Frequency-domain analysis of memoryless nonlinearities having large-signal, almost periodic excitations

Abstract

Numerical frequency-domain techniques are widely used for the a.c. steady-state analysis of nonlinear electric circuits. Such techniques require that one compute the Fourier series for the response of each nonlinear circuit element, given a known excitation.

Current approaches to this computation encounter difficulty when the response is almost periodic (that is, when the frequencies in its Fourier series are not all harmonically related), especially when the nonlinear characteristic is abrupt and the Fourier series for the response contains many significant terms.

This dissertation develops an alternative approach that is theoretically sound and computationally efficient, for the important special case of a memoryless nonlinearity described by a continuous, bounded function. To begin the development, basic properties of almost periodic functions are presented. It is proven that the response of a memoryless nonlinearity is almost periodic whenever the excitation is. Next, the concept of a basis for a set of frequencies is introduced. The frequency content of the response is investigated, and it is proven that the frequencies in the response have the same basis as those in the excitation.

The Fourier series for an almost periodic function is discussed, and its coefficients are expressed as mean values taken over an infinite interval. Results are given for the summability of the series.

Starting with a theorem from Diophantine Approximation, it is proven that the normalized (Hertzian) phases corresponding to a set of M basis frequencies have their fractional parts uniformly distributed in an M-dimensional unit cube. This property of uniform phase distribution is then used to convert the single-dimensional integral for the Fourier series coefficients into a multiple integral over the unit cube, with the dimension of the integral equal to the number of basis frequencies in the Fourier series.

A multi-dimensional extension of the Discrete Fourier Transform is used to evaluate the multiple integral, and expressions for aliasing are derived. It is shown that the multiple integral formulation compares favorably with existing approaches, and several numerical examples are presented to illustrate this formulation's capabilities.