9-8] ANGLE TRACKING LOOP POSITION ERRORS 489 



Bias Errors. As shown in Fig. 9-5, the bias error is obtained from a 

 segment of an infinite series — an error series, the sum of line-of-sight 

 angular derivatives^^ attenuated by the velocity constant A'„ and accelera- 

 tion constant of the loop. Although it is not shown, the higher-order loop 

 constants of the assumed track loop become larger and the input derivatives 

 become smaller. Therefore, sufficient accuracy is obtained by using a series 

 with only two terms. For inputs with larger, higher-order derivatives, more 

 terms would have to be used as discussed in the footnote of Fig. 9-5. As 

 shown, the bias rate error eb is much less than the value specified. If it were 

 not for transient conditions, omitted from the equations, the bandwidth 

 and error constants could have much smaller values without causing the 

 bias error to become excessive. 



9-8 ANGLE TRACKING LOOP POSITION ERRORS 



As discussed in Chapter 8, the position errors are generally not as 

 significant as the errors in rate measurements. However, this is not always 

 true and the position errors should be calculated. Again the errors are 

 divided into random and bias categories. 



Random Errors. The components of the random errors in the position 

 signal and the methods of computing them are shown in Fig. 9-7. 



The most significant random errors in position are caused by {a) mechan- 

 ical components and misalignments Cc — including boresight mechanical 

 transducers, servo noise, etc. — {b) antenna platform motion e^p not 

 completely counteracted by the stabilization loop, and (<r) radar noise Cnp 

 transmitted through the control loop. Note that the formula for radar noise 

 in the position signal is different from that used to calculate the noise in 

 the rate signal. 



For purposes of comparison with Fig. 9-5 the allowable stabilization 

 errors in position are calculated. Although the magnitude of this error 

 appears to be one-half of the allowable rate error, the loop gain required of 

 the stabilization loop is only slightly lower. ^^ Of course, if more accurate 

 components and alignments could be obtained, the stabilization loop gain 

 could be less. On the other hand, if the stabilization loop gain could be 



i^The input derivatives, e, "d, and "e are the first, second, and third derivatives, respectively, 

 of Als in Fig- 9-4. They are components in one channel only. 



15 e,^ = 2^^ S 0.0452 (from Fig. 9-7) 



G2G3 



\G,gL > "-^^^^ at c = 3.14, Atl = 4.76° peak; \G,Gz\, ^ 73.8 



' 0.0452 



Increasing this by 75 per cent to allow for loop gain variations, \G2G3\p = 129. The \G2G3\p 

 for other frequencies are plotted in Fig. 8-42. 



