70 Hr.l.l. SYSTEM TllCIISICAL JOL'RSAL 



Ill.l^;nitlulc. If Ri and R- are the olTcclive rosislanri-s of tin- iiuiiictancc 

 iliiiunis Li aiul L-,, rcsiK-ctively, the series imix-ilance, Zi, of a series- 

 shunt recurrent structure CDniposed of sections of the type shown 

 in Fig. 13 is 



Z, = /.,+,(./,.-^;.). (24) 



The impedance of the siiunt arm is 



Z, = R,+jLl,— \]. (25) 



In substituting for /?i its vahie l.\wd and for R: its vahie L-mI, the 

 ratio Zi '4Zi becomes 



1 -J'l - 7^r-r^ 



Assuming d to be zero, the ratio Z\ 4Zj is 



Zi ^ L\{^LxCi-\) 

 AZt 4C,(u.'LjC,-l)' 



(27) 



Referring to Tal)le 11, the structure shown in Fig. 13 has two dis- 

 tinct attenuation and phase characteristics. These are, respectively, 

 characteristics Nos. 9 and 10 of I'ig. 7. These two sets of cliaracter- 

 istics arise from the fact that liie shunt arm may be resonant at a 

 frequency less than, or greater than, the resonant frequency of the 

 series arm. The two attenuation characteristics are inverse with 

 resf)ect to fret|uency. We shall, therefore, discuss only one of the two 

 cases, namely, that in which the shunt arm resonates at a frequency 

 greater than the resonant fre(|uency of the series arm (that is, LiC\ 

 is greater than L^C;)- The fre<iuency at which the shunt arm is 

 resonant will be <lesignated as f,, due tf) the fact that in a non-dis- 

 sipative filter the attenuation constant is infinite at this point. In 

 other words, 



(28) 



It is evident that the fre<|iiency at which Z, is resonant is a cut-ofT 

 fre(|uency since Zi, and therefore Z| IZn, is zero at this point. An 

 insjx'ction of graphical curves '"drawn for Zi and 4Z2, under the above 



'• For an illuMr.ition i)f llie conslriulion of siuli ciirvi-s sec Uibliography 12, Fig. 7, 

 also UililioKrapliy 13. Fik. i- 



