TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 113 



i = ii the output current. We let ^i = iiZo- Eliminating either Vq 

 or io, we have 



Vi = Voe~^ or ii = ioe~^. (8) 



In the following work it is necessary to know the impedance of a 

 short circuited line and that of an open circuited line. For the short 

 circuited line, the voltage at the far end is zero, so putting z; = in 

 the last of equation (6), we have 



vo/io = Zo tanh P. (9) 



For the open circuited line we put i = 0, obtaining 



Vo/io = Zo coth P. (10) 



So far we have discussed the general transmission line. For the 

 distortionless line there exists the relation 



- = -• (11) 



Substituting this relation in equation (7), these parameters reduce to 

 Zo = Ro = ^J^ = ^J^ and P = 4RG -\- j^4LC = A +jo:D. (12) 



This equation shows that the characteristic impedance becomes a 



resistance Rq, while the propagation constant equals a real constant A 



plus jco times the constant D. To show how wave transmission 



takes place in an infinite line, these values are substituted in equation 



(8), giving 



v^ = z;oe-(^+^"^) ; ii = «og-(-^+^"^). 



To find the real solution, we take the real part of this symbolic solution, 

 obtaining 



or, taking the real part, 



Vi = voe-^ cos [aj(/ — D) -{- 62 

 and (13) 



ii = he-^ cos [a)(/ - D) + e^, 



where the dash over Vo and io indicate the maximum amplitude of 

 these quantities. Since Vq = Vo cos (cot + d), these equations show 

 that either Vi or ii has the same form as Vo or in respectively, attenuated 

 by a factor g"-* and delayed in time by an amount D, 



