High Precision Methods Based on Pade Approximation of Matrix Exponent for Numerical Analysis of Stiff-Oscillatory Electrical Circuits

There are many numerical methods for the analysis of processes in electrical circuits and also many computer programs based on these methods, but these programs often reveal considerable differences between simulation results and physical reality. This is primarily due to violations of stability area borders as well as to the low precision of most of the popular methods. This is why there is need for new methods which require little computational work, have high precision and have no limitations on integration step (A-stable). Besides this, they must be able to provide effective damping of a stiff part of the solution, if the number of integration points for its adequate calculation is insufficient (be L-stable). They should be sufficiently simple for widespread use in engineering practice. The author proposed a new set of implicit numerical-analytical methods for simulating transient processes in linear electrical circuits. They are based on the conversion of Pade approximations of the matrix exponent into a form of the simplest fractions. These approximations can be obtained simply and immediately from the scheme of the circuit, like equation systems of the Nodal Voltage Method. In fact, it is a synthesis of the matrix exponent, Nodal Voltage Method and the Method of State Variables. New methods in terms of steadiness and precision are equivalent to Radau and Lobatto methods from the implicit Runge-Kutta group. They are much more simple and require several times less computational work.