484 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Methodthe Bode diagram. Figure 7–85(a) shows the G(jv)locus together with the MandNloci.
The closed-loop frequency-response curves may be constructed by reading the magni-
tudes and phase angles at various frequency points on the G(jv)locus from the Mand
Nloci, as shown in Figure 7–85(b). Since the largest magnitude contour touched by the
G(jv)locus is 5 dB, the resonant peak magnitude Mris 5 dB. The corresponding reso-
nant peak frequency is 0.8 radsec.
Notice that the phase crossover point is the point where the G(jv)locus intersects
the–180° axis (for the present system,v=1.4radsec), and the gain crossover point is
the point where the locus intersects the 0-dB axis (for the present system,
v=0.76radsec). The phase margin is the horizontal distance (measured in degrees)
between the gain crossover point and the critical point (0dB,–180°). The gain margin
is the distance (in decibels) between the phase crossover point and the critical point.
The bandwidth of the closed-loop system can easily be found from the G(jv)locus
in the Nichols diagram. The frequency at the intersection of the G(jv)locus and the
M=–3dB locus gives the bandwidth.
If the open-loop gain Kis varied, the shape of the G(jv)locus in the log-magnitude-
versus-phase diagram remains the same, but it is shifted up (for increasing K) or down
(for decreasing K) along the vertical axis. Therefore, the G(jv)locus intersects the M
201612840- 16
- 12
- 8
- 4
- 240 ° – 210 ° – 180 ° – 150 ° – 120 ° – 90 °
(a) (b)G1 dB3 dB0.25 dB5 dB12 dB
|G| in dB- 1 dB
- 5 dB
- 12 dB 1.8
1.41.210.80.60.40.2- 30 °
- 20 °
- 10 °
- 60 °
- 150 ° – 120 ° – 90 °
v in rad/secG
1+GG
1+Gin dB- 270 °
- 180 °
- 90 °
- 15
- 10
- 5
- 90 °
05100 °0.1 0.2 0.4 0.6 0.8 1 2Figure 7–85
(a) Plot of G(jv)superimposed on Nichols chart; (b) closed-loop frequency-response curves.Openmirrors.com