Inductance Formula of an Inductor – Explanation and Example

An inductance formula is quite similar to the resistance formula. The way we calculate inductance of a single inductor and resistance of a single resistor is related to the cross-section area and material.

Not only that, we also can calculate series and parallel inductors easily like what we do with series and parallel resistors.

We will mention both inductors and coils a lot here, but don’t get confused. Both of them still have the same equation and formula.

What is an Inductor

An inductor is one of the most popular passive elements for an electrical circuit. Why is it called a passive element instead of a passive component? Because an inductor provides inductance in the circuit, but an inductance may be generated without a single inductor in the circuit.

Keep in mind, inductances can be found in a single conductor wire, especially if it is wired in a core like a coil. Every coil will likely produce inductance in the circuit. A conductor wire will produce a magnetic field when electric current passes through it.

Inductors will produce self-induced EMF with opposite polarity as current flows through it (that is why the EMF is known as back-EMF). Inductor will have a changing magnetic field as long as there is change of current flowing through it.

When EMF is induced in an electrical circuit where the inductor is used, it is called Self Induction (L). Self-induction can be found in an inductor used in an electrical circuit, where there is no inductor used in the same magnetic field. 

When EMF is induced in an adjacent pair of inductors placed in the same magnetic field, it is called Mutual Induction (M). Mutual-induction is mainly found in a transformer, relay, electric motor, and everything that has a pair of coils wrapped together.

Inductance which we have talked about until now is self-induction. We will talk about mutual inductance later.

What is an Inductance

If a resistor provides a resistance against current in the circuit, inductors are quite similar to resistors.

An inductor is a conductor wire wrapped around a core. It may be air, ferrite, etc. Of course, a coil of conducting wire is also considered as an inductor.

An inductor is a passive element that stores energy in the form of a magnetic field and can be found almost everywhere in electronic circuits, power supply circuits, communication systems, and especially transformers.

Moving on, an inductor provides inductance in the circuit. Any conducting wire that is inductive in the circuit is also considered as an inductor.

What is an inductance?

An inductance is the opposition rate against a change of current by an inductor when a current flows through it.

inductance formula

From the illustration above, inductance is calculated from its length, cross-section area, material of the core, and number of turns.

Mathematically, we can use the equation:

    \begin{align*} L=\frac{N^2\mu A}{l} \end{align*}

Where:

L = inductance, measured in Henry (H)
N = number of turns
μ = permeability of the core
A = cross-section area
l = length of the inductor

Looking from the equation above, the core material which has specific permeability is a key role of an inductance value. There will be different values for air core and ferrite core.

The measurement unit Henry (H) for inductance is taken from Joseph Henry, an American physicist who contributed greatly to electromagnets. Another measurement unit for inductance is Weber per Ampere and it is equal to Henry, 1 H is equal to 1 Wb/A.

Inductance of an Inductor

Why is the EMF produced by self-inductance called back-EMF? We can answer this from Lenz’ Law.

According to Lenz’s law:

The direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the changing initial magnetic field.

Furthermore, we can define that:

One Henry will be generated by a single coil when EMF is produced as a result of one volt induced in the coil, where the change of current rate 1 Ampere per second is flowing through that coil.

Summarized,

An inductance (L) of one Henry is generated while the change of current 1 A/s is flowing through it. This change of current induces a voltage (VL) at one volt. 

Mathematically, the change of current in a time for a coil is

    \begin{align*}\frac{di}{dt}(\mbox{A/s})\end{align*}

Where:
di = change of current (A)
dt = time needed to achieve the di, measured in seconds

Combined with inductance (L) and voltage (V) we get,

    \begin{align*}v=L\frac{di}{dt}\end{align*}

Where:
V = induced voltage in the coil (V)
L = inductances (H)

Doing a little positioning and we get,

    \begin{align*}L=\frac{v}{(di/dt)}=\frac{1\mbox{volt}}{1\mbox{A/s}}=1\mbox{ Henry}\end{align*}

Where:
L = inductance (Henry)
v = voltage across the inductor (V)
di/dt = change of current per second (A/s)

Just like a resistor that “resist” currents in the circuit, an inductor “resists” the change of current in the circuit. The bigger the Henries, the lower the change of current rate, and vice versa.

Self Induction of Inductor Formula

We can say that an inductor is a looped conductor wire wrapped around a core. This device can store energy in the form of a magnetic field. We can increase the inductance by increasing its loops or turns of the wire for the coil or inductor.

If the inductance increases, the magnetic flux also increases with the same amount of current.

Observe the self induction equation below:

    \begin{align*}L=N\frac{\Phi}{I}\end{align*}

Where:
L = inductance (H)
N = number of turns
Φ = magnetic flux
I = current (A)

The equation above is also known as magnetic flux linkage divided by the current flowing in each loop of the coil (NΦ/I).

Let’s do simple example of self-inductance an inductor below:

Assume that we have an air-core inductor with:

  • 100 turns of copper wire.
  • 5 mWb of magnetic flux.
  • 2 Ampere DC current flowing through it.

Then using the self-inductance

    \begin{align*}L=N\frac{\Phi}{I}\end{align*}

Substituting the known variable into the equation results in:

    \begin{align*}L&=N\frac{\Phi}{I}\\&=100\frac{0.005}{2}\\&=0.25H\\&=250mH\end{align*}

Inductance Formula of an Inductor

There will be another inductance formula besides the self-inductance. We will find it step-by-step to make sure you understand where it comes from (even it is not that important to most of us who only need to know how to use it properly).

The magnetic flux that we used earlier is determined by the construction and characteristic of the coil or inductor. The construction is built from the length of the inductor, size, number of turns, materials, cores etc.

Among all the factors, the permeability of the core and number of turns will be the key factors here. Using a different core will make the coil’s dimension changed, especially the number of turns.

High permeability core and high number of turns produce high self-induction coefficient of an inductor.

The magnetic flux produced by its core is equal to the flux density and cross-sectional area.

    \begin{align*}\Phi=B.A\end{align*}

Where:
Φ = magnetic flux
B = flux density
A = cross-sectional area

Going deeper, the flux density depends on the permeability of the core, number of turns, flowing current, and its length.

    \begin{align*}B=\mu\frac{N.I}{l}\end{align*}

Substituting the flux density into inductance formula we knew before produces:

    \begin{align*}L=N\frac{\Phi}{I}=N\frac{B.A}{I}=N\frac{\mu .N.I}{l.I}.A\end{align*}

Simplifying the equation above into an inductance formula consists of core material, number of turns, cross-sectional area, and length.

    \begin{align*}L=\mu\frac{N^2A}{l}\end{align*}

Where:
L = inductances (H)
μ = permeability of the core
N = number of turns
A = cross-sectional area
l = length of the inductor

Inductance Formula Summary

Before closing our study here, let us mention some important things:

1. Just as the inductance formula above where it depends on the rate of change of current.

    \begin{align*}v=L\frac{di}{dt}\end{align*}

2. The value v will be zero if the current is steady. It means since the voltage is zero, an inductor acts as a short circuit in a DC circuit.

3. An instantaneous change of current is not allowed. It means a sudden discontinuity of current can’t be calculated properly. But the opposite behavior is possible for its voltage.

4. An ideal inductor doesn’t dissipate energy.

5. Inductors store energy by taking power from the circuit.

6. Inductors return energy when delivering power to the circuit.

7. Actual inductors have a resistive element since they are made from conductors such as copper wire.

Going deeper, we will continue this topic to series and parallel inductors formula.

Frequently Asked Questions

How do you find inductance?

Calculate inductance can be done using the formula

L = μ N^2 A / l

Where L is inductance, μ is permeability, N is number of turns, A is cross-sectional area, and I is the current.

What is N in the inductance formula?

N in the inductance formula indicates the number of turns the inductor or coil has while l is its length of coil or inductor.

What is the unit of inductance?

Inductance is measured in Henry as an honor to Joseph Henry. One Henry is the value of self-inductance, a coil or inductor where one volt is produced by inducing one ampere per second.

Why is L used for inductance?

L is used as an honor to Heinrich Lenz who introduced electromagnetism.

What is the symbol for inductance?

Inductance is presented by the symbol of L (Lenz) while its measurement unit is H (Henry).

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