18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

Peak Search Algorithm of Frequency Characteristics
with Unstable Region
Hiroshige Kataoka

Yoshihiro Yamagami

Yoshifumi Nishio

Dept. Electrical and Electronic Eng., Dept. Electrical and Electronic Eng., Dept. Electrical and Electronic Eng.,
Tokushima University,
Tokushima University,
Tokushima University,
Tokushima 770-8506, JAPANTokushima 770-8506, JAPANTokushima 770-8506, JAPANhiroshige@ee.tokushima-u.ac.jp
yamagami@ee.tokushima-u.ac.jp
nishio@ee.tokushima-u.ac.jp

Abstract—For designing PCBs (printed circuit boards),
it is very important to find out the locations and thefrequencies giving large peak values of the voltages.
Electrostatic capacity exists between the wire lines ofPCBs. The characteristics of the capacitors depend on
the distance between the wire lines. From this reason, we
have to analyze circuits including nonlinear capacitors. In
that case, the frequency characteristics become distorted,compared with linear circuits. If the distortion becomeslarge, the characteristics has unstable region. In thisarticle, we propose a SPICE-oriented algorithm to analyze
frequency characteristics with unstable region and to findthe peaks of the frequency characteristics.

I. INTRODUCTION
In this study, we propose a SPICE-oriented algorithm
to analyze frequency characteristics of nonlinear circuits
and to find more exact peak voltages of the frequency
characteristics. Although they may be found by thestandard transient analysis of SPICE, it is difficult tofind the exact peaks when the quality factor (Q) is verylarge. In the SPICE, analysis is every fixed spacing.
Because of this, we may pass over them if we choosea large step size. Furthermore, the frequency characteristics of nonlinear 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][2] from the nonlinear circuit. Next, the Fouriertransformation circuit [3] is used to obtain the responseof nonlinear elements. When we analyze this circuit withthe transient analysis of SPICE, we may pass over theexact peaks. In order to avoid this problem, we apply
the differentiator and the nonlinear limiter [4]. Finally,
we apply the STC (solution trace circuit) [5] to obtainthe frequency characteristics even when the curve has
unstable regions.
Section 2 shows how to use the sine-cosine circuits

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and the Fourier transformation circuit. Section 3 explains
the peak search algorithm with the differentiator and thenonlinear limiter. Section 4 explains the tracing of thefrequency characteristics curve by using STC. Illustrated
example of the proposed algorithm is shown in Sec. 5and Sec. 6 concludes the article.
II. SPICE-O RIENTED H ARMONIC BALANCE
A LGORITHM
A. Sine-cosine transformation
Sine-cosine transformation based on the HB (harmonic balance) method such that the determining equation is solved by transient analysis of SPICE. We discuss
the sine-cosine circuit corresponding to the determining
equation of the HB method. If we set the voltage througha capacitor CvC = VCS sin ωt + VCC cos ωt,(1)

the current iC is given by
iC = C

dvC= −ωCVCC sin ωt + ωCVCS cos ωt. (2)dt

Thus, the coefficients of sin ωt and cos ωt are describedbyICS = −ωCVCC , ICC = ωCVCS .
(3)
Namely, the capacitor is replaced by coupled voltagecontrolled current sources in the sine-cosine transformation of the HB method. In the same way, let the current
through an inductor L be
iL = ILS sin ωt + ILC cos ωt.

(4)Then, the voltage vL is given by114vL = L

diL= −ωLILC sin ωt + ωLILS cos ωt.
dt(5)

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

Thus, the coefficients of sin ωt, cos ωt are described byVLS = −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 a LRC ladder circuit andFig. 2 shows the corresponding sine-cosine circuits.

Fig. 1.

LRC ladder circuit.

Fig. 3.

Fourier transformation circuit.

and obtain the coefficients of the voltages by using thetrapezoidal formula as follows.
∫Fig. 2.

Sine-cosine circuit for Fig.1.

B. Fourier transformation circuit
In this study, we use the Fourier transformation circuitin order to realize the characteristics of nonlinear capacitors. Figure 3 shows the Fourier transformation circuit.
Suppose the input and output waveforms as follows:{ i(t) = I cos ωt + I sin ωt
12v(t) = V1 cos ωt + V2 sin ωt.

(7)

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

(8)

From Eq. (8), the coefficients for electric charge q(t) ofsin ωt, cos ωt are described by
1
1q(t) = − I2 cos ωt + I1 sin ωt.
(9)ωω
From this, input of Fourier transformation circuitmodel i(t) is changed asq(t) = Q1 cos ωt + Q2 sin ωt,(10)

and from Eq. (8), the characteristics of a nonlinear
capacitor is expressed with an equation using q(t) andv as v = G(q). We expand G(q) to Fourier series,

26-28 May 2010, Dresden, Germany

1 2π
V1 =(G(q) cos ωt)d(ωt)π 011π=(G0 + G2K ) + (G1 cos2KKK(2K − 1)π2π+ . . . + G2K−1 cos),+G2 cos
KK∫(11)1 2π
V2 =(G(q) sin ωt)d(ωt)π 0π2π1(G1 sin+ G2 sin=KKK(2K − 1)π). (12)+ . . . + G2K−1 sin
Kwhere∫ bhG(q)d(ωt) =(G0 + G2K )2a+h(G1 + G2 + . . . + G2K−1 ),π, Gi = G(q(ti )),h =K
ωti = 0, π/K, ..., (2K − 1)π/K, 2π. (13)
K is a number which divides the region of a to b.

III. P EAK S EARCH A LGORITHM
We search the peaks by using differentiator and nonlinear limiter. The frequencies ω at the peak voltages
satisfyd|v(ω)|
= 0,
(14)
dω115

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

on the characteristics curve. Hence, |v(ω)| need to be
firstly differentiated by a differentiator. In order to findthe exact peak points, the output is limited and expandedwith a nonlinear limiter, which consists of a limiter andnonlinear diode as shown in Fig. 4.

Fig. 6.

Fig. 4.

Nonlinear limiter.

We suppose the characteristics of the limiter as follows: −Vmaxva =kvin Vmax:::

for vin < −VL
for − VL ≤ vin ≤ VL , (15)for VL < vin

The output of the nonlinear limiter is given by{iout =Is exp(λva ):−Is exp(−λva ) :for va > 0,for va < 0(16)as shows in Fig. 5.

STC (Solution trace circuit).

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 and transformed to the current
source. We haveIs =)p (∑dvi 2
i=1dt(17)as shown in Fig. 6. If we set the voltage of node a asv̇ω , Iω = v̇ω2 can be obtained by multiplier and voltage
controlled current source VCCS (MVCCS). Thus, theadditional constant current source in Fig. 6 realize tosatisfying the arc-length method by Eq. (17). Then, thenode voltage v̇ω is integrated to obtain vω . Note that Rin Fig. 6 is a very large resistance used only to avoid
the L-J cut-set.
In this study, vω in Fig. 6 is equal to ω in the main
circuit. The value of new vω depends on voltages of main
circuit, and the voltages depend on the last vω . Repeating
the calculations, the STC traces the value of ω .
V. I LLUSTRATIVE EXAMPLE

Fig. 5.

Characteristics of nonlinear limiter.

The expansion factor of k is large enough. In order tocomplicate the analysis, we include nonlinear diodes innonlinear limiter. Thus, the analysis near the zero points(an input of Fig. 5) is executed with a very small step
size. In our algorithm, the nonlinear limiter is connected
to differentiator. Input of the nonlinear limiter is equal
to the slope of the voltage wave. From this reason, wecan analyze the curve finely around peak voltages.

Fig. 7.

LRC ladder circuit including nonlinear capacitor.

IV. T RACE FOR UNSTABLE REGION
Since we set time as frequency, we can not analyze
unstable region. In this section, we explain STC (solutiontrace circuit) for change a horizontal axis into a voltagevω from time (namely, frequency).
STC is based on the arc-length method [6][7]. Thosevoltages are differentiated with respect to the time tFig. 8.

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Sine-cosine circuit for Fig. 7.

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

As an example we consider the LRC ladder circuit
including nonlinear capacitors as Fig. 7. Figure 8 is thesine-cosine circuit for Fig. 7. In Fig. 7, L1 = L2 =L3 = L4 = 0.1H , and the nonlinear characteristics ofC1 , C2 , C3 , C4 are the same. We set K = 10 and thecharacteristics of the nonlinear capacitors are G(q) =q+0.8q 3 . The voltages of the CCVS in Fig. 7 are inputtedto the STC. We set vω in Fig. 6 as ω for the frequency
characteristics.

respectively. The horizontal axis is ω and the verticalaxis is the voltage through the nonlinear capacitor C4 .
We name the peaks of the curves as peak1, peak2, peak3
and peak4 from the left. We can see that the peaks of thecurves become inclined as reducing the resistance (thiscorresponds to increase the effect of the nonlinearity).
Although the unstable region appears in the case of
R = 0.01Ω, we could trace the curve successfully as
shown in Fig. 11. We can also notice that the step sizearound the peak becomes smaller by the effect of thenonlinear limiter. We should mention that each frequency
characteristics curve could be obtained by a single runof the transient analysis of SPICE.
VI. C ONCLUSIONS

Fig. 9.

We have proposed a SPICE-oriented algorithm toanalyze frequency characteristics with unstable region
and 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, the frequency characteristics curve canbe obtained even if the curve has unstable region. The
simulation results of the LRC ladder circuit with nonlinear capacitors showed the efficiency of the proposed
method. The analysis of printed circuit boards using ourproposed method is our future research.

Plot of frequency characteristics for R=0.1Ω.

R EFERENCES

Fig. 10.

Plot of frequency characteristics for R=0.05Ω.

Fig. 11.

Plot of frequency characteristics for R=0.01Ω.

[1] T. Kinouchi, Y. Yamagami, Y. Nishio and A. Ushida, “SpiceOriented Harmonic Balance Volterra Series Methods,” Proc. ofNOLTA’07, pp.513-516, 2007.
[2] T. Tang and M.S. Nakhla, “Analysis of High-Speed VLSI Interconnect Using the Asymptotic Waveform Evaluation Technique,”IEEE Trans. Computer-Aided Design, vol.11, pp.341-352, 1992.
[3] 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.
[4] A. Kusaka, T. Kinouchi, Y. Yamagami, Y. Nishio and A. Ushida,“A Spice-Oriented Frequency Domain Analysis of Electromagnetic Fields of PCBs,” Proc. of NCSP’09, pp.526-529, 2009.
[5] Y. Inoue, “DC Analysis of Nonlinear Circuits Using SolutionTracing Circuits,” Trans. IEICE(A), vol.J74-A, pp.1647-1655,
1991.
[6] E. Ikeno and A. Ushida, “The Arc-length Method for theComputation of Characteristic Curves,” IEEE Trans. Circuits Syst.,
vol.23, pp.181-183, 1976.
[7] 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.

The simulated results are shown in Figs. 9, 10 and
11 for the cases of R = 0.1Ω, 0.05Ω and 0.01Ω,

26-28 May 2010, Dresden, Germany117