Abstract :
A theory of N-component operators is given for steady-state calculations in linear circuits with inputs of periodic, but non-sinusoidal, form. If the input waveforms have a finite number, M, of harmonic components, N may be chosen such that an exact theory results. The theory supplements an approximate one of the same nature recently published.4 The central feature of the method is the decomposition of any periodic function with M harmonic components into a number of samples of the same form, but of magnitudes equal to the ordinates of the function at equal intervals ? throughout the period, and displaced in the time-axis by successive intervals ?. By appropriate choice of the sampling function, an exact representation may be obtained. A shift operator u is now introduced which translates any periodic function one interval ? to the right. By this means it is possible to express a periodic function with M harmonics as a product of a polynomial in u and the basic sampling function. This polynomial in u, of order N ? 1, gives an N-component operator representing the waveform. The representation is, in a sense, analogous to that used in vector algebra, where an arbitrary sinusoidal wave may be written as the product of a polynomial in j, such as a + jb, with a basic sinusoidal reference wave. The theory gives rise to the fundamental results uN = 1 and uN = ?1 for general, and odd harmonic only, waveforms respectively, and using these a standard form for an N-component operator may be obtained. Differential and integral operators are developed and rules are given for obtaining the N-component operator corresponding to an impedance, admittance or transfer function. The algebra of N-component operators is summarized and the solution of a simple circuit examined by way of illustration. The process of inverting an operator requires the solution of N simultaneous equations, and an alternative method of procedure is considered. Simple expressions result for power and r.m.s. va- lues. The application to linear circuits is advantageous in the case when the input is known graphically or numerically and a similar description of the output is desired. It is therefore an alternative method of procedure to Fourier analysis, calculation of the responses to the harmonic components, and subsequent point-by-point plotting of the output. The theory may also be applied to the solution of non-linear circuits by continued approximation, but, as this technique is the same as for the approximate operators already dealt with, it is not considered in the paper.
