LINEAR SERVO THEORY 635 



of ±90^ degrees is approached more and more closely as the length of span 

 is increased. 



This may be observed qualitatively from the transfer characteristic of 

 Fig. 10. For oj <<C 1, the gain slope is —6 db/octave, and the phase shift 

 approaches — 90 degrees. For 1 < co < 10, the average gain slope is about 



— 10 db/octave, and the phase shift near oj = 3 is — 148 degrees (instead of 



— 90 X 10/6 = —150 degrees). As co increases toward 200, the phase 

 shift increases rapidly due to the asymptotic slope of — 24 db/octave, finally 

 approaching -360 degrees (-90 X 24/6) for w » 200. 



Foreknowledge of the inevitable gain-phase relationship is of great value to 

 the servo designer, in making clear the comparatively small class of realiz- 

 able gain-phase combinations and thus averting attempts at non-physical 

 designs. For example the design use of too-rapidly falling loop gain charac- 

 teristics in the region of the high-frequency gain cross-over (that is, near 

 zero db loop gain) is not permissible because of the large negative phase 

 shifts which must accompany the steep gain slopes. Another way of stat- 

 ing the advantage of an early realization of the gain-phase laws is to say 

 that the designer is assured in advance that any paired gain and phase 

 characteristics which he chooses within the basic restrictions will be achiev- 

 able with stable physical networks.^^ 



3.2 Dynamic Error 



A serv^o system is usually designed to transmit some class of input func- 

 tions with a required degree of fidelity. This class of functions may reduce 

 substantially to one specific input signal whose time variation or whose fre- 

 quency spectrum is known, or it may include a great variety of signals 

 which have certain properties in common. In the latter case it is conceiv- 

 able that definite limits may be placed upon the allowable amplitude ranges 

 of the input signal and its various time derivatives, or certain limiting fre- 

 quency spectrum characteristics may be specified for the input function. 



Servomechanisms are subject to several types of transmission error. 

 The systematic error, or dynamic error, is predictable from knowledge of the 

 noise-free input signal and of the transfer frequency characteristic of the 

 servo system. For simplicity, the discussion of error will be limited to the 

 case where the output signal is desired to be a replica of the input, and 

 where /3 = — 1. Thus the loop transmission /xjS becomes simply — /i. The 

 input-output relationship as given by (6) is therefore 



F2 = ^ F, , (9) 



1 + M 



^^ With some necessary reservations as to practicable dissipation constants and para- 

 sitic circuit constants. 



