2009 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 11-12, 2009

Peak Search and Tracing Algorithm for Frequency
Analysis of Nonlinear Circuits
Hiroshige Kataoka

Yoshihiro Yamagami

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: hiroshige@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: yamagami@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this article, we propose a SPICE-oriented algorithm to analyze frequency characteristics with unstable regionand to find the peaks of the frequency characteristics. By
combining the sine-cosine circuits, the Fourier transformation
circuit, the nonlinear limiter and the solution tracing circuit, thefrequency characteristics curve can be obtained even if the curve
has unstable region.

Fig. 1.

LRC ladder circuit.

I. I NTRODUCTION
For designing PCBs (printed circuit boards), it is very
important to find out the locations and the frequencies giving
large peak values of the voltages. Electrostatic capacity exists
between the wire lines of PCBs. The characteristics of thecapacitors depend on the distance between the wire lines. From
this reason, we have to analyze circuits including nonlinear
capacitors. The SPICE can treat nonlinear elements, however
the standard method of SPICE cannot obtain frequency characteristics of nonlinear circuits easily.
In this study, we propose a SPICE-oriented algorithm toanalyze frequency characteristics of nonlinear circuits and tofind the peaks of the frequency characteristics. Although they
may be found by the standard transient analysis of SPICE, it
is difficult to find the exact peaks when the quality factor (𝑄)is very large. Because we may pass over them if we choose alarge step size. Furthermore, the frequency characteristics ofnonlinear circuits often have unstable regions. Such regions
cannot be obtained the standard methods using SPICE. In
our algorithm, we derive the sine-cosine circuit [1] from thenonlinear circuit. Next, the Fourier transformation circuit [2]is used to obtain the response of nonlinear elements. When
we analyze this circuit with the transient analysis of SPICE,
we may pass over the exact peaks. In order to avoid thisproblem, we apply the differentiator and the nonlinear limiter[3]. Finally, we apply the STC (solution trace circuit) [4] toobtain the frequency characteristics even when the curve has
unstable regions.
II. SPICE-O RIENTED H ARMONIC BALANCE A LGORITHM
A. Sine-cosine transformation
Sine-cosine transformation based on the HB (harmonicbalance) method such that the determining equation is solved
by transient analysis of SPICE. We discuss the sine-cosine

Fig. 2.

Sine-cosine circuit for Fig.1.

circuit corresponding to the determining equation of the HBmethod. If we set the voltage through a capacitor 𝐶𝑣𝐶 = 𝑉𝐶𝑆 sin 𝜔𝑡 + 𝑉𝐶𝐶 cos 𝜔𝑡,(1)

the current 𝑖𝐶 is given by𝑑𝑣𝐶= −𝜔𝐶𝑉𝐶𝐶 sin 𝜔𝑡 + 𝜔𝐶𝑉𝐶𝑆 cos 𝜔𝑡. (2)𝑑𝑡
Thus, the coefficients of sin 𝜔𝑡 and cos 𝜔𝑡 are described by
𝑖𝐶 = 𝐶𝐼𝐶𝑆 = −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.

(4)Then, the voltage 𝑣𝐿 is given by
𝑑𝑖𝐿= −𝜔𝐿𝐼𝐿𝐶 sin 𝜔𝑡 + 𝜔𝐿𝐼𝐿𝑆 cos 𝜔𝑡.
𝑑𝑡
Thus, the coefficients of sin 𝜔𝑡, cos 𝜔𝑡 are described by𝑣𝐿 = 𝐿

𝑉𝐿𝑆 = −𝜔𝐿𝐼𝐿𝐶 ,𝑉𝐿𝐶 = 𝜔𝐿𝐼𝐿𝑆 .

(5)

(6)

Namely, the inductor is replaced by coupled current-controlled
voltage sources in the sine-cosine transformation.

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As an example, Fig. 1 shows a LRC ladder circuit and Fig.
2 shows the corresponding sine-cosine circuits.
B. Fourier transformation circuit
In this study, we use the Fourier transformation circuitin order to realize the nonlinear characteristics of nonlinear
capacitors.
Suppose the input and output waveforms as follows:{ 𝑖(𝑡) = 𝐼 cos 𝜔𝑡 + 𝐼 sin 𝜔𝑡
12𝑣(𝑡) = 𝑉1 cos 𝜔𝑡 + 𝑉2 sin 𝜔𝑡.

Fig. 3.

Nonlinear limiter.

(7)

The characteristics of the electric current which flows througha capacitor can be indicated as𝑖 = 𝑑𝑞/𝑑𝑡 = (∂𝑞/∂𝑣)(𝑑𝑣/𝑑𝑡).

(8)

From Eq. (8), the coefficients for electric charge 𝑞(𝑡) of sin 𝜔𝑡,cos 𝜔𝑡 are described by
11𝑞(𝑡) = − 𝐼2 cos 𝜔𝑡 + 𝐼1 sin 𝜔𝑡.
𝜔𝜔(9)From this, input of Fourier transformation circuit model 𝑖(𝑡)is changed as𝑞(𝑡) = 𝑄1 cos 𝜔𝑡 + 𝑄2 sin 𝜔𝑡,==where∫ 𝑏𝑎∫
1 2𝜋(𝐺(𝑞) sin 𝜔𝑡)𝑑(𝜔𝑡)𝜋 01(𝐺1 sin 𝜃1 + 𝐺2 sin 𝜃2 + . . . + 𝐺2𝐾−1 sin 𝜃2𝐾−1 )𝐾(12)𝐺(𝑞)𝑑(𝜔𝑡) =ℎ(𝐺0 + 𝐺2𝐾 )2+ℎ(𝐺1 + 𝐺2 + . . . + 𝐺2𝐾−1 ), (13)ℎ=𝜋,𝐾Characteristics of nonlinear limiter.

III. P EAK S EARCH A LGORITHM
We search the peaks by using differentiator and nonlinearlimiter. The frequencies 𝜔 at the peak voltages satisfy𝑑∣𝑣(𝜔)∣= 0,𝑑𝜔(10)and from Eq. (8), the characteristics of a nonlinear capacitor
is expressed with an equation using 𝑞(𝑡) and 𝑣 as 𝑣 = 𝐺(𝑞).
We expand 𝐺(𝑞) to Fourier series, and obtain the coefficientsof the voltages by using the trapezoidal formula as follows.
∫1 2𝜋
𝑉1 =(𝐺(𝑞) cos 𝜔𝑡)𝑑(𝜔𝑡)𝜋 011=(𝐺0 + 𝐺2𝐾 ) + (𝐺1 cos 𝜃12𝐾
𝐾+𝐺2 cos 𝜃2 + . . . + 𝐺2𝐾−1 cos 𝜃2𝐾−1 ), (11)𝑉2Fig. 4.

on the characteristics curve. Hence, ∣𝑣(𝜔)∣ need to be firstlydifferentiated by a differentiator. In order to find the exact peakpoints, the output is limited and expanded with a nonlinear
limiter, which consists of a limiter and nonlinear diode asshown in Fig. 3.
We suppose the characteristics of the limiter as follows.
⎧for 𝑣𝑖𝑛 < −𝑉𝐿⎨ −𝑉𝑚𝑎𝑥 :𝑘 𝑣𝑖𝑛:for − 𝑉𝐿 ≤ 𝑣𝑖𝑛 ≤ 𝑉𝐿 ,𝑣𝑎 =(18)⎩𝑉𝑚𝑎𝑥:
for 𝑉𝐿 < 𝑣𝑖𝑛
The output of the nonlinear limiter is given by{𝐼𝑠 exp(𝜆𝑣𝑎 ):for 𝑣𝑎 > 0𝑖𝑜𝑢𝑡 =,−𝐼𝑠 exp(−𝜆𝑣𝑎 ) :for 𝑣𝑎 < 0(15)𝜔𝑡𝑖 = 0, 𝜋/𝐾, ..., (2𝐾 − 1)𝜋/𝐾, 2𝜋.

(16)

(19)

as shows in Fig. 4.
The expansion factor of 𝑘 is large enough. In order tocomplicate an analysis around the peaks, we include nonlinear
diode in nonlinear limiter. Thus, the transient analysis aroundthe zero points is executed with a very small step size, andwe can find out exact peak points.
IV. T RACE FOR UNSTABLE REGION(14)𝐺𝑖 = 𝐺(𝑞(𝑡𝑖 )),(17)

Fig. 5.

- 35 -STC (Solution trace circuit).

Since we set time as frequency, we can not analyze unstableregion. In this section, we explain STC (solution trace circuit)for change a horizontal axis into a voltage 𝑣𝜔 from time(namely, frequency).
STC is based on the arc-length method [5][6]. Thosevoltages are differentiated with respect to the time 𝑡 instead of
the arc-length 𝑠 by using differentiators. They are transformed
to the corresponding voltage sources with current controlled
voltage source (CCVS). Next, each voltage is squared andtransformed to the current source. We have)2𝑝 (∑𝑑𝑣𝑖
𝐼𝑠 =(20)𝑑𝑡
𝑖=1

become inclined as reducing the resistance (this corresponds toincrease the effect of the nonlinearity). Although the unstableregion appears in the case of 𝑅 = 0.01[Ω], we could trace thecurve successfully as shown in Fig. 8. We can also notice thatthe step size around the peak becomes smaller by the effect of
the nonlinear limiter. We should mention that each frequency
characteristics curve could be obtained by a single run of thetransient analysis of SPICE.

as shown in Fig. 5. If we set the voltage of node 𝑎 as 𝑣˙ 𝜔 ,𝐼𝜔 = 𝑣˙ 𝜔2 can be obtained by multiplier and VCCS (MVCCS).
Thus, the additional constant current source in Fig. 5 realize tosatisfying the arc-length method by Eq. (20). Then, the nodevoltage 𝑣˙ 𝜔 is integrated to obtain 𝑣𝜔 . Note that 𝑅 in Fig. 5 isa very large resistance used only to avoid the L-J cut-set.
V. I LLUSTRATIVE EXAMPLE
Fig. 8.

Plot of frequency characteristics for 𝑅=0.01[Ω].

VI. C ONCLUSIONS

Fig. 6.

We have proposed a SPICE-oriented algorithm to analyze
frequency characteristics with unstable region and to findthe peaks of the frequency characteristics. By combining thesine-cosine circuits, the Fourier transformation circuit, thenonlinear limiter and the solution tracing circuit, the frequency
characteristics curve can be obtained even if the curve has
unstable region. The simulation results of the LRC laddercircuit with nonlinear capacitors showed the efficiency of theproposed method.
The analysis of printed circuit boards using our proposed
method is our future research.

LRC ladder circuit including nonlinear capacitor.

R EFERENCESFig. 7.

Sine-cosine circuit for Fig. 7.

As an example we consider the LRC ladder circuit including
nonlinear capacitors as Fig. 6. Figure 7 is the sine-cosine
circuit for Fig. 6. In Fig. 6, 𝐿1 = 𝐿2 = 𝐿3 = 𝐿4 = 0.1[𝐻],and the nonlinear characteristics of 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 are thesame. We set 𝐾 = 10 and the characteristics of the nonlinear
capacitors are 𝐺(𝑞) = 𝑞 + 0.8𝑞 3 . The voltages of the CCVSin Fig. 7 are inputted to the STC. We set 𝑣𝜔 in Fig. 5 as 𝜔for the frequency characteristics.
The simulated results are shown in Fig. 8 for the casesof 𝑅 = 0.01[Ω], respectively. The horizontal axis is 𝜔 andthe vertical axis is the voltage through the nonlinear capacitor
𝐶1 . We name the peaks of the curves as peak1, peak2,peak3 and peak4 from the left. The peaks of the curves[1] T. Kinouchi, Y. Yamagami, Y. Nishio and A. Ushida, “Spice-Oriented
Harmonic Balance Volterra Series Methods,” Proc. of NOLTA’07, pp.513516, 2007.
[2] J. Kawata, Y. Taniguchi, M. Oda, Y. Yamagami, Y. Nishio and A. Usida,“Spice-Oriented Frequency-Domain Analysis of Nonlinear Electronic Circuits,” IEICE Trans. Fundamentals, vol.E90-A, no.2, pp.406-410, 2007.
[3] A. Kusaka, T. Kinouchi, Y. Yamagami, Y. Nishio and A. Ushida, “A
Spice-Oriented Frequency Domain Analysis of Electromagnetic Fields ofPCBs,” Proc. of NCSP’09, pp.526-529, 2009.
[4] Y. Inoue, “DC Analysis of Nonlinear Circuits Using Solution-Tracing
Circuits,” Trans. IEICE(A), vol.J74-A, pp.1647-1655, 1991.
[5] E. Ikeno and A. Ushida, “The Arc-length Method for the Computationof Characteristic Curves,” IEEE Trans. Circuits Syst., vol.23, pp.181-183,
1976.
[6] A. Ushida and L.O. Chua, “Tracing Solution Curves of Nonlinear
Equations with Sharp Turning Points,” Int. J. Circuit Theory Appl., vol.12,
pp.1-21, 1984.

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