8-33] DYNAMIC STABILITY REQUIREMENTS 465 



Consequently, the peak overshoot to a step input may be as high as 

 40 per cent.^^ Actually, this overshoot never appears in the antenna signals 

 during actual operation because the input signals are never step functions 

 and the output signals are heavily filtered. Therefore, the primary objective 

 is to design a stabilization loop with the necessary isolation charactersitics 

 and then to obtain the minimum bandwidth possible with a stability margin 

 that is not seriously affected by normal component tolerances, gain changes, 

 or environmental effects. Essentially, reasonable stability margin in a 

 minimum lead system for a stabilization loop would be: phase margin 

 greater than 40°, gain margin greater than a factor of 4 (or 12 db). How- 

 ever, if precision components are used which are not greatly affected by 

 environmental conditions, these margins may be reduced to: phase margin, 

 35°; gain margin, 6 db. 



The attenuation characteristic Ks of the stabilization loop with respect 

 to aircraft disturbances ^gl may be derived from Fig. 8-37a as 



j^ ] + G2CJ8 



!^^I=^tVi i^ G2G3»1.0" (8-47) 



The magnitude of Ks is determined as a function of frequency from one 

 of the criteria in Paragraph 8-32 and plotted as shown in Fig. 8-39. The 

 magnitude of 1 /G2G3 must be below these points as shown, and conse- 

 quently, the magnitude of G2G3 must be greater than the reciprocal values 

 of Ks. The magnitude of ^26*3 corresponding to the values of Ks in Fig. 

 8-39 but increased by 175 per cent is shown in Fig. 8-42.*^ This charac- 

 teristic need not be extended to frequency regions below 3 rad/sec as 

 discussed at the beginning of Paragraph 8-30. As shown in Fig. 8-42, the 

 magnitude of G2G3 is 250 at 3 rad/sec to make the actual gain equal that 

 desired. This becomes the d-c gain in the rate gyro loop shown in Fig. 8-37a. 

 In an integrating gyro loop, shown in Fig. 8-37b, the d-c gain is infinite, but 

 the velocity constant would be 750 sec~^ as indicated in Fig. 8-42. 



As in the search loop design, the other characteristics of the open-loop 

 transfer function, for a minimum bandwidth, are determined directly from 

 stability considerations, using the following equations relating the loop 



■^^This corresponds to a maximum phase margin of about 40° or a frequency response peak 

 of about 1.6 in a minimum lead system such as that discussed in Paragraph 8-28. 



4'' Actually this approximation holds very well for IG2G3I ^ 3 in most cases. 



^*The 175 per cent increase in gain is made to counteract changes in the stabilization loop 

 gain. One source of gain change is the normal variation in production tolerances, and ±20 

 per cent may be allowed for this. The other large gain change is in the gyro (azimuth only) 

 because the azimuth gyro gain changes in proportion to the cosine of the antenna elevation 

 angle when it is mounted on the antenna dish. Since the antenna may have elevation angles 

 of 50°, this causes a 55 per cent gain change, a total of 175 per cent. 



