LINEAR SERVO THEORY 



647 



Equation (21) also may be written as 



- (loop closed) = ^° . .. , ,s 



(21.1) 



where co,,, = (R -\- iJ.oRt)/J is the new corner frequency. 



The change in over-all transfer ratio clue to the tachometer feedback is 

 shown in Fig. 18. The solid line diagrams A and B are the transfer gains 

 without feedback and with feedback, respectively.-' At low frequencies 

 such that a; « w,,, , the feedback reduces the transfer ratio by the factor 

 o^m/ccm , the ratio of the two corner frequencies. In order to restore this 

 iow-frequency loss in transmission, it is necessary to provide an added 



COm CJm 



LOG OJ 



Fig. 18 — Effect of tachometer feedback on motor characteristic. 



amplification w„,/wm • If this is accomplished by increasing no and decreas- 

 ing Rt so that the product jjoRt remains constant,^' the resulting transfer 

 ratio will be that shown by the dotted lines C in Fig. 18. Comparing A and 

 C, it may be seen that the net result of applying tachometer feedback and 

 increasing the amplifier gain is to widen the transfer bandwidth by the 

 factor com/wm • The required increase in amplification is the cost of widening 

 the transfer bandwidth either by tachometer feedback or by non-feedback 

 means, such as the use of a "forward-acting" equalizer in the amplifier. 

 (However, such forward acting equalization fails to provide the increased 

 over-all linearity and mechanical impedance obtained by the feedback 

 method.) At frequencies sufiicicntly high that w ^ oj^ , the change in 

 transfer ratio due to the feedback disappears, the mechanical inertia be- 

 coming the controlling element. 



^' The straight Une asymptotes have been drawn instead of the actual gain curves. 

 '" This is also the factor by which the feedback reduces the output speed obtained for a 

 steady input voltage, neglecting circuit non-linearites and coulomb friction. 

 '1 This ensures a fixed loop transmission, and thus an unchanging value for um . 



