Resonant frequency f₀
LC = 10×10⁻³ × 100×10⁻⁹ = 1×10⁻⁹ s², √(LC) = 3.162×10⁻⁵ s, so f₀ = 1/(2π × 3.162×10⁻⁵) ≈ 5033 Hz (5.03 kHz). R plays no part in this step.
Solve a series or parallel RLC network in one pass. Enter resistance, inductance, capacitance and the operating frequency — the calculator returns impedance Z, both reactances, phase angle, resonant frequency f₀, Q factor, −3 dB bandwidth and the damping state, plus an inductive-vs-capacitive verdict at your chosen frequency.
Series or parallel R, L, C at any frequency: Z, φ, f₀, Q, bandwidth and damping in one pass.
One loop: Z = √(R² + (X_L − X_C)²), the same current flows through every element.
Verdict at f = 10000 Hz: Inductive at this frequency: X_L is greater than X_C, so the net reactance is positive and the current lags the voltage.
📌 Core formula: Z = √(R² + (X_L − X_C)²) with φ = atan2(X_L − X_C, R). At resonance X_L = X_C, so Z collapses to exactly R.
An RLC circuit combines the three passive elements — resistance R, inductance L and capacitance C — in a single network, wired either in series (one loop, one shared current) or in parallel (one node pair, one shared voltage). On its own, that description sounds innocent enough. What makes the combination interesting is that the three elements respond to frequency in opposite ways: resistance ignores it, inductive reactance grows with it, and capacitive reactance shrinks with it. Somewhere in between, the two reactive effects can cancel each other exactly, and that single frequency — resonance — is where an RLC circuit does its most useful and sometimes its most destructive work.
The R is what separates a real circuit from the ideal LC tank you meet in first-year textbooks. An ideal tank has no loss at all: charge the capacitor, let it discharge into the inductor, and the energy sloshes back and forth between the magnetic field and the electric field forever, ringing at f₀ = 1/(2π√(LC)) with undiminished amplitude. That never happens in hardware. Every real network has resistance hiding in the copper of the winding, in the equivalent series resistance of the capacitor, in the source impedance of whatever drives the circuit, and in the load resistance of whatever consumes its output. That resistance bleeds a little energy on every cycle, and it is the reason a struck tuning-fork-style circuit decays instead of singing forever.
When the calculator asks for a single R value, it is really asking you to fold every loss mechanism into one number. In a wound inductor the dominant term is the DC resistance of the copper wire, typically a few ohms for a small choke and rising with frequency because of skin effect, which pushes current toward the conductor surface. In a capacitor the loss appears as ESR — a few milliohms for a good film part, tens of milliohms for an electrolytic, and strongly frequency-dependent in both cases. Around them sit the Thevenin resistance of the source (a 50 Ω signal generator, say), the load resistance of the next stage, and any damping resistor you added on purpose to tame the peak. With the calculator's defaults — R = 100 Ω, L = 10 mH, C = 100 nF — that 100 Ω could realistically be a 50 Ω generator impedance, a 47 Ω damping resistor and a few ohms of winding resistance all stacked together.
💡 Tip: Measure or estimate R at the operating frequency, not at DC. The DC resistance of a winding understates its AC resistance because of skin and proximity effects — for a small ferrite choke, the effective resistance at 100 kHz can easily be 30–50% higher than the multimeter reading, which directly lowers your real Q.
⚠️ Warning: Do not reuse a capacitor's datasheet ESR from one frequency in a calculation at another. ESR curves bend hard — an electrolytic rated 50 mΩ at 100 kHz may present several ohms at 50 Hz, enough to reshape both the damping ratio and the resonant peak of a mains-frequency RLC network.
❌ Don't assume that changing R shifts the resonant frequency. f₀ = 1/(2π√(LC)) contains no R term for the classical series and parallel RLC networks treated here — resistance flattens, broadens and lowers the peak, but the peak stays parked at the frequency set by L and C alone.
This RLC circuit calculator treats R as a first-class input and reports the damped-circuit picture end to end: impedance, phase angle, damping ratio ζ and bandwidth alongside f₀ and Q. If you only need the frequency of an ideal lossless tank, the dedicated LC resonant frequency calculator handles the L-and-C-only math, and the calculators hub on the CapacitorCalculator.com homepage lists every tool in the series.
Impedance is the frequency-dependent opposition a circuit offers to AC current, and in an RLC network it is a phasor — it has both a magnitude and an angle. The two reactive pieces move in opposite directions as frequency changes: X_L = 2πfL starts near zero at DC and climbs without limit, while X_C = 1/(2πfC) starts enormous at DC and falls toward zero. Because they sit 180° out of phase with each other, they subtract rather than add, and which one is left standing at a given frequency decides everything about how the circuit behaves.
Call a circuit inductive at frequency f when its inductive reactance outweighs its capacitive reactance, and capacitive when the reverse holds. The consequences are about timing: in an inductive circuit the current lags the voltage (the magnetic field resists changes in current), while in a capacitive circuit the current leads the voltage (the capacitor's voltage lags its own charging current). The phase angle φ returned by the calculator quantifies that lead or lag in degrees. At 10 kHz with the default values, X_L ≈ 628 Ω against X_C ≈ 159 Ω, so a series RLC built from those parts is firmly inductive and shows φ ≈ 78° — the current trails the voltage by more than three-quarters of a right angle. Note one subtlety the calculator handles for you: in a series circuit the bigger reactance wins, but in a parallel circuit it is the smaller branch impedance that grabs the current, so the same comparison of X_L against X_C produces the opposite verdict depending on topology.
In the series configuration the same current is forced through R, L and C, and the element voltages add as phasors. The resistive voltage is in phase with the current, the inductive voltage leads it by 90°, and the capacitive voltage lags it by 90°, so the two reactive voltages fight each other and what remains is their difference. The result is the familiar magnitude Z = √(R² + (X_L − X_C)²) with phase φ = atan2(X_L − X_C, R). Run the defaults: at 10 kHz the net reactance is 628 − 159 = 469 Ω, which dwarfs the 100 Ω of resistance, giving Z ≈ 480 Ω and φ ≈ 78°. Far below resonance the capacitor blocks everything and Z climbs toward X_C; far above resonance the inductor chokes the loop and Z climbs toward X_L. In between, Z dives to exactly R.
The parallel circuit flips the logic. All three branches share the same voltage, and the branch currents add as phasors — which is why the calculator works in admittance, Y = √((1/R)² + (1/X_L − 1/X_C)²), and then inverts it to get Z = 1/Y. Current in the resistive branch is in phase with the voltage, inductor current lags by 90°, capacitor current leads by 90°, and the two reactive currents cancel where they are equal. The practical rule of thumb is the mirror image of the series case: the smaller branch reactance dominates. Feed the same default values at 10 kHz into parallel form and the capacitor branch, at 159 Ω, hogs the current: the calculator returns Z ≈ 90 Ω with a capacitive verdict, even though the identical parts in series looked strongly inductive at the same frequency. At resonance the two reactive branch currents become equal and opposite — they circulate entirely inside the L–C loop, and from the source's viewpoint the network reduces to the resistor alone.
Resonance is the frequency at which the two reactances (series view) or the two susceptances (parallel view) are exactly equal and cancel. Everything the calculator reports rearranges itself around that point, so it is worth walking through the four linked ideas one at a time: where f₀ comes from, what physically changes at resonance, how the −3 dB bandwidth and Q describe the same peak from two directions, and how the damping ratio ζ grades the circuit's behavior between ringing and sluggishness.
Setting X_L = X_C gives 2πf₀L = 1/(2πf₀C), and solving for f₀ yields the most quoted expression in AC theory:
For the default values, L·C = 10 × 10⁻³ H × 100 × 10⁻⁹ F = 1 × 10⁻⁹ s², so √(LC) = 3.162 × 10⁻⁵ s and f₀ = 1/(2π × 3.162 × 10⁻⁵) ≈ 5033 Hz — about 5.03 kHz. Notice that R appears nowhere in the formula. Resistance shapes the peak (its height, width and the ringing that follows a transient) but never moves its center for the ideal networks treated here. That is a genuinely useful design lever: you can pick L and C to place the channel wherever the application needs it, then choose R afterwards to decide how sharp the channel is.
In the series circuit, resonance is the point of minimum impedance and maximum current: the reactances cancel completely, Z collapses to exactly R (100 Ω in the default network), and if you drive the circuit from a fixed voltage source the current is limited only by that resistance. The individual element voltages can be far larger than the source — each is Q times the applied voltage — yet they are equal and opposite across L and C, so they cancel around the loop. This voltage magnification is why a careless series-resonant test rig can put hundreds of volts across a component rated for tens. In the parallel circuit the roles reverse: resonance is the point of maximum impedance and minimum line current. The reactive branch currents are equal and opposite, so they circulate entirely inside the tank while the source only supplies the resistive loss. With the same 100 Ω, 10 mH and 100 nF wired in parallel, Q is only ≈ 0.32 (R is shunting the tank hard), but if the 100 Ω sits in the coil's branch instead, the tank impedance at resonance rises to L/(C·R) ≈ 1000 Ω — ten times the resistance, which is the whole appeal of tank circuits.
Q measures how sharply the circuit selects its resonant frequency. Formally it is 2π times the energy stored divided by the energy dissipated per cycle; practically, it is the ratio of the resonant frequency to the bandwidth: Q = f₀/BW. The bandwidth is measured between the two half-power points, the frequencies at which a series circuit's current (or a parallel circuit's impedance) has fallen to 1/√2 — about 70.7% — of its peak, which is 3 dB down on a logarithmic scale. For a high-Q circuit those edges sit at approximately f₀ ± BW/2. With the defaults, Q = (1/100)·√(10 × 10⁻³ / 100 × 10⁻⁹) = 316.2/100 ≈ 3.16, so BW = 5033/3.16 ≈ 1592 Hz and the half-power edges land near 4237 Hz and 5829 Hz. A Q of 3 is a workmanlike, broad filter; the same L and C with R = 10 Ω would reach Q ≈ 31.6 and squeeze that window down to about 159 Hz.
The damping ratio ζ grades how a circuit returns to rest after a disturbance, and it is simply half the reciprocal of Q: ζ = 1/(2Q). Written out per topology, series ζ = (R/2)·√(C/L) and parallel ζ = (1/(2R))·√(L/C). When ζ is below 0.99 the circuit is underdamped — disturbed, it rings at f₀ with each cycle decaying by a fixed fraction; the default network's ζ ≈ 0.158 rings roughly e^(−π/(2Q)) per cycle, visibly for several cycles. Between 0.99 and 1.01 it is critically damped — the fastest possible return to rest with no overshoot, the target for meters, servos and stepper damping. Above 1.01 it is overdamped — no ringing at all, but a slow, sluggish crawl back to the final value. And here is the beautiful asymmetry: raising R lowers Q and raises ζ in a series circuit (more series resistance, more loss), while in a parallel circuit raising R does the opposite, because a bigger shunt resistor steals less current from the tank. The calculator's badge shows which regime you are in for the values on screen — flip the topology toggle and watch the same components change personality from ringing to overdamped.
You can sketch an RLC circuit's response from the numbers the calculator already gave you, without any simulation. The trick is to treat f₀ as the hinge of the whole story and ask, on each side of it, which element is winning and by how much.
Plot the series circuit's impedance against frequency and you get a V-shaped valley with its floor at Z = R exactly at f₀. Seen from a fixed voltage source, the current is the mirror image: a hill peaking at V/R at resonance and rolling off on both sides. That current-vs-frequency hill is the classic band-pass response — the series RLC passes a band of frequencies centered on f₀ with a width set by BW. The element voltages tell a richer story: across the capacitor the voltage is largest at low frequencies (a low-pass shape with a resonant hump near f₀), across the inductor it is largest at high frequencies (a high-pass shape with the same hump), and across the resistor it tracks the input only near resonance where the reactive voltages cancel. This is why a single series RLC stage can be tapped as a band-pass, low-pass or high-pass filter depending on where you take the output.
The −3 dB points are the frequencies where the output power has fallen to half of its resonant value, which corresponds to the output voltage or current falling to 1/√2 ≈ 0.707 of its peak. They mark the boundaries of the passband, and the distance between them is the bandwidth BW the calculator reports. For a high-Q circuit the edges sit at approximately f₀ ± BW/2 — with the defaults that is roughly 4237 Hz and 5829 Hz around 5033 Hz, though at Q ≈ 3 the true edges pull slightly inward because the response is not perfectly symmetric at low Q. The name comes from decibel bookkeeping: 20·log₁₀(0.707) = −3.01 dB, and because power goes as the square of voltage, −3 dB in voltage is exactly half power. When an engineer says a crossover is "3 dB down at 4.2 kHz," they are quoting exactly this edge.
The parallel circuit is the photographic negative. Its impedance forms a mountain peaking at f₀ rather than a valley, so when you place a parallel tank in series with a signal path it acts as a band-stop (notch) filter — it blocks the frequency it resonates at, which is precisely how mains-hum traps and RF notch filters work. Driven from a current source, its voltage response is band-pass, peaking where the impedance peaks. Below resonance the inductor branch carries the heavier current and the circuit looks inductive; above resonance the capacitor takes over and it looks capacitive; at the peak the line current collapses toward V/R and the reactive currents just chase each other around the loop. The width of that mountain is the same BW = f₀/Q as before, and the damping badge means the same thing — an overdamped parallel tank has such a broad, low mountain that calling it a filter at all becomes a stretch.
Once you can read Z, f₀, Q and ζ, you can see RLC thinking at work in almost every frequency-selective piece of electronics. These are the places the formulas on this page earn their keep.
Second-order filters are RLC networks wearing a purpose. A series L–C shunted across a line with R provided by the load makes a band-pass; tapped across C it is low-pass, across L it is high-pass, and a parallel tank in the signal path is a notch. The Q you compute here is the design parameter: an audio equalizer wants Q of roughly 1–5 per band, a crystal-like IF filter wants hundreds, and a power-line harmonic trap wants just enough Q to notch 50 Hz without ringing when switched. The damping ratio badge doubles as a stability check — a filter with ζ well below 0.5 will peak before it rolls off, adding ripple right where you wanted flatness.
Every LC oscillator — Colpitts, Hartley, Clapp — is an RLC tank whose losses an amplifier replenishes each cycle. The tank's L and C set the oscillation frequency through f₀ = 1/(2π√(LC)); its Q sets how clean the output is, because a high-Q tank filters the amplifier's harmonics down to a near-sinusoid and holds the frequency steady against component drift. The amplifier's job, viewed through this page's lens, is to inject exactly enough energy to cancel the effective series R — drive it with less and the oscillation dies out at a rate ζ dictates; drive it with more and the amplitude grows until nonlinearity clamps it. Radio receivers, induction heaters and metal detectors all run tanks the same way, just at very different f₀.
A 12 dB-per-octave passive crossover is a pair of second-order RLC sections — an inductor and capacitor working against the driver's impedance — sending lows to the woofer and highs to the tweeter. Here the R in the equations is not a part you add; it is the driver itself, whose nominal 4 Ω or 8 Ω rating is really a frequency-dependent impedance with resonant peaks. That is why the "inductive vs capacitive at f" verdict matters: the crossover's corner frequency follows from the actual driver impedance at the crossover point, and the damping of each section decides whether the two drivers sum flat through the hand-off region or leave a peak and a hole. Getting Q near 0.707 (Butterworth alignment) is the classic target, and it falls straight out of the ζ formulas above.
Unwanted resonance is an EMI problem wearing a disguise. A switching converter's input leads have stray L, the input capacitor has ESR, and together they form an underdamped RLC that can ring at a frequency an EMC test will find. The fix is damping engineering with the very tools on this page: add series resistance (or choose a lossier capacitor) to push ζ from 0.1 toward 0.5–1, trading a slightly higher loss for a murdered oscillation. The same logic sizes RC snubbers across switch nodes and ferrite-bead-plus-capacitor filters on supplies. In every case the workflow is identical — compute f₀ from the strays, compute Q and ζ with the intended R, and if the badge says underdamped, add loss until it does not.
These are exactly the calculator's defaults, so you can verify every number below by loading the tool and reading it back. Follow the five steps in order — each one uses only the values before it.
LC = 10×10⁻³ × 100×10⁻⁹ = 1×10⁻⁹ s², √(LC) = 3.162×10⁻⁵ s, so f₀ = 1/(2π × 3.162×10⁻⁵) ≈ 5033 Hz (5.03 kHz). R plays no part in this step.
√(L/C) = √(10×10⁻³ / 100×10⁻⁹) = √100000 = 316.2, so Q = 316.2/100 ≈ 3.16. Roughly three cycles of stored energy for every radian of loss.
BW = 5033/3.16 ≈ 1592 Hz. The −3 dB edges sit near 5033 ± 796, i.e. about 4237 Hz and 5829 Hz.
ζ = (100/2)·√(100×10⁻⁹ / 10×10⁻³) = 50 × 3.162×10⁻³ ≈ 0.158, or simply 1/(2 × 3.16). Since 0.158 is well below 0.99, the circuit is underdamped — it rings at f₀ when disturbed.
At 10 kHz: X_L = 2π × 10000 × 0.01 ≈ 628 Ω and X_C = 1/(2π × 10000 × 100×10⁻⁹) ≈ 159 Ω. Since X_L exceeds X_C, the series circuit is inductive at this frequency — click the 10k preset in the calculator and the verdict card confirms it. The net reactance is 628 − 159 = 469 Ω, so Z = √(100² + 469²) ≈ 480 Ω and φ = atan2(469, 100) ≈ 78°, with the current lagging the voltage. Switch the topology toggle to Parallel with the same values and the capacitor branch — at only 159 Ω — takes over: Z drops to about 90 Ω and the verdict flips to capacitive, which is the series-vs-parallel contrast from the sections above made concrete.
Every quantity the calculator computes, in both topologies, side by side. The formulas are the exact ones implemented above — where the two columns match, it is because the physics genuinely does not care how the parts are wired.
| Quantity | Series Circuit | Parallel Circuit |
|---|---|---|
| Impedance Z | Z = √(R² + (X_L − X_C)²) | Z = 1 / √((1/R)² + (1/X_L − 1/X_C)²) |
| Resonant frequency f₀ | f₀ = 1/(2π√(LC)) | f₀ = 1/(2π√(LC)) |
| Q factor | Q = (1/R)·√(L/C) | Q = R·√(C/L) |
| Bandwidth BW | BW = f₀/Q | BW = f₀/Q |
| Phase angle φ | φ = atan2(X_L − X_C, R) | φ = −atan2(1/X_L − 1/X_C, 1/R) |
| Damping ratio ζ | ζ = (R/2)·√(C/L) | ζ = (1/(2R))·√(L/C) |
Where X_L = 2πfL and X_C = 1/(2πfC) in every row. Note the mirrored Q and ζ formulas: series Q divides by R while parallel Q multiplies by it, which is the mathematical statement of the damping asymmetry described in the resonance section — the same resistor that kills a series circuit's ring barely touches a parallel tank's.
Use f₀ = 1/(2π√(LC)) — resistance does not appear in the formula. For the calculator's default values, L = 10 mH and C = 100 nF give LC = 1 × 10⁻⁹ s², so √(LC) = 3.162 × 10⁻⁵ s and f₀ = 1/(2π × 3.162 × 10⁻⁵) ≈ 5033 Hz, i.e. 5.03 kHz. R changes the sharpness of the peak, not its position: with R = 100 Ω the series circuit has Q ≈ 3.16 and a bandwidth of ≈ 1592 Hz centred on 5033 Hz.
The R. An ideal LC tank loses no energy, so its Q is infinite and any voltage you inject rings forever at f₀. A real RLC circuit dissipates energy in R every cycle, so oscillations decay. With the default values (R = 100 Ω, L = 10 mH, C = 100 nF) the series circuit has Q ≈ 3.16 and damping ratio ζ = (R/2)·√(C/L) ≈ 0.158 — underdamped, so it rings near 5033 Hz but each swing is smaller than the last. The same L and C with zero R would oscillate at the same 5033 Hz without ever decaying.
Q is the ratio of energy stored to energy dissipated per radian, and it sets the magnification at resonance. In a series RLC circuit the voltage across the inductor — and across the capacitor — at f₀ is Q times the source voltage: with Q ≈ 3.16, a 1 V source puts about 3.16 V across L and the same 3.16 V across C, and they cancel because they are 180° out of phase. Q also fixes selectivity: Q = f₀/BW, so this circuit passes roughly 1592 Hz around its 5033 Hz centre.
They are two views of the same number: BW = f₀/Q. Bandwidth is the gap between the two −3 dB (half-power) frequencies, which for a high-Q circuit sit at f₀ ± BW/2. For the defaults, f₀ ≈ 5033 Hz and Q ≈ 3.16, so BW ≈ 1592 Hz and the half-power edges fall near 4237 Hz and 5829 Hz. Halving R doubles Q and halves the bandwidth — resistance is the knob that trades peak height for selectivity, which is why low-Q wideband filters need more R and radio IF filters need very little.
LCR is just another name for RLC, and the governing equation is the impedance relation Z = R + j(X_L − X_C), written in magnitude form as Z = √(R² + (X_L − X_C)²) with X_L = 2πfL and X_C = 1/(2πfC). Equivalently, the free response satisfies the second-order differential equation L·d²q/dt² + R·dq/dt + q/C = 0, whose damping ratio ζ = (R/2)·√(C/L) decides whether the solution rings (ζ < 1), decays as fast as possible without overshoot (ζ = 1) or crawls back slowly (ζ > 1).
Opposite things. In the series circuit the reactances cancel, impedance collapses to exactly R (100 Ω here) and line current peaks — a minimum-impedance, maximum-current point, with Q-times voltage magnification across L and C. In the parallel circuit the branch susceptances cancel, line current dips to a minimum and impedance peaks: for this ideal parallel case the peak equals R itself, while the L and C branch currents circulate between them, amplified by Q. Wire the same 100 Ω as the coil's own series resistance instead and the tank impedance peaks at L/(C·R) ≈ 1000 Ω.
All formulas on this page — X_L = 2πfL, X_C = 1/(2πfC), the series and parallel impedance expressions, f₀ = 1/(2π√(LC)), the Q, bandwidth, phase-angle and damping-ratio relations — are classical AC circuit theory as presented in any standard circuits textbook. No external numeric datasets, vendor tables or measured data are used anywhere on the page; every worked number is derived arithmetically from the user's own inputs.
Scope note: this page takes R as a core input and computes the full damped-circuit picture — impedance Z, phase angle φ and damping ratio ζ, plus the resonant figures. The separate LC resonant frequency calculator covers the ideal lossless LC tank, using L and C only. Use that page when you only need the oscillation frequency of a perfect tank, and this page when you need to know how sharp, how selective and how well-behaved the real, lossy circuit will be.
Maintained by the CapacitorCalculator.com editorial team · Last updated: November 2, 2026.
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