Romaniuk R. Advanced Frequency Symbolic Method for Analyzing LPTV Circuits in MATLAB

In this dissertation, a relevant problem is solved, namely the elimination of the significant drawback of exponential growth in analysis time for complex LPTV circuits with periodically time-varying parameters as their complexity increases. Approaches for solving the resulting lower-order subproblems are proposed. The methods developed within this dissertation provide the possibility of mathematical modeling of LPTV circuits with tens, hundreds, and, in some cases, thousands of nodes. One of the factors contributing to the exponential increase in the complexity of analyzing LPTV circuits is the transition from a symbolic system of linear differential equations to its representation in the frequency domain as a symbolic system of linear algebraic equations. For the first time, polyharmonic matrix models of LPTV circuit elements are employed, where each matrix describes an individual element in the frequency domain. These mathematical models are generated programmatically only once, stored in a model library, and reused according to the modeling task for various LPTV circuits without requiring regeneration. Owing to the use of polyharmonic matrix models, a method for forming a system of linear algebraic equations directly in the frequency domain has been developed. The block matrix method has been improved to form the system of equations describing the entire LPTV circuit directly in the frequency domain using the polyharmonic matrix models of the elements. This approach makes it possible to reduce (practically equalize) the computational time required to form the system of equations of a complex LPTV circuit to that required for forming the equation system of the same circuit assuming time-invariant parameters. Another factor contributing to the exponential growth of computation time in the analysis of complex LPTV circuits is the presence of high-order matrices whose dimensions depend on the number of circuit nodes. To reduce the analysis complexity of LPTV circuits and the computational time required to form the transfer function, the subcircuit method was applied, in particular the D-Trees method. Since the D-Trees method can only be applied to systems of linear algebraic equations formulated using the nodal voltage method, the element matrices in such systems are conductance matrices. In circuits containing inductances, this formulation leads to systems of linear integrodifferential equations with integral terms. However, Zadeh’s transformation can be applied only to systems of linear differential equations. It is shown that the traditional differentiation of the LPTV circuit equation system is ineffective, as it usually increases the system order and introduces additional difficulties associated with the impossibility of eliminating the integrals. Instead, the dissertation considers the application of a variable substitution to the symbolic system of linear integrodifferential equations, which makes it possible to eliminate undesirable integral terms without increasing the complexity of the equation system. For validation purposes, several software modules have been developed: • a function for generating polyharmonic matrix models for conductance, capacitance, and inductance, with or without applying the variable substitution method; • a function for forming the system of linear algebraic equations of a complex LPTV circuit in the frequency domain using the block matrix method; • a function for converting trigonometric approximations into exponential form and computing the output voltage using partial solutions of Cramer’s method; • a function for applying the D-Trees method to a model of a long parametric transmission line; A model of a lossy transmission line with parametric elements has been investigated. The influence of different numbers of sections on the harmonic content of the signal has been analyzed: an even number of sections leads to the suppression of certain harmonic components, whereas an odd number results in their amplification. A wide range of applications using a single model has also been demonstrated, including frequency mixing and amplification. A computational study of a parametric long transmission line model with up to 128 sections was performed, exceeding the capabilities of the UDF MAOPCs 2018 system. It was found that increasing the number of harmonic components in the transfer function approximation reduces the deviation of the results from those obtained using the numerical simulation program Micro-Cap 12 (to less than 0.02% with three harmonic components). Parametric modulation of inductances provides an 18% increase in the output signal compared to a constant-parameter long-line model. The amplitude–frequency characteristic of the parametric long-line model was obtained, enabling analysis of the influence of the input signal frequency on the transfer properties of the circuit.