Library / Electrical / ELE-36
Electrical
Half-power bandwidth of a series RLC circuit
Width of the frequency band, between the two half-power points of the current response, around the resonance of a series RLC circuit.
- Ref
- ELE-36
- Rev
- v1
- Field
- Electrical
- Result
- Δf [Hz]
A series RLC circuit has a resonance peak whose width between the half-power points (where the power is half its peak) is Δf = R/(2πL), or R/L in rad/s. It depends only on R and L, not on C. The quality factor is Q = f₀/Δf: a low resistance gives a narrow, sharp peak. The same R/L also sets the decay rate of the free response, with an envelope e−Rt/2L.
Width of the frequency band, between the two half-power points of the current response, around the resonance of a series RLC circuit. A smaller resistance gives a narrower, sharper resonance.
Calculator
- Bandwidth in rad/s, R/L1,000 rad/s
- Time constant of the free-response envelope, 2L/R2 ms
- Resonant frequency that would give Q = 10 with this bandwidth1,592 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A circuit with 10 Ω and 10 mH has a half-power bandwidth of 159.2 Hz. If it resonates at 1,590 Hz, its Q is 10; at 159 Hz, Q = 1.
- Free oscillations die away with a time constant 2L/R = 2 ms: a narrow bandwidth means the circuit rings for a long time after being excited.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Inductance | 0.1 to 1,000 mH | 1.592 | 15,920 | 15,910 | +10% | −9.09 % |
| R Series resistance | 0.1 to 1,000 Ω | 1.592 | 15,920 | 15,910 | +10% | +10 % |
Sweep of Series resistance
| Series resistance (Ω) | Half-power bandwidth (Hz) |
|---|---|
| 0.1 | 1.592 |
| 0.3162 | 5.033 |
| 1 | 15.92 |
| 3.162 | 50.33 |
| 10 | 159.2 |
| 31.62 | 503.3 |
| 100 | 1,592 |
| 316.2 | 5,033 |
| 1,000 | 15,920 |
Sweep one variable
Variables and how each one contributes
Half-power bandwidth
What the equation gives you. Everything else feeds it.
Series resistance
The total series resistance, in ohms, including the coil and source resistance. Raises the bandwidth in direct proportion: more resistance flattens the peak and widens it. Raising it by 1 Ω (to 11 Ω) moves the result by +10 %, from 159.2 to 175.1 Hz.
Inductance
The inductance, in mH. Lowers the bandwidth in inverse proportion: a larger inductance sharpens the resonance. Raising it by 2 mH (to 12 mH) moves the result by −16.7 %, from 159.2 to 132.6 Hz.
Worked example with the values above
- Inductance in henries: L = 10 mH = 0.01 H.
- Denominator: 2πL = 2π × 0.01 = 0.06283 H.
- Δf = 10 / 0.06283 = 159.2 Hz.
Assumptions and limits
- A series RLC circuit with a high enough Q that the bandwidth formula holds (Q above about 3).
- Frequency-independent resistance.
Common mistakes
- Looking for a dependence on C: the bandwidth does not depend on it, only the centre frequency does.
- Using the formula in a parallel circuit, where the bandwidth is 1/(2πRC).
Statistics and uncertainty
Both R and L carry tolerance, and R also includes the loss in the coil, which rises with frequency (skin effect) and temperature, so a measured bandwidth is usually wider than the nominal value. Measure Δf directly with a frequency sweep for a critical filter.
Where it is used
- Selectivity of a tuned circuit or filter, and the bandwidth of a receiver stage.
- Estimating how long a resonant circuit rings.
Maths you need
- Resonance: the frequency where the reactances cancel
- Half-power points: frequencies where the power falls to half its peak