Free Phase Margin Calculator
Calculate the phase margin of a second-order open-loop system from damping ratio as a quick stability check.
Approximate Phase Margin
51.83 °
Approximate Phase Margin vs Damping Ratio (zeta)
Phase Margin
Phase margin is the additional phase lag that would bring the system to the verge of instability. It is measured at the gain crossover frequency (where the open-loop gain equals 1).
Approximation for Second-Order Systems
For a standard second-order system, the phase margin can be approximated from the damping ratio. A useful rule of thumb is PM ≈ 100 × zeta (in degrees) for zeta between 0.1 and 0.7.
Adequate phase margin (typically 30-60°) ensures robustness against modeling errors and parameter variations.
Example Calculation
A system with damping ratio zeta = 0.5.
- 014*zeta^4 = 4 × 0.0625 = 0.25
- 02sqrt(1 + 0.25) = sqrt(1.25) = 1.1180
- 03-2 × 0.25 + 1.1180 = 0.6180
- 042 × 0.5 / sqrt(0.6180) = 1.0 / 0.7861 = 1.2721
- 05PM = atan(1.2721) × 180/pi = 51.8°
- 06Simple approximation: 100 × 0.5 = 50°
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