The Transfer function of the Lead Compensator is.

The conventional approach used to design a lead compensator uses the following steps:-

Consider the default parameters (20.0, 0.5, 0.3, 0.12, 0.2, 1.0).

The data on the applet shows :-

The frequency of max phase lead w

The uncompensated cross over frequency cofu is ~6.17rad/s with slope ~38.17dB/Decade.

The uncompensated phase margin upm is 17.95° while the uncompensated gain margin is undefined because the phase does not become less than -180°.

For the given value of a the lead gain dBLead is 9.206dB and the estimated new crossover frequency cofe is 10.65rad/sec with phase margin 10.62°.

The compensated system crossover frequency cofc is 11.4rad/s with a slope of 23.86 dB/dec.

The phase margin of the compensated system cpm is 61.33°.

To get a phase margin of ~60° change a to 0.13 to get a f

TL is now adjusted to make w

The slope at the new crossover frequency is 23.91dB/dec down from 38.17dB/dec.

The 3dB bandwidth is 16.55rad/s up from 9.659rad/s and the peak of the closed loop response is 0.838dB, down from ~10dB.

eg.

For Kv = 20.0, T1 = 0.3, Phase Margin = 50°, Set TL = 0.18, a = 0.28.

For Kv = 20.0, T1 = 0.6, Phase Margin = 50°, Set TL = 0.26, a = 0.21.

For Kv = 15.0, T1 = 0.6, Phase Margin = 50°, Set TL = 0.29, a = 0.25.

For Kv = 20.0, T1 = 0.5, Phase Margin = 60°, Set TL = 0.26, a = 0.14.

The applet below show the Nyquist plot of a system with a lead compensator. The information is similar to the Bode plot but shown differently.

The parameters on the plot show the values of the curves where the "x" is located. The position of the "x" is changed by w on the applet.

Setting the parameters to (20.0, 0.5, 0.28, 0.1, 0.2, 6.2 or 11) shows that the compensator changes the phase margin by ~47° and an increase in crossover frequency by ~ 5rad/s.

The applet below show the root locus for the system.

For the parameter values (K, 0.5, 0.3, 0.12, 0.2, 1.2), increasing K from zero first produces real poles up to K ~ 0.7. Furthur increase in K does not produce large overshoot because the closed loop poles move furthur to the left. For the parameter values (10.0, 0.5, 0.3, α, 0.2, 1.0), increasing α up to ~0.18 moves the poles furthur to the left, thereafter they start moving to the right.

For parameters (0.0, 0.5, 0.3, α, 0.2, 1.0). setting α = 1 show the locus of the uncompensated system. Reducing α pulls the locus to the left. At α = 0.16 it comes back to the real axis and forms a circle for smaller α.

This is similar to what happens when Td in a PD control system is increased.

Design by the root locus method requires finding TL and a which produces a given w

eg for Kv = 15, T1 = 0.5, Mag = ~6, angle ~ 35°. Set TL = 0.23 and a = 0.1. With these parameters the crossover frequency is 7.578rad/s with slope 24.37dB/dec, the 3dB frequency is 10.75rad/s and the phase margin is 65.05°.

Gif images below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.