Trigonometric Components of a Frequency-Modulated Wave

The exact solution of the differential equation of a variable-capacitance (or variable-inductance) resonant circuit is given, in a form having a clear physical meaning, and allowing an accurate numerical computation. The explicit expression of the output voltage, as well as the expressions of the charge, and of the current, are written in terms of the two parameters of the nondissipative circuit. The stability of the solutions is discussed, and it is noted that the regions of instability are in number only one-half of those which might be presumed in an investigation of the problem by an approximating Mathieu equation. From the rigorous solutions, approximate expressions are deduced which are valid in the case of small percentages of modulation. The exact results are compared, in a numerical discussion with those of the approximate formulas, as well as with the usual expressions, involving Bessel Functions. The case of the dissipative circuit is discussed briefly, both in the general case and in the case where the dissipative term is comparatively small.