Inductive Reactance

Equivalent forms

Z_L = j\omega L

V_L = I X_L

A single linear law: double the frequency, double the opposition — the cleanest possible frequency dependence.

Symbol	Name	SI unit	Dimension	Range
$X_L$	Inductive reactanceoutput Opposition of the inductor to AC	Ω	M·L²·T⁻³·I⁻²	0 – 100000
$f$	Frequency AC drive frequency	Hz	T⁻¹	1 – 1000000
$L$	Inductance Coil inductance	H	M·L²·T⁻²·I⁻²	0.0001 – 1

Unit systems

Symbol	SI
$X_L$	Ω
$f$	Hz
$L$	H

Where it holds

Valid for sinusoidal steady state and an ideal linear inductor. Real coils add series resistance and self-capacitance, so X_L deviates near self-resonance and at high frequency.

Dimensional analysis

$[\omega ][L] = s^{-1} \cdot H = s^{-1}\cdot (V\cdot s$ /A $) = V$ / $A = \Omega \checkmark$

Discovery

Charles Proteus Steinmetz · 1893

Steinmetz tamed AC engineering by representing reactances as imaginary numbers, turning differential equations into simple algebra and making X_L = ωL a one-line design tool.

Try this

An inductor is just a coil of wire — so why does it choke high-frequency AC while passing DC freely?

Find the reactance of a 10 mH inductor at 50 Hz and again at 50 kHz, and explain the thousand-fold difference.

Research status: stable

Real-world applications

RF chokes blocking high-frequency noise
AC motor and ballast design
Crossover networks in speakers
Impedance matching

Common misconceptions

Reactance dissipates power like resistance — it stores and returns energy each cycle; average power in a pure reactance is zero
X_L depends on the applied voltage — it depends only on f and L

Experimental verification

Drive a known inductor with a sine source, measure V and I amplitudes, and

X_L = V/I

plots as a straight line through the origin versus frequency — slope

2\pi L

.

Derivation

For

I = I_{0} \sin (\omega t)

,

V = L dI/dt = \omega L I_{0} \cos (\omega t)

.

The voltage leads current by

90^{\circ} and

its amplitude

is \omega L\cdot I_{0}

, so the magnitude of the opposition (reactance) is

X_L = V_{0}/I_{0} = \omega L = 2\pi fL

.

Limiting cases

f \to 0 (DC)

⟶

X_L \to 0

To steady current a coil is just resistance-less wire — it offers no reactive opposition.

f \to \infty

⟶

X_L \to \infty

At very high frequency the inductor behaves like an open circuit — the basis of RF chokes.

What if…

What if you feed it pure DC?

$f = 0$ makes $X_L = 0$ — the inductor is just a wire and current is set only by resistance.

What if you double both f and L?

Reactance quadruples — X_L is proportional to the product fL.

1

10 mH at 50 Hz vs 50 kHz

Given ·

L:: 0.01
f1:: 50
f2:: 50000

Find · X_L at both frequencies

Steps

$X_L = 2\pi fL$
50 Hz: $2\pi (50)(0.01) \approx 3.14 \Omega$
50 kHz: $2\pi (50000)(0.01) \approx 3.14 k\Omega (1000\times$ , exactly the frequency ratio)

Result ·

3.14 \Omega vs 3.14 k\Omega

— reactance scales linearly with frequency.

reactance capacitiveimpedance complexRLC Series Resonance→Self-Inductance→