For a lead compensator, whose transfer function is given by \(k \
A. a < b
B. a > b
C. a ≥ kb
D. a =0
Please scroll down to see the correct answer and solution guide.
Right Answer is: A
SOLUTION
In general, the lead and lag compensator is represented by the below transfer function
\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = k\frac{{s + a}}{{s + b}}\)
If a > b then that is lag compensator because pole comes first.
If a < b then that is lead compensator since zero comes first.
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Compen- sator |
Pole zero plot |
Response |
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Lead |
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Lag |
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Lag-lead |
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Lead-lag |
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NOTES:
Compensator:
It is an electrical network that adds one finite pole & one finite zero to the system at the required location to achieve a good performance.
1. Lead compensator
2. Lag compensator
3. Lag-lead compensator
4. lead-lag compensator
Lead compensator:
1) When sinusoidal input applied to this it produces sinusoidal output with the phase lead input.
2) It speeds up the Transient response and increases the margin for stability.
A circuit diagram is as shown:
Response is:
\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{{R_2}\left( {1 + s{C_1}{R_1}} \right)}}{{{R_1} + {R_2} + s{C_1}{R_1}{R_2}}}\)
\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{1 + s\tau }}{{1 + \alpha s\tau }}\)
Lead constant \(\alpha = \frac{{{R_2}}}{{{R_1} + {R_2}}}\) < 1
Lag compensator:
1) If the steady-state output has phase lag then the network is called lag compensator.
2) It improves steady-state behavior without affecting the transient response.
A circuit diagram is shown
Response is
\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{\frac{1}{{s{C_2}}} + {R_2}}}{{{R_2} + {R_1} + \frac{1}{{s{C_2}}}}}\)
\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{1 + s\tau }}{{1 + \alpha s\tau }}\) where α > 1
Lag constant \(\alpha = \frac{{{R_1} + {R_2}}}{{{R_2}}}\)
Lag-lead compensator:
- Here both phase lag and lead occur at different frequencies.
- Phase lag at low frequency and Phase lead at the high frequency
- Improves both transient and steady-state response.
Trick:
To remember the response of these compensators.
Lead – High: In both the terms 4 letters are present, and in the pole-zero plot first comes zero (it also has 4 letters). The response is like a High pass filter.
Lag – Low: In both the terms 3 letters are present and the first one is a pole, the response is like Low pass filter
Lag-lead: Response is like a Bandstop filter.
Lead-lag: Response is like a Bandpass filter.
