Some applications of differential equations in modern electrical circuit problems†

Abstract

A survey is presented on the applications of differential equations in some important electrical engineering problems. A series LCK network is chosen as the fundamental circuit; the voltage equation of this circuit is solved for a number of different forcing (driving) functions including a sinusoid, an amplitude modulated (AM) wave, a frequency modulated (KM) wave, and some exponentials. Some well-known formulas such as the AM—PM conversion mechanism and the derivation of the quasi-stationary approximation in time-invariant LCR networks are discussed. Consideration is given to the circuit containing a periodically varying parameter, i.e. the capacitance of the circuit is linearly time-varying. An introduction to the Mathieu equation is presented in general terms and examples have boon worked out for a number of electrical analogues and in this process the transformation of the Mathieu equation into Hill's equation is also discussed, The mechanism of generation of sub-harmonics is discussed by solving the Mathieu equation in non-linear form. Finally, the solutions of Mathieu equations are discussed briefly in general terms.