2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

SPICE-Oriented Algorithm for Assessment of Stability for Periodic Solutions
Hiroshige Kataoka, Yoshihiro Yamagami, Yoshifumi Nishio and Akio Ushida
Dept. Electrical and Electronic Eng, Tokushima University
Tokushima 770-8506, JAPAN
Email: hiroshige@ee.tokushima-u.ac.jp, yamagami@ee.tokushima-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—The assessment of the stability for periodic
solutions is very important for designing the circuit. Thereare many method for the assessment of the stability. Inthis article, we propose a SPICE-oriented method for theassessment of the stability, that is based on the Floquet theory. By using our method, we can assess the stability of thecircuit easily. First, we obtain the periodic solutions of thecircuit by using the SPICE. Next, we calculate the eigenvalues of a Jacobian matrix by solving variational circuits
based on the Floquet theory. As an example, we assess thestability of the periodic solutions for one order resonancecircuit including nonlinear capacitors.

the current iC is given byiC = C

dvC= −ωCVCC sin ωt + ωCVCS cos ωt.
dt

Thus, the coeﬃcients of sin ωt and cos ωt are described by
ICS = −ωCVCC , ICC = ωCVCS .

(3)

Namely, the capacitor is replaced by coupled voltagecontrolled current sources in the sine-cosine transformationof the HB method. In the same way, let the current throughan inductor L beiL = ILS sin ωt + ILC cos ωt.

1. Introduction

(2)

(4)

Then, the voltage vL is given by

When we simulate the circuit, we often refer to the assessment of the stability. In this paper, we propose aSPICE-oriented algorithm to the assessment of the stability
for periodic solutions which is based on the Floquet theory [1]. In the conventional method, we have to calculate
the Jacobian matrix for the periodic solutions by solving
the variational equations. In this study, we obtain the Jacobian matrix by the transient analysis of SPICE for variational circuits which is easily derived from the originalcircuit.
Section 2.1 shows how to use the sine-cosine circuits [2],which is based on the HB (harmonic balance) method. We
use the sine-cosine circuit to obtain the value of the voltages which are required in order to solve variational circuits. Section 2.2 shows the solution curve-tracing circuit.
It is based on the arc-length method [3][4]. Section 2.3
shows the Floquet theory. Section 3 shows an illustrative
example and how to solve the variational circuits by using
SPICE. Section 4 shows the results and confirms the eﬀectiveness of the proposed method. Section 5 concludes thisarticle.

vL = L

diL= −ωLILC sin ωt + ωLILS cos ωt.
dt(5)Thus, the coeﬃcients of sin ωt, cos ωt are described by
VLS = −ωLILC , VLC = ωLILS .

(6)

Namely, the inductor is replaced by coupled currentcontrolled voltage sources in the sine-cosine transformation.
As an example, Fig. 1 shows an RLC ladder circuit andFig. 2 shows the corresponding sine-cosine circuits.

Figure 1: RLC ladder circuit

2.2. Solution curve-tracing circuit
2. Frequency analysis and assessment of stability
2.1. Sine-cosine circuit
We introduce the sine-cosine circuit corresponding to thedetermining equation of the HB method. If we set the voltage through a capacitor CvC = VCS sin ωt + VCC cos ωt,(1)Even we use our sine-cosine circuits, we can not obtain
unstable periodic solutions, because we set the frequency
as time in SPICE. In this section, we explain the STC (solution trace circuit) realizing the arc-length method.
First, we can express the ark length in (n+1) dimensional
euclidean space as Eq. (7)√(7)
ds = (dx1 )2 + (dx2 )2 + (dx3 )2 + . . . + (dxn+1 )2- 374 -and substitute Eq. (10) to Eq. (9). We obtain the equationas ˙ []
 ∆x ∂f∂f∂f
˙ x̂, ŷ, ωt) +f ( x̂,| x= x̂,y=ŷ  ∆x  = 0. (11)∂ ẋ ∂x ∂y∆yIn Eq. (11), the first term is regular period solution andsecond term is variational equation. We change the secondterm as
˙ = A(t)∆x.
∆x(12)

Figure 2: Sine-cosine circuit for Fig.1

In order to trace the solutions curve by SPICE, we replace
the diﬀerentiation with respect to the arc-length s by thetime t. We assume xk as voltages in SPICE. From this, wecan obtain Eq. (8).
)2 ()2p (∑dvidvω
+=1(8)dt
dt
i=1
In this paper, where vi (i = 1, 2, . . . , p) are the coeﬃcientof voltages in Eqs. (1) and (4) and vω corresponds to ω.
They are realized by using diﬀerentiators (simply realizedby capacitors with 1(F) in SPICE). The circuit in Fig. 3
realizes to satisfy the arc-length method Eq. (8). In thisIn Eq. (12), A(t) is the periodic function. We apply theFloquet theory for this periodic function. We write the Jacobian matrix of the periodic solution as Φ(t). From this,the solution after one period from initial value of ∆x(0) isgiven as follows;∆x(T ) = Φ(T )∆x(0).

(13)

Hence, when the eigenvalues (λ1 , λ2 , . . . , λn ) of Φ(T ) satisfy |λk | < 1 (k = 1, 2, . . . , n), the regular periodic solutionx̂ is stable.
3. Illustrative example

Figure 4: Example model
Figure 3: STC (Solution curve-tracing circuit)
circuit, the voltages corresponding to the coeﬃcients areinputted after diﬀerentiated and squared via the voltagecontrolled current source (VCCS). If we set the voltage ofnode a as v̇ω , Iω = v̇2ω can be obtained by multiplier andvoltage-controlled current source VCCS (MVCCS). Then,the node voltage v̇ω is integrated to obtain vω . Note that Rin Fig. 3 is a very large resistance used only to avoid theL − J cut-set.
2.3. Stability of periodic solutions
We suppose that there is a circuit equation asf ( ẋ, x, y, ωt) = 0,(9)and make the variational equation for the regular period
solution of x̂. First, we assume the small change quantityas (∆x, ∆y) as{x = x̂ + ∆x
(10)y = ŷ + ∆yFigure 4 shows a circuit for an illustrative example. The
nonlinear characteristics can be solved by using the SPICE
model in Fig. 5.
We can express the circuit equations as follows;die(t) = Ri + L + αq + βq3dt
dq dt = i(14)If we write the variables as periodic solutions with smallvariations:{i = i0 + ∆i(15)q = q0 + ∆q,
we obtain the following variational equations.

- 375 -d∆ie(t) = R∆i + L+ (α + β3q0 2 )∆q


dt


d∆q



= ∆i
dt(16)

Figure 7: Frequency response of i

Figure 5: SPICE model for Fig.4

Figure 6: Variational circuit for Fig. 5

Figure 8: Frequency response of q

where we neglect higher-order small terms.
From these equations, we can make the variational circuit for Fig. 5. In Fig. 6, q0 is the steady solutions given
asq0 = Qc cos ωt + Q s sin ωt.
(17)
We analyze this circuit and calculate the values of twovariables after one period from two diﬀerent initial conditions; (∆i0 , ∆q0 ) = (1, 0) or (∆i0 , ∆q0 ) = (0, 1). We obtain
4 variational values for Φ(T ).
[]∆i(∆i0 ,∆q0 )=(1,0) ∆q(∆i0 ,∆q0 )=(1,0)Φ=∆i(∆i0 ,∆q0 )=(0,1) ∆q(∆i0 ,∆q0 )=(0,1)
We calculate the eigenvalues of Φ(T ) by using MATLAB
and assess the stability of the periodic solutions.
4. Simulation results
Figures 7 and 8 show the simulation results of the frequency response of i and q, respectively, which are obtained by the sine-cosine circuits and the STC with SPICE.
We set the parameters as follows; Em = 0.35[V], α =1.0, β = 0.8, R = 0.05[Ω], L = 0.1[H]. In this section, we compare and check the our results with the results which obtained by transient analysis of Fig. 4. We

analyze the case ω = 4.78[rad/s], ω = 4.51[rad/s] andω = 4.26[rad/s].
For the validation, we simulate the original circuit. The
simulation results of transient analysis (Fig. 9, Fig. 10 andFig. 11), indicate that ω = 4.78[rad/s] is stable, and thatω = 4.51[rad/s] and ω = 4.26[rad/s] are unstable.
Next, we show the results which obtained by ourmethod. First, we show the solutions for the case ofω = 4.78[rad/s].
[]
0.599
0.012
Φ=−13.897 0.593
Second, we show the solutions for the case of ω =4.51[rad/s].
[]
0.0447 0.149
Φ=−2.831 1.707
Lastly, we show the solutions for the case of ω =4.26[rad/s].
[]
0.4001 0.1209
Φ=0.7348 1.412
Table 1 shows the calculated eigenvalues of Φ for thetwo ω.

- 376 -

Figure 9: Transient analysis of ω = 4.78[rad/s]

Figure 11: Transient analysis of ω = 4.26[rad/s]

pacitors for three diﬀerent conditions of ω which gives both
stable and unstable solutions. Our results agree well withthe previously obtained results. We would like to improve
the proposed method more eﬀectively for the analysis oflarger scale circuit.
Acknowledgment
This work was partly supported by CASIO Science Promotion Foundation
References

Figure 10: Transient analysis of ω = 4.51[rad/s]

We can see that ω = 4.78[rad/s] is stable, because all ofeigenvalues satisfy |λ| < 1. However, for the other patterns,
the solutions are unstable, because one of the two eigenvalues does not satisfy |λ| < 1.
Namely, we can say that our method gives the same results as the results obtained by transient analysis with simpler SPICE-oriented algorithm.
5. Conclusion
We proposed a SPICE-oriented algorithm to assess thestability of periodic solutions for nonlinear circuits. We obtained periodic solutions by using SPICE and we assessed
the stability based on the Floquet theory. In detail, we analyzed the second order resonance circuit with nonlinear ca-[1] F.L. Traversa, F. Bonani and S.D. Guerrieri, “Afrequency-domain approach to the analysis of stability and bifurcations in nonlinear systems described by
diﬀerential-algebraic equations,” International Journalof Circuit Theory and Applications, vol.36, pp.421439, 2008.
[2] T. Kinouchi, Y. Yamagami, Y. Nishio and A.
Ushida, “Spice-oriented harmonic balance Volterra series methods,” Proc. of NOLTA’07, pp.513-516, 2007.
[3] Y. Inoue, “DC analysis of nonlinear circuits using
solution-tracing circuits,” Trans. IEICE, vol.J74-A,pp.1647-1655, 1991.
[4] E. Ikeno and A. Ushida, “The arc-length method forthe computation of characteristic curves,” IEEE Trans.
Circuits Syst., vol.23, pp.181-183, 1976.

Table 1: Eigenvalues of Φ
ω [rad/s]
ω = 4.78
ω = 4.51
ω = 4.26|λ1 |0.5963+0.4093i
0.358
0.319

|λ2 |
0.5963-0.4093i
1.39361.4928- 377 -