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:

[Tex]V1(s) = \frac{(R2 + R1\frac{1}{C_s})}{(R1 + \frac{1}{C_s})} [/Tex]

Transfer function G(s) = Output/Input

= [Tex]\frac{V0(s)}{V1(s)} [/Tex]

=[Tex]\frac{\frac{R_2}{R_2+\frac{R_1}{C_s}}}{R_1+\frac{1}{C_s}} [/Tex]

=[Tex]\frac{R_2(R_1C_s+1)}{R_1R_2C_s+\frac{1}{R_1+R_2}} [/Tex]

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

=[Tex]\frac{\frac{R_2}{(R_1+R_2)(R_1C_s+1)}}{\frac{R_1R_2C_s}{(R_1+R_2)+1}} [/Tex]

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

Now the Transfer function is

G(s) =[Tex] \beta\frac{(K_s+1)}{K\beta_s+1} [/Tex]

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

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

[Tex] \frac{V_0(s)}{V_1(s)}=\frac{1+K_s}{1+\beta K_s} [/Tex]

Now substitute the s =[Tex] j\omega [/Tex]

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

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

Phase angle [Tex]\phi [/Tex] =[Tex] tan^{-1}ωK – tan^{-1}ωK\beta [/Tex]

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

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.

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.