Lead compensation is a technique used in control systems to improve the stability and transient response of a system by introducing a lead compensator. The transfer function of a lead compensator is given by:
$$ G_{c}(s)=\frac{1}{\beta} \frac{s+\frac{1}{T}}{s+\frac{1}{\beta T}} $$
where $0 < \beta < 1$.
Lead compensation is called "lead" because it introduces a phase lead in the frequency response of the system. This phase lead helps to improve the system's stability and transient response.
Frequency Response: At low frequencies ($s \ll \frac{1}{T}$), the transfer function can be approximated as:
$$
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At high frequencies ($s \gg \frac{1}{\beta T}$), the transfer function can be approximated as:
$$
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This means that the lead compensator has a gain of 1 at low frequencies and a gain of $\frac{1}{\beta}$ at high frequencies. The transition between these two gains occurs at a frequency determined by the time constants $T$ and $\beta T$.
Phase Response: The phase response of the lead compensator can be derived from the transfer function:
$$ \angle G_{c}(s) = \angle \left(s+\frac{1}{T}\right) - \angle \left(s+\frac{1}{\beta T}\right) $$
At low frequencies, the phase lead introduced by the compensator is approximately:
$$
⁍ $$
At high frequencies, the phase lead introduced by the compensator is approximately:
$$
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The lead compensator introduces a maximum phase lead of $90°(\beta-1)$ at a frequency determined by the time constants $T$ and $\beta T$.
MATLAB Code: Here's a MATLAB code snippet to plot the frequency response, phase response, and zero-pole map of a lead compensator:
% Define the parameters
T = 1;
beta = 0.1;
% Create the transfer function
num = [T/beta 1/beta];
den = [beta*T 1];
G = tf(num, den);
% Plot the frequency response and phase response
figure;
subplot(2,1,1);
bode(G);
grid on;
title('Frequency Response and Phase Response of Lead Compensator');
% Plot the zero-pole map
subplot(2,1,2);
pzmap(G);
grid on;
title('Zero-Pole Map of Lead Compensator');
Explanation: