Inductive reactance: what an inductor does to AC
An inductor passes DC freely (apart from the resistance of its wire) and resists AC, more so the higher the frequency. Its reactance, XL = 2πfL, takes the place of resistance in Ohm's law — the current is V / XL — but it rises with frequency, dissipates no power, and makes the current lag the voltage by a quarter cycle. This calculator gives XL for any inductor and frequency, draws it against frequency, and with a source voltage and a series resistor works out the current, the impedance, the phase, the RL corner frequency and the time constant.
How to use the inductive reactance calculator
- Type the inductance as it is marked (10m,
100u, 4.7mH, 1 for a 1 H choke) and the frequency (50, 1k, 10M). XL appears at once and the graph shows how it changes two decades either side. - Optional: a source voltage (RMS) gives the current the inductor passes, and its phase.
- Optional: a series resistor — the coil's own winding resistance, or a load — turns it into an RL circuit: the drawing adds the resistor, the tiles show the impedance, the phase angle and the frequency where XL = R (the filter's corner), and the graph marks it.
- Other frequencies tabulates the same inductor at mains, audio and RF frequencies; Calculation writes the maths out; click the inductor in the drawing for the reverse calculation (which L gives this reactance elsewhere).
The formula
XL = 2π f L
An inductor's voltage is how fast its current changes: V = L × dI/dt. A sine-wave current of amplitude I at frequency f changes at a peak rate of 2πf × I, so the voltage needed to drive it is 2πfL × I, and the ratio of voltage to current — the thing that behaves like a resistance — is 2πfL. The 2πf is the angular frequency ω, so you will also see XL = ωL. Units: henries × hertz is ohms.
Your values XL = 2π × 1 kHz × 10 mH = 62.83 Ω.
Why it rises with frequency
Double the frequency and the current must reverse twice as often, so for the same current it must change twice as fast, which takes twice the voltage: twice the reactance. On a graph with logarithmic axes XL is a straight line rising one decade of ohms for every decade of frequency, the mirror image of a capacitor's falling line. That is why an inductor in series lets DC and low frequencies through and blocks high-frequency noise — a choke — and why the same part can be a short at 50 Hz and an open circuit at 10 MHz.
| Frequency | XL | Behaves like |
|---|---|---|
| 50 Hz | 3.14 Ω | a few ohms: mains passes almost freely |
| 1 kHz | 62.8 Ω | tens of ohms |
| 100 kHz | 6.28 kΩ | kilohms: audio gone, RF blocked |
| 10 MHz | 628 kΩ | hundreds of kilohms: an open circuit |
Current and phase
I = V / XL, lagging V by 90°
Ohm's law still works with XL in place of R, as long as both are RMS or both peak. But the voltage is largest when the current is changing fastest — as it crosses zero — so the current wave runs a quarter cycle (90°) behind the voltage, the opposite of a capacitor. The mnemonic is "ELI the ICE man": in an inductor (L) voltage E leads current I; in a capacitor (C) current I leads voltage E. Over a whole cycle the inductor takes energy into its magnetic field for half the time and gives it back for the other half, so no heat is produced; the product V × I is reactive power, in var.
With a resistor in series: impedance, the RL corner and L/R
|Z| = √(R² + XL²) φ = atan(XL / R) fc = R / (2πL)
Resistance and reactance add like the two sides of a right triangle, because the resistor's voltage is in phase with the current and the inductor's is 90° ahead of it. The result, the impedance Z, sets the current (I = V / |Z|), and the phase angle sits between 0° (resistor dominates, low frequency) and 90° (inductor dominates, high frequency). Where XL = R, at f = R / (2πL), the RL filter has its −3 dB corner: output across R for a low-pass, across L for a high-pass. The same pair has a time constant τ = L / R, how long the current takes to reach 63% of its final value when a DC voltage is switched on; the two views describe one circuit, since fc = 1 / (2πτ). Every real coil has winding resistance (DCR on the data sheet), so even an inductor on its own is an RL circuit.
Your values, step by step
- Reactance: 2π × 1 kHz × 10 mH = 62.83 Ω; at ten times the frequency it would be 628.3 Ω.
Worked example: 10 mH at 1 kHz
XL = 2π × 1000 × 0.010 = 62.83 Ω. At 10 kHz the same inductor is 628.3 Ω, at 100 Hz 6.283 Ω. Put 1 V RMS across it at 1 kHz and 15.92 mA flows, a quarter cycle behind the voltage, with no heating. Add an 8 Ω speaker in series and the corner lands at 127.3 Hz: above that the coil takes most of the voltage and the speaker goes quiet, which is exactly how a passive crossover keeps treble out of a woofer (a 1 mH coil puts the corner at 1.273 kHz).
Questions
Is reactance positive for an inductor?
In the complex notation engineers use an inductor's impedance is +jXL and a capacitor's −jXC, the sign carrying the 90° lag or lead. Put the two in series and they cancel; where they cancel exactly, at f = 1 / (2π√(LC)), the circuit is resonant.
Why does an inductor spark when switched off?
Because V = L dI/dt: interrupting the current suddenly makes dI/dt enormous, and the inductor generates whatever voltage it takes to keep the current flowing for an instant. That is the flyback spike that needs a diode across relay coils and motors.
What about DC?
f = 0 makes XL zero: an inductor passes steady current limited only by its winding resistance. That is why a choke in series with a supply lets the DC through and stops the ripple, and why a transformer or motor must never be connected to DC.
Does the core matter?
Only through L. Iron or ferrite cores multiply the inductance of a winding many times, which is how a small part reaches millihenries; but a core saturates above some current, and then L collapses and the reactance with it. Air-core coils have no such limit but far less inductance for their size.