Bandwidth⚠ unverified
Physics / Controls · Approximate the closed-loop bandwidth 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 | wb | rad/s | Closed-loop bandwidth, in radians per second (rad/s) |
The science & history
Understanding the Parameters
- $\omega_n$ — bandwidth scales with $\omega_n$.
-
$\zeta$ — affects the factor under the square root; low $\zeta$ can give peaking and $\omega_b > \omega_n$.
-
$\omega_b$ — rough indicator of speed of response (higher bandwidth ↔ faster rise).
Derivation (Approaching a Proof)
Set $|T(j\omega_b)| = T(0)/\sqrt{2} = 1/\sqrt{2}$ and solve for $\omega_b/\omega_n$. Algebra on $|1 - (\omega/\omega_n)^{2} + j 2\zeta\omega/\omega_n|^{-1}$ produces the displayed radical form.
History
Bandwidth is a primary frequency-domain specification linking to time response in classical design.
Related Concepts: Rise Time, Settling Time 2%, Bode Magnitude, Gain Crossover Frequency
Notes: Registry calculator bandwidth (unverified). Prototype second-order closed loop only.