RLC Circuit Calculator
Enter resistance, inductance and capacitance to find the resonant frequency, impedance, quality factor, damping ratio and bandwidth of a series or parallel RLC circuit. All component values accept common unit prefixes (millihenries, microfarads, kilohertz, etc.) and results update as you type. The show-your-work panel traces every formula step with your actual numbers.
Formula
Worked example
Series RLC with R = 10 Ohm, L = 100 mH (0.1 H), C = 100 uF (100e-6 F): f0 = 1 / (2 * pi * sqrt(0.1 * 100e-6)) = 1 / (2 * pi * 0.003162) = 50.33 Hz. Q = (1/10) * sqrt(0.1 / 100e-6) = 0.1 * 31.62 = 3.162. Bandwidth = 50.33 / 3.162 = 15.92 Hz.
What is an RLC circuit?
An RLC circuit contains three passive components: a resistor (R), an inductor (L) and a capacitor (C). Connected together, they form a second-order electrical system that can resonate at a specific frequency. In a series RLC circuit the same current flows through all three components in sequence. In a parallel RLC circuit all three components share the same voltage while the currents through them differ. The resonant frequency f0 is the key parameter: at f0, the inductive reactance and capacitive reactance are equal and opposite, so they cancel. In a series circuit this produces minimum impedance; in a parallel circuit it produces maximum impedance.
Resonant frequency, Q factor and bandwidth
The resonant frequency depends only on L and C and is the same for both topologies: f0 = 1 / (2 * pi * sqrt(L * C)). The quality factor Q measures the sharpness of the resonance peak. A high-Q circuit has a very narrow bandwidth and high selectivity, which is ideal for radio tuning and precision filters. A low-Q circuit responds to a broader range of frequencies. The bandwidth in hertz is BW = f0 / Q, so doubling Q halves the bandwidth. The formulas for Q differ by topology: for a series circuit Q = (1/R) * sqrt(L/C), while for a parallel circuit Q = R * sqrt(C/L). This is why adding resistance to a series circuit lowers Q but adding resistance to a parallel circuit raises it.
Damping ratio and step response
The damping ratio zeta is the reciprocal of twice the Q factor: zeta = 1 / (2Q). It determines how the circuit responds to a sudden change in input. When zeta < 1 the circuit is underdamped: the response oscillates at a frequency close to f0 before settling. When zeta = 1 the circuit is critically damped: it reaches its final value as fast as possible without oscillating. When zeta > 1 the circuit is overdamped: the response decays slowly along two separate exponential curves. Most resonant and filter applications operate in the underdamped regime; snubber circuits for suppressing voltage spikes often use overdamped designs.
Impedance and phase angle
At any given operating frequency the total impedance depends on R, XL = 2 * pi * f * L and XC = 1 / (2 * pi * f * C). For a series circuit Z = sqrt(R^2 + (XL - XC)^2). For a parallel circuit the admittances combine as Y = sqrt((1/R)^2 + (1/XC - 1/XL)^2) and Z = 1/Y. The phase angle between voltage and current is arctan((XL - XC) / R) for a series circuit. A positive phase angle means the circuit is inductive (voltage leads current); a negative angle means it is capacitive (current leads voltage). At resonance the phase angle passes through zero.
Damping regime classification
| Condition | Damping ratio (zeta) | Step response behavior | Typical application |
|---|---|---|---|
| zeta < 1 | 0 < zeta < 1 | Oscillates, decays to steady state | Filters, oscillators, LC tanks |
| zeta = 1 | zeta = 1 | Fastest settling without overshoot | Control systems, critically damped design |
| zeta > 1 | zeta > 1 | Slow exponential decay, no oscillation | Overdamped snubber circuits |
| Q > 100 | zeta < 0.005 | Very sharp resonance, high selectivity | Crystal oscillators, precision filters |
| Q = 0.5 | zeta = 1 | Critical damping boundary | Maximally flat step response |
The damping ratio (zeta) determines how a circuit responds to a step or pulse input.
Frequently asked questions
What is the resonant frequency of an RLC circuit?
The resonant frequency f0 is the frequency at which the inductive reactance (XL = 2*pi*f*L) and capacitive reactance (XC = 1/(2*pi*f*C)) are equal in magnitude. Their effects cancel, leaving only resistance. The formula is f0 = 1 / (2 * pi * sqrt(L * C)). It depends only on L and C, not on R.
How does the Q factor affect circuit behavior?
The quality factor Q tells you how peaked the resonance is. A Q of 1 gives a broad, flat response. A Q of 100 gives a very narrow, sharp peak. In filtering applications, higher Q means better adjacent-channel rejection. In oscillators, higher Q means lower phase noise. Bandwidth in hertz equals f0 divided by Q, so doubling Q halves the bandwidth at the same center frequency.
What is the difference between series and parallel RLC circuits?
In a series RLC circuit the same current passes through R, L and C in order. Resonance causes minimum impedance and maximum current. In a parallel RLC circuit R, L and C share the same voltage. Resonance causes maximum impedance and minimum total current. The Q factor formula also inverts: resistance lowers Q in series circuits but raises Q in parallel circuits.
What does the damping ratio tell me?
The damping ratio zeta = 1/(2Q) classifies the step response. If zeta < 1 (underdamped) the circuit rings before settling. If zeta = 1 (critically damped) it settles fastest without ringing. If zeta > 1 (overdamped) it settles slowly without ringing. Most filter and oscillator circuits are underdamped (high Q, low zeta). Snubber circuits for transient suppression are often overdamped.
Why is the phase angle important?
Phase angle tells you the relationship between the voltage across and current through the circuit. A positive angle (inductive regime, frequency below resonance in series) means voltage leads current. A negative angle (capacitive regime, frequency above resonance in series) means current leads voltage. At resonance the angle passes through zero and the circuit is purely resistive. Phase angle matters in power factor correction and in understanding filter roll-off.
How do I shift the resonant frequency?
Since f0 = 1 / (2 * pi * sqrt(L * C)), you can raise f0 by decreasing L, decreasing C, or both. Halving either L or C raises f0 by a factor of sqrt(2) (about 41%). To tune over a wide range, a variable capacitor (varicap diode in RF circuits) is the most common approach because it can be voltage-controlled.