RLC Circuit Calculator
Calculate the resonant frequency, quality factor (Q), bandwidth, and damping characteristics of series and parallel RLC circuits. Essential for filter design, tuned circuits, and oscillator analysis. See also our Resonant Frequency Calculator and Inductive Reactance Calculator.
How RLC Circuits Work
An RLC circuit contains a resistor (R), inductor (L), and capacitor (C). At the resonant frequency, the inductive reactance (XL) equals the capacitive reactance (XC), and they cancel each other out. In a series RLC circuit, this results in minimum impedance (just R). In a parallel RLC circuit, this results in maximum impedance. The quality factor Q determines how sharp the resonance peak is — higher Q means narrower bandwidth and more selective filtering.
RLC circuits are the foundation of radio tuning, bandpass filters, oscillators, and impedance matching networks. The interplay between energy stored in the inductor's magnetic field and the capacitor's electric field creates oscillation, while the resistor dissipates energy and determines damping.
RLC Circuit Formulas
Resonant Frequency:
f₀ = 1 / (2π√(LC))
ω₀ = 1 / √(LC)
Quality Factor (Series):
Q = (1/R) × √(L/C) = ω₀L/R = 1/(ω₀CR)
Quality Factor (Parallel):
Q = R × √(C/L) = R/(ω₀L) = ω₀CR
Bandwidth:
BW = f₀ / Q
Damping Factor (Series):
α = R / (2L)
Underdamped: α < ω₀
Critically damped: α = ω₀
Overdamped: α > ω₀
Example Calculation
Series RLC with R=100Ω, L=10mH, C=100nF:
f₀ = 1/(2π√(0.01 × 100×10⁻⁹)) = 1/(2π√(10⁻⁹))
f₀ = 1/(2π × 31.62×10⁻⁶) = 5,033 Hz ≈ 5.03 kHz
ω₀ = 2π × 5033 = 31,623 rad/s
XL = ω₀L = 31623 × 0.01 = 316.2 Ω
XC = 1/(ω₀C) = 1/(31623 × 100×10⁻⁹) = 316.2 Ω ✓ (XL = XC)
Q = (1/100) × √(0.01/100×10⁻⁹) = 0.01 × 316.2 = 3.16
Bandwidth = 5033/3.16 = 1,592 Hz
α = 100/(2×0.01) = 5000 rad/s
ω₀ = 31623 rad/s → α < ω₀ → Underdamped
RLC Circuit Reference Table (Series)
| R (Ω) | L | C | f₀ | Q |
|---|---|---|---|---|
| 10 | 1mH | 1µF | 5,033 Hz | 31.6 |
| 50 | 10mH | 100nF | 5,033 Hz | 6.3 |
| 100 | 10mH | 100nF | 5,033 Hz | 3.2 |
| 100 | 100mH | 10nF | 5,033 Hz | 31.6 |
| 50 | 1mH | 10nF | 50,330 Hz | 6.3 |
| 10 | 100µH | 100pF | 1.59 MHz | 100 |
| 50 | 100µH | 100pF | 1.59 MHz | 20 |
| 100 | 1mH | 1nF | 159.2 kHz | 10 |
| 1k | 10mH | 10nF | 15.92 kHz | 1.0 |
| 500 | 50mH | 50nF | 3,183 Hz | 2.0 |
Frequently Asked Questions
What is the quality factor Q?
Q (quality factor) measures how underdamped an RLC circuit is — how sharp its resonance peak is. High Q means narrow bandwidth and strong frequency selectivity (good for radio tuning). Low Q means wide bandwidth (good for broadband applications). Q also represents the ratio of energy stored to energy dissipated per cycle: Q = 2π × (energy stored)/(energy lost per cycle).
What is the difference between series and parallel RLC?
In series RLC, impedance is minimum at resonance (equals R), making it a bandpass filter when driven by a voltage source. In parallel RLC, impedance is maximum at resonance (equals R for ideal components), making it a band-reject (notch) filter. The Q factor formulas are inverted: series Q increases with lower R, parallel Q increases with higher R.
What does damping mean in an RLC circuit?
Damping describes how oscillations decay after a transient. Underdamped circuits oscillate with decreasing amplitude (ringing). Critically damped circuits return to equilibrium fastest without oscillation. Overdamped circuits return slowly without oscillation. The damping ratio ζ = α/ω₀ determines the behavior: ζ < 1 underdamped, ζ = 1 critical, ζ > 1 overdamped.
How do I design a bandpass filter with specific bandwidth?
For a series RLC bandpass filter: choose center frequency f₀, then Q = f₀/BW. Select L and C for the desired f₀ (f₀ = 1/(2π√LC)), then R = ω₀L/Q. For example, f₀=10kHz, BW=1kHz: Q=10, choose L=10mH → C=25.3nF, R = 2π×10000×0.01/10 = 62.8Ω.
Why does XL equal XC at resonance?
At resonance, the energy oscillates completely between the inductor's magnetic field and the capacitor's electric field. The inductive reactance (XL = ωL, increasing with frequency) and capacitive reactance (XC = 1/ωC, decreasing with frequency) are equal at exactly one frequency — the resonant frequency. At this point, they cancel in series (minimum Z) or create maximum impedance in parallel.
What are practical Q values for different applications?
Audio filters: Q = 0.5-5. Radio IF filters: Q = 20-100. Crystal oscillators: Q = 10,000-1,000,000. Microwave cavities: Q = 1,000-100,000. Higher Q requires lower-loss components — air-core inductors, low-ESR capacitors, and minimal resistive losses. Component Q limits the achievable circuit Q.
Practical Applications
- Radio tuning: Variable capacitor with fixed inductor selects radio stations
- Bandpass filters: Select specific frequency bands in audio and RF systems
- Oscillators: RLC tank circuits generate sinusoidal signals at resonant frequency
- Impedance matching: L-networks and pi-networks match source to load impedance
- Power line filters: Notch filters remove specific harmonic frequencies
- Wireless charging: Resonant coupling between transmitter and receiver coils