TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 127 



where h{Q) is a constant denoting the current when / is zero. Now i 

 at any time / is the indicial or direct current admittance, designated 

 by //(/), hence a{t) is 



«(0 =jtW)). 



The infinite integral equation (43) takes the form 



d 



z(k=[X"^^^'^^^^^"''^+'^'0 



(44) 



This integral equation does not have quite the same form as Carson's 

 integral equation but can be readily put into that form by means of 



0.1 0.2 03 0.4 Q5 0.6 0.7 0.8 Q9 1.0 I.I 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 

 VALUE OF TIME X CRITICAL FREQUENCY = -tfc 



Fig. 9 — Transient current resulting from the application of an alternating voltage, 

 E = £o_cos Icoct, on several sections of lattice network. The current plotted is the 

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

 twice the resonant frequency, fc, of the network. 



Borel's theorem ^ which is given below. Suppose that 1/Z' and 1/Z" 

 are two admittances, which when multiplied together give the admit- 

 tance l/II. The admittances 1/Z' and 1/Z" have the expansions 



^, = Lao" + a/'e-^ycoz. + 02"^-^'"^ +•••]• 

 6 See B. S. T. J., October 1925, page 722. 



