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Resonant Bandwidth, Q & Cutoff Frequencies

Relate bandwidth, quality factor and cutoff frequencies in the retained second-order resonant model.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

Lower and upper cutoffs of 100 Hz and 400 Hz give a geometric center of 200 Hz, bandwidth 300 Hz and quality factor 2/3.

Bandwidth = fH − fL; center frequency = √(fLfH); Q = center frequency/bandwidth

Worked example

Enter these known values and leave the other values blank.

Lower cutoff frequency
100 Hz
Upper cutoff frequency
400 Hz
Resonant center frequency
200 Hz
Quality factor Q
0.6667
Bandwidth
300 Hz

Assumptions and limitations

  • All quantities must be positive and the upper cutoff must exceed the lower cutoff.
  • This model uses the geometric center and half-power cutoffs of an ideal second-order resonant band-pass response. It is not a universal relationship for every filter, spectrum or arbitrary pair of band edges.
  • Q is dimensionless. The bandwidth and every frequency must refer to the same response and cutoff convention.
  • A high-Q narrow band has a geometric center close to its arithmetic midpoint, but the two means are not identical in general.

Common questions

Why is the center not (fH + fL)/2?

The retained resonant model uses √(fLfH). For 100 Hz and 400 Hz, that is 200 Hz, while the arithmetic midpoint is 250 Hz.

What does a larger Q mean here?

At fixed center frequency, a larger Q means a smaller modeled bandwidth.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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