Sii BELL SYSTEM TECHNICAL JOURNAL 



Imagine, for instance, that we represent the resonator and the mismatched 

 line as in shunt with a section of Hne N wavelengths or 6 radians long mis- 

 terminated in a frequency insensitive manner so as to give a standing wave 

 ratio <r. If Ml is the characteristic admittance of the line, the admittance 

 it produces at the resonator is 



Y,=M,f±4^^. (9.33) 



1 + ja- tan 6 



Now, if the frequency is increased, 6 is made greater and Y is changed. 



{1 -j- j(T tan&)2 

 We are interested in the susceptive component of change. If 



Vz. = Gl+JBj^ (9.35) 



we find 



»Bjm = M, " ~ "'Y'r ^ytf """ ' • (9-36) 



(1 + 0- nan^ 6) 



Now, if frequency is changed by an amount df, 9 will increase by an a mount 

 6(df/f) and Bl will change by an amount 



dB:^ = {dBJdd){2T,N){df/f). (9.37) 



We now define a parameter Mm expressing the effect of the mismatch as 

 follows 



TidB^/dd) = Mm. (9.38) 



Then 



dBj^ = INMuidf/f). (9.39) 



If the characteristic admittance of the resonator is Mr , then the characteris- 

 tic admittance of the resonator plus the line is 



M = Mji-\- NMm. (9.40) 



If, instead of a coaxial line, a wave guide is used, and Xo and X are the cutoff 

 and operating wavelengths, we have 



dB^ = 2NMM{df/f)(l - (X/Xo)2)-^ (9.41) 



and 



ikr = M« + NMm(1 - (X/Xo)2)-^ (9.42) 



In Fig. 49 contour lines for Mm constant are plotted on a Smith Chart 

 (reflection coefficient plane). Over most of the plane Mm has a moderate 



