Mutual Inductance

Mutual Inductance

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 transformers, motors, generators and any other electrical component that interacts with anothers magnetic field. 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 is 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 nearly all of the magnetic flux from the first coil will interact with the turns of the second coil inducing a large emf and therefore producing a large mutual inductance value. Likewise, if the two coils are farther apart from each other the amount of induced magnetic flux from the first coil 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 shown below.

Mutual Inductance

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 m²
•         l is the coils length in meters

We remember from our tutorials on Electromagnets that the self inductance of each individual coil is given as:

and

Then by cross-multiplying the two equations above, the mutual inductance 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 two coils as:

Mutual Inductance Between Coils

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 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, were 0 indicates zero or no inductive coupling and 1 indicates full or maximum inductive coupling. Then the equation above which assumes unity coupling can be modified to take into account this coefficient of coupling, k and is given as:

Coupling Factor Between Coils

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 other, the mutual inductance is equal to the geometric mean of the two individual inductances of the coils. So when the two inductances are equal and L ₁ is equal to L ₂, the mutual inductance that exists between the two coils can be defined as:

Example No 1

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 them.