. The Bell System technical journal . gral a{t) = f a{T)a{t - T)dr,Jo where t = k/2D, and the complete relation is 1 1 r* = r I r a\T)a{t - r)^T 1 e-^^dt + aoao. Suppose now that we let Z = jco; Z = Z(jco). We know fromHeavisides rule and from Section I that l/jco has the direct currentor indicial admittance solution i = / = h{t).Hence g(0 =J^h\t) =j^t=^ and Go =0,and the infinite integral equation J. ^ W) r^r / (^)( ~ ^)^^i ^^^+^o takes the form jwZ w=r[i«(^]^-=r«^*- TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 129 Dropping the primes, we have the Laplacian integral equation 1 J0iZ{j(X>) -f
. The Bell System technical journal . gral a{t) = f a{T)a{t - T)dr,Jo where t = k/2D, and the complete relation is 1 1 r* = r I r a\T)a{t - r)^T 1 e-^^dt + aoao. Suppose now that we let Z = jco; Z = Z(jco). We know fromHeavisides rule and from Section I that l/jco has the direct currentor indicial admittance solution i = / = h{t).Hence g(0 =J^h\t) =j^t=^ and Go =0,and the infinite integral equation J. ^ W) r^r / (^)( ~ ^)^^i ^^^+^o takes the form jwZ w=r[i«(^]^-=r«^*- 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 Carsons integral equation, if (jco) isreplaced by p. It will be noted that in deriving this equation use is made only ofthe general form of the expansion of admittances. For particularadmittances, the values of the as in equation (43) or the Jis in equa-tion (45) can be derived directly from an expansion of the admittancefunction, as shown in the foregoing work. Hence, if the solution of. 07 1,4 OF TIME X CRITICAL FREQUENCY = tfc 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 thecurrent in the termination of the network. The frequency of the applied voltage istwice the resonant frequency, fc, of the network. the integral equation is not known from a table of integrals, one methodfor obtaining its solution is the expansion method developed method may then have some application as a method for solvingintegral equations. A. Illustrative ExampleAs an illustration of the use of this method in solving integralequations we will consider the equation 1 ■y[{W+ 2\jco a> Jo {t)e-dt. (46) 130 BELL SYSTEM TECHNICAL JOURNAL The expression on the left can be written 1 V;coV2X + joi (47) Noting that the square of the first factor has the form of an indu
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