528 BELL SYSTEM TECHNICAL JOIRNAL 



We will remember that ^o is the length of line in radians, a is the standing 

 wave ratio, measured as greater than unity, and Qe is the external Q of the 

 resonator for unity standing wave ratio. 



Replacing a given length of line by the same length of wave guide, we fnd 

 that the phase angle of the reflection changes more rapidly with frequency, 

 and instead of (9.31) we have the condition for no loops as 



e < 2(3^(1 - (X/Xo)2)/(a-^ - 1) (9.32) 



'^ < Vl +2Qe(1 - (X/Xo)2)/0o- 



Here X is the free space wavelength and Xn is the cutoff wavelength cf the 

 guide. 



Equations (9.32) are for a particular phase of standing wave, tl at is, for 

 relations of Yt and 6o which, produce a loop symmetrical abcve the C axis. 

 Loops above the G axis are slightly more locped than Iccps belcw the G 

 axis because of the increase of do with frequency. For reasonably Icng lines, 

 (9.32) applies quite accurately for formation of loops in any position; for 

 short lines locps are cf no consequence unless they are near the G axis. 



An imporant case is that in which the resonant lead is ccupled to the 

 resonator by means of a line so short that it may be considered to have a 

 constant electrical length for all frequencies of interest. The resonant 

 load will be assumed to be shunted with a conductance equal to the charac- 

 teristic admittance of the line. As the multiple resonance of a long mis- 

 matched line resulted in formation of many locps, so in this case we would 

 rightly suspect the possibility of a single loop. 



If the resonant load is |, f, etc. wavelengths from the resonator, and 

 both resonate at the same frequency, a loop is formed symmetrical about the 

 G axis. Figure 43 is an admittance curve for resonator and lead placed 5 

 wavelength apart. Tuning either resonator or load moves this loop up 

 or down. 



If the distance from resonator to resonant load is varied above or below a 

 quarter wave distance, the loop moves up or down and expands. This is 

 illustrated by an eighth wavelength diagram for the same resonator and load 

 as of Fig. 43 shown in Fig. 44. 



When the distance from the resonator lo the resonant load, including 

 the effective length of the coupling loop, is 5, 1, 1^, etc. wavelengths, for 

 frequencies near resonance the resonant load is essentially in shunt with 

 the resonator, and its effect is to increase the loaded Q of the resonator. An 

 admittance curve for the case is shown in Fig. 45. In this rase the loo])s 



