The solution of a certain differential equation. Its usefulness in a particular application of silicon-carbide resistors

Abstract

The paper is concerned primarily with developing a particular method of obtaining an approximation to the periodic solution of the equation ∫ay3dx+By = sin ωx. This equation arises in the problem of a capacitor in series with a resistor having a characteristic iau3, when connected to a source of simple harmonic voltage. The problem is first converted into a modified one which is as follows: If the voltage across the resistor is V2[sin ωt+b sin(3ωt+λ)], find b and λ so that the necessary input voltage shall not contain a third-harmonic component.Preliminary consideration shows that, in either the original or the modified problem, the waveform of the voltage across the non-ohmic resistor cannot be symmetrical about its mid-ordinate (i.e. λ cannot be zero in the modified problem) and this is equivalent to the statement that in any alternating-current tests the non-ohmic resistor must appear to have a capacitor in parallel with it, in addition to any effect which may arise from the intrinsic capacitance of the resistor.The values of the constants b and λ are calculated as functions of the ratio V1/V2 where V1 is the voltage across the capacitor; they are displayed graphically in Figs. 3–6. It is shown that there is a solution to the modified problem for all capacitances in series with a given resistor.In Section 5 the waveform of the necessary input voltage is calculated for a range of values V1/V2 between 0.2 and infinity, and it emerges that, even in the most extreme cases, no harmonic in the applied voltage (from which the third harmonic is excluded by the statement of the modified problem) can be as large as 2½%: in cases which are likely to be of practical interest in a real application the fifth harmonic is unlikely to exceed 1½%, the seventh less than 0.3% and all higher harmonics very minute indeed. This establishes that the solution of the modified problem is a very close approximation to that of the original problem, in which the applied voltage is precisely simple harmonic.This method of approach has been developed in the hope that it might be fruitful in other non-linear problems which arise in engineering science and which are, for practical purposes, more worthy of close consideration than the particular one which has been chosen here to illustrate a possible method.Sections 7 and 8 are devoted to an experimental investigation of the behaviour of a capacitor in series with a non-ohmic resistor of the silicon-carbide type when connected to a source of applied voltage whose waveform was very nearly simple harmonic. The characteristic of such resistors is not precisely a cubic and is more nearly of the form i=vn, where n is likely to have a value between 4 and 5. It is of practical interest to explore whether the solution which is correct for the particular case n=3 is a reasonably close approximation in circumstances when the appropriate value of n is about 4.5. The test consisted in measuring the third-harmonic component of voltage across the diagonal of a simple bridge network which had been balanced for the fundamental component. These tests con firmed all the general phenomena which had been disclosed by the algebraic solution for the particular case n=3, and there was close quantitative agreement provided that V1/V2 was not in excess of unity, i.e. in all those cases where the impedance of the series capacitor was important but not overwhelming in comparison with that of the silicon-carbide resistor.