Calculation of Transients in Linear Electrical Circuits Using the Pade Approximation of a Matrix Exponent

The article considers the application of a new family of implicit high-precision numerical methods to the calculation of transients in linear electrical circuits. The methods are applicable for calculating processes in conventional circuits, as well as in circuits with rigid and oscillatory systems of equations and in circuits with frequent switching. The basis of the methods is the Cauchy formula, in which the matrix exponent is represented as a Pade approximation. This approximation is decomposed into the simplest matrix fractions. As a result, simple formulas are obtained in which the matrices are formed directly from the circuit diagram, similar to the way it is done in the nodal potential method. As a result, to calculate the state of the circuit at the next step of integration, only the parameters of the circuit elements, the circuit incidence matrix, as well as the voltages and currents of energy sources specified within the integration step by their values or coefficients of local polynomials are needed. By using the new methods, it is possible to calculate processes in circuits with constant parameters and with piecewise constant parameters fixed within each integration step. The methods feature simplicity, versatility and low computational costs.