Rlc Bandwidth⚠ unverified
Electrical / Filters · Compute the bandwidth of an RLC resonant circuit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| R | R | — | 1.0 | Resistance |
| L | L | H | 1.0 | Inductance |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | BW | Hz | Bandwidth, in hertz (Hz) |
The science & history
Understanding the Parameters
- $R$ widens the bandwidth (more loss → broader, less selective response).
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$L$ narrows it. Notably $C$ does not appear: the width depends on $R/L$, while the centre $f_0$ depends on $LC$ (see Rlc Series Resonant).
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BW — the span between the two half‑power ($-3\,$dB) frequencies straddling $f_0$.
Derivation (Approaching a Proof)
For series resonance the current magnitude is $|I| = \dfrac{V}{\sqrt{R^2 + X^2}}$ with $X = \omega L - 1/(\omega C)$. The half‑power points occur where the total reactance equals the resistance, $|X| = R$. Solving $\pm R = \omega L - 1/(\omega C)$ for the two roots $\omega_1, \omega_2$ and subtracting gives the angular bandwidth
$$\Delta\omega = \omega_2 - \omega_1 = \frac{R}{L},$$
so in hertz
$$\text{BW} = \frac{\Delta\omega}{2\pi} = \frac{R}{2\pi L}.$$
Dividing the resonant frequency by this recovers the quality factor: $\dfrac{f_0}{\text{BW}} = \dfrac{\omega_0 L}{R} = Q$ (see Q Factor).
History
The half‑power bandwidth and its reciprocal relation to $Q$ became standard language in radio and filter engineering as tuned circuits were characterised by how narrow a band they select — the defining figure of merit for receivers and band‑pass stages.
Related Concepts: Q Factor, Rlc Series Resonant, Resonance Frequency LC Circuit, Damping Ratio
Notes: Registry calculator rlc-bandwidth (unverified; R mislabelled dimensionless — should be ohms).