Lag Compensator

It is an electrical network that produces the sinusoidal output having a phase lag when sinusoidal input is given and it provides the phase lag at the frequencies at the low level thus is reduces the steady state error so we can say that it is meant to produce steady state sinusoidal steady state signal which have the phase lag to the applied input of that system . The effects of lag compensator are as follows:

• Increases the rise time (t_r)
• Decreases the peak overshoot (Mp)
• Enhances the stability
• Eliminates the high-frequency noise.

Mathematical Calculation for Lag Compensator

Let us consider the circuit diagram given below for the lag compensator.

In this diagram, we have two registers R1 & R2, one capacitor which is denoted in Laplace Transform (1/sc), and V0(s) and V1(s) represents the voltage in the circuit. Here, we can see that we are using the capacitor in series with the resistance R2 to obtain the phase lag. The capacitor is the major component responsible for the phase shift, in the lag compensator.

We need to calculate and obtain the transfer function of a certain component or device in a control system, thus we must also calculate the Lag Compensator’s transfer function.

[Tex]Transfer Function = \frac{Output}{Input}[/Tex]

The input voltage V1(s) and current travelling via the first branch where resistor R1 is present, as shown in the lag compensator circuit diagram. The current flowing through the series connection of resistor R2 and a capacitor is then V0(s), and the output voltage is V0(s). The output of the circuit will be:

Output

[Tex]V_{0}(s) = R_{2}+\frac{1}{sc}[/Tex] (due to series connection of resistor and Laplace capacitor 1/sc)

Let us calculate the complete transfer function:

According to the voltage divider rule:

[Tex]V_{0}(s)=\frac{R_{2}+\frac{1}{sc}}{R_{1}+R_{2}+\frac{1}{sc}}V_{i}(s)[/Tex]

[Tex]Transfer function (G(s)) =\frac{1+sR_{2}C}{s(R1+R2)C+1}[/Tex] —- equation(1)

The general transfer function for the compensator is:

[Tex]G_{c}(s)=\frac{s+z}{s+p}[/Tex]

Consider [Tex]p=\frac{z}{\beta}[/Tex]

[Tex]G_{c}(s)=\frac{s+z}{s+\frac{z}{\beta}}[/Tex]

[Tex]G_{c}(s)=\frac{s+\frac{1}{z}}{s+\frac{1}{z\beta}}[/Tex] —- equation(2)

Comparing the equation(1) and equation(2) we will get:

z = R₂C

[Tex]\beta=\frac{R1+R2}{R2}[/Tex]

Phase Angle

Since the phase is always negative to calculate the phase angle let us replace ‘s’ with ‘[Tex]j\omega[/Tex]‘ in the equation(2). The transfer function will look like:

[Tex]G_{c}(s)=\frac{j\omega+\frac{1}{z}}{j\omega+\frac{1}{z\beta}}[/Tex]

[Tex]G_{c}(s)=\frac{\beta(1+jz\omega)}{1+jz\omega\beta}[/Tex]

[Tex]Phase Angle= tan^{-1}(z\omega)-tan^{-1}(z\omega\beta)[/Tex]

As we have calculated the value of z and [Tex]\beta[/Tex]:

[Tex]Phase Angle= tan^{-1}(R_{2}C\omega)-tan^{-1}(R_{2}C\omega(\frac{R1+R2}{R2}))[/Tex]

In a transfer function, the numerator is the zeros and the denominators are the poles. The poles and zeros of the equation(1) are:

[Tex]zero=-\frac{1}{R_{2}C}[/Tex]

[Tex]zero=-\frac{1}{(R_{1}+R_{2})C}[/Tex]

The zeros and poles graph in the below image:

Lag Angle

As we have already calculated the phase angle for the lag compensator which is given below:

[Tex]Phase Angle= tan^{-1}(z\omega)-tan^{-1}(z\omega\beta)[/Tex]

[Tex]tan(\phi)=\frac{z\omega-z\omega\beta}{1+z^{2}\omega^{2}\beta}[/Tex] —– equation(1)

Magnitude response will be equal to: [Tex]|G(j\omega)| =\sqrt{\frac{1+z^{2}\omega^{2}}{1+z^{2}\omega^{2}\beta^{2}}}[/Tex]

For the maximum phase condition

[Tex]\frac{d\phi}{d\omega}|_{\omega=\omega_{m}}=0[/Tex]

[Tex]\omega_{m}=\sqrt{\omega_{c1}\omega_{c2}}=\sqrt{\frac{1}{z}*\frac{1}{z\beta}}[/Tex]

[Tex]\omega_{m}=\frac{1}{z\sqrt{\beta}}[/Tex] —– equation(2)

Substituting the value of [Tex]\omega_{m}[/Tex] in equation(1)

[Tex]tan(\phi)=\frac{1-\beta}{2\sqrt{\beta}}[/Tex]

Magnitude for maximum phase will be

[Tex]|G(j\omega)|=\frac{1}{\sqrt{\beta}}[/Tex]

The given below shows the bode plot for the phase lag compensator.

It has two corner frequencies which is shown in above diagram i.e., 1/T and 1/aT (where T=z and a=[Tex]\beta[/Tex]). The phase angle of phase lag compensator is negative which is used to provide the phase lag in the system. It helps in the refining of the transient response of the system which results in reducing the peak overshoot. This results in providing the precise results.

Compensators, which have a wide range of functionality and variants, are an essential component of Control Systems. The compensator is an electrical system that is used to obtain the desired performance of the system. It is used to stabilize the unstable system.

There are various types of compensators such as cascaded, feedback, and cascaded with feedback compensator. Furthermore, the control system is an important subject in the engineering curriculum, and it incorporates many important electronics components. There are some other types of compensators like lead, lag, lag-lead, and lead-lag compensators. The lag compensator eliminates high-frequency noise and improves stability. In this article, we will study lag compensators in detail.