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Second year Lesson 4 of 13 About 18 min

Reactance and impedance

What you’ll learn

  • What reactance is: the opposition coils and capacitors give to AC.
  • Inductive reactance, XL = 2πfL, and capacitive reactance, XC = 1 ÷ (2πfC).
  • Why XL rises with frequency and XC falls.
  • Impedance: combining resistance and reactance at right angles, Z = √(R² + X²).
  • Resonance, where the two reactances cancel.

Reactance is the opposition a coil or a capacitor gives to alternating current. It is measured in ohms, like resistance, but it depends on the frequency — and it pushes the current out of step with the voltage. Put resistance and reactance together and you get impedance, the number that actually sets the current in an AC circuit.

Inductive reactance

A coil pushes back against changing current. The faster the current changes — the higher the frequency — and the bigger the inductance, the harder it pushes back:

  • XL = 2πfLohms; f in Hz, L in henries
  • XC = 1 ÷ (2πfC)ohms; C in farads
  • 2π × 60 ≈ 377a handy number on 60 Hz

Capacitive reactance

A capacitor is the other way round. At a high frequency it barely has time to charge before the current reverses, so it hardly opposes the flow at all; at a low frequency it fills up and blocks. So XC falls as frequency rises — and at DC (zero frequency) it is infinite: a charged capacitor blocks DC completely.

resonance ≈ 50 Hz XL XC 100 50 0 Ω 30 60 90 120 Hz 0.1 H coil and 100 µF capacitor
The coil’s reactance (copper) climbs in a straight line with frequency; the capacitor’s falls away. At 60 Hz they are 37.7 Ω and 26.5 Ω. Where the lines cross, about 50.3 Hz, is the pair’s resonant frequency.

1. A coil and a capacitor on 60 Hz

  1. 0.1 H coil: XL = 2 × 3.1416 × 60 × 0.1 = 37.7 Ω.
  2. 100 µF = 0.0001 F. XC = 1 ÷ (2 × 3.1416 × 60 × 0.0001) = 1 ÷ 0.0377 = 26.5 Ω.
  3. On 50 Hz the coil would be 31.4 Ω and the capacitor 31.8 Ω — frequency matters.

Impedance: R and X at right angles

In a resistor, current and voltage are in step. In a reactance they are 90° apart. So when a circuit has both, you can’t simply add the ohms — they act in different directions, like walking 30 m east and 40 m north. You end up 50 m from where you started, not 70. That is the impedance triangle:

R = 30 Ω X = 40 Ω Z = 50 Ω θ = 53.1°
Resistance along the bottom, reactance straight up, impedance the long side. Z = √(R² + X²); the angle θ is how far the current is shifted from the voltage.
  • Z = √(R² + X²)series circuit
  • X = XL − XCthey cancel each other
  • I = V ÷ ZOhm’s law on AC

2. A coil with resistance on 120 V

  1. R = 30 Ω in series with XL = 40 Ω.
  2. Z = √(30² + 40²) = √(900 + 1,600) = √2,500 = 50 Ω.
  3. I = 120 ÷ 50 = 2.4 A.
  4. VR = 2.4 × 30 = 72 V; VL = 2.4 × 40 = 96 V. Added straight that is 168 V — but added as a right triangle, √(72² + 96²) = 120 V. ✓
  5. Angle: θ = tan⁻¹(40 ÷ 30) = 53.1° — the current lags the supply by that much.

3. R, L and C in series

  1. R = 30 Ω, XL = 80 Ω, XC = 40 Ω, on 120 V.
  2. Net reactance X = 80 − 40 = 40 Ω (inductive, because XL is bigger).
  3. Z = √(30² + 40²) = 50 Ω; I = 120 ÷ 50 = 2.4 A.
  4. VL = 2.4 × 80 = 192 V — more than the supply! VC = 96 V. They are 180° apart, so they partly cancel. This is real, and it is why you respect every part of a resonant circuit.

Resonance

For any coil and capacitor there is one frequency where XL = XC. They cancel, the only thing left limiting the current is R, and a series circuit draws its biggest current. That frequency is f = 1 ÷ (2π√(LC)). For 0.1 H and 100 µF it is about 50.3 Hz. Radios tune by resonance; in power systems an accidental resonance between correction capacitors and the supply can amplify harmonics.

Try one: a 0.5 H coil on 120 V, 60 Hz (ignore its resistance)

XL = 2 × 3.1416 × 60 × 0.5 = 188.5 Ω. I = 120 ÷ 188.5 ≈ 0.64 A.

On the job

Tips from the job

  • Ohm’s law still works on AC — use impedance in place of resistance: I = V ÷ Z. That is how a contactor coil’s current is set, not by its DC resistance.
  • A coil’s ohms on your meter is its DC resistance only. The same coil on AC shows far more opposition because of XL.
  • Harmonics are higher frequencies. Capacitors (XC falls with frequency) soak up harmonic current — one reason correction capacitors fail early in buildings full of drives.
  • Voltages across parts of an AC circuit can add up to more than the supply. Don’t assume a meter is broken; think right triangles.

Common mistakes

  • Adding R and X straight: 30 Ω + 40 Ω is not 70 Ω of impedance. It is 50.
  • Forgetting to convert µF to farads (× 0.000001) or mH to henries (× 0.001) before using the formula.
  • Using the DC resistance measured with a meter as the coil’s AC opposition.
  • Getting the direction wrong: XL goes up with frequency, XC goes down.

Check yourself

8 questions. Tap an answer — it is marked straight away and the reason opens under it. Get 80% or better and the lesson is ticked off on the year page.

Not started yet.

  1. What is the inductive reactance of a 0.1 H coil at 60 Hz?

    Answer and why

    B. XL = 2πfL = 2 × 3.1416 × 60 × 0.1 ≈ 37.7 Ω.

  2. What is the capacitive reactance of a 100 µF capacitor at 60 Hz?

    Answer and why

    A. XC = 1 ÷ (2πfC) = 1 ÷ (2 × 3.1416 × 60 × 0.0001) = 1 ÷ 0.0377 ≈ 26.5 Ω.

  3. If the frequency doubles, a coil’s reactance…

    Answer and why

    C. XL = 2πfL rises in step with f. A capacitor goes the other way: double f and XC halves.

  4. A series circuit has 30 Ω of resistance and 40 Ω of inductive reactance. Its impedance is…

    Answer and why

    C. Resistance and reactance are at right angles, so they combine like the sides of a right triangle: Z = √(30² + 40²) = √2,500 = 50 Ω. Not 70.

  5. That 50 Ω circuit is on 120 V. What current flows?

    Answer and why

    B. Ohm’s law with impedance: I = V ÷ Z = 120 ÷ 50 = 2.4 A.

  6. In that circuit the voltage across the resistor is 72 V and across the coil 96 V. Why don’t they add to 120 V?

    Answer and why

    B. The two voltages peak at different moments. Added as a right triangle: √(72² + 96²) = √14,400 = 120 V. ✓

  7. At resonance in a series RLC circuit…

    Answer and why

    A. XL and XC cancel, leaving only R to limit the current — so current is at its maximum.

  8. Which unit is reactance measured in?

    Answer and why

    C. Reactance is an opposition to current, like resistance, so it is in ohms. That lets it sit in Ohm’s law: I = V ÷ X.

Questions people ask

What is the difference between reactance and resistance?

Both are measured in ohms and both limit current. Resistance turns electrical energy into heat. Reactance stores energy — in a coil’s magnetic field or a capacitor’s electric field — and hands it back every half-cycle, so a pure reactance uses no real power. It also shifts the current 90° from the voltage, which resistance does not.

What is impedance?

The total opposition to AC current: resistance and reactance combined. Because they act 90° apart they add like the two short sides of a right triangle, Z = √(R² + X²). Impedance is what you put into Ohm’s law on an AC circuit.

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