FEEDBACK AMPLIFIER DESIGN 



427 



it is much larger near the point u = than it is in other regions. We 

 can, therefore, conclude that while the derivative of attenuation at all 

 frequencies enters into the phase shift at any particular frequency 

 f = fc the derivative in the neighborhood of /c is relatively much more 

 important than the derivative in remote parts of the spectrum. 



As an illustration of (2), let it be supposed that A = ku, which cor- 

 responds to an attenuation having a constant slope of 6 ^ db per octave. 

 The associated phase shift is easily evaluated. It turns out, as we 

 might expect, to be constant, and is equal numerically to kir/2 radians. 

 This is illustrated by Fig. 3. As a second example, we may consider 



lok 



o _J 



-lok 



w krr 

 < 



< 



EC 



i k^ 



y= 2 



0.1 



0.2 0.3 0.4 0.5 



3 4 5 



fo 



Fig. 3 — -Phase characteristic corresponding to a constant slope attenuation. 



a discontinuous attenuation characteristic such as that shown in Fig. 4. 

 The associated phase characteristic, also shown in Fig. 4, is propor- 

 tional to the weighting function of Fig. 2. 



The final example is shown by Fig. 5. It consists of an attenuation 

 characteristic which is constant below a specified frequency /& and has 

 a constant slope of 6 ^ db per octave above /&. The associated phase 

 characteristic is symmetrical about the transition point between the 

 two ranges. At sufficiently high frequencies, the phase shift ap- 

 proaches the limiting ^7r/2 radians which would be realized if the 

 constant slope were maintained over the complete spectrum. At low 

 frequencies the phase shift is substantially proportional to frequency 

 and is given by the equation 



TT fb 



(3) 



