EXPONENTIAL TRANSMISSION LINE 563 



to the resistance load. From (3) the best single reactive element is 

 found to be a condenser whose impedance is equal to the impedance 

 level at the cutoff frequency. This gives a value oi K = \ — jv which 

 when substituted in (5) shows that the input impedance is to a first 

 approximation a constant times the terminal impedance. To correct 

 for the reactive component of the input impedance an inductance 

 having an impedance jZi/;/ which is equal to the input impedance level 

 at cutoff is shunted across the input. The resulting impedance trans- 

 forming network consists of an exponential line with a series capaci- 

 tance at the high impedance end and a shunt inductance at the low 

 impedance end. When terminated in a resistance load at either end 

 equal to the impedance level at that end the input impedance, to a 

 first approximation, is a resistance equal to the impedance level at the 

 input end. In fact the deviations of the input impedance from the 

 ideal for transmission in one direction are just the reciprocal of those 

 for transmission in the other direction. 



The magnitudes of the series capacitance and shunt inductance that 

 give the improved network may be expressed in terms of the electro- 

 static capacitance and loop inductance of the line. Simple calculation 

 shows that the required series capacitance is equal to 2j{k — 1) times 

 the electrostatic capacitance of the line and the required shunt in- 

 ductance is equal to the same factor times the total inductance of 

 the line. 



There is an interesting relationship between these terminations and 

 a simple high-pass filter. The LC product of the shunt and series arms 

 of the filter resonates at /i. If an ideal transformer with transforma- 

 tion ratio k is inserted between the shunt inductance and the series 

 capacitance, the capacitance becomes Cjk and the new LC resonates 

 at /iV^. This is the same frequency at which the series capacitance 

 and shunt inductance that are added to the terminations of the ex- 

 ponential line resonate. Furthermore the reactance of the shunt 

 inductance is equal to the impedance level at the cutoff frequency and 

 the reactance of the series capacitance is equal to the impedance level 

 at the cutoff frequency exactly as in the case of the high-pass filter. 



By using the exponential line it is possible to construct a network 

 with properties that no network with lumped circuit elements possesses, 

 namely, a high-pass impedance transforming filter. 



Critical Lengths 



Besides the characteristics of the exponential line that are sub- 

 stantially independent of the length of the line, it has properties that 



