58 PELL SYSTEM TECHNICAL JOURNAL 



Impedatice and Profxttialion Charaiteristics of Non-Dissipative 

 Fillers. If thi; scries and shunt inipotlanccs of the structures shown in 

 Figs. 2 and 4 are pure reartances. as they would l)e in the case of a 

 non-dissi|)ative filter, the ratio of the quaiuit\- Z) AZ^ must be eitlier 

 a i)ositive or nejjative numeric. It has been shown b>' ('ampbell' and 

 «)thers thai the attenuation constant is zero, and that the structure 

 freely transmits at all frwiuencies at which the ratio Zi/4Zi lies 

 between and — 1. Therefore, In ploitinj; values of the ratio Zi/AZi 

 il is [xjssible to determine the attenuation characteristic of any sym- 

 metrical structure as a function of fretjuency. 



Iti the transmission ranne, the phase constant of the s\niiiutric.il 

 structure shown in Fig. 2 or Fig. 4, is 



^ = 2 ^'" \l^- (14) 



Hence, the expression for the image transfer constant of either of the 

 symmetrical structures shown in Fig. 2 or Fig. 4 is 



9 = 0+7 2 sin- J Z^. (15) 



In the attenuation region, Z.\ \Z.i may be cither negative or pnsiti\e. 

 If Zy AZt is negative anti is greater in absolute magnitude than iinii\ , 

 the attenuation constant is 



.■l=2cosh-'^Z^> (16) 



.111(1 'he phase constant, or the imaginary' component of the image 

 transfer constant, is 



B = (2K-\)ir (17) 



when- A' is an\- integer. Ili^ncc, 



e = 2cosh-' lz:|i+7(2A'-l);r. (18) 



\ AZt 



From e<|uation (S), when Zi/4Zj is positive, the altciiuatinn idii'-tam is 

 /I=2sinh-'^|_i|^ (19) 



and the ph.ise c<instant B is zero. Hence, 



" = -'-"'•' 'yJ^^-+JO- (20) 



•( .impUII, C. .\ , "l'hv>i..tl Tliciry of ihr Klnlrir \V;i\o-Kill(r," Hrll Svs Tech 

 Jour., Nov., I<J22. 



