Library / Electrical / ELE-28
Electrical
Quality factor of a series RLC circuit
Sharpness of resonance.
- Ref
- ELE-28
- Rev
- v1
- Field
- Electrical
- Result
- Q
R is the total series resistance, L the inductance in millihenries and C the capacitance in microfarads. Q = 1/R × √(L/C) is the ratio of the reactance at resonance to the resistance. It also equals f₀ divided by the bandwidth between half-power points.
Sharpness of resonance. Higher Q means a narrower, more selective response and less damping.
Calculator
- Resonant frequency f₀159.2 Hz
- Bandwidth f₀/Q (half-power points)159.2 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.
- Q = 1 at the defaults, so the response is very broad: the bandwidth equals f₀ (159 Hz). A radio front end may need Q of 50 to 200.
- At resonance the voltage across L or C is Q times the supply voltage. A circuit with Q = 100 on a 10 V supply puts 1000 V across the capacitor, which can damage it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Resistance | 0.01 to 1,000 Ω | 0.01 | 1,000 | 1,000 | +10% | −9.09 % |
| C Capacitance | 0.001 to 100,000 µF | 0.03162 | 316.2 | 316.2 | +10% | −4.65 % |
| L Inductance | 0.01 to 10,000 mH | 0.03162 | 31.62 | 31.59 | +10% | +4.88 % |
Sweep of Resistance
| Resistance (Ω) | Quality factor |
|---|---|
| 0.01 | 1,000 |
| 0.04217 | 237.1 |
| 0.1778 | 56.23 |
| 0.7499 | 13.34 |
| 3.162 | 3.162 |
| 13.34 | 0.7499 |
| 56.23 | 0.1778 |
| 237.1 | 0.04217 |
| 1,000 | 0.01 |
Sweep one variable
Variables and how each one contributes
Quality factor
What the equation gives you. Everything else feeds it.
Resistance
The total resistance of the circuit, including the coil and losses. Inverse: halving the resistance doubles Q. Resistance is the loss; less of it makes a sharper response.
Inductance
The inductance, in mH. Square root: four times the inductance doubles Q at the same C.
Capacitance
The capacitance, in µF. Inverse square root: four times the capacitance halves Q at the same L.
Worked example with the values above
- Characteristic impedance: √(L/C) = √(0.01 / 1×10−4) = 10 Ω.
- Divide by the resistance: 10 / 10.
- Q = 1.
Assumptions and limits
- A series circuit driven near resonance.
- Constant R, although the coil resistance rises with frequency.
Common mistakes
- Using the parallel-circuit formula (Q = R√(C/L)) for a series circuit.
- Forgetting that high Q means a narrow bandwidth, so a tuned receiver may miss a drifting signal.
Statistics and uncertainty
Q ∝ 1/R, so the coil’s resistance (which varies with temperature and frequency, see skin depth) sets the accuracy. A 10 % error in R gives a 10 % error in Q and in the bandwidth, but leaves f₀ unchanged.
Where it is used
- Selectivity of a filter or receiver front end.
- Estimating the voltage magnification at resonance.
Maths you need
- Resonant frequency: f₀ = 1/(2π√LC)
- Square roots: √(L/C)