Inductance of a Coil: Formula, Factors & Examples

Inductance of a coil is the property by which it opposes changes in the current flowing through it by inducing a voltage. When the current changes, the magnetic field around the coil also changes, producing an induced voltage according to Faraday’s law of electromagnetic induction. Although coils are specifically designed to provide inductance, even a straight conductor has some inductance due to the magnetic field created around it by the current.

Table of Contents

What Is the Inductance of a Coil?

The inductance of a coil is its ability to oppose changes in the current flowing through it by producing an induced voltage. This property depends on factors such as the number of turns, coil dimensions, core material, and magnetic field characteristics. Inductance is represented by $L$ and measured in henries (H).

When the current through a coil changes, its magnetic field also changes, inducing a voltage in the same coil. This phenomenon is known as self-induction. The induced voltage opposes the change in current, in accordance with Lenz’s law. It is often called back EMF because its polarity opposes the change that produces it.

The induced voltage can be expressed as:

e=-L\frac{di}{dt}

Where:

$e$ is the self-induced electromotive force (EMF), measured in volts (V).
$L$ is the self-inductance of the coil, measured in henries (H).
$\frac{di}{dt}$ is the rate of change of current, measured in amperes per second (A/s).
The negative sign indicates that the induced EMF opposes the change in current.

Mutual induction occurs when a changing current in one coil produces a changing magnetic field that induces a voltage in another nearby coil. This principle is used in transformers, induction machines, relays, and other electromagnetic devices.

Self-inductance and mutual inductance are related phenomena, but they are distinct properties. Self-inductance describes the voltage induced in a coil by changes in its own current, whereas mutual inductance describes the voltage induced in one coil by changes in the current of another coil.

Measuring Inductance in Henries

The SI unit of inductance is the henry (H), named after the American scientist Joseph Henry. Inductance can also be expressed in webers per ampere:

1\text{ H}=1\text{ Wb/A}

According to Lenz’s law, the induced EMF opposes the change in magnetic flux that produces it. This principle explains why a coil opposes changes in the current flowing through it.

A coil has an inductance of one henry (1 H) when a current changing at a rate of one ampere per second induces an EMF of one volt across the coil.

Mathematically, the relationship between induced EMF and the rate of change of current is:

e=-L\frac{di}{dt}

Rearranging the equation gives the magnitude of inductance:

L=\frac{|e|}{|di/dt|}

Where:

$L$ is the inductance, measured in henries (H).
$e$ is the induced EMF, measured in volts (V).
$\frac{di}{dt}$ is the rate of change of current, measured in amperes per second (A/s).

For example, if a current changes at a rate of $2\text{ A/s}$ and produces an induced EMF of $6\text{ V}$ , the inductance is:

L=\frac{6}{2}=\boxed{3\text{ H}}

Thus, inductance measures how much voltage a coil develops in response to a changing current.

Induced Voltage Equation for an Inductor

V_L=-L\frac{di}{dt}\,(\text{V})

Note that the negative sign indicates that the induced voltage opposes the change in current flowing through the coil over time, represented by $\frac{di}{dt}$ .

From the above equation, the inductance of a coil can be expressed as:

and di/dt is the rate of change of current in Amperes per second, A/s.

Inductance of a Coil in Henries

L=\frac{V_L}{di/dt}=\frac{1\,\text{volt}}{1\,\text{A}/\text{s}}=1\,\text{Henry}

Where $L$ represents the inductance in henries (H), $V_L$ is the voltage across the coil, and di/dt denotes the rate of change of current, measured in amperes per second (A/s).

Defining Inductance

Inductance (L) is the property of an electrical inductor or wound coil that opposes changes in the current flowing through it. It indicates how strongly a coil resists changes in current. A higher inductance value, measured in henries (H), results in a lower rate of current change for the same applied voltage.

As discussed earlier, an inductor stores energy in the form of a magnetic field. It is generally constructed by winding insulated wire into multiple loops or turns to form a coil. When current flows through the coil, it produces a magnetic field. Increasing the number of turns generally increases the magnetic flux linkage for the same current.

Therefore, the number of turns is an important factor affecting a coil’s self-inductance. For a simple, single-layer coil, the relationship between self-inductance $L$ and the number of turns $N$ can be expressed as:

Formula for the Self-Inductance of a Coil

L=\frac{N\Phi}{I}

Where:

$L$ is the self-inductance of the coil, measured in henries (H).
$N$ is the number of turns in the coil.
$\phi$ is the magnetic flux through each turn, measured in webers (Wb).
$I$ is the current flowing through the coil, measured in amperes (A).

Self-inductance can also be expressed as the magnetic flux linkage $\Phi$ divided by the current flowing through the coil, since the same current passes through each turn. This relationship applies directly to linear magnetic materials, where the magnetic flux linkage is proportional to the current.

Inductance Worked Example No. 1

A hollow air-cored inductor coil consists of 400 turns of copper wire and produces a magnetic flux of 8 mWb when carrying a DC current of 5 A. Calculate the self-inductance of the coil in milli-henries.

Solution

Given:

Number of turns, $N=400$
Magnetic flux, $\Phi=8\,\text{mWb}=8\times10^{-3}\,\text{Wb}$
Current, $I=5\,\text{A}$

The formula for self-inductance is:

L=\frac{N\Phi}{I}

Substituting the given values:

L=\frac{400\times 8\times10^{-3}}{5}

Converting henries into milli-henries:

L=0.64\times1000=640\,\text{mH}

The self-inductance of the coil is $640\,\text{mH}$

Inductance Worked Example No. 2

Calculate the value of the self-induced EMF produced in the same coil after a time period of 8 milliseconds (8 ms).

From Worked Example No. 1, the self-inductance of the coil is $L = 0.64\,\text{H}$ .

Given:

Self-inductance, $L = 0.64\,\text{H}$
Time period, $\Delta t = 8\,\text{ms} = 0.008\,\text{s}$

The formula for self-induced EMF is:

e=-L\frac{\Delta I}{\Delta t}

where $\Delta I$ is the change in current during the time interval.

Note: The change in current is not provided in the question, so the exact self-induced EMF cannot be calculated from the given information alone.

If the current changes by $5\,\text{A}\ \text{in}\ 8\,\text{ms},$ in $8\,\text{ms}$ , then:

e=-0.64\times\frac{5}{0.008}

e=-400\,\text{V}

The magnitude of the self-induced EMF would be 400 V, assuming the current changes by $5\,\text{A}\ \text{in}\ 8\,\text{ms}$ . The negative sign indicates that the induced EMF opposes the change in current.

The self-inductance of a coil, also known as its coefficient of self-inductance, depends on its construction and physical characteristics, such as its size, length, and number of turns.

A coil can achieve a high self-inductance by using a core material with high magnetic permeability and increasing the number of turns. The magnetic flux produced within the coil’s core can therefore be expressed as:

Magnetic Flux Formula

The magnetic flux through a coil can be calculated using the following formula:

\Phi = B \times A

Where:

$\Phi$ is the magnetic flux, measured in webers (Wb).
$B$ is the magnetic flux density, measured in teslas (T).
$A$ is the cross-sectional area, measured in square metres (m²).

For a long, hollow, air-cored solenoid with $N$ turns per unit length, the magnetic flux density inside its core is given by:

B = \mu_0 n I

Where $n$ is the number of turns per metre, $I$ is the current, and $\mu_0$ is the permeability of free space.

Substituting the expressions for magnetic flux and flux density into the inductance equation gives the relationship for the self-inductance of the coil.

After simplifying the equation, the self-inductance of an air-cored solenoid is:

L = \frac{\mu_0 N^2 A}{\ell}

Where:

$L$ is the self-inductance, measured in henries (H).
$\mu_0$ is the permeability of free space, approximately $4\pi\times10^{-7}\,\text{H/m}$ .
$N$ is the total number of turns in the coil.
$A$ is the cross-sectional area of the coil’s core, calculated as $A=\pi r^2$ , in square metres (m²).
$\ell$ is the length of the coil, measured in metres (m).

Note: In this formula, $N$ represents the total number of turns, not the number of turns per metre. The formula assumes a long solenoid with an air core and and negligible end effects.

Effect of a Ferromagnetic Core on Coil Inductance

The inductance of a coil can be increased by winding it around a ferromagnetic core, such as soft iron, instead of using a hollow air core or a non-ferromagnetic material.

When a coil is wound around a ferromagnetic material, such as soft iron, cobalt, or nickel, its inductance increases significantly. This happens because the core provides a path of high magnetic permeability, producing greater magnetic flux for the same current flowing through the coil.

The ferromagnetic core concentrates the magnetic field within the coil, increasing its magnetic flux linkage. This principle is also used in electromagnets and transformers.

For example, if a core material has a relative permeability of 1,000, its permeability is 1,000 times that of free space. Under ideal conditions, the coil’s inductance can increase by approximately the same factor compared with an equivalent air-cored coil.

Therefore, the inductance of a coil depends on the permeability of its core material. For a coil wound around a magnetic core, the inductance formula can be modified to include the relative permeability, $\mu_r$ , of the core material:

L=\frac{\mu_0\mu_r N^2 A}{\ell}

Where $\mu_r$ is the relative permeability of the core material.

Note: This formula is an approximation for a long solenoid with a uniform, linear magnetic core. In practical ferromagnetic materials, permeability can vary with current, frequency, and magnetic saturation, so inductance may not increase in direct proportion to relative permeability under all operating conditions.

Saturation of a Ferromagnetic Core

When a coil is wound around a ferromagnetic core, its inductance depends on the core’s magnetic permeability. As the magnetic flux density increases, the core may approach magnetic saturation, causing its effective permeability to decrease.

At low current levels, the magnetic domains within the ferromagnetic material progressively align with the applied magnetic field. As the current increases further, most of these domains become aligned. The core then becomes less effective at concentrating additional magnetic flux, and its relative permeability may decrease towards 1.

As a result, the inductance decreases as the core approaches saturation. The relationship between current and inductance becomes nonlinear because the magnetic flux no longer increases proportionally with the current.

The saturation point depends on the type of core material and the magnetic flux density produced by the current flowing through the coil. Therefore, the inductance $L$ of a ferromagnetic-core coil can vary with the current $I$ , rather than remaining constant.

Conclusion

The inductance of a coil is an important property that determines its ability to oppose changes in current and store energy in a magnetic field. It depends on several factors, including the number of turns, core area, coil length, and core material. Using a ferromagnetic core can significantly increase the inductance of a coil due to its higher magnetic permeability. However, when the core approaches magnetic saturation, its effective permeability decreases, causing the inductance to drop and the relationship between current and magnetic flux to become nonlinear. Understanding these factors is essential for designing inductors used in filters, transformers, power supplies, and electrical machines.

Read Next:

What Is the Inductance of a Coil?

Measuring Inductance in Henries

Induced Voltage Equation for an Inductor

Inductance of a Coil in Henries

Defining Inductance

Formula for the Self-Inductance of a Coil

Inductance Worked Example No. 1

Inductance Worked Example No. 2

Magnetic Flux Formula

Effect of a Ferromagnetic Core on Coil Inductance

Saturation of a Ferromagnetic Core

Conclusion

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