The RLC circuit calculator above characterises a resonant circuit built from a resistor, an inductor and a capacitor. It returns the resonant frequency, the quality factor, the bandwidth between the half-power points, the damping ratio and the resistance that would make the circuit critically damped. It handles both the series and the parallel arrangement, which behave in opposite ways as resistance changes, and it names the damping regime rather than leaving you to interpret a bare number.
Arb Digital builds free calculators that answer the whole question rather than one part of it. Most RLC pages give a resonant frequency and stop, which is the easy half — the resonant frequency ignores resistance entirely. The useful half is what resistance does to the shape of the response and to the way the circuit settles after a step, and that is what the Q factor, the bandwidth and the damping ratio on this page describe.
What This RLC Circuit Calculator Does
It computes the frequency-domain and time-domain descriptors of a second-order circuit from three component values. The hero gives the undamped resonant frequency in hertz, with the half-power frequencies underneath. The grid gives the quality factor, the bandwidth, the damping ratio and the critical resistance, and the note names the damping regime and describes what the circuit will actually do when disturbed.
The topology selector matters more than it looks. Both arrangements resonate at the same frequency, because that frequency is set by the point where inductive and capacitive reactance cancel and resistance plays no part. But Q is inversely proportional to resistance in the series case and directly proportional to it in the parallel case. Increasing a series resistor flattens the peak; increasing a parallel resistor sharpens it. Getting the topology wrong therefore does not shift the frequency, which is exactly why the mistake survives so long undetected.
Unit selectors sit next to the inductance and capacitance fields because real components are almost never quoted in henries and farads. A tuned circuit for the AM broadcast band might use hundreds of microhenries and hundreds of picofarads, while a mains-frequency filter might use whole henries and microfarads. The selectors keep you from having to count zeros.
How to Use It
- Choose the topology. Series if the three parts sit in one current path, parallel if they share the same pair of nodes. This decides which way resistance affects sharpness.
- Enter the resistance you actually have. In a series tuned circuit that means the coil's winding resistance plus any deliberate resistor, and the coil usually dominates.
- Enter L and C with their units. Check the resonant frequency against a rough expectation before reading anything else; a wild answer is nearly always a unit error.
- Read Q before bandwidth. Q is dimensionless and comparable between circuits, whereas a bandwidth of 800 Hz means something very different at 5 kHz than at 5 MHz.
- Compare your resistance with the critical value. Below it the circuit rings; above it the response creeps to its final value without overshoot.
The Formula: How RLC Resonance Is Calculated
The undamped resonant angular frequency is ω0 = 1 ÷ √(LC), and the frequency in hertz is that divided by 2π. For the series circuit, Q = (1 ÷ R)√(L ÷ C) and the damping ratio is ζ = 1 ÷ 2Q. For the ideal parallel circuit, Q = R√(C ÷ L), the reciprocal relationship. Bandwidth in both cases is f0 ÷ Q. Section 15.5 of OpenStax University Physics Volume 2, Resonance in an AC Circuit, derives the peak in current amplitude and defines Q as the ratio of resonant angular frequency to bandwidth, which is the definition used here.
The half-power frequencies are not symmetric about f0 on a linear scale; they are symmetric on a logarithmic one, and f0 is their geometric mean. The exact expressions are f0 [√(1 + 1÷4Q²) ± 1÷2Q]. For Q above about five the asymmetry is small enough to ignore, which is why textbooks often quote f0 ± BW÷2 as an approximation, but at low Q that shortcut fails visibly.
Work the defaults. With R = 50 Ω, L = 10 mH and C = 100 nF in series, √(LC) = √(0.01 × 10−7) = 3.1623 × 10−5, so f0 = 1 ÷ (2π × 3.1623 × 10−5) = 5,032.9 Hz. The characteristic impedance √(L÷C) is 316.23 Ω, so Q = 316.23 ÷ 50 = 6.32 and the bandwidth is 5,032.9 ÷ 6.32 = 795.8 Hz. The damping ratio is 1 ÷ 12.65 = 0.079, comfortably underdamped, and critical resistance is 2 × 316.23 = 632.5 Ω. The half-power points come out at 4,650.7 Hz and 5,446.5 Hz, and their difference is the bandwidth, as it must be.
Q Factor Means Three Different Things at Once
The quality factor is unusually rich because three separate definitions coincide for a simple second-order circuit. It is the ratio of resonant frequency to bandwidth, so it measures selectivity. It is 2π times the ratio of energy stored to energy lost per cycle, so it measures efficiency — and only the resistance dissipates, a point made precisely in section 15.4 of OpenStax University Physics Volume 2, Power in an AC Circuit, where the inductor and capacitor are shown to absorb and return energy without consuming any of it. And it is the voltage magnification across the reactive components at resonance in a series circuit, so it measures gain.
That last one has practical consequences people do not expect. In the default circuit, Q is 6.32, which means the voltage across the inductor at resonance is 6.32 times the applied voltage. In a high-Q tuned circuit with Q of 100, a modest few volts of drive produces hundreds of volts across the coil and the capacitor, and those two voltages are almost exactly out of phase so they cancel at the terminals. Components in a resonant circuit therefore routinely see far more voltage than the supply, which is a genuine failure mode when a capacitor is chosen on supply voltage alone.
Q also sets how long the circuit rings. A struck resonant circuit decays to about four per cent of its initial amplitude after Q cycles, so a Q of 6 means a handful of visible oscillations while a Q of 1,000 means a tone that hangs for a thousand cycles. This is the electrical version of a lightly damped bell, and the mathematics is identical to a mechanical oscillator on a spring.
Damping Regimes and What They Sound and Look Like
The damping ratio divides behaviour into three regimes, and the boundary is where resistance equals the critical value 2√(L÷C) in a series circuit. Below it, ζ is less than one and the circuit is underdamped: a step input produces overshoot followed by decaying oscillation at the damped natural frequency, which is slightly below the undamped resonant frequency by a factor √(1 − ζ²).
At exactly critical damping, ζ equals one and the response reaches its final value in the shortest time possible with no overshoot at all. This is what you want in a meter movement, a servo or a car suspension. Above it, the circuit is overdamped, the two roots of the characteristic equation are real and distinct, and the response is a slow exponential creep that takes longer to settle than the critical case despite having more resistance.
That last point is the counter-intuitive one worth remembering: adding more resistance beyond critical does not make the circuit settle faster, it makes it settle slower. Damping past critical trades oscillation for sluggishness, and the sweet spot for most step-response applications is slightly under critical, around ζ of 0.7, which gives a small overshoot and the fastest settling to within a few per cent. The time-domain half of this behaviour connects to the exponential charging described by our RC time constant calculator, which is the first-order case with no inductor and therefore no possibility of oscillation.
Real Components Are Not Ideal, and Q Is Where It Shows
The resonant frequency this tool reports is robust, because it depends only on the nominal L and C. The Q factor is not, because it depends on losses that no component data sheet fully captures. An inductor's winding resistance rises with frequency through the skin effect, and its core adds hysteresis and eddy-current losses that are not resistive in any simple sense. A capacitor has an equivalent series resistance of its own and a dielectric that absorbs energy.
The practical consequence is that a measured Q is nearly always lower than a calculated one, sometimes by a factor of two or more in the megahertz region. If you are designing a filter to a bandwidth specification, treat the calculated Q as a ceiling and design in margin. The impedance triangle behind all of this, Z = √(R² + (XL − XC)²), is set out in section 15.3 of OpenStax University Physics Volume 2, RLC Series Circuits with AC, and it is what collapses to plain R at resonance. It is also why the parallel case is trickier in reality: an ideal parallel RLC has infinite impedance at resonance when R is infinite, but a real coil's series loss transforms into a finite parallel resistance and caps the peak. The reactance calculator gives the individual inductive and capacitive reactances at any frequency, which is the fastest way to sanity-check where cancellation actually occurs.
How This Differs From the Site's Other Circuit Tools
The boundary in one sentence: this page describes what happens at and around resonance — the peak frequency, its sharpness and the circuit's damping — whereas the RLC impedance calculator gives the impedance magnitude and phase angle at any drive frequency you choose, including frequencies far from resonance where this page has nothing to say.
The LC resonant frequency calculator is the lossless case: it solves the same square root for frequency, or works backwards to a component value, but it has no resistance and therefore no Q, no bandwidth and no damping. Use it when you are picking components for a target frequency and this page when you need to know how sharp the result will be. The filter cutoff calculator handles first-order RC and RL corner frequencies, the Ohm's law calculator covers the steady-state DC relations, and the capacitance converter and inductance converter rescale component values between unit prefixes.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Expecting resistance to move the resonant frequency — it does not. Only L and C set f0; R sets how sharp and how ringy the response is.
- Applying the series Q formula to a parallel circuit — the two are reciprocals, so the error gives a Q that is wrong by a factor of Q squared.
- Ignoring the coil's winding resistance — in a series tuned circuit it is frequently larger than any resistor you deliberately added, and it is what really sets Q.
- Sizing capacitors on supply voltage — at resonance the voltage across the reactive parts is Q times the applied voltage, so a high-Q circuit stresses them far beyond the rail.
- Assuming the half-power points are symmetric — they are geometrically symmetric about f0, not arithmetically, and at low Q the difference is easy to see.
Related Free Tools From Arb Digital
For impedance away from resonance, use the RLC impedance calculator; for the lossless frequency alone, the LC resonant frequency calculator. The reactance calculator gives individual reactances, the RC time constant calculator covers first-order transients, and the filter cutoff calculator handles corner frequencies. Use the capacitance converter and inductance converter for unit prefixes and the Ohm's law calculator for DC relations. Anything involving mains wiring belongs with the breaker size calculator and a licensed electrician. The full free online tools hub lists everything.
Frequently Asked Questions
No. The resonant frequency is set entirely by inductance and capacitance, because it is the frequency at which their reactances cancel. Resistance decides the sharpness of the peak and the damping, not its location.
Because the resistance sits in a different place. In a series circuit resistance is in the current path and dissipates energy, so more resistance lowers Q. In an ideal parallel circuit resistance diverts current away from the tank, so more resistance raises Q.
It is critical damping, the boundary between oscillatory and non-oscillatory behaviour. A critically damped circuit reaches its final value in the shortest possible time without overshooting, which is why meter movements and servos are designed near it.
No. Beyond critical damping the response becomes a slow exponential creep that takes longer to settle than the critical case. Overdamping trades oscillation for sluggishness rather than buying extra speed.
Because at resonance the reactive voltages are magnified by the Q factor and are almost exactly out of phase, so they cancel at the terminals while each individually can be many times the applied voltage.
Real components have losses the ideal model omits: winding resistance that rises with frequency, core losses in the inductor and equivalent series resistance in the capacitor. Treat the calculated Q as an upper bound.
Not arithmetically. The resonant frequency is their geometric mean, so the upper point sits slightly further away in hertz than the lower one. The asymmetry becomes negligible above a Q of about five.
This tool is provided for educational and study use. It models ideal lumped components with no parasitics and no source impedance, so treat its output as a design starting point rather than a verified result. Any circuit connected to mains electricity must be designed, installed and signed off by a qualified electrician working to the applicable wiring rules.
