646 BELL SYSTEM TECHNICAL JOURNAL 



stant A, the transfer ratio of the motor and amplifier without feedback is, 

 from (3.1), 



Mr = T^ (loop open) = y ^—-. — ■ , (19) 



E J JOiijOi + w„) 



where the constant /io = Ayn . (To avoid confusion with primary loop 

 quantities, the tachometer loop will be represented by the s>Tnbols jur and 

 /3r , rather than n and )3.) The quantity oim was defined as the ratio {Rm + 

 R'm)/J (see Fig. 2b), and is the reciprocal of the motor time constant. 

 Replacing {Rm + Rn^ by R for convenience, (19) may be rewritten as 



Mr = t; (loop open) = ■ .^7 • n • ^^^'^^ 



The transfer ratio of the tachometer is 



Ep 

 ^T = -r = —jo:Rt , 



and thus the loop transmission characteristic is 



R + jooJ 



For values of a> small compared with co„ this loop transmission is constant 

 and closely given by ht^t{^) = —fioRt/R- When w ^ com , htPt ap- 

 proaches the form —isoRt/jooJ, and thus falls off at 6 db /octaxe. Conse- 

 quently the maximum phase shift of the factor —fxr^r is —90 degrees, and 

 no stability problem arises for the local tachometer loop.^* 



From (19.1) and (20), the over-all transfer ratio with feedback is 



- (loop closed) = — , 



E 1 — flTRT 



(21) 

 _ /fo 



juiR + fioRt + joiJ)' 



Comparing (21) with (19.1), it may be seen that the sole effect of the tach- 

 ometer feedback upon the over-all transfer ratio has been to add an apparent 

 "ohmic" friction or mechanical resistance ^loRt to the original value R. 

 (It will be shown that this increase in apparent mechanical resistance also 

 is effective in increasing the mechanical output impedance, although no 

 power is dissipated in the added component noRt-) 



** Actually, the effects of parasitic elements always modify this situation somewhat, 

 especially if unusually high loop transmission is sought. However tachometer loops often 

 require little or no stabilizing equalization. 



i 



