Frequency Symbolic Analysis of Linear Periodically Time-Variable Circuits by Sub-Circuits Method

frequency symbolic method (FS method) of the analysis of the steady-state mode of linear periodically time-variable (LPTV) circuits is intended to form their transmission functions in frequency domain. The transmission functions are approximated by Fourier polynomials and contain a complex variable, time variable and parameters of circuit elements in the form of symbols. The coefficients of such Fourier polynomials by the FS-method are unknown in the symbolic systems of linear algebraic equations (SSLAE), and are defined as their solutions in symbolic form. An obvious disadvantage of solving SSLAE is a significant increase in the required computer time with an increase in the number of unknowns. In research paper, the reduction of SSLAE solution time has been achieved by using one of the sub-circuit methods, namely, the d-trees method. The d-trees method was developed for circuits with constant parameters, so the paper considers its extension, the so-called matrix method of d-trees, to the circuits with variable parameters. The matrix method of d-trees provides close to optimal rendering of similar ones in formed algebraic expressions, which leads to a significant reduction in their formation time. The analysis of the given examples of chain circuits with variable parameters has shown a significant gain in time (hundreds and thousands of times) using the matrix method of d-trees than with standard MATLAB tools. This reduction of time allows to expand significantly the scope of application of the FS method in the problems of statistical research or optimization of electronic devices that are modelled by linear parametric circuits.