THEORY AND DESIGN OE WAVE-FILTERS 11 



(2) Determine the magnitudes of all inductances and capacities 

 in Z\k from the conditions, z K , = ±i2i? at all preassigned critical fre- 

 quencies, where R ( = k) is the given mean line resistance; 



(3) Derive a structure, in addition to the inductance and capacity 

 magnitudes, of the shunt impedance, z 2 ., considering the latter as an 

 inverse network to z ik , where Zi k z V; = R 2 - 



2. General Wave-Filters Having Any Preassigned Trans- 

 mitting and Attenuating Bands and Propagation Constants 

 Adjustable Without Changing One Mid-Point Character- 

 istic Impedance. 



It was shown above how a " constant k " wave-filter may always 

 be designed so as to have any preassigned transmitting and attenu- 

 ating bands. A method will now be given for deriving the two most 

 general ladder types, each having one mid-point characteristic im- 

 pedance equivalent at all frequencies to the corresponding mid-point 

 characteristic impedance of the known " constant k " wave-filter; 

 one of them has such equivalence at mid-series, and the other at mid- 

 shunt. Because of this equivalence, these general wave-filters must 

 necessarily have the same transmitting and attenuating bands as the 

 " constant k " wave-filter which they include as a special case. Their 

 propagation constants will be found to be adjustable over a wide 

 range. 



Mid-Series Equivalent Wave- Filter 



Assume the known " constant k " wave-filter has n attenuating 

 bands and that its series impedance derived by the second method 

 has the form of n simple anti-resonant components in series, repre- 

 sented as 



Zik = Z a i + Za2+ ■ ■ ■ Zan- (10) 



Its mid-series characteristic impedance is 



K lk =VR' 2 + lztk. (11) 



Let the series and shunt impedances of the desired general wave- 

 filter be Zn and 021, respectivel y, where the second subscript 1 in- 

 dicates that these impedances belong to the wave-filter which is to 

 have mid-series equivalence with the " constant k " wave-filter. Then 



and the fundamental relation is that 



K lx =K lk . (13) 



