平成22年度電気関係学会四国支部連合大会

326

18-27
SPICE-Oriented Stability Assessment
Using Limited Number of Variables
Hiroshige Kataoka

Yoshihiro Yamagami(Tokushima University)

1. Introduction

3. Illustrative example and result

The assessment of the stability is very importantfor designing the circuit. In this article, we propose aSPICE-oriented method for the assessment of the stability, which is based on the Floquet theory. We haveto choose all of variables in the conventional method.
If the circuit has too many variables, it is diﬃcult
to assess the stability. We choose the variables fromthe necessary minimum variables. By combining ourmethod with the conventional spice-oriented frequency
analysis, we can assess the stability of the solution easily with SPICE. First, we obtain the periodic solutionsof the circuit by using the SPICE. Next, we calculate the eigenvalues of an elementary matrix by solving variational circuits based on the Floquet theory.
As an example, we assess the stability of the periodic
solutions for a ﬁfth order resonance circuit.

2. Stability of periodic solutions
We suppose that there is a circuit equation asf (ẋ, x, y, ωt) = 0,(1)and make the variational equation for the regular period solution of x̂. If we assume the small change quantity as (∆x, ∆y), we obtain the equation as˙∆x
˙ x̂, ŷ, ωt) + ∂f ∂f ∂f |x=x̂,y=ŷ  ∆x  = 0.
f (x̂,∂ ẋ ∂x ∂y∆y(2)In Eq. (2), the ﬁrst term is regular period solution
and second term is variational equation. We change
the second term as[Yoshifumi Nishio]

˙ = A(t)∆x.
∆x

(3)

In Eq. (3), A(t) is the periodic function. We apply theFloquet theory for this periodic function. We write
the elementary matrix of the periodic solution as Φ(t).
From this, the solution after one period from initialvalue of ∆x(0) is given as follows;∆x(T ) = Φ(T )∆x(0).

(4)

Hence, when the eigenvalues (λ1 , λ2 , . . . , λn ) of Φ(T )satisfy |λk | < 1 (k = 1, 2, . . . , n), the regular periodic
solution x̂ is stable.
First, we develop the variational circuit from original circuit. We perform the transient analysis of
the variational circuit for one period T , with diﬀerent initial conditions. In this study, we choose a small
number of variables compared with the conventional
method.

R1L1

eR2C1L2L3C2R3

Figure 1: Example model.
We assess the stability of the solutions in Fig. 1 with
3 variables. In Fig. 1, we assume a sinusoidal voltage
source and third order nonlinear characteristics for thecapacitors as follows: e = Em sin ωt
C1 ; v1 = α1 q1 + β1 q1 3(5)C2 ; v2 = α1 q2 + β2 q2 3
We set the parameters as follows; Em = 0.35[V ], α1 =2.4, β1 = 12.0, α2 = 1.0, β2 = 5.0, R1 = 0.1[Ω], R2 =R3 = 0.02[Ω], L1 = 0.2[H], L2 = 0.8[H], L3 =0.10101[H]. We assess the point of ω = 5.00[rad/s],
which is in unstable region.
First, we show the results when we choose 5 variables in Table 1, namely the elementary matrix solution has 25 components. Next, we show the results
when we choose only 3 variables in Table 1, namely
the elementary matrix solution has 9 components.
Table 1: Eigenvalues of Φ.
|λ1 | |λ2 | |λ3 | |λ4 |(i1 , i2 , i3 , q1 , q2 ) 0.53 0.53 0.41 0.85
(i1 , i2 , i3 )0.35 0.85 1.62
(i2 , i3 , q1 )1.50 0.60 0.46
(i1 , i3 , q2 )0.45 0.26 1.20
-

|λ5 |
1.04
-

From the Table 1, we succeed the assessment of stability with small number of variables. If we choose 2variables, namely the elementary matrix solution has
4 components, wrong results are obtained.

4. Conclusions
We proposed SPICE-oriented method for the assessment of the stability, which is based on the Floquet
theory. We succeed the assessment of stability withsmall number of variables. As a future work, we would
like to apply our proposed method to circuits withmore variables.

References[1] Donald O. Pederson and Kartikeya Mayaram:
Analog Integrated Circuits for Communication,
Kluwer Academic Publishers, 1991.