A new method for introducing relaxed initial conditions in transient problems

In the calculation of transients in linear electrical and mechanical systems by the classical methods of differential equations, the constants of integration which arise in the solution are ordinarily evaluated from the initial values of the dependent variable and its derivatives.^1,2 The process of finding the initial derivatives is sometimes difficult for even rather simple systems, and the difficulties are particularly pronounced when the system of equations has undergone a change of variable to remove variable coefficients, as for example in Park´s equations for the synchronous machine.³ The purpose of this paper is to present a new method by which the initial derivatives can be found for an initially relaxed system (all currents in inductances and all charges on condensers initially zero) by a manipulation of the final differential equation itself. The solution for a nonrelaxed system can be found from the relaxed solution by superposition or by means of a systematic change in the original differential equations themselves, as is commonly done in operational calculus.^1,4,5