Bandwidth Calculator
Calculate the -3 dB bandwidth of a second-order closed-loop system from natural frequency and damping ratio.
Bandwidth (omega_BW)
20.201 rad/s
Bandwidth (omega_BW) vs Natural Frequency (omega_n)
Formule
## Closed-Loop Bandwidth Bandwidth is the frequency at which the closed-loop magnitude response drops to -3 dB (70.7% of its DC value). It measures how fast the system can track inputs. ### Formula **omega_BW = omega_n × sqrt(1 - 2*zeta² + sqrt(2 - 4*zeta² + 4*zeta⁴))** Higher bandwidth means faster tracking but also more susceptibility to high-frequency noise. Good control design balances bandwidth against noise rejection.
Exemple Résolu
A second-order system with omega_n = 20 rad/s and zeta = 0.7.
- 012*zeta² = 2 × 0.49 = 0.98
- 02Inner sqrt: sqrt(2 - 1.96 + 0.9604) = sqrt(1.0004) = 1.0002
- 031 - 0.98 + 1.0002 = 1.0202
- 04omega_BW = 20 × sqrt(1.0202) = 20 × 1.0100 = 20.20 rad/s
- 05f_BW = 20.20 / 6.283 = 3.215 Hz
Questions Fréquentes
What is the relationship between bandwidth and rise time?
Bandwidth and rise time are inversely related: omega_BW ≈ 1.8 / t_r for a second-order system. Higher bandwidth means faster rise time. This is a fundamental tradeoff in control design.
Does higher bandwidth always mean better performance?
Not necessarily. Higher bandwidth makes the system track commands faster, but it also amplifies sensor noise and may excite unmodeled high-frequency dynamics (structural modes).
How is bandwidth related to stability margins?
Bandwidth, gain margin, and phase margin are all interconnected. A well-designed system typically has bandwidth well below the frequencies where phase drops to -180°, ensuring adequate stability margins.
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