Library / Electrical / ELE-44
Electrical
Damping rate of a series RLC circuit
Rate at which the oscillations of a series RLC circuit die away.
- Ref
- ELE-44
- Rev
- v1
- Field
- Electrical
- Result
- α [1/s]
α = R/(2L), in 1/s, is the “neper frequency”: the rate at which the oscillation envelope decays as e^(−αt). After 1/α seconds the amplitude has fallen to 37 %; after 3/α seconds, to 5 %. Compare it with ω₀ = 1/√(LC): the circuit rings if α < ω₀.
Rate at which the oscillations of a series RLC circuit die away. The charge and current envelope falls as e^(−αt), so the amplitude drops to about 37 % in 1/α seconds. More resistance or less inductance damps faster.
Calculator
- Time for the amplitude to fall to 37 %, 1/α0.002 s
- Time to fall to 5 %, 3/α0.006 s
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.
- The ringing decays with a time constant of 2 ms; it is below 5 % of its starting amplitude after 6 ms.
- More resistance or less inductance damps the ringing faster. When α reaches 1/√(LC) the circuit is critically damped and stops ringing.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Resistance | 0.01 to 100,000 Ω | 0.5 | 5,000,000 | 5,000,000 | +10% | +10 % |
| L Inductance | 1e-6 to 10 H | 0.5 | 5,000,000 | 5,000,000 | +10% | −9.09 % |
Sweep of Resistance
| Resistance (Ω) | Damping rate (1/s) |
|---|---|
| 0.01 | 0.5 |
| 0.07499 | 3.749 |
| 0.5623 | 28.12 |
| 4.217 | 210.8 |
| 31.62 | 1,581 |
| 237.1 | 11,860 |
| 1,778 | 88,910 |
| 13,340 | 666,800 |
| 100,000 | 5,000,000 |
Sweep one variable
Variables and how each one contributes
Damping rate
What the equation gives you. Everything else feeds it.
Resistance
The total series resistance, in Ω. Raises the damping rate in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 500 to 550 1/s.
Inductance
The inductance, in H. Lowers the damping rate in inverse proportion. Raising it by 10 % (to 0.011) moves the result by −9.09 %, from 500 to 454.5 1/s.
Worked example with the values above
- Twice the inductance: 2L = 2 × 0.01 = 0.02 H.
- α = R/(2L) = 10 / 0.02.
- α = 500 1/s.
Assumptions and limits
- A series RLC loop with no source, ideal L and C, and R the total series resistance.
- Linear components with constant values.
Common mistakes
- Using R/L instead of R/(2L) (R/L is the decay rate of the energy, twice that of the amplitude).
- Leaving out the inductor’s winding resistance.
- Using the formula for a parallel RLC circuit, where α = 1/(2RC).
Statistics and uncertainty
α is proportional to R and to 1/L, so a 5 % tolerance on a resistor gives a 5 % error in the decay rate. Real inductors add their own series resistance, often more than the resistor, so use the total R.
Where it is used
- Predicting how long a switching transient rings.
- Selecting a damping resistor.
Maths you need
- Series RLC circuit: R, L and C in a loop
- Exponential decay: e^(−αt)