A New Method for Obtaining Transient Solutions of 

 Electrical Networks 



By W. P. MASON 



Synopsis: A new method for obtaining transient solutions of electrical 

 networks is developed in this paper which depends upon the fact that a 

 distortionless line can be made to approach as a limit all three of the circuit 

 elements, resistance, inductance and capacity. The process of solution 

 consists in solving for the current in a distortionless line — which is ordinarily 

 a simple process — and then proceeding to the limiting case of the distortion- 

 less line which approaches the element or elements of interest. Some ex- 

 amples are worked out and a derivation of the Laplacian integral solution 

 is given. It is interesting to note that this method gives a formal solution 

 of the Laplacian integral equation. 



THE following paper sets forth a new method for obtaining the 

 transient solutions of electrical networks, which it is believed 

 has some advantages over other methods of solution, in that the 

 operations required for solution are quite simple, and also because this 

 method presents a more definite physical picture of the processes 

 involved. By means of this method, the current at any time / can 

 be obtained, due to an applied voltage which is zero when / is less than 

 zero, and is Eq cos {wt + 6) when / is greater than zero. This type 

 of voltage includes as a special case the applied voltage, which is 

 zero when t is less than zero, and is unity when / is greater than zero, 

 and hence the solutions obtained by this method reduce to the cases 

 discussed by Heaviside,^ when co and d are taken equal to zero. 



This method gives directly the more compact Laplacian integral 

 equation solution, first obtained by Carson, and in addition gives a 

 method for solving this integral equation, if its solution is not already 

 known. 



I. Method of Solution 



All practical schemes for solving the transient type of circuit 

 problem, including the Laplacian integral equation, and the generalized 

 Fourier integral solution, are made to depend on the known and easily 

 determined steady state solution. This implies that all circuits which 

 have the same steady state solution, have also the same transient or 

 time solution. The method described in the present paper rests on 

 the same basis. 



The method of solution used here depends upon the fact that the 



1 Heaviside, "Electromagnetic Theory," Volume II. 



109 



