636 BELL SYSTEM TECHNICAL JOURNAL 



where Fi and F-i are again t}'pical sinusoidal components of the input and 

 output respectively. Thus the corresponding sinusoidal error component 

 may be written as 



A = Fi - F2 = -^ . (9.1) 



The methods which may be used to determine the actual dynamic error 

 A(/) from (9.1) depend both upon the nature oifi(f) and the type of informa- 

 tion available about /i(/). If the input signal is a known periodic function, 

 A(/) may be found by appl^ang (9.1) for each sinusoidal component of the 

 input and summing the resulting terms. If the input is non-periodic in 

 character, then the error may be calculated from the Fourier integral ex- 

 pression 



A(/) = i- f " ^ .^-^ dco, (10) 



where Fi{cci) represents the continuous frequency spectrum oi fi(t), as ob- 

 tained from 



FM = r Mt)e''''-" df. (10.1) 



J— 00 



The problems of calculating Fi{u) from/i(/) and A(/) from Fi(w) often may 

 be avoided by consulting well-known tabular lists of paired time and fre- 

 quency functions. 



Equation (9.1) may be used as a broad guide in selecting the type of /z 

 characteristic best suited to a particular input signal. It has been men- 

 tioned previously that because of input noise and parasitic circuit elements, 

 the servo transfer bandwidth usually should be kept as narrow as possible, 

 consistent with dynamic error requirements. The transfer characteristic 

 /i/(l -J- /x) will be closely equal to unity while | /x | ^ 1, will rise slightly 

 in the region where \ n \ ^=^ 1, and fall off as n when /x is small compared 

 with unity. The "cross-over" frequency, for which [ m 1= 1, may be taken 

 as a rough measure of the transfer bandwidth. Thus, the requirement of 

 minimizing the bandwidth may be restated as that of minimizing the cross- 

 over frequency, while holding the dynamic error within specified limits. 

 Reasoning in a general way, this requirement may be met by designing /x 

 so that the amplitudes of the sinusoidal error components, as given by 



''" An excellent list is given by G. A. Campbell and R. M. Foster in a Bell Sj'stem mono- 

 graph "Fourier Integrals for Practical Application," September, 1931. A table of Laplace 

 Transforms, which also may be used, is given by M. F. Gardner and J. L. Barnes in 

 "Transients in Linear Systems," John Wiley and Sons Inc., 1942. 



21 Assuming a phase margin of the order of 60 degrees. 



