New Results in the Calculation of Modulation Products

Abstract

A new method of computing modulation products by means of multiple Fourier series is described. The method is used to obtain for the problem of modulation of a two-frequency wave by a rectifier a solution which is considerably simpler than any hitherto known.T HE problem of computing modulation products has 10Rg been recognized as being of fundamental importance in communication engineering. Heretofore certain quite fundamental modulation problems have been attacked by methods which are difficult to justify from the standpoint of mathematical rigor and some of the solutions obtained have been in the form of complicated infinite series that are not easy to use in practical computations. In this paper these problems are solved by means of a new method which is mathematically sound and whi~h yields results in a form well suited for purposes of computation.The analysis here given applies specifically to the case of two frequencies applied to a modulator of the" cut off" type; i.e., a modulator which operates by virtue of its being insensitive to input changes throughout a particular range of values. A simple rectifying characteristic forms a convenient basis of approximation for study of such modulators, and hence we consider in detail methods of calculating modulation in rectifiers when two frequencies are applied. Applications to certain other types of modulation problems and to the case of more than two applied frequencies are discussed briefly at the close. HALF WAVE LINEAR RECTIFIER-Two ApPLIED FREQUENCIESWe shall define a half wave linear rectifier as a device which delivers no output when the applied voltage is negative and delivers an output wave proportional to the applied voltage when the applied voltage is positive. We may take the constant of proportionality as unity since its only effect is to multiply the entire solution by a constant. Assume the input voltage e(L) to be specified by e(i) = P cos (pt + Op) + Q cos (qt + 0,).(1)The output wave will then consist of the positive lobes of the above function with the negative lobes replaced by zero intervals. It is 228

Voltage and Current Spectra for Matrix Power Converters

“…Our method provides a rather more direct alternative to the usual approach of Black's multiple Fourier series [2,3], and appears to be the first published calculation of the full spectrum. The mathematical expressions involved in the present calculations are considerably more compact than would be the equivalent expressions using Black's method.…”

“…In the engineering literature, spectra for switching devices are generally computed by a multiple Fourier series method usually ascribed to Black [3], but acknowledged to go at least as far back as Bennett [2] (see, for example, [4]). The method involves introducing separate independent variables representing time scaled by each of the input, output, and switching frequencies, then writing the required quantities as multiple Fourier series, in terms of each of these variables separately.…”

Intrinsic Distortion of a Fully Differential BD-Modulated Class-D Amplifier With Analog Feedback

“…in the second line of (23) is more complicated; for this we use the method of Bennett [21], [22]. The detailed derivation is given in the Appendix, with the final expression for g a (t) being…”

Multiple Fourier analysis in rectifier problems. II

“…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. …”