IDEAL FILTERS 249 



At these frequencies the image impedance changes sign, and therefore 

 also the constant term of equation (25). Thus, although the phase 

 slope is uniform throughout the attenuating range, the phase charac- 

 teristic itself suffers discontinuities of x radians at each impedance 

 controlling frequency. Whether the discontinuity is an increase or a 

 decrease of x radians is not distinguishable for a non-dissipative net- 

 work. When parasitic dissipation is taken into account the peaks are 

 finite and the phase increases or decreases according as the line- or 

 cross-arm of the lattice has the smaller resistance component at the 

 peak frequency. The infinite peak at this frequency, and the associ- 

 ated abrupt change in phase, can evidently be restored by adding 

 additional resistance to the smaller impedance so as to bring the arms 

 into balance. 



This observation is of importance in considering the effect of 

 dissipation on the phase shift. A counterpart of Mayer's theorem can 

 be found which relates the change in phase shift resulting from uniform 

 dissipation in the network elements to the slope of the loss curve. 

 The formula is 



Ai> = — oid -r— > 



00} 



where d is the dissipation constant, and where A and B are in nepers 

 and radians respectively. In the transition interval, where the slope 

 of the loss curve is great, the effect of uniform parasitic dissipation may 

 reduce the phase appreciably. This effect can be compensated by 

 small modifications in the theoretical frequency spacings, or by the 

 introduction of a lumped resistance to balance the bridge at the first 

 impedance controlling frequency, according to the plan suggested 

 above. 



Example 

 To illustrate the performance of this sort of network, we may con- 

 sider a low-pass filter containing four evenly spaced critical frequencies 

 in the practical transmission band. Subsequent natural frequencies 

 will then occur at 4.75a:, 5.5q:, 6.5q:, etc., according to the rule for three- 

 quarter spacing adjacent to the cut-off. We may suppose that the 

 requirement for linearity of phase shift does not extend above 7.5a, 

 so that the sequence of uniformly spaced impedance controlling fre- 

 quencies may be terminated after this point according to the scheme 

 proposed in the case of the high-pass filter. In the frequency range of 

 interest, we can replace the omitted chain of uniformly spaced fre- 

 quencies by a single natural frequency at double spacing. The trans- 

 fer constant and image impedance expressions can then be written as: 



