Intermodulation Analysis by Spice-Oriented
Harmonic Balance MethodJunji Kawata

Yoshihiro Yamagami

Dept. Mechanical and Electronic Eng.,
Tokushima Bunri University
Email: kawata@fe.bunri-u.ac.jp

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

Yoshifumi Nishio

Akio Ushida

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

Dept. Mechanical and Electronic Eng.,
Tokushima Bunri University
Email: ushida@fe.bunri-u.ac.jp

S UMMARY
Frequency-domain analysis of nonlinear electronic circuits excited by multiple frequency inputs is very important for
designing integrated circuits and communication systems. Volterra series methods are widely used for the analysis [1–3] because
they give the solutions in analytical forms. The algorithms are based on bilinear theorem [1], and they can be effectively applied
to the intermodulation analysis, where nonlinear elements must be described by the polynomial functions.
HB (harmonic balance) method is also well-known in the frequency-domain analysis, which will give good results even for
relatively strong nonlinear circuits. The classical HB methods [4–6] have been applied to relatively small scale circuits andfound many interesting nonlinear phenomena such as bifurcations and chaos. In the HB method, nonlinear elements must be
also described by polynomial functions to obtain the Fourier coefficients. Therefore, the applications of HB method to large
scale circuits is complicated task. For example, if the circuit equation has “n” of polynomial functions and “K” of frequency
components are considered in the analysis, then the dimension of determining equation is “n(2K + 1)”.
In this paper, we propose Spice-oriented HB method combining with MATLAB. Firstly, the nonlinear elements such as bipolartransistors and MOS FET are modeled by the exponential function. Then the functions are approximated by the Taylor series
in the polynomial forms. The input-output relation of Fourier coefficients for the polynomial function can be obtained in thesymbolic forms with the use of MATLAB, and then modeled by ABMs (Analog Behavior Models) of Spice. Therefore, it
is further possible to formulate the determining equation of HB method in the schematic form or net-list of Spice. It can beeasily solved by DC analysis of Spice, and we can get the frequency-domain solutions, efficiently. Our HB method requires
no troublesome tasks such as formulation of the circuit equations and many transformations required to get the determining
equation.
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[4] Y.Ueda, The Road to Chaos-II, Aerial Press. Inc., 2001.
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[6] R.J.Gilmore and M.B.Steer, “Nonlinear circuit analysis using the method of harmonic balance-A review of the Art. Part I. Introductory concepts,” Int.
Jour. of Microwave and Millimeter-Wave Computer-Aided Eng. vol.1, pp.22-37, 1991.
[7] A.Ushida, Y.Yamagami and Y.Nishio, “Frequency responses of nonlinear networks using curve tracing algorithm,”ISCAS 2002, vol.I, pp.641-644, 2002.
[8] A.Ushida, J.Kawata, Y.Yamagami and Y.Nishio, “ Intermodulation analysis of nonlinear circuits using two-dimensional Fourier transformation,”
NOLTA’05, Budges, pp.282-285, 2005.
[9] R.L.Geiger, P.E.Allen and N.R.Strader, VLSI: Design Techniques for Analog and Digital Circuits, McGraw-Hill, 1990.
[10] A.S.Sedra and K.C.Smith, Microelectronic Circuits, Oxford Univ. Press, 2004.
[11] K.K.Clarke and D.T.Hess, Communication Circuits: Analysis and Design, Addison-Wesley Pub. Co., 1971.

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