Spectral matrix for analysis of time-varying networks

Develops the analysis of networks containing controlled R(t), L(t) and C(t) which can be represented by a Fourier series and may have more than one period. The general solution in the time domain, viz., the solution by linear differential equation with periodic coefficients, and the differential cascading equations for instantaneous component values are discussed. With sinusoidal input voltage the steady state solution of the differential equation leads to an infinite number of equations and unknowns and although a solution can be approximated by considering only a finite number of equations, frequency domain analysis leads to more manageable computational data. Analysis in the frequency domain (steady state conditions) and with multiple periods (yielding modulation products) produces a set of equations similar in form to the cascade equations of a time-invariant passive two-port and the matrix analogous to the cascade matrix is termed the spectral matrix. The latter can be considered as the cascade matrix of a network having infinite input and output ports. Similarity exists between operations on both types of system but since the elements of the spectral matrix are also matrices, the sequence of terms within the elements cannot be changed. After discussing further transmission aspects and considering the determination of the output function for arbitrary input functions the author states that all properties of time-varying networks cant therefore be analysed in the frequency domain by the known methods of network theory. Spectral matrices of various two-ports are tabled.