Free 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)
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.
Example Calculation
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
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