Bode plot and Step Response for 3 Real Poles

Cuthbert Nyack

This applet shows the Bode plot of a 3 real pole system:-
The phase(green) varies from zero to -1.5p and the slope of the magnitude(red) plot becomes -60dB/decade at high frequencies. The magnitude vs phase plot is shown in yellow and the step response in pink. Gain and phase margins are indicated by approximate values.

The eg parameters (2.19, 0.2, 1.0, 2.0, 1.0, 117) show the step response when the phase margin is ~45° at ~ 0.78rad/s and (0.87, 0.2, 1.0, 2.0, 1.0, 96) when the phase margin is ~90°, at 0.35rad/s and a gain margin of ~ 19dB.
Parameters (7.92, 0.2, 1.0, 2.0, 1.0, 136) show one case where the phase margin is zero and the system is oscillatory with period ~3.89s and angular frequency 1.61rad/s. At this frequency the magnitude curve crosses the 0dB line, the phase curve crosses the -180° line and the mag vs phase passes over the intersection of the 0dB and -180° lines.
eg parameters (8.3, 0.2, 1.0, 2.0, 4.6, 137) show the step response when the closed loop poles have crossed over the imaginary axis and are now in the right half of the s plane.
eg parameters.
(2.88, 1.0, 1.0, 0.2, 1.0, 130) w = 1.183rad/s.
(6.48, 1.0, 1.0, 0.8, 1.0, 135) w = 1.612rad/s.
(13.53, 1.0, 1.0, 1.6, 1.0, 145) w = 2.05rad/s.
(23.14, 1.0, 1.0, 2.4, 1.0, 149) w = 2.41rad/s.
show the effect, on K required for oscillation, of moving pole c to the left.

Gif image below shows how applet should appear.

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