246 BELL SYSTEM TECHNICAL JOURNAL 



ignored if we assume the loss to be reasonably high. The variation of 

 the phase shift with frequency must therefore be attributed to re- 

 flection effects, which we can write as 



^ R 



6"^ = 



4^^ 

 R 



where Or is the sum of the reflection effects at the two ends of the 

 structure. 



Since Z/ is reactive in the attenuating band, the angle of the 

 denominator in this equation is ± 7r/2, while that of the numerator is 

 2 arctan Z jjiR. Thus 



Br^^ | + 2arctan^'. (25) 



Except for the constant term, which we will consider presently, this is 

 a function of precisely the type we have been considering. Hence if 

 the impedance controlling factors are spaced at the same uniform 

 interval that was used in the pass-band, the phase slope will be constant 

 and equal in both bands. 



Phase Characteristics in Transition Intervals 



The transition factors — or rather, factor, since clearly we have to 

 rule out the solutions for m > 1— must be determined so that these 

 linear parts of the phase characteristic are joined by a chord of the 

 same slope. If we suppose the transition interval to be bounded by 

 the last uniformly spaced frequencies of the transfer constant and 

 image impedance chains, and to contain only the cut-off factor, it is 

 easily shown that it must include a net change in phase of 37r/2 radians. 

 The interval must therefore contain 3/2 uniform spaces if the average 

 slope is to be correct. Considerations of symmetry to be described 

 later require that the cut-off be the center of the interval, which thus 

 comprises two three-quarter spaces. The behavior of the several 

 components of the total insertion phase is exhibited by Fig. 19, in 

 which B, Br, and Bi refer respectively to the phase shifts contributed 

 by the transfer constant, by reflection effects, and by the interaction 

 factor. The mutually annulling discontinuities of 7r/2 radians in B, 

 and Br at the cut-off are noteworthy. 



The fact that this choice of parameters is sufficient as well as 

 necessary to obtain the desired linearity of phase shift is not easily 

 shown analytically. It can, however, be verified by direct computa- 



