Inductive Reactance

In one sentence: Inductive reactance is how strongly a coil pushes back against alternating current, and that push gets stronger as the frequency goes up.

$X_L = 2\pi f L$

Read it out loud: “X sub L equals two times pi times f times L. Pi is about 3.14159, so two pi is about 6.28. f is the frequency in hertz, and L is the inductance in henries. The answer, X sub L, is the inductive reactance in ohms.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Alternating current and frequency in hertz (Hz): Alternating current, or AC, keeps reversing its direction, unlike the steady current of a battery, which is called direct current or DC. The frequency is how many complete cycles happen each second, measured in hertz (Hz). One hertz is one cycle per second. Mains power is 60 Hz in the US and 50 Hz in much of Europe, so at 50 Hz the current reverses direction 100 times every second.
Pi, radians and the factor $2\pi$: Pi, written $\pi$ , is about 3.14159. It is the distance around a circle divided by its width across. A radian is the angle unit that goes with circles, and one full turn is $2\pi$ radians, about 6.28 radians. Each cycle of a sine wave, the smooth wave shape of AC, covers one full turn. So $2\pi f$ tells you how many radians the wave turns through each second. Physics calls this number the angular frequency. The Greek letter omega, written $\omega$ and said ‘oh-MAY-guh’, is the usual symbol for this angular frequency, measured in radians per second.
Inductance, the henry and the millihenry: Inductance, written $L$ , measures how strongly a coil opposes changes in its current. Its unit is the henry, written H, which is one volt-second per ampere. A millihenry, written mH, is one thousandth of a henry, so 100 mH is 0.1 H. A coil with more turns, or one wound around an iron core, usually has a bigger inductance.
Powers of ten, like $10^{-3}$: A power of ten is 10 with a small raised exponent that says how many times to multiply by ten. $10^{-3}$ means one divided by 1,000, which is 0.001, or one thousandth. So $100 \times 10^{-3}$ is 100 thousandths, which is 0.1. The prefix milli, written m, means $10^{-3}$ . When powers of ten are multiplied, their exponents add. The sign $\approx$ means approximately equal. It is used when a result has been rounded, so the number after it is close to the exact value but not exact.
Reactance and the ohm: Reactance is an opposition to current that comes from storing energy in a magnetic or electric field and giving it back, instead of turning energy into heat. It is measured in ohms, the same unit as resistance, written $\Omega$ . The reactance of a coil grows with frequency, but the resistance of an ordinary resistor does not change with frequency. The letter X marks reactance, which is a different quantity from resistance R.

What each symbol means

Symbol	Meaning
$X_L$	Inductive reactance, the opposition a coil gives to alternating current, measured in ohms ( $\Omega$ ). It is not resistance, because the coil stores energy in its magnetic field and gives it back on each cycle instead of turning it into heat. In the worked example it is about 31.4 $\Omega$ at 50 Hz.
$\pi$	The number pi, about 3.14159. It has no unit, because it is a ratio: the distance around a circle divided by its width across.
$f$	The frequency of the alternating current, measured in hertz (Hz). One hertz is one cycle per second. The example uses 50 Hz, the mains frequency in much of the world, and then a higher 500 Hz.
$L$	The inductance of the coil, measured in henries (H). It measures how strongly the coil opposes changes in its current. The worked example uses 0.1 H, which is 100 millihenries.

The formula piece by piece

$f$: The frequency in hertz: how many complete cycles happen each second. The faster the current reverses, the faster it is changing, and a coil opposes faster changes more strongly.
$2\pi f$: The frequency multiplied by 2π, which converts cycles per second into radians per second. One full cycle is $2\pi$ radians, so this product counts the radians the wave turns through each second.
$L$: The inductance in henries. It measures how strongly the coil opposes changes in current, and it sets how big the opposing voltage is for a given rate of change.
$2\pi f L$: Frequency times inductance, times 2π. Both the frequency and the inductance multiply the result, so a rise in either one raises the reactance in step. The product is the opposition in ohms.
$X_L = 2\pi f L$: The whole formula in one sentence: the inductive reactance in ohms equals two pi times the frequency times the inductance, so the reactance grows as either the frequency or the inductance grows.
$X_L = \omega L$: A standard form that uses the angular frequency $\omega = 2\pi f$ , measured in radians per second. It gives the same number as the form above, and many textbooks use it.

What the formula is doing, and why

A coil is a length of wire, often wound into loops. When the current through it changes, the coil makes a voltage that opposes the change. Steady direct current does not change, so the coil barely affects it. Alternating current is always changing, so the coil pushes back on every cycle. That push-back is the inductive reactance $X_L$ , measured in ohms. The inputs are the frequency $f$ and the inductance $L$ , and the output is $X_L$ .

Why multiply? For a sine wave, the fastest rate of change of the current is $2\pi f$ times its peak value. The coil’s opposing voltage is $L$ times that rate, so the peak opposing voltage is $L \times 2\pi f \times I$ . The opposition is that voltage divided by the current, which leaves $2\pi f L$ . Both $f$ and $L$ multiply, so a rise in either one raises the reactance in step.

Why 2π? A sine wave turns through $2\pi$ radians in each cycle. The factor $2\pi f$ turns cycles per second into radians per second, which is the rate at which the wave’s angle grows. That is why the formula carries $2\pi$ , and dropping it gives an answer about six times too small.

What frequency and inductance do. Double the frequency and the reactance doubles, because the current changes twice as fast and the opposing voltage follows the rate of change. At zero frequency, which is steady direct current, the reactance is zero, so an ideal coil acts like a plain wire. Inductance sets the scale: a coil with more turns, or with a core that concentrates its magnetic field, has a bigger inductance and a bigger reactance at the same frequency.

When it applies. The formula is for a sine wave through an ideal coil with a fixed inductance. Real coils have some wire resistance, and at very high frequencies their own small capacitance starts to matter, so the formula becomes less exact there.

An everyday picture

Think of a heavy flywheel that you try to speed up and slow down by hand. The flywheel resists every change in its speed, and the faster you try to change it back and forth, the harder it resists. A coil resists changes in current in the same way, so a fast alternating current meets more opposition than a slow one. The flywheel’s weight plays the part of the inductance, and how quickly you swing it back and forth plays the part of the frequency.

Worked example: A 0.1 henry coil at 50 Hz and at 500 Hz

Imagine a 0.1 H coil in a simple filter circuit, first connected to a 50 Hz supply, which is the mains frequency in much of the world, and then to a 500 Hz supply. The 500 Hz case is a made-up value chosen to make the comparison easy. Find the reactance in each case.

Write the formula and list the inputs. The formula is $X_L = 2\pi f L$ . The coil has $L = 0.1$ H, which is already in henries, and the first frequency is $f = 50$ Hz, which is already in hertz. Because both inputs are in base units, the formula gives the answer directly in ohms.
Multiply the frequency by $2\pi$ . The number $2\pi$ is about 6.2832, so $2\pi \times 50$ Hz is about 314.16 radians per second. This factor turns cycles per second into radians per second, which is the natural rate for a sine wave.
Multiply by the inductance. Now $X_L = 314.16 \times 0.1\ \text{H} \approx 31.42\ \Omega$ . Multiplying by 0.1 is the same as dividing by 10, so the result is one tenth of 314.16. The units are checked in a later step.
Repeat the steps at 500 Hz. The frequency is ten times bigger, so $2\pi \times 500$ Hz is about 3141.6 radians per second, and $X_L = 3141.6 \times 0.1 \approx 314.16\ \Omega$ . The reactance rose ten times, to about 314 ohms, because the frequency rose ten times.
Check the units. A henry is a volt-second per ampere, and a hertz is one cycle per second, or one per second. So $\text{Hz} \times \text{H} = \frac{1}{\text{s}} \times \frac{\text{V s}}{\text{A}} = \frac{\text{V}}{\text{A}} = \Omega$ . The seconds cancel, and volts per ampere is the ohm, so the reactance comes out in ohms as an opposition should.
Check the direction of the change. A coil should oppose a fast current more than a slow one. The reactance rose from about 31.4 ohms to about 314 ohms, so the coil passes far less of the faster current, which matches the rule.

Answer: At 50 Hz the coil has a reactance of about 31.4 Ω. At 500 Hz it rises to about 314 Ω, ten times larger, so the coil passes far less alternating current at the higher frequency.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Mixing up inductance and inductive reactance. Inductance $L$ is a property of the coil and is measured in henries, while reactance $X_L$ is the opposition at one frequency and is measured in ohms.
Expecting a coil to block direct current. A coil opposes changes in current, not steady current, so steady direct current passes through it almost unaffected.
Dropping the $2\pi$ factor. Without it the answer is about six times too small, because $f L$ is only about one sixth of $2\pi f L$ .
Entering millihenries as henries. A 100 mH coil has $L = 0.1$ H, which is $100 \times 10^{-3}$ H, not 100 H. Skipping that conversion makes the answer a thousand times too big.

Where you will meet it

Radio tuning circuits, where a coil and a capacitor are chosen to pick out one station.
Ballasts in fluorescent lights, where a coil limits the current through the lamp when it runs on AC mains.
Audio crossover filters, where a coil in series with a woofer lets low frequencies through and blocks high ones.
Smoothing chokes in power supplies, which resist the ripple current that would otherwise pass through.

Questions people ask

Why does a coil resist alternating current?

The current in an AC coil keeps changing, and a changing current induces a voltage in the coil that opposes the change. That opposing voltage limits the current.

How is inductive reactance different from resistance?

A resistor turns electrical energy into heat. An ideal coil stores energy in its magnetic field and gives it back on each cycle, so on average it does not heat up. Both reactance and resistance are measured in ohms, but they limit the current in different ways.

What is a henry?

A henry (H) is the unit of inductance. A coil has an inductance of one henry when a current changing at one ampere per second induces one volt across it. In symbols, $1\ \text{H} = 1\ \text{V} \cdot \text{s} / \text{A}$ .

Why are coils used in filters?

Their reactance rises with frequency, so they pass low frequencies more easily than high frequencies. That behavior makes them useful in crossovers and other filters.

Capacitive Reactance: Capacitive reactance is how strongly a capacitor pushes back against alternating current, and that push gets weaker as the frequency of the current goes up.
Ohm's Law: Ohm’s law says the push on a wire, called voltage, equals the flow through it, called current, times how hard the wire resists that flow, called resistance.
RC Time Constant: The RC time constant tells you how long a capacitor takes to fill up through a resistor. You get it by multiplying the resistance in ohms by the capacitance in farads, and the answer comes out in seconds.