Inductance

Inductance is a physical characteristic of an inductor, which is an electrical device that produces a voltage proportional to the instantaneous change in current flowing through it. In a typical inductor, whose geometry and physical properties are fixed, the voltage generated is as follows:

$v=-L{\frac {di}{dt}}$,

where v is the voltage generated, di/dt is the rate of change of current, and L is a property of the device called inductance. The symbol L is used for inductance in honour of the physicist Heinrich Lenz. The SI unit of inductance is the henry (H).

Strictly speaking, the quantity just defined is called self-inductance, because the voltage is induced in the same conductor that carries the current. If the voltage is induced in another nearby conductor, the property is called mutual inductance, which has the symbol M. The above equation, with either L or M as the constant, applies to both cases.

The operation of an inductor can be understood using a simple loop of wire as an example. The current flowing through the loop of wire produces a magnetic field by Ampere's law. A change in current (di/dt) results in a change in this magnetic field. This changing magnetic field causes an electromotive force in the conductor under Faraday's law of induction, which results in a voltage (v) forming in a such a direction as to oppose the change in current (see Lenz's law). The constant of proportionality L, which tells us for a particular device how big a voltage should be expected for a given change in current, is called the inductance.

The inductance L of a solenoid (an idealization of a coil) can be calculated from

$L={\mu N^{2}A \over l}$,

where μ is the permeability of the core, N is the number of turns, A is the cross sectional area of the coil, and l is the length. This, and the inductance of more complicated shapes, can be derived from Maxwell's equations.

Mutual Inductance

Final Expression

The mutual inductance (in SI) by circuit i on circuit j is given by the double integral

$M_{ij}={\frac {\mu _{0}}{4\pi }}\oint _{C_{i}}\oint _{C_{j}}{\frac {\mathbf {ds} _{i}\cdot \mathbf {ds} _{j}}{|\mathbf {R} _{ij}|}}$

Derivation

$\Phi _{i}=\int _{S_{i}}\mathbf {B} \cdot \mathbf {da} =\int _{S_{i}}(\nabla \times \mathbf {A} )\cdot \mathbf {da} =\oint _{C_{i}}\mathbf {A} \cdot \mathbf {ds} =\oint _{C_{i}}(\sum _{j}{\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathbf {ds} _{j}}{|\mathbf {R} |}})\cdot \mathbf {ds} _{i}$

where $\Phi _{i}\ \!$ is the magnetic flux through the ith surface by the electrical circuit outlined by C_j, C_i is the enclosing curve of S_i, B is the magnetic field vector, A is the vector potential, and Stokes' theorem has been used.

$M_{ij}\equiv {\frac {\Phi _{ij}}{I_{j}}}={\frac {\mu _{0}}{4\pi }}\oint _{C_{i}}\oint _{C_{j}}{\frac {\mathbf {ds} _{i}\cdot \mathbf {ds} _{j}}{|\mathbf {R} _{ij}|}}$

so that the inductance is a purely geometrical quantity independent of the current in the circuits.

Self-Inductance

Self-inductance, denoted L, is a special case of mutual inductance where, in the above equation, i ==j. Thus,

$M_{ij}=M_{jj}=L_{jj}=L_{j}=L={\frac {\mu _{0}}{4\pi }}\oint _{C}\oint _{C'}{\frac {\mathbf {ds} \cdot \mathbf {ds} '}{|\mathbf {R} |}}$

Physically, the self-inductance of a circuit represents the back-emf described by Faraday's law of induction.

Usage

The flux $\Phi _{i}\ \!$ through the ith circuit in a set is obviously given by:

$\Phi _{i}=\sum _{j}M_{ij}I_{j}=L_{i}I_{i}+\sum _{j\neq i}M_{ij}I_{j}$

so that the induced emf, ${\mathcal {E}}$, of a specific circuit, i, in any given set can be given directly by:

${\mathcal {E}}=-{\frac {d\Phi _{i}}{dt}}=-{\frac {d}{dt}}(L_{i}I_{i}+\sum _{j\neq i}M_{ij}I_{j})=-({\frac {dL_{i}}{dt}}I_{i}+{\frac {dI_{i}}{dt}}L_{i})-\sum _{j\neq i}({\frac {dM_{ij}}{dt}}I_{j}+{\frac {dI_{j}}{dt}}M_{ij})$

See also

Inductor, alternating current, electricity, RLC circuit, RL circuit, LC circuit

References

Wangsness, Roald K. (1986). Electromagnetic Fields (2nd Ed.). Wiley Text Books. ISBN 0471811866.