Inductor reactance:
formula and worked example

Why an inductor resists changing current

An inductor is a coil, and a coil resists any change in the current through it by generating a back voltage. A steady DC current meets only the wire's own resistance, but a current that keeps reversing meets a lasting opposition, because it is never allowed to settle.

That opposition is the inductive reactance, written XL, and it is measured in ohms. It is what makes chokes useful for blocking noise, what limits current in a fluorescent ballast, and what makes a transformer's primary draw only a small magnetising current when nothing is connected.

• Frequency: the AC frequency in hertz.
• Inductance h: the coil's inductance in henries; millihenries must be divided by 1,000.
• Result: reactance in ohms.

The formula and how it differs from capacitors

Here both frequency and inductance multiply, so reactance climbs linearly with each. Double the frequency and XL doubles. This is the mirror image of the capacitor, whose reactance falls with frequency, and the two cancel each other at a particular frequency, which is the idea behind resonant circuits.

At DC the frequency is zero and the ideal reactance is zero, so a coil is a plain piece of wire. This is why a choke passes the DC supply and blocks the high-frequency ripple riding on it.

Worked example: 0.1 H on 50 Hz

A 0.1 H choke is connected across a 50 Hz supply. What opposition does it present to the alternating current?

1. 1Compute the angular frequency: 2 × π × 50 = 314.16 rad/s.
2. 2Multiply by the inductance: 314.16 × 0.1 = 31.416 Ω.
3. 3Estimate the ideal current at 230 V: 230 ÷ 31.416 ≈ 7.3 A.

That 7.3 A is an ideal figure. The real coil also has wire resistance, which adds to the opposition, and heats the winding, so the actual current and power are different from the ideal reactive picture.

Reactance plus winding resistance

A real inductor is an ideal inductance in series with the resistance of its wire. The two do not add as plain numbers, because the reactive part is a quarter-cycle out of step with the resistive part. The combined opposition, called impedance, is the square root of R² plus XL².

• With 31.4 Ω of reactance and 10 Ω of winding resistance, impedance is about 33 Ω, only slightly above the reactance.
• With 31.4 Ω of reactance and 1 Ω of resistance, the impedance is 31.4 Ω, effectively the reactance alone.
• At DC the reactance disappears and only the winding resistance limits current, a classic cause of burned-out coils connected to the wrong supply.

Frequency matters too. The same 0.1 H coil at 1 kHz presents about 628 Ω, so a choke that is almost invisible at mains frequency is a real obstacle to higher-frequency noise.

Moving frequency or inductance

Reactance is directly proportional to frequency: a 10% rise from 50 to 55 Hz takes XL from 31.416 Ω to 34.558 Ω, exactly 10% more. Inductance behaves the same way, so a 10% larger coil gives a 10% higher reactance.

The linearity is convenient for estimation. If you know XL at one frequency, you know it at any other by simple proportion: halve the frequency, halve the reactance. It also means that mains frequency variation of a percent or two has only a minor effect on coils.

Mistakes with inductive reactance

• Entering millihenries as henries, which makes the answer a thousand times too large.
• Ignoring winding resistance and treating a coil on DC as having a high opposition, when in fact it is nearly a short.
• Adding reactance and resistance as plain numbers instead of combining them as a square root of the sum of squares.
• Forgetting that the core can saturate: the inductance of an iron-cored coil falls at high current, so the formula's L is not constant.

A scenario: a series choke to limit current

A 230 V, 50 Hz load needs its current limited to about 1 A through an inductor with negligible resistance. The reactance required is 230 ÷ 1 = 230 Ω, so L = 230 ÷ (2π × 50) ≈ 0.73 H. Such a large coil is heavy, which is why capacitors or electronic limiters are often chosen instead.

The idea does carry over in smaller form: chokes in lamp ballasts and in filter lines work on exactly this principle, limiting current without burning power as heat in a resistor.