248 BELL SYSTEM TECHNICAL JOURNAL 



controlling frequencies with respect to the cut-oflf. On both sides of 

 the transition interval, in the regions of uniform spacing of poles and 

 zeros, the non-resonant reactance approximates R tan irfjla or 

 R cot irfjla and at the middle of each space, where 7r//2a is an odd 

 multiple of 7r/4, is numerically equal to R. Hence, by symmetry, 

 this must also be the value approximated at the middle of the non- 

 uniform interval between the two chains, i.e., at the cut-off frequency. 



Nature of the Approximation 

 The argument of Part I shows that the three-quarter spacing 

 between the cut-off and the chain of transfer constant controlling 

 factors results in poorer approximations to phase linearity in the trans- 

 mission band and to complete suppression in the attenuating band 

 than would the half-spaced cut-off solution. The three-quarter 

 spacing between the cut-off and the chain of impedance controlling 

 frequencies also leads to less perfect uniformity of the impedance 

 characteristic. This is the price we pay for the larger range of phase 

 linearity. Nevertheless, the error of approximation for both 6 and 

 Zi if we follow the sense of equation (22) can be shown to be 



-a\ ^. 1. when a is small, and hence can be made as 



8 \1-/ 1+// 



small as we please by a suitable choice of a}^ So far as the phase and 

 impedance characteristics are concerned, experience shows that satis- 

 factory precision can be obtained with a moderate value of a. The 

 situation with respect to the attenuation characteristic is more serious. 

 As we have already seen, the best approximation in the attenuating 

 band is obtained by a cut-off spacing which is, if anything, slightly 

 less than, rather than slightly greater than, a/2. Furthermore, it 

 appears from the above formula that with the three-quarter cut-off 

 spacing, the approximation error at a given frequency in the attenu- 

 ating band is proportional only to the first power of a. Hence cutting 

 a in two, which substantially doubles the number of elements in the 

 structure, adds but 6 db to the attenuation at this frequency. It is 

 clear that a practical limit is thus set upon the suppression which can 

 be provided. 



Since the attenuation of the structure is relatively low, the con- 

 tribution of reflection effects to the total loss is correspondingly im- 

 portant. A peak of loss occurs at each impedance controlling fre- 

 quency, where the lattice impedances are zero or infinite together. 



2'' It is not true that the error in dfijdw vanishes with a. However, in the following 

 example, which may be taken as typical, the variation of d^ldu is still only about 

 1 per cent of its average value. 



