Multiple reactions in inductors

In this paper, the importance of the high-order nested (recursive) reactions that can take place in lossy inductors is investigated. The inquisitive student may ask, “What about the Faraday/Lenz reaction to the ‘first’ Faraday/Lenz reaction?”—a matter about which textbooks are generally silent. It is shown here that the sinusoidal steady-state results for the current in a lossy inductor normally obtained by solving differential equations or doing phasor calculations can be reproduced exactly using Faraday’s/Lenz’s laws recursively, to infinite order. Thus no knowledge of differential equations nor of phasors is required to find the current, and the result concretely demonstrates the connection between Faraday’s/Lenz’s laws and circuit analysis. The case of a single sinusoid is analyzed, which by Fourier’s theorem and linear superposition can in principle be generalized to any excitation. Moreover, the dynamic equations developed can be used directly for any excitation without appeal to Fourier decomposition. A typical nonsinusoidal initial-value problem, involving both transient and driving-function components, is solved as a further demonstration of the technique.