TRANSFORMER COUPLING CIRCUITS 611 



With the assumption that the elements of the couphng circuit are 

 pure reactances and that the two anti-resonant circuits are at resonance 

 at the same frequency, /o, the following expression for Zi. 2 as a function 

 of frequency/ was derived with the use of equations 1 and 2. 



■^1, 2 — 



K--'_^\l^[i-l-K'^ ^^^ 



and Z3, 4 is the same expression with C2 substituted for Ci or 



■^3, 4 — Zi, 2 X TT == Zi, 2 73 , 

 C2 -r 



where K is the coefficient of coupling between inductances P and S, 

 and /o is the common resonance frequency of the two anti-resonant 

 circuits in Fig. 1. 



The lower and upper cut-off frequencies /: and fi are related to the 

 resonance frequency /o as follows : 



/o = /wrTx 



The geometrical mean frequency 



fm = V/52 



_ /o 



<ll - K" 



Substituting the above expressions for fm in place of / in equation 

 (3), we find that at the geometrical mean frequency the image im- 

 pedance in terms of fi and fi has the value 



(5) 



27r(/2-/i)Cr 



which will be denoted by Zi, 2- 



It will be noted upon examining equation (3) that Zi, 2 and Z3, 4 

 are resistive between /i and f^ and are minimum at the geometrical 

 mean frequency. They increase uniformly on either side of the 

 geometrical mean frequency and are infinite at /i and f^. For fre- 

 quencies below /i and above /2 they are reactive. It can therefore be 

 seen that this type of coupling circuit inherently possesses the char- 

 acteristics of a band-pass filter. However, in order that the band 

 characteristic may be properly developed, the magnitudes of the 



