LIXEAR SERVO THEORY 631 



the system is seen to be stable.^ From a practical standpoint it is necessary 

 to know not only that a design is stable, but that it has sufficient margin 

 against instability. The need for proper stability margin arises from two 

 general considerations. First, the loop transmission of the physical system 

 will vary with time due to aging, temperature changes, line voltage fluctua- 

 tions, etc. Also the physical embodiment will depart from the paper de- 

 sign due to errors of adjustment and measurement, and to the effects of 

 unallowed-for parasitic elements. Second, a design which is too near 

 instability will have an undesirable transient response — large overshoots 

 and persistent oscillations — and will unduly enhance noise in the input 

 signal. 



Stability margin is measured in a sense by the minimum displacement 

 between the polar plot and the point —1,0. In feedback amplifier design, 

 two numbers often are taken as a measure of margin against instability. 

 These are called the gain margin and the phase margin. The gain margin, 

 Gm , measures the amount, in decibels, by which the magnitude of n0 falls 

 short of unity, at a phase angle of ±180 degrees. The numerical value of 

 gain margin for the system of Fig. 8 is about 18 db, which is the required 

 increase in amplifier gain to make the servo unstable. That is, this in- 

 crease in amplifier gain would multiply the cur\-e of Fig. 8 by a constant 

 factor such that it would intersect the point —1,0. The phase margin, 

 Bm , is equal to the absolute magnitude of the angle between — /u/3 and the 

 negative real axis, at | iu/3 | = 1. Figure 8 illustrates a phase margin of 

 about 50 degrees. That is, if the points on the curve where | /i/3 | = 1.0 

 were swung toward the negative real axis by about 50 degrees they woirid 

 coincide with the point —1,0, and the ser\'o would be unstable. 



The t>^e of transient response obtained with reasonable gain and phase 

 margins is indicated in Fig. 9, which shows the response of the illustrative 

 ser\-o system to an input step. The initial overshoot is about 25%, and 

 the oscillation damps out very quickly. For the general case, (6) ma}- be 

 rewritten in the form 



-M/3 



^' (8) 



LI - M^J -/3 



The relation Fo = Fi/ — i3 ma}- be regarded as the desired one, with the 

 bracketed factor acting as the inevitable modifier. Then if the quantity 



' With more complicated systems it may not be obvious whether or not the plot en- 

 circles — 1,0. A simple test employs a vector with its origin at the —1,0 point and its 

 tip on the curve. If the vector undergoes zero net rotation as it traces along the curve 

 from o) = to CO = =c , the curve does not encircle the critical point. 



'" In some servo systems a decrease in amplifier gain also may cause instability. Such 

 systems are still covered by the polar plot criterion of stabihty, and are commonly called 

 "Nyquist stable," or "conditionally stable." 



