TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 



129 



Dropping the primes, we have the Laplacian integral equation 



1 



J0iZ{j(X>) 



-f 



h{t)e-^'^Hl. 



(45) 



Hence (43) is equivalent to Carson's integral equation, if (jco) is 

 replaced by p. 



It will be noted that in deriving this equation use is made only of 

 the general form of the expansion of admittances. For particular 

 admittances, the values of the a's in equation (43) or the Jis in equa- 

 tion (45) can be derived directly from an expansion of the admittance 

 function, as shown in the foregoing work. Hence, if the solution of 



0.1 0.2 0.3 0.4 0.5 0.6 07 0.8 0.9 1.0 I.I 1.2 1.3 1,4 1.6 

 VALUE OF TIME X CRITICAL FREQUENCY = tfc 



1.6 1.7 1.8 1.9 



Fig. 10 — Current resulting from the application of an alternating voltage, 

 E = Eo cos Icjct, 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. 



the integral equation is not known from a table of integrals, one method 

 for obtaining its solution is the expansion method developed above. 

 This method may then have some application as a method for solving 

 integral equations. 



A . Illustrative Example 

 As an illustration of the use of this method in solving integral 

 equations we will consider the equation 



1 



■y[{W+ 2\jco 



a 

 > Jo 



{t)e-'"'dt. 



(46) 



