平成19年度電気関係学会四国支部連合大会

391

17-61

Spice-Oriented Harmonic Balance
Method in Frequency Domain
T. Kinouchi1 Y. Yamagami1 Y. Nishio1 J. Kawata2 A. Ushida2(Tokushima University1 Tokushima Bunri University2 )1. Introduction
For designing integrated circuits and communication
systems, it is very important to analyze the frequencydomain characteristics of nonlinear circuits. There are two
typical types of frequency-domain analysis of nonlinear circuits. One is Volterra series method which is widely usedin European countries, because the algorithm is based on
the bilinear theorem and the higher order solutions can beobtained in the analytical forms. However, it can be only
applied to weakly nonlinear systems, and, furthermore, thenonlinear characteristics should be described by approximated polynomial functions at the DC operation points.
The other is the harmonic balance (HB) method. In thisstudy, we analyze a base modulation circuit using Spiceoriented HB method. The proposed HB method is thetechnique that Fourier coefficients can be obtained in symbolic forms by using MATLAB. The determining equationof HB circuit is given by the net-lists, which can be solvedby the DC analysis of Spice. Thus, we can obtain a goodsolution for the strong nonlinear systems.

Fig. 1 Base modulation circuit.
R1 = 50 [kΩ] , R2 = 10 [kΩ] , R = 1 [kΩ] ,C = 1 [nF ] , C0 = 1 [nF ] , L = 10 [µH] , E = 10 [V ] ,e1 (t) = 0.01 sin ω1 t, e2 (t) = 0.01 sin ω2 t.
Table 1 Frequency components for input.

2. Proposed harmonic balance method
We consider a base modulation circuit driven by two inputs as shown in Fig. 1. We consider here 8 intermodulation frequencies given in Table 1. In our harmonic balancemethod, we have modeled the transistor by the Ebers-Mollmodel where
id = 10eλvd , for Is = 10−12 [A], λ = 40, α = 0.99.

(1)Firstly, we approximate the nonlinear characteristics ofthe diode by Taylor expansion at vd0 as follows:

3. Simulation result
We show the analysis results in Fig. 2.
6[V]
52

4
3
2

23iG = k0 (vd0 ) + k1 (vd0 )vG + k2 (vd0 )vG+ k3 (vd0 )vG(2)   

11

The coefficients are functions of the unknown operating
DC point vd,0 . We approximate the input voltage of thediode using Fourier expansion as follows:

02010(Vd,2k−1 cos νk t + Vd,2k sin νk t)

(3)

k=1

[V]

id (t) = Id,0 +8X

(Id,2k−1 cos νk t + Id,2k sin νk t)(4)

These calculations are so troublesome by hand. Hence,
we apply MATLAB and obtain the terms in the formulasymbolic forms. Thus, we have the Sine-Cosine circuit thatis equivalent to the HB determining equation. It can beconstructed in the schematic forms using ABMs (AnalogBehavior Models) of Spice. Thus, the circuit can be solvedby the DC analysis of Spice, and the frequency response isobtained.

30

2

12



12

1

2

12

2

00Fig. 2

k=1[MHz]0.04

0.02

Substituting (3) into (2), the Fourier series are describedas follows:20

Frequency characteristic
Harmonic balance
1vd (t) = Vd,0 +2

12

0.08

0.06

8X

Frequency characteristic
Harmonic balance1020[MHz]30

Frequency characteristic

4. Conclusions and remarks
In this study, we proposed a Spice-oriented HB method
which could efficiently trace the characteristic curves suchas frequency responses. As the future work, in order to apply our method to large scale networks, we need to develop
device HB modules for bipolar transistors and MOSFETs.
References[1] A. Ushida, Y. Yamagami and Y. Nishio, “Frequency responses of nonlinear networks using curve tracing algorithm,” Proc. of ISCAS’02, vol.I, pp.641-644, 2002.