Harmonic balance (HB) method is well known principle for analyzing periodic oscillations on nonlinear networks and systems. Because the HB method has a truncation error, approximated solutions have been guaranteed by error bounds. However, its numerical computation is very time-consuming compared with solving the HB equation. This paper proposes an algebraic representation of the error bound using Grobner base. The algebraic representation enables to decrease the computational cost of the error bound considerably. Moreover, using singular points of the algebraic representation, we can obtain accurate break points of the error bound by collisions.

This paper presents efficient applications of invariants to harmonic balance (HB) methods using Grobner base. The Grobner base is a powerful tool based on ideal theory. Using the Grobner base, we can obtain the solutions of the HB equation. However, its computation is very time-consuming when the equation has equivalent different solutions based on symmetries of the system. We show that invariants enable to transpose the equivalent different solutions to a unique solution. The bifurcation diagram of the invariant is simpler than the original bifurcation diagram, and its computation is considerably decreased. Further, we can obtain the relation among the amplitudes of each frequency component using the invariants. We propose a method for finding the circuit parameters using the amplitude relation.

Distortion analysis of nonlinear circuits is very important for designing analog integrated circuits and communication systems. In this letter, we propose an efficient frequency-domain approach for calculating frequency response curves, which is based on HB (harmonic balance) method combining with ABMs (Analog Behavior Models) of Spice. Firstly, nonlinear devices such as bipolar transistors and MOSFETs are transformed into the HB device modules executing the Fourier transformations. Using these modules, the determining equation of the HB method is formed by the equivalent sine-cosine circuit in the schematic form or net-list. It consists of the coupled resistive circuits, so that it can be efficiently solved by the DC analysis of Spice. In our algorithm, we need not to derive any troublesome circuit equations, and any kinds of the transformations.

We consider oscillators consisting of a reactance circuit and a negative resistor. They may happen to have multi-mode oscillations around the anti-resonant frequencies of the reactance circuit. This kind of oscillators can be easily synthesized by setting the resonant and anti-resonant frequencies of the reactance circuits. However, it is not easy to analyze the oscillation phenomena, because they have multiple oscillations whose oscillations depend on the initial guesses. In this paper, we propose a Spice-oriented solution algorithm combining the harmonic balance method with Newton homotopy method that can find out the multiple solutions on the homotopy paths. In our analysis, the determining equations from the harmonic balance method are given by modified equivalent circuit models of "DC," "Cosine" and "Sine" circuits. The modified circuits can be solved by a simulator STC (solution curve tracing circuit), where the multiple oscillations are found by the transient analysis of Spice. Thus, we need not to derive the troublesome circuit equations, nor the mathematical transformations to get the determining equations. It makes the solution algorithms much simpler.