Lead Compensator in control system

Compensators, which have a wide range of functionality and variants, are an essential component of Control Systems. Furthermore, the control system is an important subject in the engineering curriculum, and it incorporates many important electronics components. To understand the Lead Compensator, we must first understand the compensator and its variations, as well as how to apply it in a control system.

What is a Compensator?

The compensator is a device or component that is used to obtain the desired performance, stability, and behaviour of the system.

• The compensator is a part of the feedback device in a control system
• It is used to stabilize the unstable system
  

  

In the image R(s) is the input signal, G(s) is the gain in the system, here we usually have a transfer function and C(s) is the output signal.

Types of Compensators

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

The first three compensators are the most commonly utilized in between.

Here we will only learn about the Lead Compensator.

What is Lead Compensator?

Lead compensator is a type of compensator or device which produces a sinusoidal output having the phase lead when sinusoidal input is applied.

Note: A sinusoidal input, often known as a sine wave or sinusoidal signal.

Consider the following Lead compensator diagram.

In this diagram, we have two registers R1 & R2, one capacitor C, and V0(s) and V1(s) represents the voltage in the circuit.

Here, we can see that we are using the capacitor C in parallel with the first resistance R1 to obtain the phase lead, while the second branch of the circuit just has the second resistance R2. The capacitor is the major component responsible for the phase shift, in the lead 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 Lead Compensator's transfer function.

Transfer function = Output/Input

The input voltage V1(s) and current travelling via the first branch where resistor R1 and capacitor C are in parallel connection, as shown in the lead compensator circuit diagram. The current flowing through the resistor R2 is then V0(s), and the output voltage is V0(s).

As a result, the circuit's output should be,

Output:

V0(s) = R2 (as only one resistor in the second branch of the circuit)

Let us also determine the input. Here, the resistor R1 and capacitor C are connected in parallel, so we must compute that first, and then the resistor R2 is connected in series, so we will add it to what we got from the parallel connection.

Input:

V1(s) = \frac{(R2 + R1\frac{1}{C_s})}{(R1 + \frac{1}{C_s})}

Transfer function G(s) = Output/Input

= \frac{V0(s)}{V1(s)}

=\frac{\frac{R_2}{R_2+\frac{R_1}{C_s}}}{R_1+\frac{1}{C_s}}

=\frac{R_2(R_1C_s+1)}{R_1R_2C_s+\frac{1}{R_1+R_2}}

We get by dividing and multiplying the transfer function by R1 + R2.

=\frac{\frac{R_2}{(R_1+R_2)(R_1C_s+1)}}{\frac{R_1R_2C_s}{(R_1+R_2)+1}}

let us consider, K = R_1C and B = \frac{R_2}{(R_1+R_2)}

Now the Transfer function is

G(s) = \beta\frac{(K_s+1)}{K\beta_s+1}

\frac{V_o(s)}{V_1(s)}=G(s)=\beta\frac{K_s+1}{K\beta_s+1}

Phase Angle

For calculating Phase Angle we have to ignore the constant part of the equation and proceed with the imaginary part, now the equation will be

\frac{V_0(s)}{V_1(s)}=\frac{1+K_s}{1+\beta K_s}

Now substitute the s = j\omega

\frac{V0(j\omega)}{V1(j\omega)} = \frac{1+j\omega K}{1+\beta Kj\omega}

Now, calculate the magnitude by square rooting the whole R.H.S.

Phase angle \phi = tan^{-1}ωK - tan^{-1}ωK\beta

In a transfer function, the numerator is the zeros and the denominators are the poles, in this equation zeros = -1/K and poles = -1/KB

The zeros and poles graph in the below image.

Characteristics and Usage of Lead Compensator

Given Below are Some of the Characterstics of Lead Compensator

• The primary characteristics of the lead compensator is phase lead in the system.
• The lead compensator also used to control the frequency of a system.
• When the input to the system is quickly changed, the behaviour is known as transient response, and the lead compensator helps to reduce it.
• Lead compensator helps to increase stability in the system, also helps to increase bandwidth of the system.
• Lead compensators have numerous applications in a variety of fields, including aerospace, control systems, communication, and power systems.

Things to keep in mind

• Designing: While designing the lead compensator, we must ensure that it provides appropriate phase margin; additionally, it is used to manage frequency, thus it must be designed in such a manner that it does not cause any disturbance in the system.
• When using a lead compensator in an electronics system, we must guarantee that the engineer choose the correct gain crossover frequency; otherwise, the overall output would suffer.
• It helps to reduce transient response, but we must be cautious when choosing unsuitable settings, as this might lead to overshooting and instability.
• To get the desired effects, the design may grow too complex, making it unsuitable for all sorts of control systems. So there are certain benefits and drawbacks, which we will explain in the section below.

Advantages and Disadvantages of Lead Compensator:

There are some list of Advantages and Disadvantages of Lead Compensator given below

Advantages

• The lead compensator assists electronics and electrical engineers in adjusting the system's phase margin to meet their needs.
• The lead compensator improves the overall stability of the system, by increasing the phase margin.
• The system's increased damping allows for reduced overshoot, as well as shorter rising and settling time. Consequently, the transitory response improves.
• The primary advantage of the lead compensator is reducing the transient response in the system.
• It maximizes the system’s velocity constant.

Disadvantages

• Some times getting precise results can necessitate a more complex design of a lead compensator in a control system.
• The lead network boosts bandwidth, but it also makes the system more prone to noise.
• A single lead network provides a lead angle of approximately 60°. Thus, for higher leads ranging from 70 to 90°, a multiple lead compensator must be added to the system.
• Engineers must use caution when selecting a lead compensator since it may not be appropriate for all types of control systems.

Conclusion

One of the most important components for electronics and electrical engineers is the lead compensator. The lead compensator is in charge of injecting a phase lead into the system. As seen in the diagram above, we utilize a capacitor to compensate for phase shift in the compensator. By introducing phase advancement, we may improve system stability while also controlling the frequency of the system. However, attenuation can occur during frequency regulation, resulting in noise in the system. Aside from that, it is useful for decreasing transient response and overshoot in the system. Because of these factors and traits, it has emerged as a key and vital component.