556 BELL SYSTEM TECHNICAL JOURNAL 



taper. As the frequency is increased the transfer constant * ap- 

 proaches the propagation constant of the equivalent uniform Hne. At 

 sufficiently low frequencies the only effect of the line is to connect the 

 input to the load. 



Above cutoff the magnitudes of the characteristic impedances at any 

 point are approximately equal to the nominal characteristic imped- 

 ance * at that point but their phase angles (in radians) differ by an 

 amount which at the higher frequencies is equal to the cutoff frequency 

 divided by the frequency in question. The ratio of input impedance 

 to the input impedance level * of an exponential line terminated in a 

 resistance equal to the impedance level at the output always remains 

 within the range from 1 — fi/fto 1/(1 — fi/f) for frequencies,/, greater 

 than the cutoff frequency, /i. For a 2 : 1 transformation this means 

 that the input impedance remains within ± 6 per cent of the desired 

 value for all frequencies above that for which the line is a wave-length 

 long. For a 4 : 1 transformation under the same conditions the 

 irregularities are twice as great. 



A transforming network having deviations from the ideal of the 

 order of ± (fi/fY may be made by connecting an inductance in parallel 

 with the low impedance terminal and a capacitance in series with the 

 high impedance terminal. The magnitudes of these reactances are such 

 that their impedances are equal to the impedance levels of the line at 

 their respective ends at the cutoff frequency. Or expressed in another 

 way the capacitance is equal to 2/(k — 1) times the electrostatic 

 capacitance of the line and the inductance is the same factor times the 

 total loop inductance of the line where k is the impedance transforma- 

 tion ratio of the line. 



Figure 1 shows the theoretical input impedance-frequency charac- 

 teristics for 2 to 1 step-up and step-down exponential lines. Curve 1 

 is for the line with a resistance termination. At low frequencies the 

 input impedance is equal to the load impedance while at high fre- 

 quencies the line approaches an ideal transformer. Curve 2 is the 

 input impedance of the line terminated with the appropriate resistance- 

 reactance combination. The improvement in the input impedance 

 characteristic for frequencies above the cutoff frequency is evident. 

 At the lower frequencies the input impedance does not approach the 

 terminal reactance but approaches the reactance of the capacitance of 

 the line in parallel with the series terminal capacitance for the step-up 

 line and the reactance of the inductance of the line in series with the 

 shunt terminal inductance for the step-down line. The improvement 

 is not as great as apparent from the figures because the phase angle is 



* See appendix for definition of terms. 



