m
1.11-1.95
720.03m
722.05m∆v2-6.39m705.24m-2.17
248.94m
2.63-349.93m∆v1-279.42µ3.53m-642.02µ14.70m-18.74m-4.58m∆i1
1.40m-17.92m
8.29m-87.00m
92.17m24.44m∆v2-369.76µ12.36m-127.32m
233.17m
58.22m-25.35m∆i2
4.08µ172.8µ-38.07m
90.25m13.00m-19.05m∆v3
1.67m-5.09m33.00m4.13m-122.78m
15.57m∆i3-1.01m2.79m
18.88m-2.55m
6.98m
8.71m

∆v3
229.98µ-2.66-26.39573.54m-15.55-542.72m∆i3
573.26µ
-6.89
1.07
1.85
-464.38m
18.68

0.993
−0.202−0.112
0.093
0.034
0
0.101

2.003
−38.9470−18.424
18.12039.23033.1334.002-0.4110.467
0.365-0.186
0
0.122

4.999
5.670
-4.154-1.717
0
2.889
0.885

5.980
0.402
-0.3590.181-0.0040.050-0.196(b) ω=2.0028∆i2-3.16m
37.94
1.12
-8.97
2.41
8.60

(c) ω=4.0022∆i2-3.05m
44.94m-247.64m
107.90m
224.77m
74.93m∆v3
14.41m-140.34m
103.94m
203.78m
272.48m-54.25m∆i3-16.17m
150.30m-24.97m
119.56m-95.58m
19.11m∆v3-5.72m
579.10m
103.94m
143.10m
59.38m-365.69m∆i3-11.89m
1.28-24.97m
800.73m-803.17m
190.94m∆v3-24.20m
300.38m
130.64m-115.71m
21.73m-18.70m∆i3-15.52m
209.46m-149.45m
61.94m-49.11m
12.03m(d) ω=4.999∆i2
12.28m-1.32
282.53m
5.25
116.79m
304.82m[3] A. Ushida, Y. Yamagami, I. Kinouchi, Y. Nishio and Y. Inoue, “An
Efficient Algorithm for Finding Multiple DC Solutions Based on the
SPICE-Oriented Newton Homotopy Method,” IEEE Trans. ComputerAided Design of Integrated Circuits Syst., vol.21, no.3, pp.337-348, Mar.
2002.
[4] Y. Yamagami, Y. Nishio and A. Ushida, “Analysis of Reactance oscillators
Having Multi-Mode Oscillations,” IEICE Trans. Fundamentals., vol.E89A, no.3 Mar. 2006.
[5] J. Kawata, Y. Taniguchi, M. Oda, Y. Yamagami, Y. Nishio and A. Ushida,“SPICE-Oriented Frequency-Domain Analysis of Nonlinear Electronic
Circuits,” IEICE Trans. Fundamentals, vol.E90-A, no.2, Feb. 2007.
[6] P. Wambacq and W. Sansen, Distortion Analysis of Analog Integrated
Circuits, Kluwer Academic Publishers, 1979.
[7] D.O. Pederson and K. Mayaram, Analog Integrated Circuits for Communication, Kluwer Academic Publishers, 1991.
[8] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, Wiley-Inter Science,
1998.
[9] E. Ikeno and A. Ushida, “The Arc-Length Method for the Computationof Characteristic Curves,” IEEE Trans. Circuits Syst., vol.23, pp.181-183,
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[10] F.L. Traversa, F. Bonani and S.D. Guerrieri, “A Frequency-Domain
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(e) ω=5.9797∆i2
12.82m-192.80m
356.41m
186.32m-78.73m
16.01m

V. C ONCLUSION
We proposed the SPICE-oriented algorithm to assess thestability of periodic solutions for oscillator. We obtained
periodic solutions by using harmonic balance method andNewton homotopy method. We assessed the stability basedon the Floquet theory. In detail, we analyzed the Cauer oscillator for 5 different frequencies which gives both stable andunstable solutions. Our results agree well with the previously
obtained results. For the next step of our research, we need towork on assessment of stability of oscillator having complex
characteristics.
R EFERENCES[1] H. Kataoka, Y. Yamagami and Y. Nishio, “SPICE-Oriented Algorithm
for Assessment of Stability for Periodic Solutions,” Proc. of NOLTA’10,
pp.374-377, 2010.
[2] Y. Inoue, “DC Analysis of Nonlinear Circuits Using Solution-Tracing
Circuits,” Trans. IEICE, vol.J74-A, pp.1647-1655, 1991.

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