2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Peak Search for Frequency Analysis of Nonlinear Circuits
Hiroshige KATAOKA, Yoshihiro YAMAGAMI and Yoshifumi NISHIO
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {hiroshige, yamagami, nishio}@ee.tokushima-u.ac.jp

Abstractcharacteristics of the circuits. Computer simulation results
show that the circuits have several peaks on the frequency
For designing PCBs (printed circuit boards), it is very im- characteristics curves depending on the number of the nonportant to find out the locations and the frequencies giving linear elements.
large peak values of the voltages. The case frequency is verylarge, we have to treat the wire line in the PCBs as transmission line. We apply the nonlinear elements for that transmis- 2. Simulation Results of Frequency Responsesion line. In that case, the frequency characteristics become
distorted, compared with linear circuits. If the distortion beIn this article we set the value of resistor, inductor and lincomes large, the characteristics has unstable region. In this ear capacitor as R = 0.01[Ω], L=0.1[H], C1 = C2 = C3 =article, the SPICE-oriented algorithm based on HB (harmonic C4 =1[F]. We set the characteristics of nonlinear capacitorbalance) method, which has been proposed by the authors, is C1 = C2 = C3 = C4 as v(q) = q + 0.8q3 . The value ofapplied to some LRC ladder circuits including nonlinear ca- R = 0.01[Ω] is enough to be high Q circuits in our models.
pacitors in order to show the applicability of the algorithm The results of frequency response in the followings are voltand to investigate complex frequency characteristics of the ages of C4 in all examples.
circuits.
2.1. Ladder circuit with one nonlinear capacitor

1. Introduction
In our algorithm, we apply the sine-cosine circuit [1][2]based on HB (harmonic balance) method. The HB method iswell-known method for the frequency domain analysis. The
Fourier transformation circuit [3] is used to obtain the response of nonlinear elements. Although they may be foundby the standard transient analysis of SPICE, it is diﬃcult tofind the exact peaks when the quality factor (Q) is very large.
Because we may pass over them if we choose a large step
size. In order to avoid this problem, we apply the diﬀerentiator and the nonlinear limiter [4]. Furthermore, the frequency
characteristics of nonlinear circuits often have unstable regions. Such regions cannot be obtained the standard methods
using SPICE. In order to avoid this problem, we apply theSTC (solution trace circuit) [5][6]. By combining the sinecosine circuits, the Fourier transformation circuit, the nonlinear limiter and the solution tracing circuit, we can analyze thefrequency response even if the curve has unstable region andobtain more exact peaks compared with common analysis ofSPICE [7].
In this article, we apply the SPICE-oriented algorithm,
which has been proposed in [7], to some LRC ladder circuits
including nonlinear capacitors in order to show the applicability of the algorithm and to investigate complex frequency

First, we analyze LRC ladder circuit including only one
nonlinear capacitor as shown in Fig. 1. Note that four circuits
in Fig. 1 have nonlinear capacitors in diﬀerent positions. Figure 2 shows the simulation results of peak search and tracing
curve of the frequency characteristics for four ladder circuits
in Fig. 1. The simulation results show that each circuit has
at least one resonance. By combining nonlinear limiter in thealgorithm, we can analyze exact peaks even when the resonance is too sharp. There are not unstable regions in thesecurves.
2.2. Ladder circuit with two nonlinear capacitors
Next, we analyze LRC ladder circuit including two nonlinear capacitors as shown in Fig. 3. The simulation results areshown in Fig. 4. In this case, the curves have two resonances
at least. Although it is unclear in the figure, the resonanceoccurs between 5[rad/s] and 12[rad/s] of ω in Fig. 4(c). All
curves in Fig. 4 have unstable regions. It is interesting thattwo diﬀerent types of resonances exist in the curve. Namely,
almost resonance curves have positive second derivative toreach the peaks, while some have negative like the second
resonance in Fig. 4(e).

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(a)
(b)

(c)

(b)

(d)

Figure 1: LRC ladder circuit with one nonlinear capacitor.
(a) C1 is nonlinear capacitor. (b) C2 is nonlinear capacitor.
(c) C3 is nonlinear capacitor. (d) C4 is nonlinear capacitor.

(c)2.3. Ladder circuit with three nonlinear capacitors
Next, we analyze LRC ladder circuit including three nonlinear capacitors as shown in Fig. 5. The simulation results
are shown in Fig. 6. In this case, the curves have three resonances at least. In Fig. 6(b), the resonance occurs between
8[rad/s] and 10[rad/s] of ω. It is very interesting that thecircuit in Fig. 5(b) has very complex frequency characteristics curve like Fig. 6(b).

(d)

Figure 2: Simulation results for ladder circuit in Fig 1. (a) Results of Fig 1(a). (b) Results of Fig 1(b). (c) Results ofFig 1(c). (d) Results of Fig 1(d).
3. Conclusions
We analyzed 14 patterns of fourth order LRC ladder
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(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: LRC ladder circuit with two nonlinear capacitors. (a) C1 and C2 are nonlinear capacitors. (b) C1 and C3 arenonlinear capacitors. (c) C1 and C4 are nonlinear capacitors. (d) C2 and C3 are nonlinear capacitors. (e) C2 and C4 arenonlinear capacitors. (f) C3 and C4 are nonlinear capacitors.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Simulation results for ladder circuit in Fig 3. (a) Results of Fig 3(a). (b) Results of Fig 3(b). (c) Results of Fig 3(c).
(d) Results of Fig 3(d). (e) Results of Fig 3(e). (f) Results of Fig 3(f).

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Figure 5: LRC ladder circuit with three nonlinear capacitors. (a) C1 , C2 and C3 are nonlinear capacitors. (b) C1 , C2 and C4
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(a)

(b)

(c)

(d)

Figure 6: Simulation results for ladder circuit in Fig 5. (a) Results of Fig 5(a). (b) Results of Fig 5(b). (c) Results of Fig 5(c).
(d) Results of Fig 5(d).

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