Вісники та науково-технічні збірники, журнали
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Item On the Adequacy of the Frequency-Symbolic Method for Linear Parametric Circuits Analysis(Видавництво Львівської політехніки, 2020-02-24) Шаповалов, Юрій; Shapovalov, Yuriy; Lviv Polytechnic National UniversityЧастотний символьний метод (ЧС-метод) аналізу усталеного режиму лінійних параметричних кіл призначений для формування їх передавальних функцій у частотній області. Передавальні функції апроксимуються поліномами Фур’є та містять комплексну змінну, змінну час та параметри елементів кола у вигляді символів. Коефіцієнти таких поліномів Фур’є за ЧС-методом виступають невідомими у символьних системах лінійних алгебраїчних рівнянь (ССЛАР), і визначаються як їх розв’язки у символьному вигляді. Подано спосіб формування апроксимаційного виразу, який забезпечує адекватність обчислень. Наведено приклади та результати комп’ютерних експериментів. Основана на частотному символьному методі система функцій MAOPCs використовується при оптимальному проектуванні електронних пристроїв завадостійких скритних радіотехнічних систем з використанням кодових сигналів.Item Дослідження стійкості параметричних підсилювачів у середовищі MAOPCs(Видавництво Львівської політехніки, 2015) Шаповалов, Ю. І.; Мандзій, Б. А.; Бачик, Д. Р.У пропонованій роботі наведено результати дослідження асимптотичної стійкості одно- та двоконтурного параметричних підсилювачів. Оцінка стійкості у системі функцій MAOPCs основана на визначенні коренів знаменника параметричної нормальної передавальної функції кола, що має вигляд степеневого полінома від комплексної змінної. The paper considers the question of the research of assessment of the stability of linear periodically time-variable circuits by the frequency symbolic method. The function system MAOPCs, which is based on the frequency symbolic method, is an effective tool of investigation of linear periodically time-variable circuits and in particular parametric amplifiers. The assessment of circuit stability in the systemMAOPCs is carried out by the real parts of the denominator roots of a normal parametric transfer function of the inertial part of circuit, which is also defined by the frequency symbolic method in the form of approximation by the Fourier trigonometric polynomials. If the real parts of roots are negative, the circuit is asymptotically stable and is not stable if the real part of at least one of the roots is equal to zero or positive. This criterion of asymptotic stability deservedly gained great importance with the appearance of such an effective method of formation of parametric transfer functions as frequency symbolic method. This paper presents the results of research of asymptotic stability of one- and double-circuit parametric amplifiers. For a single-circuit parametric amplifier with two parametric elements maps of stability for different phase differences of parametric elements were built. When the phase difference is 0 ° zone of stability is the biggest and when the phase difference 180 ° zone of stability is the smallest. These two cases are called in literature a synchronous and asynchronous modes and it is shown that energies brought into the circuit by change of capacitance and inductance in the first case are deducted, and in the second case are attached. This fact has received full confirmation in experiments carried out in the system MAOPCs. In this paper it is shown that the formation of parametric transfer function by the frequency symbolic method and determining the roots of the denominator in which parameters of the circuit set in symbolic form nowadays is the most effective tool of assessment of the asymptotic stability of electronic devices which are represented by linear periodically time-variable circuits. This approach allows you to build the trajectories of roots and maps of stability under multiple change of numerical values of symbolic parameters of circuit. The results of computer experiments presented in this paper made it possible to draw the following conclusions: – complete coincidence of the results between programs MAOPCs and Micro-Cap proves the adequacy of transfer functions formed by frequency symbolic method and high accuracy of assessment of stability through the roots of the polynomial; – the frequency symbolic method allows you to effectively assess stability and to form trajectories of roots or map of stability of circuit by the change of its random parameters that it is sufficiently comfortable at stability control in tasks of statistical character and optimization of parametric devices; – computer experiments have shown that the formation of parametric transfer function by the frequency symbolic method and determining the roots of its denominator nowadays is the most effective tool of assessment of the asymptotic stability of electronic devices which are represented by linear periodically time-variable circuits.Item Tolerance analysis and optimization of linear periodically time-variable circuits based on the frequency symbolic method(Publishing House of Lviv Polytechnic National University, 2014) Shapovalov, Yuriy; Mandziy, Bohdan; Bachyk, DariyaThe paper presents the procedures of tolerance analysis and optimization of linear periodically time-variable circuits, based on the frequency symbolic method and realized by the system of functions MAOPCs in an environment MATLAB. The peculiarity of tolerance analysis and circuit optimization consists in the fact that previously the assessment of asymptotic stability of the investigated linear periodically time variable circuit within the given limits of changes of circuit parameters is carried out and the permissible areas of their change in which the circuit is stable are determined. Exclusively in these areas tolerance analysis and optimization is realized. Розглянуто процедури аналізу допусків та оптимізації лінійних параметричних кіл, які основані на частотному символьному методі та реалізовані системою функцій MAOPCs у середовищі MATLAB. Особливість аналізу допусків та оптимізації полягає у тому, шо попередньо проводиться оцінка стійкості (асимптотичної) досліджуваного лінійного параметричного кола в заданих межах зміни параметрів та визначення областей допустимої їх зміни, у яких коло є стійким. Виключно у цих областях відбувається аналіз допусків та оптимізація.Item Про зменшення громіздкості математичної моделі лінійного параметричного кола(Видавництво Львівської політехніки, 2014) Шаповалов, Ю. І.; Мандзій, Б. А.; Бачик, Д. Р.Досліджено вплив вибору змінних диференціального рівняння, що описує лінійне параметричне коло у часовій області, на зменшення громіздкості такого рівняння. Запропоновано правила формування системи лінійних диференціальних рівнянь кола, що забезпечують її прийнятну громіздкість. 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.