LEAD LAG COMPENSATION IN CONTROL SYSTEM
In this post we are going to learn about types of compensations, why compensation of control system is required, lead and lag compensation in control system.
Series Compensation:
If we connect compensation circuit between plant and error detector it is called series compensation. The circuit for series compensation is as follows.
Parallel compensation:
If we connect compensation circuit as feedback type it is called parallel compensation. The circuit for parallel compensation is as follows.
Load compensation:
If we combine series and parallel feedback circuits it results in Load compensation. The circuit for load compensation is as follows.
Why compensation of control system is required?
1.We do compensation for a control system in order to achieve best performance.
2. To make an unstable system stable.
3. It increases steady state accuracy of the system.
4. Reduces overshoot of a system.
5. Generally compensation is done in the form of feed forward path gain adjustment.
6. When we apply compensation network to a control system it introduces poles and zeros in to the system which results in change of transfer function, this makes the performance of the system to alter.
Note: Increase in the steady state accuracy brings instability to the system.
By substituting
in above equation. we get transfer function.
here T and β are respectively the time constant and DC gain.
In the lag compensator pole is more dominating than the zero and because of this lag network introduces negative phase angle to the system when connected in series.
Substitute s = j𝛚 in the transfer function , also we have ɑ < 1 then we get transfer function as,
Where, θm is the maximum phase lead angle.β is generally chosen to be greater than 10.
Phase Lead Compensation:
A lead network has one pole and one zero, zero must be near to the origin and also the poles and zeros must lie on the negative real axis. By seeing the pole zero plot of lead compensation network you can get a clear idea, its figure is shown below.
Pole Zero Plot Of Lead Compensation Network :
So to add a dominating zero( zero near to origin) to a control system we use phase lead compensation network. The network is shown below.
Phase Lead Compensation Network:
From this circuit we have,
Now on equating above two I values we get,
Determining Transfer Function Of Lead Compensation Network:
Here transfer function is the ratio of output voltage to input voltage.
By applying Laplace transform on both sides of above equation we get,
we know,
By substituting this equation in the above equation we get transfer function,
Here T and ɑ are time constant and attenuation constant respectively.
This lead network when connected in series gives positive phase angle to the system which makes the zero to be close to origin .
Substitute s = j𝛚 in the transfer function , also we have ɑ < 1 then we get transfer function as,
In order to know at what frequency we get maximum phase lead we need to differentiate the above equation and equate to zero. Now we get,
Here θm is the maximum phase lead angle, corresponding magnitude of the transfer function at maximum θm is 1/a.
Advantages Of Phase Lead Compensation:
1. As the gain crossover frequency shifts to a higher value because of phase lead compensation the speed of the system increases.
2. Maximum overshoot of the system decreases because of phase lead compensation.
Disadvantages Of Phase Lead Compensation:
No improvement in steady state error.
Effects Of Phase Lead Compensation:
1. Increase in phase margin.
2. Faster Response.
3. Increase in velocity constant Kv.
4.The slope of the magnitude plot reduces at the gain crossover frequency so that relative stability improves and error decrease due to error is directly proportional to the slope.
Phase Lag Compensation:
A lag network has one zero and one pole, pole must be near to the origin and also the poles and zeros must lie on the negative real axis. By seeing the pole zero plot of lag compensation network you can get a clear idea, its figure is shown below.
Pole Zero Plot Of Lag Compensation Network :
So to add a dominating pole(pole near to origin)to a control system we use phase lag compensation network. The network is shown below.
From this circuit we have,
Determining Transfer Function Of Lag Compensation Network:
Here transfer function is the ratio of output voltage to input voltage.
By applying Laplace transform on both sides of above equation we get,
here T and β are respectively the time constant and DC gain.
Substitute s = j𝛚 in the transfer function , also we have ɑ < 1 then we get transfer function as,
In order to know at what frequency we get maximum phase lead we need to differentiate the above equation and equate to zero. Now we get,
Advantages Of Phase Lag Compensation:
1.Low and high frequencies can be attenuated.
2. Increase in steady state accuracy.
Disadvantages Of Phase Lag Compensation:
Decrease in speed of system.
Effects Of Phase Lag Compensation:
1. Increase in phase margin.
2. Decrease in bandwidth.
3. Increase in gain cross over frequency.
4.Response will be slower before due to decreasing bandwidth, the rise time and the settling time become larger.
Phase Lag Lead Compensation:
By cascading both lead and lag networks we can obtain the benefits of both lead network and lag network which makes the given system to work better. Given below is circuit diagram of lead lag network.
Circuit Diagram Of Lead Lag Network:
Determining Transfer Function Of lead lag compensation network:
Here transfer function is the ratio of output voltage to input voltage.
By applying Laplace transform on both sides of above equation we get,
By substituting,
in above equation we get,
Pole Zero Plot Of Lead Lag Network:
In the lag-lead compensation pole is more dominating than the zero and because of this lag-lead network may introduces positive phase angle to the system when connected in series.
Advantages Of Phase Lead Lag Compensation:
1. Improved accuracy.
2. Speed of the system increases because it shifts gain crossover frequency to a higher value.
In this way Lead Lag compensation networks are used in order to increase the performance of a system.
In this post we have learnt about lag lead compensators in control system.
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June 14, 2017 at 11:05PM by EEE, ADBU
