Inductance

Inductance is a physical characteristic of an inductor, which is an electrical device that produces at any time a voltage proportional to the instantaneous rate of change in current flowing through it. The symbol L is used for inductance in honour of the physicist Heinrich Lenz. The SI unit of inductance is the henry (H). The term inductance was coined by Oliver Heaviside in February 1886.

In a typical inductor, whose geometry and physical properties are fixed, the voltage generated is as follows:

[itex]v = -L \frac {di} {dt}[itex]

where

v is the voltage generated, measured in volts.
L is the inductance of the device, measured in henries.
di/dt is the rate of change of current, measured in amperes/second.

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, that some refer to as a counter-electromotive force because it runs against the current that induces it, in the conductor under Faraday's law of induction, which results in a voltage (v) forming in 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 self-inductance L of a solenoid (an idealization of a coil) can be calculated from

[itex] L = {\mu N^2 A \over l}[itex],

where

μ is the permeability of the core, measured in henries per metre
N is the number of turns.
A is the cross sectional area of the coil, measured in square metres.
l is the length, measured in metres.

This, and the inductance of more complicated shapes, can be derived from Maxwell's equations.

 Contents

Mutual inductance

Final expression

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

[itex] M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|} [itex]

Derivation

[itex] \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} \left(\sum_{j}\frac{\mu_0 I_j}{4\pi} \oint_{C_j} \frac{\mathbf{ds}_j}{|\mathbf{R}|}\right) \cdot \mathbf{ds}_i [itex]


where

[itex]\Phi_i\ \,[itex] is the magnetic flux through the ith surface by the electrical circuit outlined by Cj

Ci is the enclosing curve of Si.

B is the magnetic field vector.

A is the vector potential.

Stokes' theorem has been used.

[itex] 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}|} [itex]

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,

[itex] 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}|}[itex]

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

Usage

The flux [itex]\Phi_i\ \![itex] through the ith circuit in a set is obviously given by:

[itex] \Phi_i = \sum_{j} M_{ij}I_j = L_i I_i + \sum_{j\ne i} M_{ij}I_j \,[itex]

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

[itex] \mathcal{E} = -\frac{d\Phi_i}{dt} = -\frac{d}{dt}(L_i I_i + \sum_{j\ne i} M_{ij}I_j) = -(\frac{dL_i}{dt}I_i +\frac{dI_i}{dt}L_i) -\sum_{j\ne i}(\frac{dM_{ij}}{dt}I_j + \frac{dI_j}{dt}M_{ij})[itex]

References

Wangsness, Roald K. (1986). Electromagnetic Fields (2nd Ed.). Wiley Text Books. ISBN 0471811866.es:Inductancia fr:Inductance it:Induttanza ja:インダクタンス pl:Indukcyjność pt:Indutância sl:induktivnost

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