12m , 1 I I 



/ 



732 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



having equal Q's and equal resonant frequencies is given by, 



Y = Gig+jb), 



where 



(QkV 



, QiQ'm - 1) / , \^^/ 



^^ "^ '^" [ Qi^'m- D TI- 



L ^m J/ 



(3.1) 



\ 2 ^ ^ 



^^^"^: ^0(Q^m-l) t' 



"^ L fi^ J 



G = shunt conductance of primary and secondary resonators. 



y, 6 = total input conductance and'susceptance respectively normalized 

 with respect to G, 



12 = normalized frequency = f/fo . 



/o = resonant frequency of both primary and secondary resonators. 



k = coefficient of coupling. 



m = 1 - k\ 



This expression for input admittance, though accurate, is rather un- 

 wieldy because of the many variables involved. A few obvious approxi- 

 mations, however, will change equation (3.1) into a much simpler and 

 more meaningful expression, yet sufficiently accurate for the range of 

 Q's and bandwidths of interest here. Let, 



Jo Jo Jo 



where 6 denotes the normalized frequency deviation from resonance, 

 Af/fo . If we further assume that k^ « 1 and the range of Q's is such 

 that {Qk) may vary from zero to about five and that the maximum 

 value of 5 is small enough so that its higher powers may be neglected, 

 then equation (3.1) may be simplified to. 



Y 

 G 



-b-T^,]*H'-rf^l « 



