The compensators in the control system are classified as:
Lead compensator
Lag compensator
The differences between the phase lead and phase lag compensators are as follows:
Lag-lead compensatorAdvantages of the lag-lead compensator
Applications of the lag-lead compensator
Example 1. Find the phase shift provided by the lead compensator with the transfer function given by:
Transfer function = (1 + 6s) / (1 + 2s).
(a) 30 degrees
(b) 15 degrees
(c) 45 degrees
(d) 60 degrees
Correct Answer is option (a)
The transfer function of the lead compensator is in the form:
= s + 1/T divided by s + 1/AT
Comparing,
(1 + 6s) / (1 + 2s) or with (s + 1/T) / (s + 1/AT), we get:
A = 1/3
The maximum phase shift of the lead compensator is given by:
Phase angle = sin-1(1 - A / 1 + A)
= sin-1(1 - 1/3 / 1 + 1/3)
= sin-1(2/4)
= sin-1(1/2)
= 30 degrees
Example 2. The transfer function given below represents which type of compensator.
Transfer function = 1 + 0.5s/ 1 + s
(a) Lead compensator
(b) Lag compensator
(c) Lag-lead compensator
(d) Proportional Integral Controller
Correct Answer is option (b)
The transfer function of the lag compensator is in the form Vo(s)/Vi(s) = 1 + Ts/ 1 + BTs.
After comparing, we can find out the given transfer function is a type of lag compensators.
Example 3. The phase-lead compensator for a gain constant K,
(a) Increases the phase margin
(b) Increases the gain crossover frequency
(c) Reduces the slope magnitude curve in the entire range of frequency domain
(d) All of the above
Correct Answer is option (d)
The phase margin increases due to the addition of phase angle near the gain crossover point. It also increases the gain crossover frequency due to the increase in the gain at high frequencies. It also reduces the slope magnitude curve in the entire range.
Example 4. A lead compensator used for a closed loop system controller has the following transfer function K(1 + s/a)/(1 + s/b). The condition for such as lead compensator is given by:
(a) a < b
(b) b < a
(c) a < Kb
(d) a > Kb
Correct Answer is option (a)
The phase angle for the phase lead compensator is generally positive.
It is given by:
Tan-1w/a - tan-1w/b
For the angle to be positive,
Tan-1w/a should be greater than the tan-1w/b.
Hence, a < b
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