488 



-(^X 



AUTOMATIC TRACKING CIRCUITS 



180^ 



en = (0.047) (6.28) 



fl (9 + 3 + 1) ]'^^ 

 [3(3 + 3+ 1)J 



0.047° rms 

 1 (9 + 3 + 1) 



eb 



3 (3 + 3 

 0.0578 Vsec rms 



0.13 (0.0246)3 



(1 + 3)- 



(20)(6.28) (6.28)2 



eg = (0.0I)(1.05) = 0.0105 Vsec 

 es"" = <TT^ - (eg' + ej) = (0.105) 

 = 0.011025 - 0.00343 (7sec) = 

 es = 0.0870 °/sec rms 



= 0.00292 Vsec 



[(0.0100)2 + (0.0578)2] 



"This is actually part of an infinite series. Higher-order terms are neglected because the 

 higher derivatives of 6 are negligible in most cases. If the higher derivatives have large 

 magnitudes, they must be included with more terms in the series or the series must be termi- 

 nated with simple transient terms as described in a paper "A Simple Method for Calculating 

 the Time Response of a System to an Arbitrary Input" by G. A. Biernson, MIT Servo Lab 

 Report #7138-R-3, Jan. 1954. 



The parameter k (the ratio of coc to C02) was chosen as 3.0. A larger ratio 

 would produce long transient settling times and larger steady-state errors 

 due to accelerations. A smaller ratio would cause a substantial increase in 

 rate noise. This is shown in Fig. 9-6. For the simplified track loop shown, 



Simplified Track 



Loop Transfer 



Ratio 



Hi*^) 





Noise Power 

 Density Function 



IKJ 



_ao_ ^ h \ k^+ kr +7 T 

 ""^"''hlk +kr + ri 



Fig. 9-6 Variation of Rate Noise with Tracking Loop Transfer Function. 



the rate noise formula in the figure has been derived without simplification. 

 From the curves, it can be seen that the choice of ^ = 3 is a rather arbitrary 

 compromise. Actually k = 2 could have been selected, but, although not 

 indicated in the figure, it would have produced a less stable system. 



