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Electrical
Damping ratio of a series RLC circuit
Dimensionless measure of how strongly a series RLC circuit is damped: below 1 it rings, at 1 it is critically damped, above 1 it is overdamped.
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- ELE-47
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- v1
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- Electrical
- Result
- ζ
ζ = (R/2)√(C/L) is the damping ratio of the series loop. ζ < 1 means ringing, ζ = 1 critical damping, ζ > 1 overdamped. It is related to the quality factor by Q = 1/(2ζ) and to the other quantities by ζ = α/ω₀.
Dimensionless measure of how strongly a series RLC circuit is damped: below 1 it rings, at 1 it is critically damped, above 1 it is overdamped. It equals α/ω₀ and is the reciprocal of 2Q.
Calculator
- Quality factor, 1/(2ζ)3.162
- Overshoot of the step response60.47 %
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.
- ζ = 0.1581, so the circuit is underdamped and rings; the quality factor is 3.162.
- A step into this circuit overshoots by 60.47 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Resistance | 0.01 to 100,000 Ω | 1.581e-4 | 1,581 | 1,581 | +10% | +10 % |
| L Inductance | 1e-6 to 10 H | 0.005 | 15.81 | 15.81 | +10% | −4.65 % |
| C Capacitance | 1e-9 to 0.01 F | 0.001581 | 5 | 4.998 | +10% | +4.88 % |
Sweep of Resistance
| Resistance (Ω) | Damping ratio |
|---|---|
| 0.01 | 1.581e-4 |
| 0.07499 | 0.001186 |
| 0.5623 | 0.008891 |
| 4.217 | 0.06668 |
| 31.62 | 0.5 |
| 237.1 | 3.749 |
| 1,778 | 28.12 |
| 13,340 | 210.8 |
| 100,000 | 1,581 |
Sweep one variable
Variables and how each one contributes
Damping ratio
What the equation gives you. Everything else feeds it.
Resistance
The total series resistance, in Ω. Raises the damping ratio in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 0.1581 to 0.1739.
Inductance
The inductance, in H. Lowers the damping ratio in inverse proportion to the square root of L. Raising it by 10 % (to 0.011) moves the result by −4.65 %, from 0.1581 to 0.1508.
Capacitance
The capacitance, in F. Raises the damping ratio in proportion to the square root of C. Raising it by 10 % (to 1.1×10−5) moves the result by +4.88 %, from 0.1581 to 0.1658.
Worked example with the values above
- Ratio: C/L = 1×10−5 / 0.01 = 0.001.
- Square root: √(C/L) = 0.0316228.
- ζ = (R/2) × 0.0316228 = 5 × 0.0316228 = 0.158114.
Assumptions and limits
- A series RLC loop with ideal L and C and total series resistance R.
- Linear, time-invariant components.
Common mistakes
- Using the parallel RLC formula, ζ = (1/2R)√(L/C).
- Confusing ζ with the quality factor (Q = 1/2ζ).
- Leaving out the inductor’s winding resistance.
Statistics and uncertainty
ζ is proportional to R, so a 1 % error in the resistance gives a 1 % error in ζ. Overshoot is much more sensitive when ζ is small: a 20 % error in ζ near 0.1 changes the overshoot from about 73 % to 68 %.
Where it is used
- A quick check on whether a circuit will ring and by how much it overshoots.
- Converting between damping ratio and Q.
Maths you need
- Second-order systems: damping ratio ζ
- Quality factor: Q = ω₀L/R