Step-by-Step Calculation (Using Bode Plot)

 

Key Definitions

• Gain Crossover Frequency (${\omega }_{gc}$)

  • The frequency at which the magnitude of the open-loop transfer function is unity (0 dB).

  • Found from the Bode magnitude plot.

• Phase Crossover Frequency (${\omega }_{pc}$)

  • The frequency at which the phase of the open-loop transfer function is –180°.

  • Found from the Bode phase plot.

• Gain Margin (GM)

  • Defined at the phase crossover frequency.

  • It is the reciprocal of the magnitude at ${\omega }_{pc}$.

  • Expressed in dB:

$$GM=-20{\log }_{10}\mathrm{\mid }G(j{\omega }_{pc})\mathrm{\mid }$$

• A positive GM indicates stability.

• Phase Margin (PM)

  • Defined at the gain crossover frequency.

  • It is the difference between the actual phase at ${\omega }_{gc}$ and –180°.

  • Expressed as:

$$PM={180}^{\circ }+\mathrm{\angle }G(j{\omega }_{gc})$$

• A larger PM means better stability.

📊 Step-by-Step Calculation (Using Bode Plot)

1. Draw/obtain the Bode plot of the open-loop transfer function $G(s)$.

2. Find ${\omega }_{gc}$:

   • Locate the frequency where the magnitude curve crosses 0 dB.

3. Find ${\omega }_{pc}$:

   • Locate the frequency where the phase curve crosses –180°.

4. Calculate Gain Margin (GM):

   • At ${\omega }_{pc}$, read the magnitude.

   • Convert to dB using the formula above.

5. Calculate Phase Margin (PM):

   • At ${\omega }_{gc}$, read the phase.

   • Subtract from –180° to get PM.

📌 Example

Suppose the Bode plot of a system shows:

• Magnitude crosses 0 dB at 10 rad/s → ${\omega }_{gc}=10$.

• Phase at 10 rad/s is –135° →

$$PM={180}^{\circ }-{135}^{\circ }={45}^{\circ }$$

• Phase crosses –180° at 20 rad/s → ${\omega }_{pc}=20$.

• Magnitude at 20 rad/s is –6 dB →

$$GM=6\text{dB}$$

Thus, the system has Phase Margin = 45° and Gain Margin = 6 dB, indicating good stability.

⚠️ Stability Notes

• PM > 45° → system is generally stable and well-damped.

• GM > 6 dB → system can tolerate moderate gain increase.

• If either margin is negative, the system is unstable.