562 



BELL SYSTEM TECHNICAL JOURNAL 



for frequencies below and above cutoff respectively. Here r? = — j2Yl 

 is twice the electrical length of the line in radians, sin 2f = Ijv, 

 sin 2^ = V and cos 2^ is ratio of the electrical length of the line to that 

 of a uniform line of the same physical length. For the step-down line 

 the corresponding ratios are the reciprocal of the above expressions. 

 These ratios are plotted in Fig. 1. 



When /— ^ 0, £i = kZi = Zi — Zi and the only effect of the line is 

 to connect the load to the input. Above cutoff the magnitude of the 

 input impedance oscillates about the nominal characteristic impedance 

 and the phase angle oscillates about the value — 2^(« — /i// for 

 />J> /i) which goes from — 7r/2 to as the frequency increases indefi- 



0.1 



0.5 1.0 



FREQUENCY IN MEGACYCLES 



Fig. 3 — Input impedance characteristics. 

 Curve 1 — 150 : 600 ohm line, 100 meters long. 

 Curve 2—300 : 600 ohm line, 200 meter? long. 



Both lines have the same rate of taper. 



nitely from cutoff. The variation of the input impedance with fre- 

 quency is shown for two lines of different length but the same rate of 

 taper in Fig. 3. The magnitude of the oscillations depends only on 

 the rate of taper and decreases with increase in frequency. The 

 impedance varies between (1 + /i//) and 1/(1 + /i//). The positions 

 of the maxima and minima, however, are determined by the length of 

 the line. They occur respectively at those frequencies for which the 

 line is approximately 1/8 of a wave-length more than an even or an odd 

 number of quarter wave-lengths long. The phase angle is usually 

 negative but has a small positive value when the line is approximately 

 a half wave-length long. 



The locations of these maxima and minima are the same as would 

 result from terminating a uniform line in an impedance whose magni- 

 tude is the same as the characteristic impedance but has a small 

 reactive component. This suggests adding a compensating reactance 



