Editing Inductance (section)

=== Mutual inductance and magnetic field energy ===
Multiplying the equation for ''v<sub>m</sub>'' above with ''i<sub>m</sub>dt'' and summing over ''m'' gives the energy transferred to the system in the time interval ''dt'',
<math display=block>
  \sum \limits_m^K i_m v_m \text{d}t = \sum\limits_{m,n=1}^K i_m L_{m,n} \text{d}i_n
    \mathrel\overset{!}{=} \sum\limits_{n=1}^K \frac{\partial W \left(i\right)}{\partial i_n} \text{d}i_n.
</math>

This must agree with the change of the magnetic field energy, ''W'', caused by the currents.<ref>The kinetic energy of the drifting electrons is many orders of magnitude smaller than W, except for nanowires.</ref> The [[symmetry of second derivatives|integrability condition]]

<math display=block>\displaystyle\frac{\partial^2 W}{\partial i_m \partial i_n} = \frac{\partial^2 W}{\partial i_n \partial i_m}</math>

requires ''L<sub>m,n</sub>&nbsp;=&nbsp;L<sub>n,m</sub>''. The inductance matrix, ''L<sub>m,n</sub>'', thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,
<math display=block>\displaystyle W\left(i\right) = \frac{1}{2} \sum \limits_{m,n=1}^K i_m L_{m,n} i_n.</math>

This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. A [[impedance analogy|mechanical analogy]] in the ''K''&nbsp;=&nbsp;1 case with magnetic field energy (1/2)''Li''<sup>2</sup> is a body with mass ''M'', velocity ''u'' and kinetic energy (1/2)''Mu''<sup>2</sup>. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage).

[[File:Mutually inducting inductors.PNG|thumb|300px|right|Circuit diagram of two mutually coupled inductors.  The two vertical lines between the windings indicate that the transformer has a [[magnetic core|ferromagnetic core]] . "n:m" shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows the [[dot convention]].]]

Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which [[transformer]]s work, but it can also cause unwanted coupling between conductors in a circuit.

The mutual inductance, {{nowrap|<math>M_{ij}</math>,}} is also a measure of the coupling between two inductors. The mutual inductance by circuit <math>i</math> on circuit <math>j</math> is given by the double integral ''[[Franz Ernst Neumann|Neumann]] formula'', see [[#Calculating inductance|calculation techniques]]

The mutual inductance also has the relationship:
<math display=block>M_{21} = N_1\ N_2\ P_{21} \!</math>
where

{{plainlist|1=
* <math>M_{21}</math> is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.
* <math>N_1</math> is the number of turns in coil 1,
* <math>N_2</math> is the number of turns in coil 2,
* <math>P_{21}</math> is the [[permeance]] of the space occupied by the flux.
|indent=1}}

Once the mutual inductance <math>M</math> is determined, it can be used to predict the behavior of a circuit:
<math display=block> v_1 = L_1\ \frac{\text{d}i_1}{\text{d}t} - M\ \frac{\text{d}i_2}{\text{d}t} </math>
where

{{plainlist|1=
* <math>v_1</math> is the voltage across the inductor of interest;
* <math>L_1</math> is the inductance of the inductor of interest;
* <math>\text{d}i_1\,/\,\text{d}t</math> is the derivative, with respect to time, of the current through the inductor of interest, labeled 1;
* <math>\text{d}i_2\,/\,\text{d}t</math> is the derivative, with respect to time, of the current through the inductor, labeled 2, that is coupled to the first inductor; and
* <math>M</math> is the mutual inductance.
|indent=1}}

The minus sign arises because of the sense the current <math>i_2</math> has been defined in the diagram. With both currents defined going into the [[dot convention|dot]]s the sign of <math>M</math> will be positive (the equation would read with a plus sign instead).<ref>{{cite book|first1=Mahmood|last1=Nahvi |first2=Joseph|last2=Edminister |url=https://books.google.com/books?id=nrxT9Qjguk8C&pg=PA338|title= Schaum's outline of theory and problems of electric circuits|page=338|publisher=McGraw-Hill Professional|year=2002|isbn=0-07-139307-2}}</ref>