Page 521 - Mechanical Engineers' Handbook (Volume 2)

512 Control System Performance Modification

Figure 11 Family of constant-N circles.

and constant-M loci derived earlier cannot be directly applied. However, with a slight mod-
ification constant-M and constant-N loci can still be applied to systems with nonunity feed-
back. We modify Eq. (5) as

C(s) 1 G(s)H(s)

R(s) H(s)1 G(s)H(s)
The magnitude and phase angle of G (s)/[1 G (s)], where G (s) G(s)H(s), may be
1
1
1
obtained easily by plotting the G ( j ) locus and reading the values of M and N at various
1
frequency points. The closed-loop frequency response C( j )/R( j ) may then be obtained
by multiplying G ( j )/[1 G ( j )] by 1/H( j ).
1
1
3.4 Closed-Loop Amplitude Ratio
In obtaining suitable performance, the adjustment of gain is usually the first consideration.
The adjustment of gain is usually based on the maximum closed-loop gain or the resonant
peak. That is the gain K which must be chosen so that over the entire frequency range the
closed-loop amplitude ratio M is not exceeded.
r
Consider first isolating the circle corresponding to M as shown in Fig. 12. Then a
r
tangent line to the M circle is drawn from the origin, which makes an angle with the real
r
line.