THEORY AND DESIGN OF WAVE-FILTERS 



critical frequencies separating these bands are the frequencies at which 

 the series impedance equals ±i2k. 



With these known facts and the properties of reactance networks, the 

 determination of the physical structure and design of any " constant 

 k " wave-filter is a relatively simple matter. At the series-resonant 

 frequency of any transmitting band both characteristic impedances 

 Kik and K 2k , using the same notation as above, have by (1) the 

 value k. This indicates that if k has been chosen equal to the im- 

 pedance of the line (assumed as a constant resistance) with which 

 the wave-filter is to be associated there will be no impedance irregu- 

 larity at the junction of the mid-terminated wave-filter and the line 

 for any of these series-resonant frequencies. We shall put, therefore, 



k == \Zz lk Z2k = Mean Line Resistance = R, (6) 



which is assumed given, and R will have this meaning thruout the 

 remainder of this paper. At the critical frequencies we then have 

 to satisfy the conditions 



z lk = ±i2R, ( 7 ) 



where also 



Kik = and K 2 k = °° • 



If there are to be n transmitting bands z ik may be designed out of 

 n simple resonant components, all in parallel, wherein each component 

 accounts for only one band. For example, with resonant components 



Zy\ . . . z rn , we have 



1 



Z\k 



Zf i Z r j z rn 



This is sufficient since, owning to the positive slope of reactance, there 

 is bound to be but one anti-resonant frequency and attenuating 

 band between every adjacent pair of resonant frequencies. It is 

 obvious that the component corresponding to the zero frequency 

 transmitting band is an inductance, l\ k ; the component corresponding 

 to any (j) internal transmitting band is an inductance, l ik , and capac- 

 ity, c{ k in series; and the component corresponding to the infinite 

 frequency transmitting band is a capacity, c n Xk . 



The magnitudes of the inductances and capacities will be uniquely 

 determined by satisfying the relations (7) at all the critical frequen- 

 cies. For at the critical frequency of the zero frequency transmitting 

 band z ik = -\-i2R; at the lower critical frequency of any internal 

 transmitting band z ik — —i2R and at the higher critical frequency 

 Z\k— -\-i2R\ at the critical frequency of the infinite frequency trans- 

 mitting band z u .= —i2R. Hence, no matter what J' constant k " 



