Lead Compensator Phase⚠ unverified
Physics / Controls · Compute the alpha parameter for a lead compensator
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| max_phase | maxphase | — | 1.0 | Maximum phase lead to be provided by the compensator, in degrees |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | α | — | Alpha parameter (dimensionless) of the lead compensator |
The science & history
Understanding the Parameters
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$\phi_m$ — maximum phase the lead can contribute (typically 30–70 deg in design). Larger $\phi_m$ → smaller $\alpha$ (more aggressive lead).
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$\alpha$ — pole/zero separation parameter; zero at $-1/T$, pole at $-1/(\alpha T)$.
Derivation (Approaching a Proof)
The phase of a lead is $\arg G_c(j\omega) = \tan^{-1}(\omega T) - \tan^{-1}(\alpha\omega T)$. Maximising vs $\omega$ gives $\sin\phi_m = (1-\alpha)/(1+\alpha)$, which rearranges to the displayed $\alpha(\phi_m)$.
History
Lead compensation for phase margin is a classical Bode loop-shaping tool (alongside lag and lead-lag).
Related Concepts: Lag Compensator Attenuation, Phase Margin, Gain Crossover Frequency
Notes: Registry calculator lead-compensator-phase (unverified). Confirm degree vs radian for
max_phase. Output is $\alpha$, not the phase itself (despite the calculator name focusing on phase).