Capacitive Reactance

Q: What unit is reactance measured in?

Reactance is measured in ohms ($\Omega$), the same unit as resistance. It is a different quantity, though, because it comes from storing energy rather than turning it into heat, and in a circuit it combines with resistance in a different way.

In one sentence: 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.

$X_C = \frac{1}{2\pi f C}$

Read it out loud: “X sub C equals one divided by two times pi times f times C. X sub C is the capacitive reactance, measured in ohms. Pi is about 3.14159. Two pi means two times pi, which is about 6.28. f is the frequency in hertz, and C is the capacitance in farads.”

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, is current that keeps reversing its direction, unlike the steady direction of a battery’s current, 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, which is 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.
Capacitance, farads and microfarads: Capacitance, written $C$ , is how much charge a capacitor holds for each volt across it. Its unit is the farad, written F. Practical capacitors are usually given in microfarads, written $\mu$ F, where micro means one millionth. A 100 microfarad capacitor therefore holds 100 millionths of a farad, which is 0.0001 F.
Powers of ten, like $10^{-6}$: A power of ten is 10 with a small raised exponent that says how many times to multiply by ten. $10^{-6}$ means one divided by ten six times, which is 0.000001, one millionth. So $100 \times 10^{-6}$ is 100 millionths, which is 0.0001. Multiplying powers of ten adds their exponents, and dividing subtracts them. 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 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 capacitor changes with frequency, but the resistance of an ordinary resistor does not. That is why reactance gets its own letter, X, to show that it is a different quantity from resistance R.

What each symbol means

Symbol	Meaning
$X_C$	Capacitive reactance, the opposition a capacitor gives to alternating current, measured in ohms ( $\Omega$ ). It is not resistance, because the capacitor stores energy and gives it back on each cycle instead of turning it into heat. In the worked example it is about 31.8 $\Omega$ 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. Mains power is 50 or 60 Hz, and radio signals are often millions of hertz, which is written in megahertz (MHz).
$C$	The capacitance of the capacitor, measured in farads (F). The worked example uses 100 microfarads, which is $100 \times 10^{-6}$ F, or 0.0001 F.

The formula piece by piece

$f$: The frequency: how many complete cycles of alternating current happen each second. A higher frequency means the current reverses direction more often, so the capacitor charges and discharges more often each second.
$2\pi f$: The frequency multiplied by 2π. This converts cycles per second into radians per second, which is the natural rate for a sine wave. One full cycle is $2\pi$ radians, so $2\pi f$ counts how many radians pass each second.
$C$: The capacitance in farads, which says how much charge the capacitor can hold per volt. A bigger capacitance can move more charge with each reversal of the current, so it opposes the current less.
$2\pi f C$: Frequency times capacitance, times 2π. This product measures how easily the capacitor passes current, in siemens (S), where one siemens is one over an ohm. It grows with both the frequency and the capacitance, so both make the capacitor pass more current.
$\frac{1}{2\pi f C}$: One divided by that product. Taking the reciprocal turns the ease of passing current into the opposition to it, and that opposition is the reactance, in ohms.
$X_C = \frac{1}{2\pi f C}$: The whole formula in one sentence: the capacitive reactance in ohms equals one divided by two pi times the frequency times the capacitance, so the reactance gets smaller as either the frequency or the capacitance gets bigger.
$I = \frac{V}{X_C}$: A standard form that works like Ohm’s law for reactance: the current equals the voltage divided by the capacitive reactance. Both the voltage and the current must be rms values, or both must be peak values.

What the formula is doing, and why

A capacitor is two conductors separated by an insulator, so steady direct current cannot pass through it once it is charged. Alternating current seems to pass, because it charges and discharges the plates on every cycle, moving charge back and forth. The capacitor still limits that flow, and the limit is the capacitive reactance $X_C$ , measured in ohms like resistance. The inputs are the frequency $f$ and the capacitance $C$ , and the output is $X_C$ .

Why one divided by? A bigger capacitance holds more charge for each volt, so it can move more charge on each cycle, and more current flows. That means less opposition, so $C$ goes in the bottom of the fraction. Frequency works the same way: more reversals each second means more charge moved each second, so $f$ also goes in the bottom. The one divided by is how a bigger product gives a smaller answer.

Why 2π? A sine wave is a smooth wave shape, and one full cycle is a full turn of $2\pi$ radians. The factor $2\pi f$ turns cycles per second into radians per second, which is the rate at which the wave’s angle grows. The formula uses the natural units of a wave, and that is why you must multiply by $2\pi$ rather than use $f$ alone.

What frequency does. Double the frequency and the reactance halves, because the current reverses twice as often and moves twice as much charge each second. At zero frequency, which is steady direct current, the reactance is infinite, and the capacitor blocks the current once it has charged. At very high frequencies the reactance becomes very small, and the capacitor passes the signal almost freely.

When it applies. The formula is for a sine wave through an ideal capacitor of fixed value. Real capacitors lose a little energy as heat, and at very high frequencies their own small inductance starts to matter, so the formula becomes less exact there.

An everyday picture

Think of a rubber sheet stretched across a pipe. Water cannot pass straight through the sheet. But if you push the water back and forth, the sheet bulges in and out on each push, and the water seems to flow through. A slow push barely moves the sheet, so very little water moves. A bigger sheet moves more water on each push, which matches a bigger capacitance, and a faster push moves more water each second, which matches a higher frequency. Both let more flow through, so the reactance is smaller.

Worked example: A 100 microfarad capacitor at 50 Hz and at 500 Hz

Imagine a 100 $\mu$ F capacitor in a power supply circuit, first connected to 50 Hz mains and then to a higher frequency of 500 Hz. 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 $X_C = \frac{1}{2\pi f C}$ . For the first case we have $C = 100\ \mu\text{F}$ and $f = 50$ Hz. The formula gives ohms only when $C$ is in farads and $f$ is in hertz, so the first job is to convert the capacitance.
Convert the capacitance to farads. Micro means one millionth, $10^{-6}$ . So $C = 100 \times 10^{-6}$ F. Writing 100 as $10^{2}$ and adding the exponents gives $10^{2} \times 10^{-6} = 10^{-4}$ , so $C = 10^{-4}$ F, which is 0.0001 farads.
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. The factor $2\pi$ turns cycles per second into radians per second, which is the natural rate for a sine wave.
Multiply by the capacitance. Next, $2\pi f C \approx 314.16 \times 10^{-4}$ S $= 0.031416$ S. Multiplying by $10^{-4}$ moves the decimal point four places to the left, from 314.16 to 0.031416. The unit S is the siemens, which is one over an ohm, so this product measures how easily the capacitor passes current.
Take the reciprocal to get ohms. Now $X_C = \frac{1}{0.031416\ \text{S}} \approx 31.83\ \Omega$ . Dividing one by the ease of passing current gives the opposition in ohms. So at 50 Hz the capacitor has a reactance of about 31.8 ohms.
Repeat the steps at 500 Hz. The frequency is now ten times bigger, so $2\pi \times 500$ Hz is about 3141.6 radians per second. Multiplying by $10^{-4}$ gives $0.31416$ S, and the reciprocal is $X_C = \frac{1}{0.31416} \approx 3.183\ \Omega$ . The reactance fell ten times, to about 3.18 ohms, because the frequency rose ten times.
Check the direction of the change. A higher frequency should let the capacitor pass more current, so its reactance should drop. The numbers agree, because 31.8 ohms falls to 3.18 ohms. A very slow signal, such as direct current, would give an even larger reactance.

Answer: At 50 Hz the capacitor has a reactance of about 31.8 Ω. At 500 Hz it drops to about 3.18 Ω, ten times smaller, because the frequency is ten times higher.

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

Common mistakes

Leaving out the conversion to farads. $100\ \mu\text{F}$ is $100 \times 10^{-6}$ F, not 100 F, and the answer is off by a factor of a million without the conversion.
Dropping the $2\pi$ factor. Without it the answer is about six times too big, because $2\pi$ is about 6.28 and dividing by a smaller number gives a larger result.
Treating reactance as resistance. A capacitor stores energy and gives it back on each cycle, so an ideal capacitor does not heat up the way a resistor does.
Assuming the reactance is fixed. The same capacitor has a very different reactance at 50 Hz and at 500 Hz, so always state the frequency along with the answer.

Where you will meet it

Filter circuits, where a capacitor can be arranged to pass high frequencies and block low ones, or the reverse.
Audio crossover networks, which help send low notes to a woofer and high notes to a tweeter.
Power factor correction banks, which add capacitors to offset the current drawn by motors and other inductive loads.
Radio tuning circuits, where a capacitor and a coil together select one frequency.

Questions people ask

Why does a capacitor let AC through but not DC?

Once a capacitor charges to the supply voltage, steady direct current stops flowing. Alternating current keeps reversing, so the capacitor charges and discharges on every cycle, and the current appears to flow through it.

What unit is reactance measured in?

Reactance is measured in ohms ( $\Omega$ ), the same unit as resistance. It is a different quantity, though, because it comes from storing energy rather than turning it into heat, and in a circuit it combines with resistance in a different way.

Why does reactance go down when frequency goes up?

A faster signal reverses direction more often, so it moves more charge through the capacitor each second. More current for the same voltage means less opposition.

Does a capacitor use up energy in an AC circuit?

An ideal capacitor stores energy during one part of the cycle and gives it back in the next, so it uses no net energy over a full cycle. Real capacitors lose a little energy as heat.

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.
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.
Resistors in Series and Parallel: Resistors in a line add up, and resistors side by side always give a total smaller than the smallest one.