平成18年度電気関係学会四国支部連合大会

Volterra Series Method versus Harmonic Balance
Method in Frequency-Domain Analysis

17 - 13

T. Kinouchi1
Y. Yamagami1
Y. Nishio1
J. Kawata2(Tokushima University1 Tokushima Bunri University2 )1. Introduction
Distortion analysis of nonlinear circuits is very important for
designing analog integrated circuits and/or communication systems. In this paper, Volterra method is compared with theharmonic balance method. The Volterra series is based on thebilinear theorem, so that the higher order solutions can be foundin a sequential manner, and the harmonic distortions (HDs) aregiven in the analytical forms. However, it can be only applied tothe weakly nonlinear systems, and, furthermore, the nonlinear
characteristics should be described by approximate polynomial
functions at the DC operation points. Our harmonic balancemethod can get the Fourier coefficients in symbolic forms using
MATLAB, and whole the harmonic components are consideredas the variables in the analysis, all together. Thus, the harmonicbalance method can be applied to the strong nonlinear systems.
We will compare the results of the above two methods with asimple example.
2. Volterra series method
To understand the properties of Volterra method, we consider
a simple example shown in Fig.1, where nonlinear resistor isdescribed by the following polynomial function:
23iG = k1 vG + k2 vG+ k3 v G(1)

The first term in (1) is includedin the linear network, so thatthe circuit for computing thefirst order-kernel is given by
the transfer function K(jω) asfollows:

iGEm+vG-LinearNetworkFig.1 Nonlinear resistivecircuit.
V1 (jω) = H1 (jω),

for H1 (jω) = K1 (jω)

(2.1)

. Next, the current for computing the 2nd-order kernel is givenby [1]I2 (jω, jω) = −k2 (H1 (jω))2(2.2)
and we have the 2nd-kernel as follows:
V2 (jω, jω) = H2 (jω, jω),

(2.3)

for H2 (jω, jω) = K2 (jω, jω)I2 (jω, jω). Finally, the current for computing the 3rd-kernel is given byI3 (jω, jω, jω) = k3 (H1 (jω))3 + 2k2 H1 (jω)H2 (jω, jω)(2.4)

V3 (jω, jω, jω) = H3 (jω, jω, jω)(2.5)

for H3 (jω, jω, jω) = K3 (jω, jω, jω)I3 (jω, jω, jω)
Thus, the combining complex solutions up to 3rd-order kernel
is given by2(jω)
V (jω, jω, jω) = H1 (jω)Xin (jω) + H2 (jω, jω)Xin3(jω)+H3 (jω, jω, jω)Xin
K (j1

V1 (j V2 (jK 2 (j j(3)V3 (j  I (j j j32(b) 2nd oder

Therefor, we have g1 = 0.0288, g2 = 0.576, g3 = 7.68
RFig.3 An example
id +id = 10−8 exp(40vd ),vde(t)CR = 1000[Ω], C = 1[µF ],E = 1[V ], e(t) = Em cos ωtEThe Volterra kernels are decided by Fig.4 as follows:REmRRgC

1

V1 (jCV (jg12Cg   j(a) 1st order(b) 2nd oderI 2(jV3 (j1(c) 3rd order

I 3(j j jFig.4 Computations of Volterra kernels.
Then, the harmonic distortions are given byg2
HD2 =|K1 (jω)||K2 (jω, jω)|Em(5.1)
2
12
HD3 = |K1 (jω)|2 |2g2 K2 (jω, jω) + g3 K3 (jω, jω, jω)|Em4(5.2)
where1RK1 (jω) =, K2 (jω, jω) =1 + g1 R + jωCR
1 + g1 R + j2ωCRRK3 (jω, jω, jω) =1 + g1 R + j3ωCR
Note that the computations of HD2 and HD3 are obtained by
Fig. 4(b) and (c), respectively.
3. Our harmonic balance method
In our harmonic balance method, the nonlinear characteristics are described by polynomial forms in symbolic manners.
Namely, the coefficients are functions of the unknown operating
DC point v0 s as follows:
23iG = k0 (v0 ) + k1 (v0 )vG + k2 (v0 )vG+ k3 (v0 )vG(6)Therefore, the Fourier coefficients are also described in the following forms with MATLAB :MXvG (t) = VG,0 +(VG,2k−1 cos kωt + VG,2k sin kωt)

(7.1)

k=0M
XiG (t) = IG,0 +(IG,2k−1 cos kωt + IG,2k sin kωt)

(7.2)(c) 3rd orderFig.2 Computation of Volterra kernels of Fig.1.
Example2.1 Now, consider an example of nonlinear circuit
shown in Fig.3. The DC operating point of diode is v0 = 0.2796,
and the current is expanded by the Taylor expansion as follows:
id = 7.19 × 10−4 + 0.0288vd + 0.576vd2 + 7.68vd3

Using these relations, the Sine-Cosine circuits [2] can be constructed in the schematic forms using ABMs (Analog BehaviorModels) of Spice, which are corresponding to the determining
equations of the harmonic balance method. Thus, it can besolved by the DC analysis of Spice. In this case, we can get
the frequency response curve if we choose the frequency as anadditional variable.
4. Conclusions and remarks
In this paper, we discussed the comparisons between theVolterra series method and our harmonic balance method, andfound that latter can be applied to the strong nonlinear system,
safely. We will show the numerical examples at our presentationof the conference.
References  

K3 (j j j I (j j(a) 1st order

A. Ushida2

k=0

and we have the 3rd-kernel as follows:

Em309

(4)

[1] P.Wambacq and W.Sansen, Distortion Analysis of Analog
Integrated Circuits, Kluwer Academic Pub., 1998.
[2] 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.