Frequency Bandwidth Calculator (−3 dB bandwidth from Q, or Q from the cut-off frequencies)

Understanding the frequency bandwidth calculator — concept and principle

This calculator links the centre frequency, quality factor Q and −3 dB bandwidth of a resonant circuit or band-pass filter. Give it the centre frequency and Q to get the bandwidth and the exact half-power frequencies, or give it the two cut-off frequencies to get the centre frequency and Q. It is used for radio IF filters, LC tank circuits and audio filters.

The two half-power frequencies are slightly asymmetric, but their geometric mean is exactly 455 kHz.

How it is calculated

Q = f₀ ÷ Δf, so Δf = f₀ ÷ Q

Exact half-power frequencies (series RLC): f± = f₀ [√(1 + 1/(4Q²)) ± 1/(2Q)]

From measured cut-offs: f₀ = √(f_low × f_high), Q = f₀ ÷ (f_high − f_low)

Fractional bandwidth = Δf ÷ f₀ × 100%

All frequencies use the same unit, so kHz can also be read as Hz or MHz.

The principle

At resonance the energy stored in the inductor and capacitor swaps back and forth while resistance removes a little each cycle. Q measures how many cycles the energy lasts, so a high-Q circuit responds only to a narrow band of frequencies. The bandwidth is measured between the two frequencies where the delivered power falls to half its peak, which is −3 dB. These points are not symmetric about f₀ on a linear scale, but their geometric mean is exactly f₀.

Worked example

Default inputs: centre frequency and Q, f₀ = 455 kHz, Q = 50.

Δf = 455 ÷ 50 = 9.1 kHz.
1/(2Q) = 0.01 and √(1 + 1/(4 × 2,500)) = 1.00005.
f_low = 455 × (1.00005 − 0.01) = 450.473 kHz; f_high = 455 × (1.00005 + 0.01) = 459.573 kHz.
Fractional bandwidth = 9.1 ÷ 455 = 2%.

Each point is the calculator re-run with only “Quality factor Q” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Quality factor Q	Bandwidth (kHz)
12.5	36.4
25	18.2
37.5	12.1333
50	9.1
62.5	7.28
75	6.06667
100	4.55

How to read the result

High Q gives a narrow, selective response but makes the circuit sensitive to component tolerance and temperature. Low Q gives a broad, more forgiving response. A 455 kHz IF stage with about 9 kHz bandwidth suits AM broadcast signals, whose channels are spaced 9 or 10 kHz apart depending on the region.

Common mistakes

Averaging the two cut-offs arithmetically when the geometric mean is correct.
Mixing units between f₀ and the cut-offs.
Applying a single-resonator Q formula to multi-pole filters, whose skirts are steeper.

Keep in mind: The exact cut-off formula is for a single series RLC resonance. Coupled or multi-stage filters need their own analysis or measurement.

Frequently asked questions

What does −3 dB mean?

The power has fallen to half its peak; the voltage or current is about 70.7% of its peak.

Is higher Q always better?

No. It improves selectivity but narrows the passband and increases sensitivity to tuning.

Why use the geometric mean for the centre?

For a resonant circuit the product of the half-power frequencies equals f₀².

Sources: OpenStax, University Physics Volume 2 (2016), section 15.5 "Resonance in an AC Circuit"

Checked 9 Nov 2026. OpenStax, University Physics Volume 2 (2016), §15.5 “Resonance in an AC Circuit”: ω₀ = 1/√(LC) (Eq. 15.17), bandwidth as the range where the average power exceeds half its maximum, Q = ω₀/Δω (Eq. 15.18) and Q = ω₀L/R (Eq. 15.19). Half-power frequencies of the series RLC circuit — standard result.

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