Multiple Fourier analysis in rectifier problems. II

Abstract

Abstract.The nonlinear problem of the multiple Fourier analysis of the output from a cut-off power-law rectifier responding to a two-frequency input, reviewed in general in Part I of this study [1], is further scrutinized here for the special case of a zero-power-law device; i.e., a bang-bang device or a total limiter. Solutions for the modulation product amplitudes or multiple Fourier coefficients as in Part I appear as Bennett functions, and line graphs of the first fifteen basic functions for the problem are given. The new functions Amnw(h, 7c) studied, being based on a discontinuous device, then, together with the functions Amnn)(h, k) studied in Part I, provide approximate solutions to the two-frequency modulation product problem for an arbitrary piecewise continuous nonlinear modulator, and the solution for this general problem is outlined. Finally, numerical tables of the zeroth-kind functions Amnm (h, k) graphed have been prepared and arc available separately in the United States and Great Britain. As before, the entire theory is based on the original multiple Fourier methods introduced by Bennett in 1933 and1947.

Multiple Fourier analysis for bang-bang controls with memory

“…Thus, for the first four orders of the functions A {^n {h,k) we obtain the following double series expansions: where again c = cos" 1^ -A: cos y), and integrate by elementary methods much as before, but without the necessity to expand in series since these integrals reduce to simple polynomials in /; and k. Thus, for the first four orders of the functions B^n{h,k) we obtain the following polynomial expressions 2(1-/0 In case additional series representations for the Bennett functions AZ Vu k) and A™ n (/?, k) of higher order m + n are needed, the same methods of expansion in series used above can be applied, and the various recurrence relations satisfied by Bennett functions can be used to check the results obtained. More directly, those relations can be used to develop the series for the higher order Bennett functions by assembling suitable additive combinations of the series given here for the first four orders of the functions AZ, (h, k) and A { " n {h, k) (see Bennett [1,2], Sternberg et al [3,[8][9][10] and Kaufman [13] for the required recurrence formulas). No similar recurrence relations for the modified Bennett functions BZ (K k) and B^n\ (h, k) are as yet readily available, but it is worth noting that since the functions BZ,{h,k) = 0 for m < n and the functions B\^n{h,k) = 0 for m <n-\, it follows that more than one-third of all of these functions are zero, thereby substantially reducing the number of additional functions BZ(h,k) or B^n{h,k) required to be computed beyond those evaluated above for any particular applications.…”