TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 125 



the network. In addition, the sohuion for infinite frequency is 

 readily obtained from (39) since for this frequency <p = 90°. Then 



E 

 ^ ^ 2R '-^^^ (w/ + — imr)']. 



01 0.2 0.3 0.4 0.5 Q6 0.7 0.8 0.9 1.0 II 1.2 1.3 1.4. 1.5 1.6 

 VALUE OF TIME X CRITICAL FREQUENCY = tfc 



Fig. 7 — Transient current resulting from the application of an alternating voltage, 

 E = Eo cos coj, for several sections of lattice network. The current plotted is the 

 current in the termination of the network. The frequency of the applied voltage is 

 the resonant frequency, fc, of the network. 



III. Laplacian Infinite Integral Equation and Its Formal 



Solution 



The solution of circuit problems by means of the Laplacian integral 

 equation has been used by Carson ^ to a large extent. It is interesting 

 to note that this integral form can be derived in a simple manner by 

 means of this expansion method, and that this method provides a 

 means for solving the Laplacian integral equation. 



Any impedance Z is made up of resistances, inductances and capaci- 

 ties, and hence the current i can be represented by a series 



i =§= Elao + aie-^2-^ + aoe"''*"^ + • • • + a„e-'-""^+ • • •]• (-11) 



The interpretation of this expansion from a physical standpoint 

 is that the current is JSao, for the first inter^-al of time ID, 

 £[ao + aie~-''2'*'^] for the next interval of time 2D, etc. Hence at 

 the time / = n{2D), the current i will be given by the sum of n terms 



^ See "Electric Circuit Theory and the Operational Calculus," B. S. T. J., 

 October 1925, and following. 



