Inductors: What Are They? (Worked Examples Included)

Contents

💡

Key learnings:

Inductor Definition: An inductor is a component that stores energy in a magnetic field when electric current flows through it.
Series Inductors: When inductors are connected in series, their total inductance is affected by mutual inductance.
Parallel Inductors: Inductors in parallel can either aid or oppose each other, influencing the overall inductance.
Voltage Across an Inductor: The voltage across an inductor is proportional to how quickly the current through it changes.
Example Problems: Example problems help in understanding how to calculate inductance in series and parallel configurations.

What is an Inductor?

Series and parallel inductor networks combine two or more coils to obtain an equivalent inductance. Each inductor is a two-terminal passive electrical element that stores energy in the form of a magnetic field when electric current flows. Depending on its use, it may be called a coil, choke or reactor.

For uncoupled inductors, inductance adds in series and follows the reciprocal rule in parallel. The component itself is a coil of conducting material, usually insulated copper, wound around air, a non-magnetic former or a ferromagnetic material.

Inductors are made across a wide range of values, including microhenry values where 1 µH equals 10^-6 H. A ferrite or iron core can concentrate the magnetic field and raise inductance, although the core also sets saturation, loss and frequency limits.

Under Faraday’s law of electromagnetic induction, a changing current produces a changing magnetic flux and an induced electromotive force. The resulting voltage is proportional to the rate of change of current.

Inductance (L) relates induced voltage to the rate at which current changes. For the same current, a larger inductance stores more magnetic energy.

How Do Inductors Work?

An inductor responds to a change in current by developing a voltage whose polarity opposes that change. The following circuit illustrates this behaviour.

inductor working — Inductor Working in a Circuit

The circuit contains a lamp, an inductor and a switch connected to a battery. Without the inductor branch, the lamp follows the switch directly. The inductor adds a current transient.

The inductor winding has much less resistance than the lamp. After DC conditions settle, most current takes the low-resistance coil branch and the lamp becomes dim.

When the switch closes, the lamp first glows brightly and then dims as current builds in the coil. When the switch opens, stored magnetic energy briefly drives current through the lamp before it goes out.

Current in the coil establishes a magnetic field. As that field changes, it induces a voltage with the polarity required by Lenz’s law to oppose the change in current.

The induced voltage opposes both an increase and a decrease in current. Once the DC current reaches a steady value, the ideal inductive voltage falls to zero.

When the switch opens, the collapsing magnetic field maintains current for a short time. That discharge current keeps the lamp glowing briefly.

The inductor stores magnetic energy while current flows. In an ideal circuit, its current cannot change instantaneously because an instantaneous change would require unbounded voltage.

Inductor Circuit Symbol

The following images show the schematic symbol for an inductor.

Inductor Equation

Voltage Across an Inductor

For a constant ideal inductance, voltage is proportional to the rate of change of current:

$\begin{align*} v_L = L \frac{di_L}{dt} \end{align*}$

where $v_L$ is the instantaneous voltage across the inductor in volts,

$L$ is inductance in henries,

$\frac{di_L}{dt}$ is the rate of change of current in amperes per second.

The sign of the voltage depends on the reference direction and whether current is rising or falling.

For steady d.c. current, $\frac{di_L}{dt}$ is zero and the voltage across an ideal inductor is zero. A real winding still has resistance, so only the ideal inductor becomes a short circuit in DC steady state.

Current Through an Inductor

Integrating the voltage relation expresses inductor current as:

$\begin{align*} i_L = \frac{1}{L} \int v_L dt \end{align*}$

The integration limits account for the current history or initial condition from $-\infty \,\, to \,\, t(0^-)$ .

If switching occurs at t=0, the current equation becomes:

$\begin{align*} i_L = \frac{1}{L} \int v_L dt \end{align*}$

Split the integral into $-\infty \,\, to \,\, 0$ and $0 \,\, to \,\,t$ . The symbol $0^-$ is the instant before switching, while $0^+$ is the instant after switching. Therefore:

$\begin{align*} i_L = \frac{1}{L} \int_{-\infty}^{t} v_L dt \end{align*}$

Therefore,

$\begin{align*} i_L = \frac{1}{L} \int_{-\infty}^{0^-} v_L dt + \frac{1}{L} \int_{0^-}^{t} v_L dt \end{align*}$

The term $\frac{1}{L} \int_{-\infty}^{0^-} v_L dt$ represents the current $i_L$ established before switching. Denote its initial value by $i_L(0^-)$ .

$\begin{align*} i_L = i_L(0^-) + \frac{1}{L} \int_{0^-}^{t} v_L dt \end{align*}$

At $t=0^+$ , write:

$\begin{align*} i_L (0^+)= i_L(0^-) + \frac{1}{L} \int_{0^-}^{0^+} v_L dt \end{align*}$

For finite voltage over an interval that shrinks to zero, the integral from $0^-$ to $0^+$ is zero.

Therefore,

$\begin{align*} i_L(0^-) = i_L(0^+) \end{align*}$

Therefore, ideal inductor current is continuous through the switching instant: its value immediately before switching equals its value immediately after.

Inductor at t=0

At $t = 0$ , an inductor does not automatically produce a voltage of $\infty$ . Such a result would require a finite current change during zero time $dt$ . Its current instead remains continuous. In a zero-current DC energisation model it may initially resemble an open circuit, while at $t = \infty$ an ideal inductor resembles a short circuit.

If the inductor carries initial current I₀, its current at $t=0^+$ is $I_0$ . A constant-current source is a useful equivalent at that instant, while the ideal inductor becomes a short circuit in DC steady state at $t=\infty$ .

Series and Parallel Inductors

Uncoupled inductors combine like resistors: inductances add in series and their reciprocals add in parallel. Magnetic coupling changes those rules. Consider coils 1 and 2 with self-inductance $L_1$ and $L_2$ , plus mutual inductance M in henries.

Equivalent inductance depends on whether the coils are in series or parallel and whether their mutual fluxes aid or oppose.

Inductors in Series Formula

How to add inductors in series

Two mutually coupled inductors have two series phasing arrangements.

In a series-aiding or cumulative connection, the mutual flux contributions act in the same direction.
In a series-opposition or differential connection, one winding is reversed so the mutual flux contributions oppose.

Let inductor 1 have self-inductance $L_1$ and inductor 2 have self-inductance $L_2$ . Their mutual inductance is M.

Series-aiding (Cumulative) Connection (mutually induced emf assists the self-induced EMFs)

The following diagram shows the series-aiding, or cumulative, connection.

With this dot orientation, the mutual terms add to the self-induced EMFs.

For each coil:

Self-induced EMF in inductor 1: $e_s_1 = -L_1\frac{di}{dt}$
Mutually induced EMF in inductor 1: $e_m_1 = -M\frac{di}{dt}$
Self-induced EMF in inductor 2: $e_s_2 = -L_2\frac{di}{dt}$
Mutually induced EMF in inductor 2: $e_m_2 = -M\frac{di}{dt}$

The total induced EMF is:

$\begin{align*} e=-(L_1\frac{di}{dt}+M\frac{di}{dt}+L_2\frac{di}{dt}+M\frac{di}{dt}) \end{align*}$

(1) $\begin{equation*} e = -(L_1+L_2+2M) \frac{di}{dt} \end{equation*}$