Using symbolic-topological method for stability analysis of linear circuits

Stability analysis of a linear circuit is an important task. The stability theory of linear circuits historically went parallel to the mathematical concept of stability, but later the stability theory of linear circuits was consolidated in the automatic control theory (ACT) and is used practically unchanged in the electrical circuits theory (ECT). The ACT using, on the one hand, allows the application of stability theory to a wide range of problems (mechanical, thermal, economic, etc.), but ACT, unfortunately, does not take into account all the features of an electrical circuit. Therefore, it is very important to consider the linear circuit stability from the position of ECT, i.e. to consider the use of topological methods for stability analysis. This will allow us to expand the stability theory of linear electric circuits. The article discusses the possible advantages of symbolic-topological methods using (the circuit determinants method and the «Bode surface» using) in the stability analysis of linear electrical circuits with negative feedback, using various stability criteria. The possibility of numerical determination of natural frequencies of an electric circuit using the «Bode surface» is shown. The possibility of measuring the frequency characteristics of the Mikhailov hodograph in a linear electric circuit is shown. Additional capabilities of the Routh–Gurvits and Mikhailov criteria are shown when using the circuit determinants method – the stability criteria are converted from numerical to analytical. A complete formula for finding the circuit determinant of a linear electric circuit with feedback has been obtained. Various criteria in the selection of a cross-section for breaking the negative feedback loop for the correct loop gain determination are shown. A formula for loop gain measuring in any place of linear electric circuit is obtained. A new meaning of the basic formulas of the circuit determinants method is shown from the point of view of circuit stability.