Control Systems Compensators - Control Systems

What are compensators in control systems?

You can find there are three types of mostly used compensators including lag, lead and lag-lead compensators.

Lag Compensator

The Lag Compensator is known as an electrical network which provides a sinusoidal output including the phase lag when a sinusoidal input is applied. The lag compensator circuit in the‘s’ domain is mentioned as in the following figure.
lag_compensator
While the capacitor is in series includes the resistor R2R2 and the output is calculated with this combination.
The transfer function of this lag compensator is -

lag

From the above equation, αα is always greater than one.
We can find that the lag compensator includes one pole at s and one zero at s. Which means the the pole will be equal to origin in the pole-zero configuration of the lag compensator.

cap

We can mention that the phase of the output sinusoidal signal is equal to the sum of the phase angles of input sinusoidal signal and the transfer function.
To produce the phase lag at the output of this compensator, the transfer function should be negative at the phase angle of the transfer function, which will be happen when α>1α>1.

Lead Compensator

The lead compensator is known as an electrical network to provide a sinusoidal output which includes a phase lead when a sinusoidal input is applied. The lead compensator circuit in the ‘s’ domain is mentioned in the following figure.
lead_compensator
If the capacitor is parallel to the resistor R1R1 and the output is measured with $R_2.
The transfer function of this lead compensator is known as -

cap

So that we can know that the phase of the output sinusoidal signal is equal to the sum of the phase angles of input sinusoidal signal and the transfer function.
To produce the phase lead at the output of this compensator, the phase angle of the transfer function should be positive. It happens when 0<β<10<β<1. So that zero will be equal to origin in pole-zero configuration of the lead compensator.

Lag-Lead Compensator

Lag-Lead compensator is an electrical network that provides a phase lag at one frequency region and phase lead at other frequency region. It is known as the mix of both the lag and the lead compensators. The lag-lead compensator circuit in the‘s’ domain is mentioned in the following figure.
lag_lead
This circuit appears as both the compensators are cascaded. So that the transfer function of this circuit can be the product of transfer functions of the lead and the lag compensators.

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