Bandwidth Calculator Formula

Understand the math behind the bandwidth calculator. Each variable explained with a worked example.

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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

VariableDescriptionDefault
wnNatural Frequency (omega_n)(rad/s)20
zetaDamping 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.

wn = 20zeta = 0.7
  1. 012*zeta² = 2 × 0.49 = 0.98
  2. 02Inner sqrt: sqrt(2 - 1.96 + 0.9604) = sqrt(1.0004) = 1.0002
  3. 031 - 0.98 + 1.0002 = 1.0202
  4. 04omega_BW = 20 × sqrt(1.0202) = 20 × 1.0100 = 20.20 rad/s
  5. 05f_BW = 20.20 / 6.283 = 3.215 Hz

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