In practical electrical circuits, there is almost always some mutual inductance between or among coils when they are wound in a cylindrical shape. The magnetic fields extend significantly outside solenoidal coils, and mutual effects are almost inevitable. The same is true between and among lengths of wire, especially at very-high, ultra-high, and microwave radio frequencies. Sometimes, mutual inductance is all right, and doesn't have a detrimental effect on the behavior of a circuit. But it can be a bad thing.

Mutual inductance can be minimized by using shielded wires and toroidal inductors. The most common shielded wire is coaxial cable.

**Coefficient of Coupling**

The coefficient of coupling, specified by the letter k, is a number ranging from 0 (no coupling) to 1 (maximum possible coupling). Two coils that are separated by a sheet of solid iron would have essentially k = 0; two coils wound on the same form, one right over the other, would have practically k = 1.

**Mutual Inductance**

The mutual inductance is specified by the letter M and is expressed in the same units as inductance: henrys, millihenrys, microhenrys, or nanohenrys. The value of M is a function of the values of the inductors, and also of the coefficient of coupling.

For two inductors, having values of L_{1} and L_{2} (both expressed in
the same size units), and with a coefficient of coupling k, the mutual inductance M is
found by multiplying the inductance values, taking the square root of the result, and
then multiplying by k. Mathematically,

M = k (L_{1}L_{2})^{1/2}

**Effects of Mutual Inductance**

Mutual inductance can operate either to increase the inductance of a pair of series connected inductors, or to decrease it. This is because the magnetic fields might reinforce each other, or they might act against each other.

When two inductors are connected in series, and there is reinforcing mutual inductance between them, the total inductance L is given in the formula:

L = L_{1} + L_{2} + 2M

where L_{1} and L_{2} are the values of the individual inductors, and
M is the mutual indutance. All inductances must be expressed in the same size units.

**Problem 10-5**

Suppose two coils, having values of 30 µH and 50 µH, are connected in series so that their fields reinforce, as shown in Fig. 10-5, and that the coefficient of coupling is 0.5. What is the total inductance of the combination?

First, calculate M from k. According to the formula for this, given above,
M = .5(50 × 30)^{1/2} = 19.4 uH. Then the total inductance is equal to
L = L_{1} + L_{2} + 2M + 30 + 50 + 38.8 = 118.8 uH, rounded to 120 uH
because only two significant digits are justified.

When two inductors are connected in series and the mutual inductance is in opposition, the total inductance L is given by the formula

L = L_{1} + L_{2} - 2M

where, again, L_{1} and L_{2} are the values of the individual inductors.

**Problem 10-6**

There are two coils with values L1 = 835 uH and L2 = 2.44 mH. They are connected in series so that their coefficient of coupling is 0.922, acting so that the coils oppose each other, as shown in Fig. 10-6. What is the net inductance of the pair?

First, calculate M. Notice that the coil inductances are specified in different units.
Convert them both to microhenrys, so that L_{2} becomes 2440 uH. Then
M = 0.922(835 x 2440)^{1/2} = 1316 uH. The total inductance is therefore
L = L_{1} + L_{2} - 2M = 835 + 2440 - 2632 = 643 uH.

It is possible for mutual inductance to increase the total series inductance of a pair of coils by as much as a factor of 2, if the coupling is total and if the flux reinforces. Conversely, it is possible for the inductances of two coils to cancel each other. If two equal-valued inductors are connected in series so that their fluxes oppose, the result will be theoretically zero inductance.

**About the Author**

Stan Gibilisco is one of McGraw-Hill's most prolific and popular authors, specializing in electronics and science topics. His clear, reader-friendly writing style makes his science books accessible to a wide audience, and his background in research makes him an ideal editor for professional references and course materials. He is the author of The Encyclopedia of Electronics; The McGraw-Hill Encyclopedia of Personal Computing; and several titles in the popular Demystified library of home-schooling and self-teaching books. His published works have won numerous awards. The Encyclopedia of Electronics was chosen a "Best Reference Book of the 1980s" by the American Library Association, which also named his McGraw-Hill Encyclopedia of Personal Computing a "Best Reference of 1996." Stan Gibilisco maintains a Web site at www.sciencewriter.net.

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