4 BELL SYSTEM TECHNICAL JOURNAL 



wherein 7 and k have the significance of being the propagation con- 

 stant and characteristic impedance of the corresponding smooth line, 

 i.e., a line having series and shunt impedances Z\ and z 2 , respectively, 

 per unit length uniformly distributed along the line. 



When Zi and z 2 are dissimilar reactances, as in a non-dissipative 

 wave-filter, currents of frequencies within continuous frequency bands 

 can be transmitted without attenuation and the location of these bands 

 on the frequency scale may be found from the conditions which must 

 there be satisfied. As derived from the first equation of (1) the 

 latter are 



A=0, 



and cos2J = l+A (2) 



z 2 



Since the cosine limits are = t 1 this shows that free transmission may 

 occur at all frequencies corresponding to impedance ratio values 

 satisfying the relation 



a result which may be stated as follows: 



The transmitting bands in a ladder type wave-filter having series and 

 shunt impedances Z\ and z?, respectively, include all frequencies at which 

 these impedances are of opposite signs and the absolute value of Z\ is not 

 greater than that of Az%. This statement is useful in roughly determin- 

 ing the relative positions of such bands on an impedance diagram 

 where Z\ and 4z 2 have been plotted as functions of frequency. 



In the attenuating bands corresponding to the remainder of the 

 frequency range we have for the non-dissipative case 



cosh A = 



1+ v)- 



ZiJ 



and Lm-i A - = ^.-rijy, (4) 



sin B = 0, 



the sign above being taken such as to make the right member positive. 

 While the above formulae contain the necessary relations for wave- 

 filter action they do not specify the physical structure of the reactance 

 network. Hence we need to combine with them certain properties 

 of physical reactances to arrive at a structure having desired char- 

 acteristics. In wave-filter design all resistances in the physical re- 

 actance elements, although being unavoidable in construction, are 



