THEORY AND DESIGN OF WAVE-FILTERS 9 



A general comparison of these two methods of designing a " con- 

 stant k " wave-filter shows that the series impedances in the two 

 cases have the same number of inductances and the same number 

 of capacities. Since the total number of these elements is the same 

 in both and the two impedances are made equal at a number of 

 critical frequencies equal to this total number, these impedances 

 are identical at all frequencies. Similarly for the shunt impedances; 

 all of which agrees with the first reactance theorem and leads to the 

 following conclusion. 



As regards propagation constant and impedance characteristics, only 

 one "constant &" wave-filter exists in each class, and the magnitudes of 

 its series and shunt impedances, each of which contains elements equal 

 in number to the critical frequencies, are uniquely determined by the 

 preassigned critical frequencies and the magnitude of k. Its physical 

 structure, however, is in general not unique. 



The structure of these impedances may in all but the lowest classes 

 be given a variety of different forms, the number of inductances re- 

 maining fixed as well as the number of capacities. In the low pass, 

 high pass, low-and-high pass, and band pass classes, the above two 

 modes of derivation give the same designs for their respective series 

 and shunt impedances. In those of a higher class the designs so 

 obtained are different and for more than three elements per im- 

 pedance may be put in even other forms. 



Taking the "constant k", low-band-and-high pass wave-filter with 

 critical frequencies f , f\, / 2 , and f 3 , as an example, the first method 

 gives the series impedance as an inductance in parallel with both a 

 resonant component and a capacity, and the shunt impedance as a 

 capacity in series with both an anti-resonant component and an 

 inductance. The second method gives the structure shown in Fig. 2. 

 Two other equivalent structures for the series impedance are possible; 

 one is an inductance in parallel with the series combination of a 

 capacity and an anti-resonant component, the other is a capacity in 

 parallel with the series combination of an inductance and an anti- 

 resonant component. Similarly the shunt impedance may have two 

 other structures; one is a capacity in series with the parallel com- 

 bination of an inductance and a resonant component, the other is an 

 inductance in series with the parallel combination of a capacity and 

 a resonant component. Relations between the element magnitudes 

 are given in Appendix III, which contains general equivalent im- 

 pedances. There being four equivalent structures for each of the 

 series and shunt impedances this would mean a total of sixteen pos- 

 sible structures for this one "constant &" wave-filter. The impedance 



