EXPONENTIAL TRANSMISSION LINE 561 



is the circular arc. At and below cutoff it is a pure reactance. If the 

 load is a resistance equal to the norninal characteristic impedance at 

 the terminal as indicated at 2 of Fig. 2, there will be no reflection at 

 infinite frequency, but as the frequency is lowered there will be an in- 

 creasing impedance mismatch with its accompanying reflected wave. 



This reflection may be materially reduced by inserting a condenser 

 in series with the resistance load as shown by curve 3. Further im- 

 provement results from more complicated networks. Curve 4 shows 

 the effect of adding an inductance in shunt with the resistance load of 

 the resistance-capacitance combination. The arrows indicate the 

 resulting impedance mismatch which is a measure of the reflected wave. 



The characteristic impedance looking toward the low impedance end 

 is the inverse of that looking in the other direction as shown by 

 curve 5. Shunting the resistance load with an inductance gives the 

 impedance curve 7. Adding a capacitance element gives curve 8. 



Division of (1) by (2) and substitution of the result of (3) gives the 

 following ratio for the impedance looking into the line at the point x 

 to the impedance level at that point, 



Z. _ K{^\ - v'-jv) + 1 + [J^(Vr^^7^ +» - lle-'-^rc/-.) 

 Zx K + jV + Vl - v' - [K - Vr- v' + jV>-2i^<'--^ 



where K = ZijZi is the ratio of the load impedance to the impedance 

 level at the terminal. Here as before the indicated root is in the 

 fourth quadrant. 



Network Characteristics 



Three parameters are required to specify the characteristics of an 

 exponential line of negligible loss: (1) the cutoff frequency, /i, (2) the 

 length of the line which is perhaps best specified as the frequency, 

 /o = velocity of light/length of line, for which the line is one wave- 

 length long, and (3) the impedance level at some point along the line. 

 We will designate the impedance levels at the low and high impedance 

 ends of the line by Zi and Z^ respectively, and their ratio Z2/Z1 by k. 



When the line is terminated in a resistance equal to the impedance 

 level at the output (5) reduces to 



Zi J „ ., , 1 + j tan f k-""^ 2f 



Z, „,, l+tan{e<'^-^l-) ^ , ,_, 



zr' \^^ .<-«-!-,)' •'<' (^) 



1 -f- tan § e ^ 2 / 



