LINEAR SERVO THEORY 627 



As usual, specification of the form of ix^ijoi) is beset by a series of per- 

 formance objectives on the one hand and a set of physical hmitations and 

 restrictions on the other. Assuming the relationship expressed by (6.1) 

 to be the required one, it would seem desirable to make ii^{j<ji) very large 

 compared to unity at all frequencies. However, there are reasons why this 

 is neither possible nor actually desirable. As the value of oj is increased, 

 the loop transmission is eventually controlled by parasitic circuit elements 

 such as distributed capacity and inductance in the electrical circuits, and 

 distributed inertia, compliance, and backlash in the mechanical circuits. 

 The effect of these parasitic elements at the higher signal frequencies is to 

 cause I Mi3 I to decrease as a very high order of l/oj with increasing frequency. 

 It will be shown, however, that feedback stability considerations require 

 the loop transmission to be decreasing comparatively slowly through the 

 frequency region where | m/3 | is of the order of unity. Thus ju/3 must be 

 reduced below unity at a frequency sufficiently low to avoid excessive con- 

 tribution from the parasitics. 



The presence of "noise" or undesired disturbances in the servo input 

 signal is another compeUing factor in the design of the loop characteristic. 

 Input noise is harmful both in causing spurious output fluctuations and in 

 overloading the power stages of the serv'o system. Both of these effects 

 are reduced by narrowing the frequency band of the servo transfer charac- 

 teristic. Referring to the expression for the transfer characteristic given 

 by (6), it may be seen that a restricted transfer bandwidth may be obtained 

 by reducing y. and n^ well below unity at a small value of signal frequency.^ 



On the other side of the picture is the requirement of fidelity in main- 

 taining the desired input-output relationship. Undue narrowing of the 

 transfer bandwidth of the servo results in large dynamic error, the mag- 

 nitude of which depends both upon the character of the input signal and 

 upon the chosen transfer characteristic. 



The optimum design of a servo system, for a specified input signal and 

 noise, thus is a compromise between dynamic error and output noise fluc- 

 tuations, with stability considerations and parasitic circuit elements re- 

 stricting the possible choice of loop transmission characteristics, 



3.1 Stability of Single Loop Systems 



The word stable as applied to a servo system is used here to imply a sys- 

 tem whose transient response decreases with increasing time. It is possible 



* When the /3 characteristic is under suitable design control, another method is available* 

 Thus if is made to rise in the frequency region of the desired transfer cut-ofT, and if nfi 

 is maintained large beyond this region, (6.1) shows that the desired restriction is elTected. 

 For a given transfer characteristic, this cut-off method requires a \Yider frequency range 

 for A(/3 and is thus more vulnerable to the elTects of parasitic circuit elements. However, 

 shaping of both the /z and /3 circuits permits a more rapid cut-off of the servo transfer 

 characteristic than is possible with /x circuit shaping alone. 



