Inductor Charging and Discharging: Current and Voltage

An inductor stores energy in its magnetic field. When you connect it to a DC source, the current cannot jump instantly from zero to its final value. The inductor generates a voltage that opposes the change in current, so the current increases gradually. This process is often called inductor charging.

When the source is disconnected, the magnetic field releases its stored energy and drives current through the remaining circuit. The current then decreases exponentially. This is called inductor discharging.

The standard model is a resistor $R$ in series with an inductor $L$. Real circuits also include the source resistance and the inductor’s winding resistance; these can be included in the total $R$.

The inductor law

For an ideal inductor, the voltage is related to the rate of change of current by

$$v_L = L\frac{di}{dt}.$$

Here, $L$ is the inductance in henries, $i$ is current in amperes, and $v_L$ is the inductor voltage in volts.

This equation explains the key behavior:

• A constant current produces zero inductor voltage.
• A rapidly changing current produces a large inductor voltage.
• An instantaneous change in current would require an infinitely large voltage in the ideal model.

Therefore, the current through an ideal inductor is continuous. It may change quickly, but it cannot change discontinuously unless an idealized infinite voltage is applied.

Charging an inductor from a DC source

Suppose a source of voltage $V$ is connected in series with $R$ and $L$. Kirchhoff’s voltage law gives

$$V - Ri - L\frac{di}{dt}=0.$$

If the initial current is zero, the solution is

$$i(t)=\frac{V}{R}\left(1-e^{-t/\tau}\right),$$

where the RL time constant is

$$\tau=\frac{L}{R}.$$

The final, steady-state current is

$$I_{\infty}=\frac{V}{R}.$$

The current approaches this value but, mathematically, does not reach it at any finite time. After several time constants, it is very close:

Time after switching on	Current as a fraction of final current
$t=\tau$	$63.2\%$
$t=2\tau$	$86.5\%$
$t=3\tau$	$95.0\%$
$t=4\tau$	$98.2\%$
$t=5\tau$	$99.3\%$

The resistor and inductor voltages are also useful. With the source polarity chosen so that the current is positive,

$$v_R(t)=V\left(1-e^{-t/\tau}\right),$$

and the inductor voltage is

$$v_L(t)=Ve^{-t/\tau}.$$

At the instant of connection, $i=0$, so the resistor voltage is zero and the inductor takes essentially the full source voltage. At long times, $di/dt$ approaches zero, so the inductor voltage approaches zero and the inductor behaves like a short circuit in this ideal DC model.

Worked charging example

A $12\text{-V}$ source is connected to a $6\,\Omega$ resistor and a $2\text{-H}$ inductor in series. Find the current after $0.50\text{ s}$.

First calculate the time constant:

$$\tau=\frac{L}{R}=\frac{2}{6}=0.333\text{ s}.$$

The final current is

$$I_{\infty}=\frac{V}{R}=\frac{12}{6}=2\text{ A}.$$

Now use the charging equation:

$$\begin{aligned} i(0.50)&=2\left(1-e^{-0.50/0.333}\right)\\ &\approx 2(1-e^{-1.5})\\ &\approx 2(1-0.223)\\ &\approx 1.55\text{ A}. \end{aligned}$$

Thus, the current is approximately $1.55\text{ A}$. The inductor voltage at that moment is

$$v_L=12e^{-1.5}\approx 2.68\text{ V}.$$

The resistor voltage is $12-2.68=9.32\text{ V}$, which also equals $iR=(1.55)(6)\approx9.32\text{ V}$.

Discharging an inductor

Now suppose the source is removed after the inductor has reached an initial current $I_0$. The resistor remains connected so that the inductor’s stored energy has somewhere to go. Kirchhoff’s law becomes

$$Ri+L\frac{di}{dt}=0.$$

The current is

$$i(t)=I_0e^{-t/\tau},$$

with the same time constant

$$\tau=\frac{L}{R}.$$

The current falls to $36.8\%$ of its initial value after one time constant, $13.5\%$ after two, and about $0.67\%$ after five.

The inductor voltage during discharge is opposite in polarity to the voltage during charging. If the current direction is kept as the reference direction, then

$$v_L=L\frac{di}{dt}=-RI_0e^{-t/\tau}.$$

The negative sign indicates that the inductor reverses its voltage polarity to keep current flowing in the same direction. This is why opening an inductive circuit can create a large voltage spike: if the current has no safe discharge path, the voltage rises until an arc, switch, diode, or other path conducts.

Energy stored in the magnetic field

The energy stored in an inductor carrying current $I$ is

$$U_L=\frac{1}{2}LI^2.$$

For the charging example, the limiting stored energy as the current approaches $2\text{ A}$ is

$$U_L=\frac{1}{2}(2)(2^2)=4\text{ J}.$$

During discharge, this energy is converted mainly into thermal energy in the resistor. At any time, the remaining magnetic energy is

$$U_L(t)=\frac{1}{2}L I_0^2e^{-2t/\tau}.$$

Notice the factor of $2$ in the exponent: energy depends on the square of current, while current itself decays with $e^{-t/\tau}$.

Common mistakes

Treating an inductor as a battery

An inductor is not a constant-voltage source. Its voltage is $L\,di/dt$, so its voltage depends on how the current changes. It can temporarily act like a source during discharge, but its voltage is not fixed.

Using $R/L$ as the time constant

For an RL circuit, the time constant is $L/R$, not $R/L$. The units confirm this: henries per ohm gives seconds.

Assuming the current instantly reaches $V/R$

The value $V/R$ is the final DC current, not the current immediately after switching. At the instant of connection, the current is still its pre-switching value.

Disconnecting an energized inductor without a discharge path

The current must continue if possible. A relay coil, motor winding, or other inductive load can produce a damaging voltage when switched off. A flyback diode, snubber, or controlled switching circuit provides a safer path, although it changes the decay time.

Ignoring resistance

An ideal inductor connected directly to an ideal voltage source would have continuously increasing current, because there would be no resistance to limit it. Real circuits always have some resistance, and that resistance determines the final current and the time constant.