About the decrease bulkiness of mathematical model of linear periodically time-variable circuit

Shapovalov Yu.I., Mandziy B.А., Bachyk D.R.

Lviv Polyechnic National University

In this paper are investigated the influence of variables the differential equation describing the linear periodically time-variable circuits in the time domain on decrease of bulkiness such equation. Rules of forming a system of linear differential equations of circuit that provide its acceptable bulkiness are proposed. Communication of voltages and currents on elements of an electric circuit looks like algebraic, differential and integral equations. As showed computer experiments, on the bulkiness of system of the equations, that describe in particular LPTV circuit and its further transformation in time domain, essentially influences presence of integrated expressions in this system. Since the input and output variables are usually given you must to choose such method of formation of system of equations, which would provide absence in it integral expressions. The first rule of formation of the system of equations of circuit is as follows: to provide absence the integral expressions in the system of equations describing a circle in the time domain, as variables in it need to choose the voltage on the capacitor and current in the inductor. One of the perspective methods of forming of equations is the tabular method. By the tabular method in system of equations as variables are selected nodal voltages, currents and voltages on the elements of the circuit, and the equations themselves may be formulated so that the integral expressions were absent. It follows the second rule of formation of system of differential equations which is as follows: in the absence of other requirements, mathematical model of circuit in the time domain advisable to form by the tabular method that provides absence the integral expressions in equations without additional action to remove them. In addition, the tabular method does not impose restrictions on the structure and elements of circuit. Under condition of performance of the presented two rules the system of the equations describing a circuit less bulky than without it. Normally in such systems variables which do not interest the researcher further are eliminated, and the equation in which there are only two variables is formed. Bulkiness of the last equation too can be different, and it depends on what variables left in it. The third rule of forming a differential equation that describes a circle in the time domain and that provides acceptable bulkiness is as follows: despite the fact which variables of circuit is an output variables, in a mathematical model of circuit that was formed by a rule 2, we eliminate all variables, except what correspond to an input signal and a parametric element. If sources of input signal or parametric elements in the circuit a few then such equations necessary to form few - one for each pair of “input - parametric element.” The differential equation that formed by the three rules using method L.A. Zadeh is transferred into the frequency domain and is solved by the
frequency symbolic method. As a result we receive symbolic parametric transfer functions which are a basis for formation of a frequency symbolic model of each parametric element. Such models in turn are a basis for creation of frequency symbolic model of circuit as a whole.
Such frequency symbolic model of circuit contains or determines output signals. Values of output signals of circuit may be are converted to the time domain.