# Phase Lag 0T1 Transient, Frequency Analysis and Root locus

Cuthbert Nyack
The Transfer function of the Lag Compensator is. This function has a lead and a lag factor with the 3dB frequency of the Lead factor being higher than that of the Lag factor. The Transfer function therefore produces a phase lag which would normally destabilise a system. However the 3dB frequency of the lag factor is made low so the increased phase lag at the gain crossover frequency is low. The stabilising factor in this compensator is the reduction in gain at frequencies around the gain cross over frquency without affecting the low frequency gain. Although this compensator reduces the transient oscillations, it slows the response because of the reduced cross over frequency. This is illustrated by the Applets below which show how the transient response, frequency response and root locus are affected by the compensator.

eg parameters (2.0, 0.5, 10.0, 1.0, 1.0, 1.0).
In this the compensated and uncompensated systems are identical.
The uncompensated and compensated cross over frequencies are 1.5726rad/s. Gradient of the mag plot at the crossover frequency is 27.38dB/decade, phase margins are 51.82°, 3dB frequencies are 2.542rea/s and the peak of the closed loop response is 1.243dB. With a Kv of 2, the velocity error of this system is large.

eg parameters (20.0, 0.5, 12.5, 11.6, 1.0, 1.0).
The increased Kv reduces the velocity error by a factor of 10. The uncompensated system now has undesirable transient behaviour with the slope at the crossover frequency = 38.11dB and a phase margin of 17.95°.
The compensated system has a slope at the crossover frequency of 26.77 dB/decade. a phase margin of 51.82° (same as for the uncompensated system with Kv = 2.0) and a peak closed loop response of 1.307dB. The transient behaviour is now acceptable with a tenfold reduction in the velocity error. The negative side of this compensator is that the crossover frequency has dropped from 1.5726rad/s for the uncompensated system to 1.411rad/s for the compensated system. The 3dB frequency has dropped from 2.542rad/s for the uncompensated system with Kv = 2 to 2.305 for the compensated system.

NB The lag compensator has reduced the gain at 6.172rad/s(crossover frequency of the uncompensated system with Kv = 20.0) by -21.2dB while the contribution of the compensator to the phase at the new crossover frequency of 1.411rad/s is -2.96° which is negligible.

The effect of the Lag compensator is also shown with the Nyquist plot below. 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.

In the root locus plot the phase lag compensator introduces a zero and pole very close to the origin. In the applets TG moves both the pole and zero while β changes the pole independently of the zero. For the parameter values (K, 0.5, 10.0, 10.0, 1.0, 1.0), the poles remain real for K up to ~ 5.5. Larger values of K produce complex conjugate poles with the angle they make to the -ve real axis increasing. For the parameter values (10.0, 0.5, 10.0, β, 1.0, 1.0), the poles remain real for β > ~18.1. Smaller values of β result in the poles becoming complex conjugate with the angle made with the -ve real axis increasing.
For the parameter values (0.0, 0.5, 10.0, β, 1.0, 1.0), changing β from 1(no compensation) to 10 produce very little change to the appearance of the root locus. However the value of K on any point of the locus is strongly affected.
Eg for β = 1, K = 1 has cc poles at -1 ± j. With β = 10, K = 1 has poles at -0.055 ± j0.0865. To get the poles close to -1 ± j, K has to be increased to 10.4(poles at -0.95 ± j0.9942).

Gif images below shows how applets should appear.   