What resonance is in an LC circuit
An inductor and a capacitor store energy in different forms — the inductor in a magnetic field proportional to current, the capacitor in an electric field proportional to voltage. Connect them and energy sloshes between the two, and it does so at exactly one frequency: the frequency at which the inductive reactance and the capacitive reactance are equal in magnitude and opposite in sign. That frequency is the resonance, and cancelling reactances is the whole reason the circuit behaves so sharply there.
Set the two reactances equal and you get the answer directly. Inductive reactance rises with frequency, X_L = 2πfL; capacitive reactance falls with frequency, X_C = 1/(2πfC). Setting them equal and solving for f gives the Thomson formula, f₀ = 1/(2π√(LC)). Notice what is absent: resistance. Loss changes how sharp the peak is and how fast the ringing dies, but for any Q above roughly 10 it does not move the frequency enough to matter.
What changes at resonance depends on how the parts are wired. In a series RLC the two reactances cancel and the loop impedance collapses to the resistance alone, so the current peaks and the voltage across L and across C each rise to Q times the source voltage. In a parallel tank the same cancellation makes the impedance seen by the source peak, so the line current dips to a minimum while a large current circulates inside the loop. Series resonance is a short circuit at f₀; parallel resonance is an open circuit. Choosing the wrong one is the fastest way to a filter that does the opposite of what you drew.
If you want the individual reactance terms rather than the resonance, the inductive reactance calculator and the capacitive reactance calculator give X_L and X_C at any frequency.
Q factor and bandwidth, and why the formula differs for series and parallel
Q is the ratio of energy stored to energy lost per radian of oscillation. The practical definition you can measure is that Q equals the resonant frequency divided by the width of the response between the two points where power has fallen to half its peak — the −3 dB points. That gives you the relation this calculator uses in reverse: BW = f₀ / Q. A 1.6 MHz tank at Q = 500 passes a 3.2 kHz window; the same tank at Q = 50 passes 32 kHz.
The quantity √(L/C) is the characteristic impedance of the tank, often written Z₀. It is the magnitude of the reactance of either element at resonance, and it is the bridge between the two Q formulas. In a series circuit the loss resistance sits in the current path, so Q = Z₀/R — more resistance, lower Q. In a parallel circuit the loss resistance sits across the tank and shunts current away from it, so Q = R/Z₀ — more resistance, higher Q. The two expressions are reciprocals of each other in units of Z₀, which is why entering the same 2 Ω under the wrong topology can be wrong by a factor of 250,000.
Once you have Q, the half-power frequencies follow exactly. They are f₀(√(1 + 1/4Q²) ∓ 1/2Q), which is slightly asymmetric about f₀ — the geometric mean of the two is exactly f₀, not the arithmetic mean. Their difference is exactly f₀/Q at any Q, which is why the simpler bandwidth formula holds even when the endpoints do not sit symmetrically. For Q above about 10 the asymmetry is under 1% and you can treat the response as centred.
Below Q = 0.5 a series RLC stops being resonant in any useful sense: the circuit is overdamped, a step input decays without overshoot, and there is no peak in the response to measure. The calculator flags that case rather than quoting a bandwidth wider than the centre frequency.
Worked example: 100 µH with 100 pF and 2 Ω of coil resistance
You have a 100 µH coil measuring 2 Ω of series resistance and you tune it with a 100 pF capacitor. Work out where it resonates and how sharp it is.
- Convert to base units. L = 100 µH = 1.0000 × 10⁻⁴ H, C = 100 pF = 1.0000 × 10⁻¹⁰ F.
- Multiply. LC = 1.0000 × 10⁻⁴ × 1.0000 × 10⁻¹⁰ = 1.0000 × 10⁻¹⁴.
- Take the square root. √(1.0000 × 10⁻¹⁴) = 1.0000 × 10⁻⁷.
- Divide into 1/2π. f₀ = 1 / (6.283185 × 1.0000 × 10⁻⁷) = 1 / 6.283185 × 10⁻⁷ = 1,591,549 Hz, or 1.5915 MHz.
- Characteristic impedance. Z₀ = √(10⁻⁴ / 10⁻¹⁰) = √(10⁶) = 1,000 Ω. Check it the other way: 2πf₀L = 6.283185 × 1,591,549 × 10⁻⁴ = 1,000 Ω.
- Q for the series case. Q = Z₀/R = 1,000 / 2 = 500.
- Bandwidth. BW = f₀/Q = 1,591,549 / 500 = 3,183.1 Hz.
- Half-power points. 1/2Q = 0.001 and √(1 + 1/4Q²) = 1.00000050, so the two multipliers are 1.00000050 − 0.001 = 0.99900050 and 1.00000050 + 0.001 = 1.00100050. That gives f₁ = 1,591,549.43 × 0.99900050 = 1,589,958.7 Hz and f₂ = 1,591,549.43 × 1.00100050 = 1,593,141.8 Hz. The multipliers differ by exactly 0.002 = 1/Q, so f₂ − f₁ = 1,591,549.43 × 0.002 = 3,183.1 Hz, matching BW = f₀/Q.
Now the design question. You want 1.000 MHz from the same 100 pF capacitor. ω = 2π × 10⁶ = 6.283185 × 10⁶ rad/s, so ω² = 3.947842 × 10¹³. Then L = 1/(ω²C) = 1/(3.947842 × 10¹³ × 10⁻¹⁰) = 1/3,947.842 = 2.5330 × 10⁻⁴ H, or 253.30 µH. Keeping the 100 µH coil instead, C = 1/(ω²L) = 1/(3.947842 × 10¹³ × 10⁻⁴) = 2.5330 × 10⁻¹⁰ F, or 253.30 pF. Both scale as the inverse square of frequency: dropping from 1.5915 MHz to 1.0000 MHz is a factor of 1.5915 in frequency and 1.5915² = 2.533 in L or C.
How to read the numbers you get back
Compare the bandwidth against the signal you intend to pass. A 455 kHz IF stage at Q = 50 has a 9.1 kHz window, which comfortably passes an AM broadcast channel; the same stage at Q = 200 has a 2.3 kHz window and will muffle the audio. Sharper is not better — it is better only until the passband gets narrower than the signal.
Check Q against what the parts can actually deliver. An air-cored coil at HF typically reaches an unloaded Q in the low hundreds; a good toroid on powdered iron reaches similar figures; a coil wound on a lossy ferrite at the wrong frequency can fall below 50. If the calculator reports a Q of several thousand from a resistance you guessed, the number is telling you about your guess, not about your circuit. Measure the coil's series resistance at the operating frequency, not with a DC ohmmeter — skin effect and core loss raise the effective resistance well above the DC value.
Watch the voltage magnification in the series case. At resonance the voltage across the inductor and across the capacitor each reach Q times the applied voltage, in antiphase, so they cancel as far as the source is concerned. A 10 V drive into a Q = 100 series tank puts 1,000 V across each reactive element. That is a genuine hazard and a genuine cause of capacitor failure, and it is invisible if you only meter the source.
Finally, treat the computed frequency as the starting point of an alignment, not the end of one. Stray capacitance from wiring, the coil's own self-capacitance and the input capacitance of whatever follows all add to C, always pulling the resonance down. That is why tuned circuits are built with a trimmer. If you need the impedance at a frequency away from resonance, the RLC impedance calculator gives magnitude and phase across the whole range.
Resonant frequency for common L and C pairs
| Inductance | C = 100 pF | C = 1,000 pF | C = 0.01 µF |
|---|---|---|---|
| 1 µH | 15.915 MHz | 5.0329 MHz | 1.5915 MHz |
| 10 µH | 5.0329 MHz | 1.5915 MHz | 503.29 kHz |
| 100 µH | 1.5915 MHz | 503.29 kHz | 159.15 kHz |
| 1 mH | 503.29 kHz | 159.15 kHz | 50.329 kHz |
| 10 mH | 159.15 kHz | 50.329 kHz | 15.915 kHz |
Read diagonally: any pairing whose LC product is the same lands on the same frequency, which is why 10 µH with 1,000 pF and 100 µH with 100 pF both give 1.5915 MHz.
Assumptions and traps
- Stray capacitance is always additive. Coil self-capacitance, socket and layout capacitance and the following stage's input capacitance all appear across the tank, so the built circuit resonates below the calculated figure. Budget a few picofarads at VHF and design with a trimmer.
- DC resistance is not the loss resistance. Skin effect, proximity effect and core loss raise a coil's effective series resistance with frequency, sometimes by an order of magnitude. A Q computed from a DC ohmmeter reading is optimistic.
- Loaded Q is what your circuit actually has. Whatever the tank drives — the next stage, an antenna, a detector — appears as extra loss and lowers Q. Unloaded Q is a property of the components; loaded Q is a property of the circuit, and only loaded Q sets the bandwidth you observe.
- The series and parallel Q formulas are not interchangeable. Entering a coil's 2 Ω winding resistance under the parallel topology models a 2 Ω short across the tank, which is not the same circuit at all.
- Above roughly 30 MHz the lumped model breaks down. Capacitor lead inductance and coil self-resonance make the real components stop behaving as pure L and C, and the true resonance departs from the ideal calculation.
- Capacitor tolerance dominates the error budget. A ±20% ceramic gives roughly ±10% of frequency error on its own, because frequency depends on the square root of C. Silver mica, C0G/NP0 ceramic or a polystyrene part is the fix where the frequency has to be repeatable.
Where resonant circuits are used, and what to use instead
Tuned LC circuits select one frequency out of many. Every superheterodyne receiver contains several: the antenna input tank, the local oscillator tank that sets which station you hear, and the IF transformers that give the receiver most of its selectivity. In transmitters the same circuits form matching networks and harmonic filters, where the tank presents the right impedance to the final device and short-circuits the harmonics. In power electronics, resonant converters exploit the fact that current through a series tank crosses zero at a predictable instant, allowing switches to turn off with no current flowing.
Where you want a gentle roll-off rather than a peak, an RC network is simpler, cheaper and unconditionally stable — the RC filter cutoff calculator covers first-order low-pass and high-pass design. Where you need selectivity far beyond what a practical coil's Q allows, a crystal or a ceramic resonator replaces the LC pair and delivers Q in the tens of thousands from a mechanical rather than an electrical resonance. Where the frequency must be programmable, a phase-locked loop or a direct digital synthesiser sidesteps tuned circuits almost entirely.
Resonance also appears where nobody wants it. A power factor correction capacitor bank and the supply transformer's leakage inductance form a parallel tank, and if that tank lands on a harmonic the plant is producing, the circulating current can destroy the capacitors — which is why detuning reactors exist. The arithmetic on this page is the arithmetic that predicts it; the power factor correction calculator handles the kVAR side of that problem.