Lag Compensator

A compensator is a device or component that is used to obtain the desired performance, stability, and behavior of the system. It is the part of the feedback device in a control system and is used to stabilize the system and regulate the other system with its ability of conditioning the input or output form of that system. We can say that it is an electrical network which produces the output in sinusoidal waveform when the sinusoidal input is applied. In simple terms, it adjusts the frequency response and thus improving the response of the system.

In the above figure R(s) is the input signal, G(s) is the gain in the system, and C(s) is the output signal.

Different Types of Compensator

There are various types of compensators such as cascaded, feedback, and cascaded with feedback compensator. There are some other types of compensators like:

• Lead Compensator
• Lag Compensator
• Lag-Lead Compensator
• Lead-Lag Compensator

Here, we will learn about Lag Compensator in detail.

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 resistors 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.

Transfer Function = Output/Input

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

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

Let us calculate the complete transfer function:

According to the voltage divider rule:

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

Transfer \, Function (G(s)) =\frac{1+sR_{2}C}{s(R1+R2)C+1} ---- equation(1)

The general transfer function for the compensator is:

G_{c}(s)=\frac{s+z}{s+p}

Consider p=\frac{z}{\beta}

G_{c}(s)=\frac{s+z}{s+\frac{z}{\beta}}

G_{c}(s)=\frac{s+\frac{1}{z}}{s+\frac{1}{z\beta}} ---- equation(2)

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

z = R₂C

\beta=\frac{R1+R2}{R2}

Phase Angle

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

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

G_{c}(s)=\frac{\beta(1+jz\omega)}{1+jz\omega\beta}

Phase Angle= tan^{-1}(z\omega)-tan^{-1}(z\omega\beta)

As we have calculated the value of z and \beta:

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

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

zero=-\frac{1}{R_{2}C}

zero=-\frac{1}{(R_{1}+R_{2})C}

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:

Phase Angle= tan^{-1}(z\omega)-tan^{-1}(z\omega\beta)

tan(\phi)=\frac{z\omega-z\omega\beta}{1+z^{2}\omega^{2}\beta} ----- equation(1)

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

For the maximum phase condition

\frac{d\phi}{d\omega}|_{\omega=\omega_{m}}=0

\omega_{m}=\sqrt{\omega_{c1}\omega_{c2}}=\sqrt{\frac{1}{z}*\frac{1}{z\beta}}

\omega_{m}=\frac{1}{z\sqrt{\beta}} ----- equation(2)

Substituting the value of \omega_{m} in equation(1)

tan(\phi)=\frac{1-\beta}{2\sqrt{\beta}}

Magnitude for maximum phase will be

|G(j\omega)|=\frac{1}{\sqrt{\beta}}

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=\beta). 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.

Phase Lag Compensator

In the frequency domain, the goal of phase lag compensator design is typically to meet requirements for phase margin and steady-state accuracy. Additionally, a closed-loop bandwidth or gain crossover frequency specification might exist. A phase margin specification can indicate a need for relative stability because of a system's pure time delay, or it can indicate properties of the intended transient response that have been converted from the time domain to the frequency domain.

The phase angle of this 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.

Process for Designing the Phase Lag Compensator

The process for designing the phase lag compensator in order to meet phase margin and steady-state error requirements is described in the following sections. The sections that follow will provide a detailed description of each stage.

1) Ascertain whether the steady-state error specification can be satisfied by increasing System Type N, and if

If necessary, add as many poles to the plant as needed at s = 0. To satisfy the steady-state error, compute Kc.

2) Create the G(s) = K_cG_p(s)/s^{(Nreq−Nsys)} Bode graphs.

3) Create the compensator's lag section:

Finding the frequency where G(jω) would satisfy the phase margin specification if it were the gain crossover frequency is step one. Step two is calculating the amount of attenuation needed to bring the magnitude of G(jω) down to 0 db at that frequency and corresponding α. Step three is computing the lag compensator's zero zc and pole pc using the selected gain crossover frequency and α.

4) Select the right resistor and capacitor values if needed to put the compensator design into practice.

Characteristic of Lag Compensator

• It provides the phase lag to the system which helps in reducing the oscillations. Phase shift helps in adjustment of the phase margin which results in stability enhancement.
• It helps in eliminating the high frequency noise. It has high gain at lower frequencies which helps in greater low frequency correction.
• Similar to to higher frequency noise attenuation, it also attenuates amplitude of the input signals.
• 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.

Advantages of Lag Compensator

• It helps in adjusting the phase margin of the system which helps in enhancing the stability.
• It improves the transient response of the system which helps in eliminating the noise and faster settling of the system.
• It helps in the improvement of performance of a system under various disturbances.
• It is easier to implement due to its simple design.

Disadvantages of Lag Compensator

• It can be applied to low or moderate frequency. It does not provide better results at high frequencies.
• It provides the phase lag to system in order to enhance the stability but on the other side it decreases the gain of the system hence affecting the performance of the system.
• It is sensitive to the external disturbances which can affect the stability.

Applications of Lag Compensator

• It helps in enhancing the stability of the system by introducing the phase lag in the system.
• It adjust the phase margin which is also the another factor for improving the stability of 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.
• In order to enhance the stability, precision of the system it is used in robotics, electrical engineering system, and biomedical engineering.