Computation of the Solutions of (1+2ϵ cos2z)y″+θy = 0; Frequency Modulation Functions

摘要

This equation pertains to a loss-free frequency modulation circuit. Floquet''s theory is applied to obtain stable solutions when θ, Є are outside the range of the approximation solution given by Carson in 1922. A method is described by means of which solutions may be computed to any degree of accuracy. The necessary formulae associated with the solution are derived, and convergence is discussed. A numerical example, illustrating the procedure to be followed, is worked out in detail. An appropriate normalization of the solutions is suggested in order to obtain standard frequency modulation functions. It is shown how the analysis may be extended to cover the case of the equation d2Q/dt2+(R/L)(dQ/dt)+[Q/LC0(1+2Є cos2ω1t)]=0, which applies to a frequency modulation circuit having constant resistance R.