Bandwidth Calculator Formula
Understand the math behind the bandwidth calculator. Each variable explained with a worked example.
Formulas Used
Bandwidth (omega_BW)
bw_rad = wn * sqrt(1 - 2 * pow(zeta, 2) + sqrt(2 - 4 * pow(zeta, 2) + 4 * pow(zeta, 4)))Bandwidth
bw_hz = wn * sqrt(1 - 2 * pow(zeta, 2) + sqrt(2 - 4 * pow(zeta, 2) + 4 * pow(zeta, 4))) / (2 * pi)Variables
| Variable | Description | Default |
|---|---|---|
wn | Natural Frequency (omega_n)(rad/s) | 20 |
zeta | Damping Ratio (zeta) | 0.7 |
How It Works
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.
Worked Example
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
Frequently Asked Questions
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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Open Bandwidth Calculator