564 BELL SYSTEM TECHNICAL JOURNAL 



depend on the length of the Hne that are analogous to those of a uniform 

 line a half wave-length or quarter wave-length long. For non- 

 dissipative lines above the cutoff frequency (5) becomes 



Xcos(|^- 2^)+jsin| 



Z, = j^ -^ Zi. (8) 



cosn-f2tj +jXsin^ 



When the line is an integral number of half wave-lengths long {l-q = tt) 

 this reduces to 



Z, = KZi = kZ2, (9) 



which says that the input impedance is equal to the impedance trans- 

 formation ratio times the load impedance. The length of exponential 

 line that corresponds to a quarter-wave uniform line differs from an odd 

 multiple of a quarter wave-length by an amount such that 



tan ( j = ^., ^ tan 2^, (10) 



for which (8) becomes 



Zr=^. (11) 



Similar expressions exist for the step-down line, but 1/K must be 

 substituted for K in (10) for the length corresponding to the quarter- 

 wave uniform line. 



With Dissipation 



An exponential line is an improvement over the uniform "iron wire" 

 line as a resistance load that will dissipate a large amount of power. 



Provided the attenuation is not too large the current and voltage 

 distribution will be the same as for a non-dissipative line except for 

 the additional power loss so that we may use the equations for an 

 exponential line even though the distributed series resistance and 

 shunt leakage do not vary exponentially with distance. 



Suppose that the conductor size and resistance that will just dissi- 

 pate the desired input power result in an attenuation constant ao for 

 a uniform transmission line. To a first approximation the conductors 

 can carry the same current irrespective of the impedance level. The 

 current wave will be given by the first term of equation (2) which 

 becomes 



