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.

To find 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 ± j0.8.

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.

COPYRIGHT © 2006 Cuthbert Nyack.