**Mutual Inductance of Two Coils**

In the previous tutorial we saw that an inductor generates an induced emf within itself as a result of the changing magnetic field around its own turns, and when this emf is induced in the same circuit in which the current is changing this effect is called **Self-induction**, ( L ).

However, when the emf is induced into an adjacent coil situated within the same magnetic field, the emf is said to be induced magnetically, inductively or by **Mutual induction**, symbol ( M ). Then when two or more coils are magnetically linked together by a common magnetic flux they are said to have the property of **Mutual Inductance**.

**Mutual Inductance** is the basic operating principal of the transformer, motors, generators and any other electrical component that interacts with another magnetic field. Then we can define mutual induction as the current flowing in one coil that induces an voltage in an adjacent coil.

But mutual inductance can also be a bad thing as “stray” or “leakage” inductance from a coil can interfere with the operation of another adjacent component by means of electromagnetic induction, so some form of electrical screening to a ground potential may be required.

The amount of mutual inductance that links one coil to another depends very much on the relative positioning of the two coils. If one coil is positioned next to the other coil so that their physical distance apart is small, then nearly all of the magnetic flux generated by the first coil will interact with the coil turns of the second coil inducing a relatively large emf and therefore producing a large mutual inductance value.

Likewise, if the two coils are farther apart from each other or at different angles, the amount of induced magnetic flux from the first coil into the second will be weaker producing a much smaller induced emf and therefore a much smaller mutual inductance value. So the effect of mutual inductance is very much dependant upon the relative positions or spacing, ( S ) of the two coils and this is demonstrated below.

The mutual inductance that exists between the two coils can be greatly increased by positioning them on a common soft iron core or by increasing the number of turns of either coil as would be found in a transformer.

If the two coils are tightly wound one on top of the other over a common soft iron core unity coupling is said to exist between them as any losses due to the leakage of flux will be extremely small. Then assuming a perfect flux linkage between the two coils the mutual inductance that exists between them can be given as.

- Where:
- µ
_{o}is the permeability of free space (4.π.10^{-7}) - µ
_{r}is the relative permeability of the soft iron core - N is in the number of coil turns
- A is in the cross-sectional area in m
^{2} - l is the coils length in meters

Here the current flowing in coil one, L_{1} sets up a magnetic field around itself with some of these magnetic field lines passing through coil two, L_{2} giving us mutual inductance. Coil one has a current of I_{1} and N_{1} turns while, coil two has N_{2} turns. Therefore, the mutual inductance, M_{12} of coil two that exists with respect to coil one depends on their position with respect to each other and is given as:

Likewise, the flux linking coil one, L_{1} when a current flows around coil two, L_{2} is exactly the same as the flux linking coil two when the same current flows around coil one above, then the mutual inductance of coil one with respect of coil two is defined as M_{21}. This mutual inductance is true irrespective of the size, number of turns, relative position or orientation of the two coils. Because of this, we can write the mutual inductance between the two coils as: M_{12} = M_{21} = M.

Then we can see that self inductance characterises an inductor as a single circuit element, while mutual inductance signifies some form of magnetic coupling between two inductors or coils, depending on their distance and arrangement, an hopefully we remember from our tutorials on Electromagnets that the self inductance of each individual coil is given as:

and

By cross-multiplying the two equations above, the mutual inductance, M that exists between the two coils can be expressed in terms of the self inductance of each coil.

giving us a final and more common expression for the mutual inductance between the two coils of:

However, the above equation assumes zero flux leakage and 100% magnetic coupling between the two coils, L _{1} and L _{2}. In reality there will always be some loss due to leakage and position, so the magnetic coupling between the two coils can never reach or exceed 100%, but can become very close to this value in some special inductive coils.

If some of the total magnetic flux links with the two coils, this amount of flux linkage can be defined as a fraction of the total possible flux linkage between the coils. This fractional value is called the **coefficient of coupling** and is given the letter k.

Generally, the amount of inductive coupling that exists between the two coils is expressed as a fractional number between 0 and 1 instead of a percentage (%) value, where 0 indicates zero or no inductive coupling, and 1 indicating full or maximum inductive coupling.

In other words, if k = 1 the two coils are perfectly coupled, if k > 0.5 the two coils are said to be tightly coupled and if k < 0.5 the two coils are said to be loosely coupled. Then the equation above which assumes a perfect coupling can be modified to take into account this coefficient of coupling, k and is given as:

or

When the coefficient of coupling, k is equal to 1, (unity) such that all the lines of flux of one coil cuts all of the turns of the second coil, that is the two coils are tightly coupled together, the resulting mutual inductance will be equal to the geometric mean of the two individual inductances of the coils.

Also when the inductances of the two coils are the same and equal, L _{1} is equal to L _{2}, the mutual inductance that exists between the two coils will equal the value of one single coil as the square root of two equal values is the same as one single value as shown.

Two inductors whose self-inductances are given as 75mH and 55mH respectively, are positioned next to each other on a common magnetic core so that 75% of the lines of flux from the first coil are cutting the second coil. Calculate the total mutual inductance that exists between the two coils.

When two coils having inductances of 5H and 4H respectively were wound uniformly onto a non-magnetic core, it was found that their mutual inductance was 1.5H. Calculate the coupling coefficient that exists between.

In the next tutorial about Inductors, we look at connecting together Inductors in Series and the affect this combination has on the circuits mutual inductance, total inductance and their induced voltages.

Error! Please fill all fields.

Is there a way to work out maximum potential distance in cm before the energy stops transferring from one coil to the other?

Good contents

Very Informative, But I have a question!!!!! when the distance between two coil increase will the mutual inductance decrease or not????? If yes, then why distance between coils parameter is not used in the formula to find mutual inductance?????

The distance between coils comes in coupling constant. That is if distance is more k is less and vice versa. So distance do matters for mutual induction and coupling constant changes according to it.

The mutual inductance (M) that exists between two coils depends upon the self-inductances of the two coils, that is, the values of L1 and L2, the spacing between the two coils, the farther apart they are, the less is the mutual inductance and the orientation to each other. As a result the coefficient of coupling, (k) determines the ratio of flux linkage and not distance.

k=k (distance, coil size)

Why is magnetic material required in a transfomer for if Faraday’s law states

“If a conductor is placed in a varying magnetic feild emf is induced in it.” ?

pls tell me the answer of this question

mutual inductance between two coils is 4H. if current in one coil changes at the rate A/sec then emf induced in other coil is

the mutual inductance between vow coils having self inductance 3 hendry and 12 hendry and coupling coefficient 0.85 is ______ hendry .

can u explain this questions answer in detail.

Please see Example No1 above, you should get 5.1H

What is meant by the coupling distance between the coil of and inductive transformer coil and a cylindrical work piece mounted at the center of the coil?

Hi Wane , I am confused about one thing and I would appreciate a lot if you clarify me on that. So my question is If there are two wires side by side and if there is a changing current flow on the first wire and it’s changing magnetic field induces a changing current on the second wire then would that induced current on the second wire again induce a current on the first wire? If it does then does this happen back and fourth forever? and what would the direction of that current on the first wire ? would it aid the direction of the current already flowing on the first wire or opposite direction?

If the currents that flow in two parallel conductors are in the same direction, the mutual inductance is positive and the two magnetic fields aid each other to make a stronger magnetic field. If the currents that flow in the two parallel conductors are in the opposite directions, the mutual inductance is negative as the two fields are opposite in direction and tend to cancel each other. Then each of the two parallel conductors lies within the magnetic field of the other conductor.

I am satisfied with the solution.

But, I have a one question,

If a toroidal coil has an inductance of 47 mH. Then what will be the maximum self-induced emf in the coil when the current in it is reversed from 15A to -15A in 0.01s ?

The self-induced or back emf of the coil is proportional to the rate of change of the current, di/dt and the inductance, L which is a constant. As the current changes from +15A to -15A in 10ms, the induced voltage will therefore be:

e = -L(i1-i2)/t = 0.047((+15) – (-15))/0.01 = 1.41/0.01 = 141 volts

Where: e = average induced emf in volts (v); L = self-inductance of the coil in henries (H); i1 and i2 are the initial and final currents respectively in amperes (A); and t = time in seconds (s)

THANK YOU VERY MUCH