THEORY AND DESIGN OF WAVE-FILTERS 17 



the single value m,\ = . . . = m n = m, leaving but one degree of freedom. 

 They are of special interest because in wave-filters having many ele- 

 ments the impedances can be determined more directly than by the 

 general methods above and, of greater importance, because the mid- 

 shunt characteristic impedance, K 2 i(m), of the mid-series equivalent 

 M-type and the mid-series characteristic impedance, K^im), of the 

 mid-shunt equivalent M-type, both functions of m, can be made 

 approximately a constant resistance over the greater part of every 

 transmitting band, a desirable property. 



In the mid-series equivalent M-type it follows from (14) and (15) 

 that, since Zi k Z2 k = R 2 , 



z n = mz lk , 



(19) 



1-ra 2 , 1 



and Z2i = — : Zik-\ — z 2k , 



4m m 



showing the shunt impedance to be expressible as a series combina- 

 tion of different proportions of the " constant k " series and shunt 

 impedances. This structure is usually different from but equivalent 

 to the mid-series equivalent wave-filter obtainable by the first method 

 in which the ra-coefficients are all equal to m. The value of the co- 

 efficient m is determined by fixing a resonant frequency of 221, that is, 

 any one frequency of infinite attenuation, /«. From (19), for 



(*«)/. = °> 



m = Ji + (l^) . (20) 



\ V zik Jj* 



The corresponding mid-shunt equivalent M-type having the same 

 propagation constant follows from (18) with impedances 



1 



2l2 : 



1 + l 



mzik 4m , ' 



iZik (21) 



1 2 

 1 — m l 



and 1 



Z20 = — ; 

 m 



Here the series impedance is expressible as a parallel combination of 



different proportions of the " constant k " impedances. 2 



2 It is worth while to point out that from the nature of (19) and (21) these same 

 relations result if z ik and z-i k are the series and shunt impedances z t and Z2 of any 

 ladder type recurrent network whatever. In order that there be a physically realiz- 

 able structure corresponding to such general relations it is sufficient that < m 4 1. 

 A change of m will change the propagation constant without changing the mid- 

 series characteristic impedance of the first network, and mid-shunt of the second. 



