📘 Bode Plot – Detailed Stability Problems (Step-by-Step Procedure)
This section explains stability analysis using Bode Plot very slowly and clearly. Each step is shown carefully for deep understanding.
🔹 Problem 1 – Simple Integrator System
Given:
G(s) = K / [s(1 + 0.5s)]
Step 1: Identify Poles
- One pole at s = 0 (Integrator)
- One pole at s = -2
Step 2: Find Corner Frequency
1 + 0.5s = 0 s = -2 ωc = 2 rad/sec
Step 3: Initial Slope
- Integrator gives -20 dB/decade
Step 4: After ω = 2
- Another pole → total slope becomes -40 dB/decade
Step 5: Phase Calculation
- Integrator → -90°
- Second pole → -90°
- Total Phase → -180° at high frequency
Step 6: Stability Condition
At phase crossover (−180°), check magnitude. If magnitude < 0 dB → Stable.
🔹 Problem 2 – Two Poles System
Given:
G(s) = 10 / [(1+s)(1+0.1s)]
Step 1: Corner Frequencies
- ω1 = 1 rad/sec
- ω2 = 10 rad/sec
Step 2: Initial Magnitude
20 log10(10) = 20 dB
Step 3: Slope Changes
- Before 1 → 0 dB/decade
- After 1 → -20 dB/decade
- After 10 → -40 dB/decade
Step 4: Phase
- Each pole contributes -90°
- Total phase approaches -180°
Step 5: Gain Margin
Find frequency where phase = -180°. Check magnitude at that frequency.
🔹 Problem 3 – Gain Margin Calculation
Given magnitude at phase crossover frequency:
Magnitude = -10 dB
Gain Margin:
GM = +10 dB
Since GM positive → Stable.
🔹 Problem 4 – Phase Margin Calculation
Given: Gain crossover frequency phase = -140°
Phase Margin:
PM = 180° + (-140°) PM = 40°
Since PM positive → Stable.
🔹 Problem 5 – Unstable Case
Given: Phase at gain crossover frequency = -210°
Phase Margin:
PM = 180° - 210° PM = -30°
Negative PM → Unstable system.
🔹 Step-by-Step Stability Procedure (Exam Method)
- Write G(jω)
- Identify poles & zeros
- Find corner frequencies
- Determine magnitude slopes
- Determine phase contributions
- Find gain crossover frequency (0 dB point)
- Calculate Phase Margin
- Find phase crossover frequency (-180° point)
- Calculate Gain Margin
- Conclude stability
🎯 GATE Strategy
- Always check PM and GM
- Positive PM & GM → Stable
- Negative PM → Unstable
- Large PM → More stable but slower response
Stability = Proper Gain + Proper Phase
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