LINEAR SERVO THEORY 



633 



network phase shifts vanish, and that the gain curve cross zero db at only 

 one frequency.'^ 



An advantage of this logarithmic diagram is that commonly encountered 

 forms of I Mi^ I vary as w'^"' for intermediate or asymptotic frequency regions, 

 and thus plot as corresponding straight line segments. From (7.4) it may 

 be seen that the illustrative form of | )u/3 | behaves, in turn, as 200a)~', 

 200a)~-, 20aj~^ and 1.6 X lO^co"'*, asw is increased. These asymptotic lines 

 are drawn in lightly in Fig. 10, the actual gain describing smooth transi- 



120 



-140 O 



1 2 05 10 20 5.0 10 20 50 100 200 500 1000 

 Fig. 10 — Loop characteristic of illustrative servo system. 



tions between adjacent asymptotes. Since the logarithmic slope of a? is 

 ±6yfe db/octave,^^ the successive asymptotic slopes in Fig. 10 are —6, —12, 

 — 6, and — 24 db/octave. The junctures of adjacent asymptotes occur at 

 values of co of 1, 10, and 200, These juncture frequencies are called "cor- 

 ner" frequencies, and may be seen from (7.2) to coincide with the real con- 

 stants or "roots" added toyco in each of the three factors in parentheses. 

 The corner associated with the factor 200/;co is of course at co = 0. The 

 corner of the last, or cubed factor is a multiple one, joining two asymptotes 

 differing in slope by 18 db/octave. From a knowledge of such corner 



12 This discussion assumes that ^/3 has a low-pass configuration; that is, only the high 

 frequency cut-off is considered. If /i/3 has a ow-frequency cut-off also, then a corre- 

 sponding requirement identical with the above must l)e added. 



"' An octave is taken to be a 2:1 span in frequency, and 6 db is of course very closely 

 equivalent to a 2 : 1 increase in | m/3 I- 



