Electric Circuit Theory and the 

 Operational Calculus' 



By JOHN R. CARSON 



CHAPTER IX 



The Finite Line with Terminal Impedances 



So far in our discussions of wave propagation in lines and wave- 

 filters, we have confined attention to the case where the impressed 

 voltage is applied directly to the infinitely long line. We have found 

 that, by virtue of this restriction, the indicial admittance functions 

 of the important types of transmission systems are rather easily 

 derived and expressible in terms of well known functions, and the 

 essential phenomena of wave propagation clearly exhibited. In 

 practice, however, we are concerned with lines of finite length with 

 the voltage impressed on the line through a terminal impedance Zi 

 and the distant end closed by a second terminal impedance Z2. We 

 now take up the problem presented by such a system. 



Let K = K{p) denote the characteristic operational impedance of 

 the line, and 7 = 7(^) the operational propagation constant of the 

 line. We have then 



V=Ae-'^'^+Be''\ 



(240) 



K. iv 



where A and B are so far arbitrary constants. To determine these 

 constants we assume an e.m.f. E impressed on the line at x = Q through 

 a terminal impedance Zi and the line closed at .t = 5 by a second termi- 

 nal impedance Z2. Kt x = s we have therefore 



whence from (240) 



^e-y^A - ^ey'B=Ae-y'-\-Bey' 

 K K. 



and 



'^"^e-^y'A (241) 



where pi^ZijK. 



'Concluded from tlie issue of January, 1926. 



336 



