EXPONENTIAL TRANSMISSION LINE 559 



+ 7 and + r refer to the values of the indicated roots that are in the 

 first quadrant. 



If these equations are compared with those for a uniform trans- 

 mission Hne it is found that the propagation constant is F — 5/2 for 

 voltage waves traveling in the positive x direction and F + 5/2 for 

 voltage waves traveling in the negative x direction. For current 

 waves the corresponding propagation constants are F + 5/2 and F — 5/2. 

 In the terminology of wave filters, F is the transfer constant and 5 is 

 the impedance transformation constant. 5/2 is the voltage transformation 

 constant and — 5/2 is the current transformation constant. The real 

 and imaginary parts of F, a and /3 are the attenuation and phase 

 constants respectively. 



An important parameter is 



. 5 



which for a non-dissipative line is the ratio of the cutoff frequency to 

 the frequency, as can be seen if we write the transfer constant as 



r = 7 Vi - v"^, 



where the indicated root is in the fourth quadrant. For a non- 

 dissipative line V is real and the transfer constant is real or imaginary 

 depending on whether v^ is greater than or less than unity. Hence the 

 exponential line is a high pass filter whose cutoff frequency, /i, is that 

 frequency for which j^ = ± 1. The transfer constant is then less than 

 that for a uniform line by the factor Vl — i'^ so that both phase velocity 

 and wave-length are larger for the exponential line than for the uniform 

 line by the reciprocal of this factor. 



If we terminate this line dit x — I with an impedance Zi = vi/ii, the 

 ratio of the reflected to direct voltage wave is found to be 



A 1 + {z,izdNi - p' -jv) ' 



where the coefficient of the exponential is the voltage reflection coefficient. 

 There will be no reflection if 



Zi = ZiK^TT^^- + jp) = z,+, (4) 



which becomes Zie~'^^^~^ " above the cutoff frequency for non- 

 dissipative lines. This is the magnitude of the forward-looking 

 characteristic impedance at x = I as can be seen by dividing the first 

 term of (1) by the first term of (2). Curve 1 of Fig. 2 gives the charac- 



