Finding Root Locus for 3 Real Poles
Applet shows the determination of the root locus using the angle
and magnitude condition.
For parameters (0.0, 0.5, 1.0, 1.5, sr, si) the root locus for different
K is shown starting from the open loop poles(sr = 0.5, 1.0, 1.5), breaking away from the real axis at sr = ~-0.71 and tending
towards the asymptotes. To find the closed loop
poles for any K sr and si are adjusted
along the locus until the desired K is obtained.
poles for K = 1.0, sr = -0.46
and si = 0.8 give K = 1.014 and f = -180.68.
Also sr = -2.09, si = 0.0 give K = 1.02 f = -540°. Therefore for K = 1, the poles are ~-2.09, ~-0.46 ±
These parameters (1.0, 0.5, 1.0, 1.5, -0.46, 0.8) show that the locus
starts from the closed loop poles. The step response is shown in white and the frequency in light magenta.
For K = 2, poles are at ~2.33, ~ -0.34 ± j1.04.
This system may have an open loop pole to the right of the
imaginary axis and still be controllable.
eg parameters (0.0, 2.0, 2.5, -0.5, sr, si) has a pole at +0.5.
Setting sr = 0.0, si = 0.0, shows K = 2.5 when the locus
crosses the axis moving to the left.
Setting sr = 0.0, si = 1.66 show K = ~13.5 when the locus
crosses the axis to the right.
Increasing K from 0 shows the system is stable for 13.5 > K > 2.5 and unstable for other K's.
The calculated values of the closed loop poles are also shown
on the applet.
Gif image below show how applet should appear.
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COPYRIGHT © 2006 Cuthbert Nyack.