8 BELL SYSTEM TECHNICAL JOURNAL 



wave-filter is considered, the number of restrictions imposed on z Xk 

 at the critical frequencies will always equal the total number of in- 

 ductances and capacities involved, whose magnitudes are therefore 

 given by the solution of the simultaneous equations (7). 



By the second reactance theorem a corresponding value of z 2 * may 

 be obtained by designing it out of n components, all in series, wherein 

 each component is the inverse network of a component in the series 

 impedance, the product of their impedances being equal to R 2 to 

 satisfy (6). The component in z 2k corresponding to the zero fre- 

 quency transmitting band is a capacity, c\ k ; that to any (j) internal 

 transmitting band is a simple anti-resonant component of inductance, 

 l J k , and capacity c\ k , in parallel; and that to the infinite frequency 

 transmitting band is an inductance, l\ k . The relations between 

 inductances and capacities of the corresponding components are 

 given by 



l\k_ _Hk_Hk_ _%k_- R 2 , R x 



which determine the elements of z ik as soon as those of z lk are found. 

 An alternative method is to focus our attention upon the attenua- 

 tion requirements. To give n attenuating bands, z ik may be de- 

 signed out of n simple anti-resonant components, all in series, each 

 component accounting for only one band. Representing these anti- 

 resonant components by z a \ . . . z an , the series impedance is 



Zik—"al "f" • • • + Zaj + • • • + Zan- 



The component corresponding to the zero frequency attenuating band 

 is a capacity, C\ k ; that to any (j) internal attenuating band is a 

 simple anti-resonant component of inductance, L\ k , and capacity 

 C{ k , in parallel; and that to the infinite frequency attenuating band is 

 an inductance, L" k . As in the previous case Z\ k must satisfy (7) at 

 all the critical frequencies, which determines its elements. The 

 corresponding shunt impedance, z 2k , may be designed out of n com- 

 ponents, all in parallel, wherein each component is the inverse net- 

 work of a component in the series impedance, their impedance prod- 

 uct being R 2 . The components in z-i k for the three typical attenu- 

 ating bands above considered in the discussion of Z\ k are in the 

 same order, an inductance, L] k , a simple resonant component of 

 inductance, Lj k , in series with a capacity, C 1 ^, and a capacity, C"k 

 We have here 



L ,k Lt x k _L 2 k J~>ik Tin /q\ 



c\k~ ••' ~a lh c[ k ~ ■•• ~ch~ ' 



