Inductive Reactance
An inductor — a coil of wire — opposes changes in current flowing through it. This opposition is rooted in a fundamental physical principle: a changing magnetic field induces a voltage that fights the change causing it (Lenz's Law). In a DC circuit, once current reaches a steady state the inductor is just a piece of wire with a little resistance. In an AC circuit, the current is always changing, so the inductor always opposes it. That opposition grows stronger as frequency increases — the exact opposite behavior from a capacitor. Inductive reactance is the second fundamental reactive quantity in AC circuit theory and is equally important for understanding real equipment.
Inductive reactance is directly proportional to frequency. At DC (f = 0) the reactance is zero — the inductor is just a wire. As frequency rises, XL rises linearly. This is the opposite behavior to capacitive reactance.
View LargerWhy Inductors Oppose AC: Lenz's Law
When current flows through a coil of wire, it creates a magnetic field around and through the coil. If the current changes — either increasing or decreasing — the magnetic field changes with it. A changing magnetic field induces a voltage in the coil itself (self-induction), and according to Lenz's Law, this induced voltage always acts in the direction that opposes the change in current that caused it.
Think of it this way: when current increases through an inductor, the inductor generates a voltage opposing that increase, acting like a temporary "brake" on the current. When current decreases, the inductor generates a voltage that tries to keep the current flowing — like a flywheel that resists stopping. The inductor has inertia with respect to current in the same way a physical mass has inertia with respect to velocity.
In a DC circuit, the inductor's opposition is only present during the brief transient moment when current is changing (when first connected, or when disconnected). Once current reaches a steady value, the magnetic field stops changing, the induced voltage disappears, and the inductor presents only its small DC resistance (the wire's resistance). This is why inductors look like low-resistance wires at DC.
In an AC circuit, the current is always changing — that is the definition of AC. So the inductor is always generating its opposing voltage, always presenting opposition to current flow. The faster the current changes (higher frequency), the stronger the rate of change, and therefore the stronger the opposing voltage. More voltage for the same current means more opposition — higher reactance.
The Inductive Reactance Formula
Inductive reactance is given the symbol XL (the subscript L for inductor/inductance) and is measured in ohms. The formula is:
XL = 2πfL
where:
XL = inductive reactance in ohms (Ω)
f = frequency in hertz (Hz)
L = inductance in henrys (H)
2π ≈ 6.2832
Note the contrast with capacitive reactance: XL = 2πfL (direct proportionality) versus XC = 1/(2πfC) (inverse proportionality). Double the frequency, double the inductive reactance. Double the frequency, halve the capacitive reactance. These opposite dependencies are precisely what creates resonance when an inductor and capacitor are combined.
You need a choke that presents at least 500 ohms of reactance at 7 MHz, to keep switching noise from a fast regulator out of a sensitive supply line. If you wind a coil with 10 µH inductance, what is its reactance at 7 MHz?
XL = 2π × 7 × 106 × 10 × 10-6
XL = 2 × 3.14159 × 7 × 106 × 10-5
XL = 6.2832 × 70 = 439.8 Ω
The 10 µH choke gives 440 ohms at 7 MHz — not quite the 500 ohms target. You would need approximately L = 500 / (2π × 7 × 106) = 11.4 µH. A 12 µH choke would exceed the requirement.
A tuned circuit for a 3.7 MHz oscillator uses a 2.4 µH inductor. What is its reactance at this frequency?
XL = 2π × 3.7 × 106 × 2.4 × 10-6
XL = 6.2832 × 3.7 × 2.4 = 6.2832 × 8.88 = 55.8 Ω
To resonate this inductor at 3.7 MHz, a capacitor with XC = 55.8 ohms at 3.7 MHz would be required. C = 1/(2π × 3.7 × 106 × 55.8) = 773 pF.
A 1 mH toroidal inductor is used in a circuit. Compare its reactance at 1 kHz (audio), 3.5 MHz, and 144 MHz.
At 1 kHz: XL = 2π × 1000 × 0.001 = 6.28 Ω (very low, nearly transparent to audio)
At 3.5 MHz: XL = 2π × 3.5 × 106 × 0.001 = 21,991 Ω ≈ 22 kΩ (high — a good choke)
At 144 MHz: XL = 2π × 144 × 106 × 0.001 = 904,779 Ω ≈ 905 kΩ (extremely high in theory; in practice the coil's own capacitance takes over well before this)
The inductor passes audio almost unimpeded while blocking high-frequency noise — an ideal property for filtering the supply to an audio amplifier or keeping switching noise off audio wiring.
How XL Varies with Frequency
Unlike capacitive reactance (which falls as a hyperbola), inductive reactance rises linearly with frequency. Plotting XL against frequency gives a straight line through the origin — doubling the frequency exactly doubles XL. This linear relationship makes calculations predictable and simple.
| Inductance | XL at 60 Hz | XL at 1 kHz | XL at 7 MHz | XL at 144 MHz |
|---|---|---|---|---|
| 1 mH | 0.377 Ω | 6.28 Ω | 43,982 Ω | 904 kΩ |
| 10 µH | 3.77 mΩ | 62.8 mΩ | 439.8 Ω | 9,047 Ω |
| 1 µH | 0.377 mΩ | 6.28 mΩ | 43.98 Ω | 904.8 Ω |
| 100 nH | 37.7 µΩ | 628 µΩ | 4.398 Ω | 90.5 Ω |
| 10 nH | 3.77 µΩ | 62.8 µΩ | 0.440 Ω | 9.05 Ω |
This table reveals an important practical point: at 144 MHz, even a 10 nH inductor (which at high frequencies could be no more than a short piece of wire) presents 9 ohms of reactance. At microwave frequencies, the inductance of a short printed circuit board trace or a component lead can become significant. This is why high-speed digital and wireless circuit layouts minimize all trace lengths and use surface-mount components with tiny leads.
Inductive Reactance and Phase
Just as capacitive reactance introduces a 90-degree phase shift between voltage and current, inductive reactance introduces a 90-degree phase shift — but in the opposite direction. In a purely inductive circuit, the voltage leads the current by 90 degrees. The inductor's opposing voltage is generated by the rate of change of current; that rate of change is greatest at the zero crossing of the current wave, which is 90 degrees ahead of the current peak.
The mnemonic "ELI the ICE man" captures this:
- ELI: In an inductor (L), voltage (E) leads current (I)
- ICE: In a capacitor (C), current (I) leads voltage (E)
These phase relationships cannot be ignored in AC circuit calculations. When you have a circuit with both resistors and reactive components, the voltages across different components cannot simply be added arithmetically — they must be added as phasors (vectors at the appropriate angles). The Impedance lesson (the Impedance lesson) covers this in full.
Inductive Reactance Calculator
Inductive Reactance — XL = 2πfL
Enter frequency and inductance to calculate XL. Select the appropriate unit for each value.
Comparing XC and XL
Placing capacitive and inductive reactance side by side reveals the complementary nature of these two components — and explains why combining them leads to resonance:
| Property | Capacitive Reactance (XC) | Inductive Reactance (XL) |
|---|---|---|
| Formula | 1 / (2πfC) | 2πfL |
| At DC (f = 0) | Infinite (open circuit) | Zero (short circuit) |
| As frequency increases | Decreases (passes high frequencies) | Increases (blocks high frequencies) |
| Phase shift | Current leads voltage by 90° | Voltage leads current by 90° |
| Power dissipated | Zero (ideal) | Zero (ideal) |
| Sign convention | Negative reactance (−jXC) | Positive reactance (+jXL) |
The sign convention is important for impedance calculations. In the complex number representation used in AC circuit analysis, inductive reactance is positive imaginary (written +jXL) and capacitive reactance is negative imaginary (written −jXC). When XL = XC, these cancel exactly — that is resonance.
Practical Applications
Chokes
A choke is an inductor specifically designed to pass DC while blocking AC above a certain frequency. Chokes keep switching noise from a DC-DC converter out of an analog or audio supply, and keep high-frequency signals from leaking onto power wiring.
A good choke for a few megahertz typically has 100–1000 µH of inductance. At 7 MHz, 100 µH gives XL = 2π × 7 × 106 × 100 × 10-6 = 4,398 Ω — more than enough to block the noise while passing the DC current the circuit needs.
Common-Mode Chokes and Ferrite Beads
A common-mode choke uses the inductive reactance of windings on a ferrite core to block noise currents that flow in the same direction on both wires of a cable (common-mode), while letting the wanted signal, which flows out on one wire and back on the other (differential-mode), pass normally. The lump on a laptop power lead or a USB cable is a ferrite doing exactly this.
Tank Circuits (Parallel LC)
A parallel combination of an inductor and capacitor forms a "tank circuit." At the resonant frequency, the inductor's positive reactance and the capacitor's negative reactance cancel, creating extremely high impedance. Tank circuits set the frequency of LC oscillators and form the selective element in many filters, effectively trapping energy at the resonant frequency.
⚖ Experiment: Comparing Inductor and Resistor Behavior with AC
This experiment uses a small transformer or inductor to demonstrate that inductors present different opposition to AC versus DC, and that the opposition grows with frequency.
- 9V battery and DC power source
- Small audio transformer (1:1 or similar) or a ferrite toroid with 20–30 turns of wire wound on it
- 1 kΩ resistor
- Audio signal source (function generator or phone with a tone generator app)
- True-RMS multimeter
- Breadboard and connecting wires
- Connect the 1 kΩ resistor in series with the inductor (use one winding of the audio transformer). Apply 9V DC and measure the voltage across the resistor. Calculate the current from V/R. The inductor's DC resistance is very low, so nearly all the 9V appears across the resistor.
- Remove the battery. Connect the audio signal source in place of the battery. Set the source to 100 Hz and a moderate level. Measure the AC voltage across the resistor.
- Increase the frequency to 1 kHz and measure again. The voltage across the resistor should decrease as the inductor's increasing reactance claims more of the voltage.
- Try 5 kHz and 10 kHz. The voltage across the resistor should continue to drop as XL grows relative to R.
- Record the results in a table: frequency, V across R, calculated XL, observed behavior.
At DC, the inductor behaves like a short wire and almost all voltage appears across R. As audio frequency rises, the inductor's increasing reactance claims a larger share of the total voltage, reducing the voltage across R. This is the direct opposite of a capacitor (where more current flows at higher frequency). The inductor passes DC easily but blocks higher frequencies with increasing effectiveness — exactly the property exploited in RF chokes.
Frequently Asked Questions
What is the difference between inductance and inductive reactance?
Inductance (L, measured in henrys) is a fixed property of the physical inductor — determined by the number of turns, the core material, and the geometry. It does not change with frequency. Inductive reactance (XL, measured in ohms) is the opposition to AC current that the inductor presents at a specific frequency. XL = 2πfL, so reactance depends on both the inductance AND the frequency. A 10 µH inductor always has 10 µH of inductance, but its reactance is 439 ohms at 7 MHz, 879 ohms at 14 MHz, and 9 ohms at 144 kHz.
Why do real inductors have resistance as well as reactance?
Because the wire used to wind the coil has finite electrical resistance. This is called the DC resistance (DCR) of the inductor, and it is always present regardless of frequency. At low frequencies, DCR may be a significant fraction of the total impedance. At high frequencies, the much larger XL dominates and DCR becomes negligible. DCR causes real power dissipation in the inductor (P = I²R). The ratio of XL to DCR at the operating frequency is the Q factor of the inductor — a measure of how "pure" the reactance is. High-Q inductors waste little energy in their resistance and are critical in high-performance filters and resonant circuits.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.