8-33] DYNAMIC STABILITY REQUIREMENTS 467 



For each system: co^ = coc cos </>„, (f)m = 40°, coi = 3 rad/sec. 

 Solving the equations: 



29.0 rad/sec 



59.6 rad/sec 



77.8 rad/sec (12.4 cps) 



273.9 rad/sec 



0)2 = 30 A rad/sec 



Wm = 56.6 rad/sec 



coc = 74.0 rad/sec (11.8 cps) 



W3 = 263.8 rad/sec 



The derived corner frequencies are plotted in Fig. 8-42. This establishes 

 the required stabilization loop transfer function. As shown in the figure, 

 there is little practical difference in bandwidth between the stabilization 

 stability margin and the same attenuation characteristic. 



The gain margin was not used to establish the stabilization loop transfer 

 function because it is more than adequate unless resonant frequencies in 

 the antenna structure are near the crossover frequency of the loop. In fact, 

 if a resonant frequency is close to ooz with a damping factor less than 0.1, 

 it will reduce the gain margin below 6 db without changing the phase margin 

 appreciably. Since the lower frequency characteristics of the loop are 

 specified as described in previous sections, it is necessary to specify that the 

 antenna resonant frequencies with low damping should be above 003 by 

 50 per cent or more. Thus, the lowest resonant frequency should be greater 

 than 400 rad/sec or 64 cps."*^ If resonant frequenci^ below this figure 

 should exist in the stabilization loop, the loop stability or the isolation 

 specifications must be reduced or precision components must be used with 

 closer stability margins. 



In practice, ojs need not be a double corner as shown in Fig. 8-42; but it 

 should be an equivalent where C03 would be the geometric mean of two 

 corner frequencies. Actually C03 should be slightly higher to allow for the 

 inevitable but unknown higher corner frequencies which are characteristic 

 of all physical equipment and which often reduce the phase margin at the 

 crossover frequency. However, if these frequencies are known, or if they 

 can be estimated, they can be included easily in the equations which are 

 used to calculate the frequencies coc, Wm, C02, and ccs- It is not always possible 

 to do this with more rigorous methods. In fact, it should be emphasized 

 that the approximate equations give excellent results, but only if all the 

 corner frequencies, up to 20 times the loop bandwidth, of the actual 

 equipment are included or estimated and used in the calculation. Actually, 



^^Theoretically, resonant frequencies can be canceled by electrical networks, and this may 

 be done in practice to increase stability margins. However, in practice, the cancellation 

 cannot be made perfectly under all conditions, and the resulting improvement is not very large. 

 For more details about resonant frequency effects on control loop performance see "Some 

 Loading Effects on Servomechanism Performance," by G. S. Axelby, Aeronautical Electronic 

 Digest, 1955, pp. 226-241, National Conference on Aeronautical Electronics, Dayton, Ohio, 

 May 1955. 



