# What is Mutual Inductance and Its Theory

In 1831, Michael Faraday explained the theory of electromagnetic induction scientifically. The term inductance is, the capacity of the conductor to oppose the current flowing through it and induces emf. From Faraday’s laws of induction, an electromotive force (EMF) or voltage is induced in the conductor due to the change in the magnetic field through the circuit. This process is stated as electromagnetic induction. The induced voltage opposes the rate of change of current. This is known as Lenz’s law and the induced voltage is called back EMF. Inductance is divided into two types. They are, Self-inductance and mutual inductance. This article is all about the mutual inductance of two coils or conductors.

## What is Mutual Inductance?

**Definition:** The mutual inductance of two coils is defined as the emf induced due to the magnetic field in one coil opposes the change of current and voltage in another coil. That means the two coils are magnetically linked together due to the change in magnetic flux. The magnetic field or flux of one coil links with another coil. This is denoted by M.

The current flowing in one coil induces the voltage in another coil due to the change in magnetic flux. The amount of magnetic flux linked with the two coils is directly proportional to the mutual inductance and current change.

### Mutual Inductance Theory

Its theory is very simple and it can be understood by using two or more coils. It was described by an American scientist Joseph Henry in the 18th century. It is referred to as one of the properties of the coil or conductor used in the circuit. The property inductance is, if the current in one coil changes with time, then the EMF will induce in another coil.

Oliver Heaviside introduced the term inductance in the year 1886. The property of mutual inductance is the working principle of many electrical components that run with the magnetic field. For example, the transformer is a basic example of mutual inductance.

The main drawback of the mutual inductance is, leakage of the inductance of one coil can interrupt the operation of another coil utilizing electromagnetic induction. To reduce the leakage, electrical screening is required

The positioning of two coils in the circuit decides the amount of mutual inductance that links with one to the other coil.

### Mutual Inductance Formula

The formula of two coils is given as

**M= ( μ0.μr. N1. N2. A) / L**

Where μ0= permeability of free space = 4π10-^{2}

μ = permeability of the soft iron core

N1= turns of coil 1

N2= turns of coil 2

A= cross-sectional area in m^{2}

L = length of the coil in meters

#### Unit of Mutual Inductance

The unit of mutual inductance is kg. m^{2}.s^{-2}.A^{-2}

The amount of inductance produces the voltage of one volt due to the rate of change of current of 1Ampere/second.

The **SI unit of mutual inductance** is Henry. It is taken from the American scientist Joseph Henry, who explained the phenomenon of two coils.

### The Dimension of Mutual Inductance

When two or more coils are linked together magnetically with the same magnetic flux, then the voltage induced in one coil is proportional to the rate of change of current in another coil. This phenomenon is referred to as mutual inductance.

Consider the total inductance between the two coils be L since M = √(L1L2) = L

The dimension of this can be defined as the ratio of potential difference to the rate of change of current. It is given as

Since M = √L1L2 = L

L = € / (dI / dt)

Where € = induced EMF = work done / electric charge with respect to time = M. L^{2}. T-^{2}/ IT = M.L^{2}.T-3. I^{-1 }or € = M. L^{-2} . T-3. A^{-1} (Since I = A)

For inductance,

ϕ = LI

L = ϕ / A=( B. L^{2} ) / A

Where B = magnetic field =( MLT-^{2}) /LT^{-1}AT = MT^{-2}A^{-1}

Magnetic flux ϕ= BL^{2} = MT^{-2}L^{2}A^{-1}

substitute value of B and ϕ is above formula L

L= MT-^{2}L^{2}.A^{-2}

The dimension of mutual inductance when L1 and L2 are the same is given as

M = L /(T-^{2}L^{2}.A^{-2})

M = LT^{2}L^{2}.A^{-2}

### Derivation

Follow the process to get the **mutual inductance derivation**.

The ratio of EMF induced in one coil and the rate of change of current in another coil is mutual inductance.

Consider the two coils L1 and L2 as shown in the figure below.

When the current in L1 changes with time, then the magnetic field also changes with time and changes the magnetic flux linked with the second coil L2. Due to this magnetic flux change, an EMF is induced in the first coil L1.

Also, the rate of change of current in the first coil induces EMF in the second coil. Hence EMF is induced in the two coils L1 and L2.

This is given as

€= M (dI1 / dt)

M = € / ( dI1 / dt) . … .. Eq 1

If € = 1 volt and dI1 / dt = 1Amp, then

M = 1 Henry

Also,

The rate of change of current in one coil produces the magnetic flux in the first coil and associates with the second coil. Then from the Faraday’s laws of electromagnetic induction (induced voltage is directly proportional to the rate of change of magnetic flux linked) in the second coil, induced EMF is given as

€= M / ( dI1 / dt) = d (MI1) / dt… .. Eq 2

€= N2 ( dϕ12 / dt) = d ( N2ϕ12) / dt…eq 3

By equating eq 2 and 3

MI1 = N2ϕ12

M= (N2ϕ12) / I1 Henry

Where M = mutual inductance

€= mutual inductance EMF

N2 = no of turns in first coil L1

I1 = current in the first coil

ϕ12 = magnetic flux linked in two coils.

The mutual inductance between the two coils depends on no of turns on the second coil or adjacent coil and the area of the cross-section

Distance between two coils.

The EMF induced in the first coil due to the rate of change of flux is given as,

**E = -M12 ( dI1 / dt)**

The minus sign indicates opposition to the rate of change of current in the first coil when EMF is induced.

#### Mutual Inductance of Two Coils

The mutual inductance of two coils can be increased by placing them on a soft iron core or by increasing the no of turns of the two coils. Unity coupling exists between the two coils when they are tightly wound on a soft iron core. The leakage of flux would be small.

If the distance between the two coils is short, then the magnetic flux produced in the first coil interacts with all the turns of the second coil, which results in large EMF and mutual inductance.

If the two coils are farther and apart from each other at different angles, then the induced magnetic flux in the first coil generates weak or small EMF in the second coil. Hence the mutual inductance will also be small.

Thus the value of this mainly depends on the positioning and spacing of two coils on a soft iron core. Consider the figure, which shows that the two coils are tightly wound one on the top of the soft iron core.

The change of current in the first coil produces a magnetic field and passes the magnetic lines through the second coil, which is used to calculate mutual inductance.

The mutual inductance of two coils is given as

M12 = (N2ϕ12) / I1

M21= (N1ϕ21) / I2

Where M12=mutual inductance of the first coil to second coil

M21= mutual inductance of the second coil to the fist coil

N2=turns of the second coil

N1=turns of the first coil

I1=current flowing around the first coil

I2=current flowing around the second coil.

If the flux linked with the L1 and L2 is the same as the current flowing around them, then the mutual inductance of the first coil to the second coil is given as M21

The mutual inductance of two coils can be defined as M12= M21 = M

So, two coils mainly depend on the size, turns, position, and spacing between the two coils.

The self-inductance of the first coil is

**L1 = (μ0.μr.N1 ^{2}.A) / L**

The self-inductance of the second coils is

**L2 = (μ0.μr.N ^{2}.A) / L**

Cross-multiply the above two formulae

Then the mutual inductance of two coils, which exists between them is given as

**M ^{2} = L1. L2**

**M=√(L1.L2) Henry**

The above equation gives magnetic flux= 0

100% magnetic coupling between L1 and L2

### Coupling Coefficient

The fraction of magnetic flux linked with the two coils to the total magnetic flux between the coils is known as the coupling coefficient and it is denoted by ‘k’. The coupling coefficient is defined as the ratio of the open circuit to the actual voltage ratio and the ratio of magnetic flux obtained in both the coils. Since the magnetic flux of one coil links with another coil.

The coupling coefficient specifies the inductance of an inductor. If the coefficient coupling k = 1, then the two coils are coupled together tightly. So, all the lines of magnetic flux of one coil cut all the turns of another coil. Hence the mutual inductance is the geometric mean of individual inductances of two coils.

If the inductances of two coils are the same (L1=L2), then the mutual inductance between the two coils is equal to the inductance of a single coil. That means,

**M= √(L1 . L2) = L**

where L = inductance of a single coil.

### Coupling Factor between Coils

The coupling factor between coils can be represented as 0 and 1

If the coupling factor is 1, then there is no inductive coupling between the coils.

If the coupling factor is 0, then there is a maximum or full inductive coupling between the coils.

The inductive coupling is represented in 0 and 1, but not in percentages.

For example, if k= 1 then the two coils are coupled perfectly

If k>0.5, then the two coils are coupled tightly

If k<0.5, then the two coils are coupled loosely.

To find the coefficient coupling factor between the two coils, the following equation should be applied,

**K = M / √(L1 . L2)**

**M = k. √(L1. L2)**

Where L1= inductance of the first coil

L2= inductance of the second coil

M= mutual inductance

K= coupling factor

### Applications

The **applications of mutual inductance** are,

- Transformer
- Electric Motors
- Generators
- Other electrical devices, which work with a magnetic field.
- Used in calculation of eddy currents
- Digital signal processing

Thus this is all about an overview of mutual inductance – definition, formula, unit, derivation, coupling factor, coefficient coupling, and applications. Here is a question for you, What is the drawback of mutual inductance between two coils?