THEORY AND DESIGN OF WAVE-FILTERS 21 



where r is the ratio between the two inductances. The magnitudes 

 of L lk , dt and r are found from the conditions (7) which z ik must 

 satisfy at the critical frequencies / , /i, and / 2 ; namely, z ik = +i2R, 

 — i2R, and -\-i2R. The resulting simultaneous equations become 



foW+ffc—fly=+l, 



hw-f\x-f\y=-h (27) 



and fiw -{-fix —fly = + 1 , 



u t y R r x * A wx 1 

 where L lk = — ; Cia = i—d7 7 ; and r== L 



The solution of (27) gives 

 R 



L\b = 



C\k = 



tt(/o— fi^rj-ij 



(/o-/ 1 + f 2 ) 2 



(28) 



'* 47T [(f o/l - f 0/ 2 +/l/2 ) (fo -/l +/») - ^0/l/ 2 ] A' 



and r = (/o-/i+/ 2 )(^- f - + > 1 )-l. 



JO Jl '2 



The corresponding shunt elements are obtained from the series ele- 

 ments by the inverse network relations, --^- = -^— = R 2 , so that 



Lik = R 2 Cik, 



(29) 



Cik = 



With the " constant & " wave-filter elements so determined we 

 shall now derive the series and shunt impedances, z n and z 2 i, of the 

 general mid-series equivalent wave-filter. Putting for convenience 



z L = i2irfLik, and z c = (r , 



formula (26) becomes 



, rz L zc 



(30) 



