LINEAR SERVO THEORY 637 



(9.1), are roughly constant with frequency over the servo band. This de- 

 mands that fx have somewhat the same frequency distribution as the input 

 signal spectrum (for | /x | )>> 1). Because of stabihty requirements and 

 complexity of the necessary apparatus, this rule can usually be followed 

 over only a part of the servo frequency band, especially when the input 

 signal spectrum falls ofif very rapidly with increasing frequency. However, 

 even a rough adherence to this desired relation is usually of real worth in 

 reducing the noise errors of the servo. An illustration of this will be given 

 in a later section. 



3.21 Approximate Calculation of Dynamic Error 



Frequently the servo requirement is to transmit, with great accuracy, a 

 type of signal whose frequency spectrum falls off very rapidly with increas- 

 ing frequency. As may be seen from (9.1) this demands very large values 

 of loop transmission ix at the lower frequencies where the input signal 

 energy is concentrated, but permits a rapidly dropping loop transmission 

 versus frequency commensurate with the falling amplitude spectrum of the 

 input signal. Such a rapid reduction in loop gain is practicable while 

 I ju I ^ 1. However stability considerations force a more gradual gain 

 reduction as the region of gain cross-over is approached. As a result, 

 contributions to the servo error from this frequency region may be neg- 

 lected compared with those from the lower frequencies. This suggests a 

 series expansion of (9.1) in the form, 



A = [(70 + fli(iw) + a2{jo:Y -f a^ijc^y J^ ...]F,, (11) 



where oo , Qi , etc. are real constants. 



Because of the assumed rapid drop in component amplitude Fi with in- 

 creasing frequency it is often unnecessary to take account of moie than a 

 few terms of the expansion. -- 



It is easy to show that (11) may be rewritten on a time basis to give the 

 total dynamic error as 



A(0 = a,fx{t) + ai/i(0 + a2fi{t) + • • • , (12) 



where (*) = d{ )/dt. Thus the coefficient co gives the error component 

 proportional to input displacement. Similarly, ci and a^ are the coefficients 

 of the error components due to input velocity and input acceleration, re- 

 spectively. For a great many motor-drive servo systems the loop trans- 

 mission n approaches infinity as l/;a) when co approaches zero. This en- 



^"^ The series may be said to converge rapidly in a practical sense, foi the following reason : 

 For small values of co the higher order terms are negligible. For values of oj sutliciently 

 large that the high order terms may no longer be neglected the coeflicient b\ has become so 

 small as to make the contribution of the entire series negligible. 



