Library / Electrical / ELE-45
Electrical
Damped angular frequency of a series RLC circuit
Angular frequency at which a source-free series RLC circuit rings when the resistance is small enough for oscillation (underdamped).
- Ref
- ELE-45
- Rev
- v1
- Field
- Electrical
- Result
- ωd [rad/s]
For a series RLC loop released from rest, the charge satisfies L q″ + R q′ + q/C = 0. If R is below 2√(L/C), the solution rings at ω_d = √(1/(LC) − R²/(4L²)) inside the envelope e^(−Rt/2L). When the quantity under the root is zero or negative, the circuit does not oscillate and the page shows no value.
Angular frequency at which a source-free series RLC circuit rings when the resistance is small enough for oscillation (underdamped). It is slightly lower than the undamped frequency 1/√(LC). If the term under the root is zero or negative the circuit does not oscillate and no result is shown.
Calculator
- Undamped angular frequency, 1/√(LC)3,162 rad/s
- Damped frequency, ω_d/2π497 Hz
- Damping rate, R/2L500 1/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 circuit rings at 3,122.5 rad/s (496.96 Hz), 1.26 % below the undamped value 3,162.3 rad/s.
- Light damping lowers the ringing frequency only slightly; it falls to zero as R approaches the critical value 2√(L/C) = 63.25 Ω.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| C Capacitance | 1e-9 to 0.01 F | 34.48 | 316,200 | 316,200 | +10% | −4.78 % |
| L Inductance | 1e-6 to 10 H | 100 | 9,782 | 9,682 | +10% | −4.54 % |
| R Resistance | 0.01 to 100,000 Ω | 1,637 | 3,162 | 1,526 | +10% | −0.27 % |
Sweep of Resistance
| Resistance (Ω) | Damped angular frequency (rad/s) |
|---|---|
| 0.01 | 3,162 |
| 0.07499 | 3,162 |
| 0.5623 | 3,162 |
| 4.217 | 3,155 |
| 31.62 | 2,739 |
| 237.1 | — |
| 1,778 | — |
| 13,340 | — |
| 100,000 | — |
Sweep one variable
Variables and how each one contributes
Damped angular frequency
What the equation gives you. Everything else feeds it.
Resistance
The total series resistance, in Ω. Lowers the damped frequency, slowly when damping is light and quickly near critical damping. Raising it by 10 % (to 11) moves the result by −0.27 %, from 3,122 to 3,114 rad/s.
Inductance
The inductance, in H. Lowers the damped frequency, mainly through the 1/(LC) term. Raising it by 10 % (to 0.011) moves the result by −4.54 %, from 3,122 to 2,981 rad/s.
Capacitance
The capacitance, in F. Lowers the damped frequency, mainly through the 1/(LC) term. Raising it by 10 % (to 1.1×10−5) moves the result by −4.78 %, from 3,122 to 2,973 rad/s.
Worked example with the values above
- Undamped term: 1/(LC) = 1/(0.01 × 1×10−5) = 1×107 rad²/s².
- Damping term: R²/(4L²) = 10² / (4 × 0.01²) = 250,000 rad²/s².
- ω_d = √(1×107 − 250,000) = 3,122.5 rad/s.
Assumptions and limits
- A series RLC loop with no source, in the underdamped case.
- Ideal inductor and capacitor, with R as total series resistance.
Common mistakes
- Using the formula when the circuit is overdamped (the square root of a negative number).
- Forgetting that ω_d is in rad/s: divide by 2π for hertz.
- Using f₀ = 1/(2π√LC) as the ringing frequency when damping is heavy.
Statistics and uncertainty
Near critical damping the frequency drops rapidly with R, so a small resistance error gives a large error in ω_d. With light damping ω_d is almost 1/√(LC), so the inductor and capacitor tolerances (often ±10 % to ±20 %) dominate the uncertainty.
Where it is used
- Predicting ringing frequency after switching.
- Estimating L or C from a measured ring-down frequency and R.
Maths you need
- Series RLC: L q″ + R q′ + q/C = 0
- Complex roots: oscillation inside a decaying envelope