446 REGULATORY CIRCUITS 



required for the peak error to decay to within a certain percentage of the 

 final value could be specified, a maximum time for the peak error to occur 

 could be given, or the percentage distortion of the overall pattern dimen- 

 sions which can be tolerated could be specified. Fig. 8-32 indicates approxi- 

 mate but useful relationships between the dynamic transient, the open-loop 

 transfer function, and the steady-state error that help determine the search 

 control-loop design characteristics. 



For example, if the allowable distortion incident to peak dynamic error is 

 to be within 10 per cent of an overall pattern sweep of 30°, the peak error 

 would be 3°. From Fig. 8-32, the upper bound is about Irl ^c above the 

 negative error, and the approximate peak error above zero is 2f/coc — r/i^„ 

 = 3 deg; and if K, = 400 sec^^ and r is 100° /sec, 100(2K - 1 /400) = 3 

 and coc = 61.6 rad/sec. This is the search-loop bandwidth. If greater 

 accuracy is required, a more sophisticated pattern command would be 

 necessary with special accelerating and decelerating controls — perhaps 

 nonlinear control for maximum effort. This is not usually necessary, how- 

 ever, to obtain a relatively constant sweep velocity. Other characteristics 

 of the open-loop transfer function are found from stability considerations, 

 and an optimum system can be determined directly from the three following 

 equations relating the corner frequencies, peak phase margin, and the loop 

 gain shown in Fig. 8-32.^^ 



SEARCH LOOP SYNTHESIS 



1. Loop gain equation: 



^(^Yco, = K, = 400 (8-35) 



C0 2\C0l/ 



CO, ^ /C. ^ 400 6.49 (8-36) 



coi coc 61. 6 



2. Phase equation (frequency response peak = 1.3,^* peak phase margin 

 </)„ = 50.3° at a frequency co^ = ^c cos 0„, = 0.64coc) : 



(-180° + 50.3°) 



r-(5-„":,) + (!-S) 



57.3 



(8-37) 



_ CO,. ^ 0.69 = ^"^-"'^ + ^' 



CO 3 CO„j CO 3 



where ^m = phase angle of G at co„(. 



2''The derivations of these equations are discussed in the paper, "Synthesis of Feedback 

 Control Systems with a Minimum Lead for a Specified Performance," by George S. Axelby 

 in IRE Transactions in Automatic Control, PGAC-1, May 1956. 



^The closed-loop frequency response peak A/,, occurs at a frequency co„i = wc cos <^m and 

 sin <^m = ^IM ,, in the optimum, minimum lead system. 



