What is the resonant frequency of an RLC circuit?
f0 = 1 / (2π √(LC)), the same for series and parallel circuits: L and C set it and R does not move it. For 1 mH and 100 nF that is 15.92 kHz.
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RLC circuit calculator
Enter R, L and C in this RLC circuit calculator to get the resonant frequency, Q, bandwidth and damping of a series or parallel circuit, or the resistor that gives the Q you want. See whether it rings, how much it overshoots and how its impedance changes with frequency.
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Examples
Circuit
Ω
H
F
Hz
Optional: |Z| and phase at this frequency.
Resonant frequency f0
15.92 kHz
- Quality factor Q
- 10
- Bandwidth (−3 dB)
- 1.592 kHz f0 / Q
- Lower −3 dB frequency
- 15.14 kHz
- Upper −3 dB frequency
- 16.73 kHz
- Damping ratio ζ
- 0.05 1 / (2Q)
- Natural response
- underdamped
- Ringing frequency fd
- 15.9 kHz f0 √(1 − ζ²)
- Step overshoot
- 85.4 %
- Envelope time constant 1/α
- 200 µs
- Neper frequency α
- 5 kNp/s R / (2L)
- Characteristic impedance √(L/C)
- 100 Ω
- Impedance at f0
- 10 Ω the minimum, = R
- Critical damping resistance
- 200 Ω R above this does not ring
Series: R is the total loop resistance, including the source resistance, the inductor’s winding resistance and the capacitor’s ESR. The step overshoot is for the voltage across C.
At resonance the voltage across L and across C is Q = 10.0 times the voltage across the whole circuit. Check their voltage ratings.
|Z| (Ω)Phase (°)
Q and damping with other resistor values (critical R = 200 Ω)
| R | Q | ζ | Natural response | Step overshoot |
|---|---|---|---|---|
| 10 Ω | 10.0 | 0.0500 | underdamped | 85.4% |
| 22 Ω | 4.55 | 0.110 | underdamped | 70.6% |
| 47 Ω | 2.13 | 0.235 | underdamped | 46.8% |
| 100 Ω | 1.00 | 0.500 | underdamped | 16.3% |
| 220 Ω | 0.455 | 1.10 | overdamped | none |
| 470 Ω | 0.213 | 2.35 | overdamped | none |
| 1 kΩ | 0.100 | 5.00 | overdamped | none |
| 2.2 kΩ | 0.0455 | 11.0 | overdamped | none |
How do you calculate the resonant frequency, Q and damping of an RLC circuit?
Start with f0 = 1 ÷ (2π √(LC)) and Z0 = √(L/C). For a series circuit Q = Z0 ÷ R; for a parallel one Q = R ÷ Z0. Then the bandwidth is f0 ÷ Q and the damping ratio is ζ = 1 ÷ (2Q). For 10 Ω, 1 mH and 100 nF in series: f0 = 15.92 kHz, Z0 = 100 Ω, Q = 10.0, bandwidth 1.592 kHz and ζ = 0.050, so it is underdamped and rings at 15.9 kHz.
RLC circuit formulas and resonance
The resonant frequency of RLC circuits is set by L and C alone; R decides how sharp the resonance is and how quickly the circuit’s natural response dies away.
f0 = 1 ÷ (2π √(L × C)) Z0 = √(L ÷ C)
- f0
- resonant frequency in Hz (ω0 = 2πf0)
- Z0
- characteristic impedance √(L/C) in Ω
Q and the decay rate α are where the series and parallel RLC circuit equations differ:
| Quantity | Series RLC | Parallel RLC |
|---|---|---|
| Quality factor Q | Z0 ÷ R = 2πf0 L ÷ R | R ÷ Z0 = R ÷ (2πf0 L) |
| Neper frequency α | R ÷ (2L) | 1 ÷ (2RC) |
| Impedance at f0 | R (minimum) | R (maximum) |
| Critical damping R | 2 √(L/C) | ½ √(L/C) |
| Step overshoot applies to | voltage across C | current in L |
Everything else follows from Q, for both circuits:
BW = f0 ÷ Q f1,2 = f0 (√(1 + 1/(4Q²)) ∓ 1/(2Q)) ζ = 1 ÷ (2Q) fd = f0 √(1 − ζ²)
- BW
- bandwidth between the half-power (−3 dB) frequencies f1 and f2
- ζ
- damping ratio, α ÷ ω0
- fd
- ringing frequency of the natural response (ζ < 1)
For ζ below 1 the step response overshoots by e^(−πζ ÷ √(1 − ζ²)), which is how far the e^(−αt) envelope has decayed after half a ringing period.
RLC circuit examples
A textbook series circuit, a parallel tank and a damped LC filter, each of which this RLC calculator reproduces.
Example 1: series RLC, 0.2 Ω, 4 mH and 2 µF
ω0 = 1 ÷ √(4 mH × 2 µF) = 11.18 krad/s, so f0 = 1.779 kHz. Z0 = √(L/C) = 44.7 Ω and Q = Z0 ÷ R = 223.6. The bandwidth is f0 ÷ Q = 7.958 Hz (Δω = R ÷ L = 50 rad/s), from 1.775 kHz to 1.783 kHz. This is OpenStax Example 15.5, which gives ω0 = 1.12 × 10⁴ rad/s, Q = 224 and Δω = 50.0 rad/s.
Example 2: a parallel tank, 10 kΩ across 10 µH and 1 nF
f0 = 1.592 MHz, Z0 = 100 Ω, so Q = R ÷ Z0 = 100 and the bandwidth is 15.92 kHz. Now give the coil 1 Ω of winding resistance. Its own Q at f0 is 2πf0 L ÷ Rs = 100, so it acts like 10 kΩ across the tank. In parallel with the 10 kΩ that leaves 5 kΩ, and the tank’s Q halves to 50.0.
Example 3: damping a 10 µH and 22 µF LC filter
f0 = 10.73 kHz and Z0 = 674 mΩ. With 0.1 Ω of total series resistance, Q = 6.74 and a step overshoots by 79.2 %. Critical damping needs 2Z0 = 1.35 Ω; with 1 Ω, Q = 0.674 and the overshoot falls to 3.10 %. A real resistor in series with the inductor wastes power at DC, which is why damping is often added as a resistor in series with an extra capacitor across the output instead.
Damping ratio, Q and step response
Every second-order RLC response is set by ζ (or Q). Bandwidth and ringing frequency are relative to f0.
| ζ | Q | Response | Step overshoot | Ringing fd / f0 | Bandwidth / f0 |
|---|---|---|---|---|---|
| 0.01 | 50.0 | underdamped | 96.9 % | 0.9999 | 0.0200 |
| 0.05 | 10.0 | underdamped | 85.4 % | 0.9987 | 0.100 |
| 0.1 | 5.00 | underdamped | 72.9 % | 0.9950 | 0.200 |
| 0.2 | 2.50 | underdamped | 52.7 % | 0.9798 | 0.400 |
| 0.3 | 1.67 | underdamped | 37.2 % | 0.9539 | 0.600 |
| 0.5 | 1.00 | underdamped | 16.3 % | 0.8660 | 1.00 |
| 0.707 | 0.707 | underdamped | 4.33 % | 0.7072 | 1.41 |
| 1 | 0.500 | critically damped | none | no ringing | 2.00 |
| 2 | 0.250 | overdamped | none | no ringing | 4.00 |
What sets Q in a real circuit
The R in these formulas is every loss in the circuit, not just the resistor you placed. In a series circuit it includes the source resistance, the inductor’s winding resistance and the capacitor’s ESR. In a parallel tank it includes the load and the source driving it, and a coil’s series resistance Rs appears across the tank as Rs (1 + QL²) near f0. Inductor winding loss usually sets the practical ceiling on Q, and it rises with frequency through skin effect and core loss, so use the inductor’s datasheet Q at your frequency. At resonance a series circuit puts Q times the drive voltage across L and C, and a parallel tank circulates Q times the drive current, so check voltage and current ratings for high-Q designs.
Glossary
- Quality factor (Q)
- Resonant frequency divided by bandwidth. At resonance it is X ÷ R in a series circuit and R ÷ X in a parallel one, where X is the reactance of L or C.
- Bandwidth (BW)
- The span between the two half-power (−3 dB) frequencies, f0 / Q.
- Damping ratio (ζ)
- α / ω0 = 1 / (2Q). Below 1 the circuit rings; above 1 it does not.
- Neper frequency (α)
- The decay rate of the natural response, in nepers per second (1/s): R / (2L) for series, 1 / (2RC) for parallel.
- Critical damping
- ζ = 1, the fastest settling with no overshoot.
- Damped natural frequency (fd)
- The ringing frequency of an underdamped circuit, f0 √(1 − ζ²).
Related guides
- LC resonant frequency calculator: f0, L or C for an ideal tank.
Frequently asked questions
What is the difference between a series and a parallel RLC circuit?
Both resonate at f0 = 1 / (2π √(LC)). A series RLC has its minimum impedance at f0 (equal to R) and passes the most current there; a parallel RLC has its maximum impedance at f0 (also R) and develops the most voltage. R works in opposite directions: more series resistance lowers Q, more parallel resistance raises it.
How do you calculate the Q factor of an RLC circuit?
Series: Q = √(L/C) ÷ R = 2πf0 L ÷ R. Parallel: Q = R ÷ √(L/C) = R √(C/L). For 10 Ω, 1 mH and 100 nF in series, √(L/C) = 100 Ω and Q = 10.0; the same parts in parallel give Q = 0.100.
What is the RLC bandwidth formula?
BW = f0 ÷ Q, measured between the half-power (−3 dB) frequencies. The two edges are not symmetric around f0: they are f0 (√(1 + 1/(4Q²)) ∓ 1/(2Q)), whose geometric mean is f0. At Q = 10 and f0 = 15.92 kHz they are 15.14 kHz and 16.73 kHz.
What resistance gives critical damping?
Series: R = 2√(L/C); anything larger is overdamped. Parallel: R = ½√(L/C); anything smaller is overdamped. For a 10 µH and 22 µF filter the series value is 1.35 Ω.
What is the damped natural frequency?
The frequency an underdamped circuit rings at after a step: fd = f0 √(1 − ζ²), with ζ = 1 / (2Q). It is close to f0 unless the damping is heavy: at Q = 5, fd = 99.50 % of f0; at Q = 1, 86.6 %.
How do I know if an RLC circuit is overdamped or underdamped?
Compute ζ = 1 / (2Q). Below 1 the circuit rings (underdamped), at exactly 1 it settles fastest without overshoot (critically damped), above 1 it creeps to its final value (overdamped). The OpenStax example of 200 Ω, 5.0 mH and 6.0 µF in series has ζ = 3.46, so it is overdamped.
Why is the voltage across the capacitor higher than the supply at resonance?
In a series RLC at f0 the inductor and capacitor voltages are equal and opposite, and each is Q times the voltage across the whole circuit. With Q = 10.0, 1 V in puts 10.0 V across C. In a parallel tank the same happens to current: the current circulating through L and C is Q times the current fed in.
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Sources
- OpenStax University Physics Volume 2, 15.5 Resonance in an AC Circuit (Q = ω0L/R, bandwidth, Example 15.5)
- OpenStax University Physics Volume 2, 14.6 RLC Series Circuits (damped oscillation, ω′ = √(1/LC − (R/2L)²))
- Lessons in Electric Circuits, Volume II, Chapter 6: Resonance (Q and bandwidth, series and parallel)
- HyperPhysics (Georgia State University), Resonant RLC Circuits and Q
- K. K. Berggren, MIT 6.200 course notes, Time Domain R-L-C (α = 1/(2RC) for R, L and C in parallel)
- M. Zywno, Introduction to Control Systems (Toronto Metropolitan University), §7.2.1 Percent Overshoot, Equation 7-10
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