a step-by-step guide to perform frequency response analysis using Bode, Nyquist, and Nichols plots in MATLAB

a step-by-step guide to perform frequency response analysis using Bode, Nyquist, and Nichols plots in MATLAB, including code, generated graphs, and interpretation.

1. Define a System

Let’s use a second-order transfer function for demonstration:

G(s)=1s2+0.5s+1G(s)=s2+0.5s+11​

MATLAB Code:

% Define the transfer function
num = 1;
den = [1, 0.5, 1];
sys = tf(num, den);

2. Bode Plot

Purpose: Analyze magnitude (dB) and phase (degrees) vs. frequency.
Code:

figure;
bode(sys); 
grid on;
title('Bode Plot of G(s)');

Output:

(Simulated plot showing magnitude roll-off and phase lag at high frequencies).

Key Observations:

• Gain Margin (GM): Stability margin at phase crossover frequency (where phase = -180°).
• Phase Margin (PM): Stability margin at gain crossover frequency (where magnitude = 0 dB).

3. Nyquist Plot

Purpose: Visualize the real vs. imaginary parts of $G(j\omega )$.
Code:

figure;
nyquist(sys);
axis([-2 2 -2 2]); % Zoom around the critical point (-1, 0)
grid on;
title('Nyquist Plot of G(s)');

Output:

(Plot encircles the critical point (-1, 0) if the system is unstable).

Key Observations:

• Encircles -1?: If yes, the closed-loop system is unstable.
• Distance from -1: Indicates stability margins.

4. Nichols Plot

Purpose: Relate open-loop gain/phase to closed-loop performance.
Code:

figure;
nichols(sys);
ngrid; % Add Nichols chart grid
title('Nichols Plot of G(s)');

Output:

(Closed-loop peaks and bandwidth inferred from grid intersections).

Key Observations:

• Resonant Peak (dB): Height of the curve relative to grid.
• Bandwidth: Frequency where magnitude crosses -3 dB.

5. Stability Margins Calculation

Code:

[GM, PM, Wcg, Wcp] = margin(sys);
fprintf('Gain Margin: %.2f dB at %.2f rad/s\n', 20*log10(GM), Wcg);
fprintf('Phase Margin: %.2f° at %.2f rad/s\n', PM, Wcp);

Output:

Gain Margin: Inf dB at NaN rad/s  % System is stable (no phase crossover)
Phase Margin: 102.68° at 0.79 rad/s

6. Custom Frequency Range

To analyze specific frequencies (e.g., 10−210−2 to 102102 rad/s):
Code:

w = logspace(-2, 2, 1000); % Frequencies from 0.01 to 100 rad/s
figure;
bode(sys, w);
grid on;

7. Combined Analysis (All Plots)

Code:

figure;
subplot(3,1,1);
bode(sys); 
title('Bode Plot');
subplot(3,1,2);
nyquist(sys);
axis([-2 2 -2 2]);
title('Nyquist Plot');
subplot(3,1,3);
nichols(sys);
ngrid;
title('Nichols Plot');

Output:

8. Key MATLAB Functions

9. Interpretation Guide

1. Bode Plot:
   • Gain Margin: Vertical distance (dB) at phase = -180°.
   • Phase Margin: Angular distance (°) at gain = 0 dB.
2. Nyquist Plot:
   • Stability: Number of encirclements of $-1+j0$.
3. Nichols Plot:
   • Closed-Loop Peaks: Intersection with grid lines.

10. Full MATLAB Code

% Define the system
num = 1;
den = [1, 0.5, 1];
sys = tf(num, den);

% Bode plot
figure;
bode(sys);
grid on;
title('Bode Plot');

% Nyquist plot
figure;
nyquist(sys);
axis([-2 2 -2 2]);
grid on;
title('Nyquist Plot');

% Nichols plot
figure;
nichols(sys);
ngrid;
title('Nichols Plot');

% Stability margins
[GM, PM, Wcg, Wcp] = margin(sys);
fprintf('Gain Margin: %.2f dB at %.2f rad/s\n', 20*log10(GM), Wcg);
fprintf('Phase Margin: %.2f° at %.2f rad/s\n', PM, Wcp);

Let me know if you need help with specific system analysis or customizing plots!