The Exponential Transmission Line * 



By CHAS. R. BURROWS 



The theory of the exponential transmission line is developed. 

 It is found to be a high pass, impedance transforming filter. The 

 cutoff frequency depends upon the rate of taper. 



The deviation of the exponential line from an ideal impedance 

 transformer may be decreased by an order of magnitude by shunt- 

 ing the low impedance end with an inductance and inserting a 

 capacitance in series with the high impedance end. The magni- 

 tudes of these reactances are equal to the impedance level at their 

 respective ends of the line at the cutoff frequency. 



For a two-to-one impedance transformer the line is 0.0551 wave- 

 lengths long at the cutoff frequency. For a four-to-one impedance 

 transformer the line is 0.1102 wave-lengths long at the cutoff 

 frequency, etc. 



The results have been verified experimentally. Practical lines 

 50 meters and 15 meters long have been constructed which trans- 

 form from 600 to 300 ohms over the frequency range from 4 to 30 

 mc. with deviations from the ideal that are small compared with 

 the deviations from the ideal of commercial transmission lines, 

 either two-wire or concentric. 



When an exponential line is used as a dissipative load of known 

 impedance instead of a uniform line it is possible to approach more 

 nearly the ideal of constant heat dissipation per unit length. 

 This makes it possible to use a shorter line. 



THE exponential line may be defined as an ordinary transmission 

 line in which the spacing between the conductors (or conductor 

 size) is not constant but varies in such a way that the distributed 

 inductance and capacitance vary exponentially with the distance along 

 the line. That is, the impedance ratio for two points a fixed distance 

 apart is independent of the position of these two points along the line. 

 A disturbance is propagated down an exponential transmission line in 

 the same manner as it would be down a uniform line with the addi- 

 tional effect that the voltage is increased by the square root of the 

 change in impedance level and the current is decreased by the reciprocal 

 of this quantity. 



The exponential line has the properties of a high pass impedance 

 transforming filter. The cutoff frequency depends upon the rate of 



* Presented before joint meeting of U. R. S. I., and I. R. E., Washington, D. C, 

 April 1938. Published in Communications , October 1938. 



555 



