Exact Solution of a Time-Varying Capacitance Problem

By means of a new method, a closed-form solution is obtained for the harmonics generated by a sinusoidally varying capacitance in series with a fixed resistor and battery. The solution describes the behavior of the condenser microphone, the vibrating-reed electrometer, a vibrating plate contact potential measuring apparatus, and a special loudspeaker improvement. With only minor modifications the solution can also apply to the case of a sinusoidally varying resistance in series with a fixed inductance and battery; thus, it may, in addition, be usedl to caculate the response of a carbon microphone. The present large-signal solution, which applies for any finite values of the modulation index and frequency, is compared with previous small-signal approximate results, and the dependence on modulation index and frequency is investigated for such quantities as output waveform, total harmonic distortion, harmonic amplitude and phase, and average input and output power. A very distorted waveshape is obtained for low relative frequencies and values of the modulation index near and including unity.