DISTORTION CORRECTION 441 



sending end at time / = 0. With (1) and (2) in the integral equation 

 of electric circuit theory ^ we obtain 



p—a'—Tp r><x> 



-^- = J e-^mdt, (3) 



whose solution is 



g(/) =0, t <r, 

 and (4) 



g{t) = e~"' = constant, / > r. 



Thus, a constant voltage, which has been attenuated by the circuit an 

 amount a' napiers, arrives suddenly at the receiving end after a time 

 T = {b' joi) seconds, and there is no distortion with respect to the unit 

 constant e.m.f. impressed on the circuit at time 1 = 0. 



If now any type of e.m.f., E{t), is impressed on this circuit which is 

 specified by the steady-state characteristics (2) or the indicial voltage 

 (4), we obtain through a general formula ^ 



v{t) = jj' E{t - y)g{y)dy = e-'^'E{t - r). (5) 



This received voltage has the same shape as the impressed e.m.f., 

 there being an attenuation, a', and a time-of-transmission, r. Hence, 

 a circuit specified as above is distortionless to any type of impressed 

 e.m.f. A further discussion involving the phase intercept is taken up 

 in Appendix I. 



It may be stated that Heaviside's theoretical distortionless smooth 

 line was that in which the line constants R', L', G' and C per unit 

 length had the relation 



R'/G' = L'lC (6) 



giving attenuation and phase constants per unit length, respectively, 



a = ylR'G' napiers, 

 and 



/3 = VL'C'oj radians; 



also an iterative (or characteristic) impedance 

 k = a/t^ = a/tt, ohms, 



\ Cr \ C 



which is a constant resistance at all frequencies. A circuit made up 



^"Electric Circuit Theory and the Operational Calculus," John R. Carson. 

 «L.c. 



