12 BELL SYSTEM TECHNICAL JOURNAL 



Certain inferences may be drawn as to the nature of the impedances 

 2n and 221. 



a. The series impedance z n is similar in form to the series impedance 

 Z\ k and is anti-resonant at the same frequencies as z v< - This follows 

 directly from a comparison of formulae (11) and (12). For whenever 

 Z\\ is anti-resonant, corresponding to an attenuating band, K\ k is 

 infinite, and to make K n also infinite z n must be anti-resonant irre- 

 spective of z 2 \ in order to maintain an attenuating band at these 

 frequencies. 



b. The shunt impedance z 2 i corresponding to the series impedance 

 2n and the given class of wave-filter may, in its most general form, be 

 taken as a parallel combination of simple resonant components (series 

 L and C) equal in number to the total number of inductances and capac- 

 ities contained in z n - This is a consequence of a general conc'usion 

 based upon formulae (2) and (4) and the properties of reactances, 

 namely that in an attenuating band corresponding to each branch of 

 the series impedance frequency curve, where the absolute value of 

 Zn passes once continuously thru all values from zero to infinity, 

 the shunt impedance 2 2 i can be resonant no more than once. Since, 

 however, the number of branches in the z n frequency curve equals 

 the number of elements which s u contains, the above statement is 

 proven. 



c. Series resonance and shunt anti-resonance coincide if both are 

 included in an internal transmitting band. Series and shunt anti- 

 resonance coincide if both are included in an internal attenuating band. 

 This is a necessary relation in either case to preserve band confluency. 



To ensure the necessary similarity between z n and z^ it will be 

 assumed that for every series component in z Vz as above expressed 

 there is one of proportional magnitude in z n which latter may be 

 written, 



Zii = WiS al + w 2 Sfl 2 + • • • m n z an , (14) 



where the coefficients, ni\, . . . m n , are positive real numerics. From 

 the formulae (11), (12), and (13) the shunt impedance becomes 



„ _ -R 2 + i(z 2 * — Zn) ( , -x 



221 • \±d) 



If in this formula the assumed form (14) for Zn corresponding to 

 any particular z xk is substituted, it will be found that the resulting 

 expression for z 2 i has exactly the requisite form to be the most general 

 shunt impedance which that wave-filter may have. This therefore, 



