Linear Servo Theory 



By ROBERT E. GRAHAM 



The servo system is a special type of feedback amplifier, usually including a 

 mixture of electrical, mechanical, thermal, or hydraulic circuits. With suitable 

 design, the behavior of these various circuits can be described in the universal 

 language of linear systems. Further, if the servo system is treated in terms of 

 circuit response to sinusoidally varying signals, it then becomes possible to 

 draw upon the wealth of linear feedback amplifier design based on frequency 

 analysis. 



This paper discusses a typical analogy between electrical and mechanical 

 systems and describes, in f'reciuency response language, the behavior of such 

 common servo components as motors, synchro circuits, potentiometers, and tach- 

 ometers. The elementary concepts of frequency analysis are reviewed briefly, 

 and the familiar Nycjuist stability criterion is applied to a typical motor-drive 

 servo system. The factors to be considered in choosing stability margins are 

 listed— system variability, noise enhancement, and transient response. The 

 basic gain-phase interrelations shown by Bode are summarized, and some of 

 their design implications discussed. In addition to the classical methods, 

 simple approximate methods for calculating dynamic response of servo systems 

 are presented and illustrated. 



Noise in the input signal is discussed as a compelling factor in the choice of 

 servo loop characteristics. The need for tailoring the servo loop to match the 

 input signal is pointed out, and a performance comparison given for two simple 

 servos designed to track an airplane over a straight line course. The use of 

 subsidiary or local feedback to hnearize motor-drive systerns, and predistortion 

 of the input signal to reduce overall dynamic error are described. 



1. Introduction 



A SIMPLE servo system is one which controls an output quantity 

 according to some required function of an input quantity. This 

 control is of the "report back" type. That is, some property of the output 

 is monitored and compared against the input quantity, producing a net 

 input or "error" signal which is then amplified to form the output. The 

 first statement defines the servo as a transmission system; the second, as a 

 feedback loop. The problem of servo design is then to fashion the desired 

 transmission properties while meeting the stability requirements of the 

 feedback loop. This is the familiar design problem of the feedback amplifier. 



2. The Servo Circuit 



The design of linear feedback amplifiers has been developed to a high 

 degree in terms of frequency response; that is, in terms of circuit response 

 to sinusoidal signals.^ The servo system is a special type of feedback 

 amplifier, and usually can be made fairly linear. Thus, it is logical to 

 analyze and design the servo circuit on a frequency response basis. Also, 



1 See "Network Analysis and Feedback Amplifier Design," by H. W. Bode, D. Van 

 Nostrand Co., 1945. 



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