KAHIBARO
Table of Contents
Inductive Reactance
An inductor resists changes in current. In an AC circuit, the current is constantly changing, so an inductor has an important effect even if its ordinary wire resistance is small.
For an ideal inductor, the voltage across it is
$$
v_L(t) = L \frac{di(t)}{dt}
$$
where $L$ is the inductance.
If the current changes rapidly, the inductor produces a larger voltage. If the current changes slowly, the voltage is smaller. This means that an inductor opposes changing current, not current itself.
For an ideal inductor,
$$
v_L = L \frac{di}{dt}
$$
An inductor opposes changes in current.
In AC analysis, the opposition of an inductor to alternating current is called inductive reactance, written as $X_L$:
$$
X_L = \omega L = 2\pi f L
$$
where $\omega$ is angular frequency and $f$ is frequency.
This looks similar to resistance, and it is measured in ohms, but it is not the same thing. Resistance dissipates energy as heat, while ideal inductive reactance stores and returns energy in a magnetic field.
Inductive reactance:
$$
X_L = \omega L = 2\pi f L
$$
It increases when frequency increases.
Frequency Dependence
Inductors behave differently at different frequencies. At low frequency, $X_L$ is small, so the inductor allows current more easily. At high frequency, $X_L$ is large, so the current is reduced more strongly.
This is the opposite trend from a capacitor, where reactance decreases as frequency increases.
The table below shows this behavior clearly.
| Frequency | Inductive Reactance $X_L$ | Effect on Current |
|---|---|---|
| Low | Small | Current is larger |
| High | Large | Current is smaller |
If $f = 0$, which means direct current in steady state, then
$$
X_L = 0
$$
for an ideal inductor. So an ideal inductor acts like a short circuit in steady DC conditions. In real circuits, the wire of the inductor has some resistance, so a real inductor is not perfectly zero ohms.
Phase Relationship
In an AC circuit with only an ideal inductor, the voltage and current are not in phase. The voltage reaches its maximum before the current does.
If the voltage is
$$
v(t) = V_0 \sin(\omega t)
$$
then the current is
$$
i(t) = \frac{V_0}{X_L}\sin\left(\omega t - \frac{\pi}{2}\right)
$$
So the current lags the voltage by $90^\circ$, or $\pi/2$ radians.
Another way to say this is that the voltage leads the current by $90^\circ$.
In a purely inductive AC circuit, voltage leads current by $90^\circ$.
Equivalently, current lags voltage by $90^\circ$.
This phase difference happens because the voltage depends on how fast the current is changing, not simply on the current value itself.
Current in a Pure Inductor
Using an AC source with rms voltage $V_{\mathrm{rms}}$, the rms current in a pure inductor is
$$
I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{X_L} = \frac{V_{\mathrm{rms}}}{\omega L}
$$
This has the same algebraic form as Ohm's law, but the opposition here is reactance, not resistance.
If either the frequency or the inductance increases, then $X_L$ increases and the current decreases.
Energy Storage in the Magnetic Field
An inductor stores energy in its magnetic field when current flows through it. The stored energy is
$$
U = \frac{1}{2}LI^2
$$
In AC operation, this energy is repeatedly stored and returned to the circuit as the current changes. In an ideal inductor, this energy is not permanently lost.
This is why an ideal inductor does not consume average power in the same way a resistor does. The energy moves back and forth between the source and the magnetic field.
Energy stored in an inductor:
$$
U = \frac{1}{2}LI^2
$$
An ideal inductor stores and returns energy, rather than dissipating it.
Comparison with a Resistor
A resistor and an inductor affect AC current in different ways.
| Component | Opposition to Current | Depends on Frequency | Phase Difference |
|---|---|---|---|
| Resistor | Resistance $R$ | No | Voltage and current in phase |
| Inductor | Reactance $X_L = \omega L$ | Yes | Current lags voltage by $90^\circ$ |
This difference is essential in AC circuits. An inductor changes both the size of the current and its timing relative to the voltage.
Simple Example
Suppose an inductor has
$$
L = 0.20\,\text{H}
$$
and is connected to an AC source of frequency
$$
f = 50\,\text{Hz}
$$
Then the inductive reactance is
$$
X_L = 2\pi f L = 2\pi(50)(0.20)
$$
$$
X_L \approx 62.8\,\Omega
$$
If the source voltage is
$$
V_{\mathrm{rms}} = 120\,\text{V}
$$
then the rms current is
$$
I_{\mathrm{rms}} = \frac{120}{62.8} \approx 1.91\,\text{A}
$$
So the inductor limits the AC current according to its inductive reactance.
Physical Picture
You can think of an inductor as having electrical inertia. A mass resists changes in motion, and an inductor resists changes in current. When the circuit tries to increase or decrease the current, the inductor produces a voltage that opposes that change.
This idea helps explain why the current does not instantly follow the applied voltage in AC circuits.
Key Results
The most important ideas for inductors in AC circuits are the relation between voltage and changing current, the frequency dependent reactance, and the $90^\circ$ phase difference.
Key formulas for an ideal inductor in AC:
$$
v_L = L\frac{di}{dt}
$$
$$
X_L = \omega L = 2\pi f L
$$
$$
I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{X_L}
$$
Voltage leads current by $90^\circ$.
These results will be used when inductors are combined with resistors and capacitors in more general AC circuits.