📘 Bode Plot – Complete Theory & Worked Examples (GATE Focus)
Bode Plot is a frequency response method used to analyze stability and performance of control systems. It consists of two plots: 1. Magnitude vs Frequency (log scale) 2. Phase vs Frequency (log scale)
🔹 1. Standard Form of Transfer Function
General form:
G(s) = K × (1 + s/z₁)(1 + s/z₂) / [(1 + s/p₁)(1 + s/p₂)]
Replace s = jω for frequency response.
🔹 2. Magnitude in dB
Magnitude (dB) = 20 log₁₀ |G(jω)|
Important slopes:
- Zero → +20 dB/decade
- Pole → −20 dB/decade
- Two poles → −40 dB/decade
🔹 3. Phase Contribution
- Zero → +90°
- Pole → −90°
- Transition occurs over 2 decades around corner frequency
🔹 4. Corner Frequency
ωc = 1/τ
Where Ï„ is time constant.
🔹 5. Worked Example 1
Given:
G(s) = 10 / [s(1 + 0.1s)]
Poles:
- s = 0
- s = -10
Corner frequency:
ωc = 1/0.1 = 10 rad/sec
Initial slope due to integrator:
−20 dB/decade
After ω = 10:
−40 dB/decade
🔹 6. Gain Margin (GM)
Gain margin is measured at phase crossover frequency.
GM = 1 / |G(jωpc)|
In dB:
GM(dB) = −Magnitude(dB at ωpc)
🔹 7. Phase Margin (PM)
Phase margin is measured at gain crossover frequency.
PM = 180° + Phase at ωgc
🔹 8. Stability Condition
- Positive Gain Margin → Stable
- Positive Phase Margin → Stable
- Negative PM → Unstable
🔹 9. Worked Example 2
Given:
G(s) = 1 / [s(1+s)(1+0.1s)]
Corner frequencies:
- 1 rad/sec
- 10 rad/sec
Slope changes:
- Start −20 dB/decade
- After 1 → −40 dB/decade
- After 10 → −60 dB/decade
🎯 GATE Important Points
- Gain crossover & phase crossover frequently asked
- Corner frequency identification important
- Integrators and differentiators affect slope significantly
- Phase margin concept directly linked to stability
Bode Plot Connects Frequency Response to Stability
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