Nyquist Plot and Step Response for 4 Real Poles and 4 Real Poles 2 Complex Conjugate Zeros

Cuthbert Nyack

This applet shows the Nyquist plot for 4 real poles and 4 real poles plus 2 complex conjugate zeros. the plot of the 4 pole system goes from 0 to -360° while that of the 4 pole 2 zero system goes from 0 to -180°.
eg parameters
(4.0, 1.0, 1.0, 1.0, 1.0, 0.8, 0.4, 2.0, 1.0) show that the 4 pole system is oscillatory at 1rad/s. (4.0, 1.0, 1.0, 1.0, 1.0, 0.8, 0.4, 2.0, 1.85) show that adding the 2 zeros at -0.8 ± 0.4j produces a system with a ~65° phase margin.

parameters (10.0, 1.0, 1.0, 1.0, 1.0, 3.1, 1.0, 1.0, 1.56) show that when the zeros are moved furthur to the left, then the 4p2ccz system can be oscillatory at 1.56rad/s.

parameters (1.7, 0.96, 1.0, 1.0, 1.0, 2.0, 1.0, 3.0, 0.57-0.59) show that the 4p system has phase margin of ~60° and gain margin of ~ 7dB at 0.57rad/s and the 4p2z system has a phase margin of ~83° at 0.59rad/s and much higher gain margin.

The 4p2ccz can have more than one range of K for stability. eg parameters (5.4, 1.0, 1.0, 1.0, 0.5, 0.9, 1.9, 2.0, 1.16) and (32.3, 1.0, 1.0, 1.0, 0.5, 0.9, 1.9, 2.0, 2.12) show that this system can have 2 oscillation frequencies with the Nyquist plot passing through the -1 point in each case. The system is stable for K < 5.4 and K > 32.3 and unstable for intermediate K's.

Gif image below shows how applet should appear.

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