Application of the Method of Analysis of Periodic Modes for the Calculation of Transients in Electronic Circuits
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Abstract
Transient analysis is one of the most computationally intensive types of electronic circuits’ analysis. Typically, circuit simulators perform a numerical integration of the mathematical model of a circuit, which consists of a set of differential and algebraic equations, in order to find the transient response of a circuit. The maximum achievable accuracy of the transient solution is limited to the accuracy of the numerical integration method. The integration is performed point-by-point, so the error of the solution is accumulated. This is not the issue for most of the methods of periodic steady-state analysis, however these methods are not suited for the transient analysis.
In our paper we develop a transient analysis technique with usage of the discrete singular convolution method, that is intended for periodic steady-state analysis. Discrete singular convolution method decomposes unknown signals in a circuit into Shannon’s series, and the derivatives of these signals are represented by the derivatives of the series. In this way, the mathematical model of a circuit is transformed into a set of nonlinear algebraic equations, which can be solved using, for example, Newton-Raphson method. The expression for the derivatives approximation can be presented in a matrix form, which leads to simple matrix equations in the mathematical model of the circuit. The derivatives matrix is independent of the unknown signals in the circuit, and it is defined only by the derivative of the Shannon kernel. The coefficients of this matrix do not change through the iterative process for a fixed number of samples. The matrix should be calculated only once, before starting the Newton-Raphson iterative procedure. It should be recalculated only if the number of samples was changed in order to improve accuracy.
In order to apply the discrete singular convolution to the transient analysis problem, an arbitrary aperiodic waveform of the input source is repeated with the period that exceeds the duration of the transient processes in a circuit. In this way, the transient solution is found simultaneously in all time points, so the error is not accumulated as if the solution is calculated point-by-point. As an example of an arbitrary waveform, the Heaviside step function can be used.
As far as the discrete singular convolution method is suitable for nonlinear circuits, the proposed technique is suitable for transient analysis of nonlinear circuits either.
Three examples are given to illustrate the application of the new technique.
Ref. 15, fig. 10.
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