Gain Crossover Frequency⚠ unverified
Physics / Controls · Approximate the gain crossover frequency of a second-order system
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | wn | rad/s | 1.0 | Undamped natural frequency |
| zeta | ζ | — | 1.0 | Damping ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | wgc | rad/s | Gain crossover frequency, in radians per second (rad/s) |
The science & history
Understanding the Parameters
- $\omega_n$, $\zeta$ — set both magnitude shape and where it crosses unity gain.
- $\omega_{gc}$ — frequency at which phase margin is read on a Bode plot ($|G|=1$ or 0 dB).
Derivation (Approaching a Proof)
Set $|G(j\omega_{gc})| = 1$ for the second-order $G$ matching the registry algebra and solve for $\omega_{gc}/\omega_n$. The radical form is the positive real solution of that magnitude equation.
History
Gain crossover is central to Bode gain/phase margin design (Bode, Nyquist stability theory).
Related Concepts: Phase Margin, Gain Margin, Bode Magnitude, Bandwidth
Notes: Registry calculator gain-crossover-frequency (unverified). Prototype second-order loop
form only — not a general $G(s)$ solver.