Phase Margin Calculator Formula

Understand the math behind the phase margin calculator. Each variable explained with a worked example.

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Formulas Used

Approximate Phase Margin

pm_deg = atan(2 * zeta / sqrt(-2 * pow(zeta, 2) + sqrt(1 + 4 * pow(zeta, 4)))) * 180 / pi

Simplified Phase Margin (100×zeta)

pm_simple = 100 * zeta

Variables

VariableDescriptionDefault
zetaDamping Ratio (zeta)0.5

How It Works

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.

Worked Example

A system with damping ratio zeta = 0.5.

zeta = 0.5
  1. 014*zeta^4 = 4 × 0.0625 = 0.25
  2. 02sqrt(1 + 0.25) = sqrt(1.25) = 1.1180
  3. 03-2 × 0.25 + 1.1180 = 0.6180
  4. 042 × 0.5 / sqrt(0.6180) = 1.0 / 0.7861 = 1.2721
  5. 05PM = atan(1.2721) × 180/pi = 51.8°
  6. 06Simple approximation: 100 × 0.5 = 50°

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