558 



BELL SYSTEM TECHNICAL JOURNAL 



the uniform line. If x is the maximum attenuation in nepers that can 

 be obtained with a uniform line without overheating, the same length 

 of exponential line will have an attenuation of (e^^ — l)/2 nepers. 



Exponential lines of the proper length have properties similar to 

 half-wave and quarter-wave uniform lines. The input impedance of 

 an exponential line an even number of quarter wave-lengths long is 

 equal to the load impedance times the impedance transformation ratio 

 of the line. When the length of the line differs from an odd multiple 

 of a quarter wave-length by an amount that depends upon the fre- 

 quency and load impedance, the input impedance is equal to the product 

 of the terminal impedance levels divided by the load impedance. 



Mathematical Formulation 



The telegraph equations for the exponential line may be solved by 

 the methods employed in the problem of a uniform line. The resulting 

 equations for the voltage and current at any point along the line are 



and 



tx 



Vx = Ae 



5] 



-(-|)^a.R+(^+l>-..-(^-|) 



+ Be 



1 + I e^^^ 



-4^ 2-(r+|> ^^ + 2;-i-(r-|> 



Zo 7 



Za 7 



(1) 



where 



Zq 7 



1 - 



B^ + 2 



r - 



(2) 



^ ^ log^o ^ log^ ^ log^o .^ ^^^ ^^^^ ^f ^^p^^^ 



Zj; = Vz/y — Z^e^"" is the surge or nominal characteristic impedance of 

 the exponential line at the point x which is equal to the 

 characteristic impedance of the uniform line that has the 

 same distributed constants as this line has at the point x, 



Y = "^zy = Vzo3'o is the propagation constant of any uniform line that 

 has the same distributed constants as this line at any 

 point. It is independent of the point along the line to 

 which it is referred, and 



r = V7^ + 5-/4 = a + j(8 is the transfer constant of the exponential 

 line. 



