THEORY AND DESIGN OF WAVE-FILTERS 5 



neglected as they produce but secondary effects. When allowed for 

 later their most pronounced effect is the introduction of small attenu- 

 ation in the transmitting bands. 



Reactance Theorems 



The properties of physical reactances which are to be utilized may be 

 stated in the following theorems: 



1. The reactance of any non-dissipative reactance network always 

 has a positive slope with frequency, as well as abrupt changes from posi- 

 tive to negative infinity at anti-resonant frequencies, and may be repre- 

 sented identically (among others) either by a number of simple (series L 

 and C) resonant components in parallel, or simple (parallel L and C) 

 anti -resonant components in series. 



2. To any non-dissipative reactance network there cgrresponds an 

 inverse reactance network which is so related that the product of their 

 impedances is a constant, independent of frequency. 



The proofs of these theorems are given in Appendix I, where with 

 reactances which are known to be any series and parallel combina- 

 tions of inductances and capacities the method of induction is readily 

 applied. In the first theorem the simple component resonant at zero 

 frequency is a single inductance and the one at infinite frequency a 

 single capacity. Similarly the simple anti-resonant components cor- 

 responding to these limiting frequencies are single capacity and single 

 inductance, respectively. In the second theorem, if we have given 

 one reactance consisting of a number of simple anti-resonant com- 

 ponents, all in series, the inverse network may be made up of the 

 same number of simple resonant components all in parallel, each one 

 of the latter corresponding to a particular one of the former. More- 

 over, any pair of these corresponding components are resonant and 

 anti-resonant, respectively, at the same frequency and the ratio of 

 inductance in one to capacity in the other is equal to the constant 

 product of the two total impedances. 



Phase Constant Theorem 



The phase constant will not play any part in the present theory of 

 design but it has this property: The phase constant in a wave-filter 

 always increases with frequency thruout each transmitting band. As 

 shown in Appendix I, this follows as a consequence of the positive 

 slope of reactances. Consideration of this theorem will later be 

 touched upon when discussing composite wave-filters. 



