640 BELL SYSTEM TECHNICAL JOURNAL 



characteristic (loop closed) from the servo input up to the motor and power 

 amplifier must rise correspondingly with frequency, out to the cross-over 

 point. Again assuming input noise of the uniform ampUtude versus fre- 

 quency type, the total noise power at the motor input is therefore, 



Jo 



1 + M 



(cO + Wm)w" du. (15) 



Again, w,„ is the reciprocal time-constant of the motor and A'l is a propor- 

 tionality constant. If co„, is less than about half the cross-over frequency, 

 then the noise power at the motor input increases as the fifth power of the 

 bandwidth of the servo transfer characteristic.'^ Thus, if the input signal/ 

 noise ratio is small, this effect may be an important design consideration. 



Still other servo errors may result from local extraneous signals or from 

 coulomb and static frictional effects. These error sources are in a some- 

 what different class from those discussed previously, in that they are more 

 nearly under the designer's control. That is, such extraneous signals 

 and friction may be kept small by proper design and the residual friction 

 effects further reduced by the use of local feedback. In the absence of 

 local feedback, the servo error resulting from frictional or other torque dis- 

 turbances at the output shaft readily is found to be 



S{jo}) 1 + M 



S{jo^) is the actual stifTness (loop opened) of the output mesh, and T is the 

 disturbing torque. T conceivably may represent static or coulomb friction, 

 load-torque irregularities due to fluctuating running-friction, or wind torque. 

 Again assuming the mechanical impedance to be resistance and inertia in 

 series, the mechanical stiffness is, from (2.2), Sijo:) = jo}{R + i"/). Thus 

 the error is 



T 1 



]ui{R -r joiJ) 1 + M 

 and the apparent output stiffness (loop closed) is 



S' = MR + jc^J) (1 + m). (16.2) 



If T is taken as the static load torque, the resulting static error is found 

 by setting w = in (16.1). Assuming that fi behaves as wo/jco when co 

 approaches zero, the static error is 



A. = -^, (16.3) 



2^ This assumes a constant functional form for the transfer characteristic. However, 

 the statement holds approximately, even with considerable variation in this form. 



