Circuits with totally-fluctuating reactive elements

Summary

Considered here first is a circuit consisting of resistanceR, inductanceL, capacitanceK, in series with a periodically variable capacitanceC(1 + cos ωt), with maximum value 2C and minimum value zero (i.e. open circuit). The exact solution of the Hill-type differential equation for the charge on the condensers is shown to be expressible in terms of associated Legendre functions of general degree and order. However, considerable analytical simplification is effected by appropriate choice of the value of capacitanceK, for then the solution (likewise exact) is in terms of more elementary functions. It is found that the value of the angular frequency ω, in relation to the circuit parameters, influences the character of the solution, which varies considerably with ω. Secondly, a lossless circuit consisting of a variable inductanceM(1+cos ωt), shunted by a fixed inductanceL and a capacitanceC, is considered. As before, appropriate choice ofL simplifies the solution functions. Actually, the circulating charge in the variable-inductance case has the same mathematical expression as that for the current in the variable-capacitance case.