# Inductance

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Inductance is a measure of the amount of magnetic flux produced for a given electric current.

$L= \frac{\Phi}{i}$

where

L is the inductance in henries,
i is the current in amperes,
Φ is the magnetic flux in webers

Compare the above definition with that for capacitance.

## HistoryEdit

The symbol L is used for inductance in honour of the physicist Heinrich Lenz [[1]]. The term inductance was coined by Oliver Heaviside [[2]]in February 1886. The SI unit of inductance is the henry (symbol: H).

## ExplanationEdit

Strictly speaking, the quantity just defined is called self-inductance, because the magnetic field is created solely by the conductor that carries the current.

When a conductor is coiled upon itself N number of times around the same axis, the current required to produce a given amount of flux is reduced by a factor of N compared to a single turn of wire. Thus, the inductance of a coil of wire of N turns is given by:

$L= \frac{\lambda}{i} = N\frac{\Phi}{i}$

where $\lambda$ is the total 'flux linkage'.

Such a coiled conductor is an example of an Inductor.

## Properties of inductance Edit

The above equation can be rearranged as follows:

$\lambda = Li \,$

Taking the time derivative of both sides of the equation yields:

$\frac{d\lambda}{dt} = L \frac{di}{dt} + i \frac{dL}{dt} \,$

In most physical cases, the inductance is constant with time and so

$\frac{d\lambda}{dt} = L \frac{di}{dt}$

By Faraday's Law of Induction we have:

$\frac{d\lambda}{dt} = -\mathcal{E} = v$

where $\mathcal{E}$ is the Electromotive force (emf) and $v$ is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:

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

or

$i(t) = \frac{1}{L} \int_0^tv(\tau) d\tau + i(0)$

These equations together state that, for a steady applied voltage v, the current changes in a linear manner, at a rate proportional to the applied voltage, but inversely proportionally to the inductance. Conversely, if the current through the inductor is changing at a constant rate, the induced voltage is constant.

The effect of inductance can be understood using a single loop of wire as an example. If a voltage is suddenly applied between the ends of the loop of wire, the current must change from zero to non-zero. However, a non-zero current induces a magnetic field by Ampere's law. Therefore, magnetic flux also has to change from zero to a non-zero value. Now, this change in the magnetic field, per Faraday's Law, induces an emf that is in the opposite direction of the change in current. The strength of this emf is proportional to the change in current and the inductance. When these opposing forces are in balance, the result is a current that increases linearly with time where the rate of this change is determined by the applied voltage and the inductance.

### Permeability Edit

The amount of magnetic flux produced by a current depends on a physical characteristic of the medium surrounding the current that is known as the permeability, $\mu$. The greater the permeability, the greater the magnetic flux generated by a given current. Certain materials have much higher permeability than air. If a conductor (wire) is wound around such a material, the magnetic flux is generally much greater so the inductance is much greater than the inductance of the wound wire in the air. The self-inductance L of such a solenoid (an idealization of a coil) can be calculated from

$L = {\mu_0 \mu_r N^2 A \over l} = \frac{N \Phi}{i}$

where

μ0 is the permeability of free space (4π × 10-7 henries per metre)
μr is the relative permeability of the core (dimensionless)
N is the number of turns.
A is the cross sectional area of the coil in square metres.
l is the length of the coil in metres.
$\Phi = BA$ is the flux in webers (B is the flux density, A is the area).
i is the current in amperes

This, and the inductance of more complicated shapes, can be derived from Maxwell's equations [[3]].

## Coupled inductors Edit

When the magnetic flux produced by an inductor links another inductor, these inductors are said to be coupled. Coupling is often undesired but in many cases, this coupling is intentional and is the basis of the transformer. When inductors are coupled, there exists a mutual inductance that relates the current in one inductor to the flux linkage in the other inductor. Thus, there are three inductances defined for coupled inductors:

$L_{11}$ - the self inductance of inductor 1
$L_{22}$ - the self inductance of inductor 2
$L_{12} = L_{21}$ - the mutual inductance associated with both inductors

## Vector field theory derivations Edit

### Mutual inductance Edit

Mutual inductance is the voltage induced in one circuit (the secondary circuit) when the current in another circuit (the primary circuit) changes by a unit amount in unit time. It is important as the mechanism by which transformers work, but it can also cause unwanted coupling between conductors in a circuit. The mutual inductance (in SI) by circuit i on circuit j is given by the double integral Neumann formula [[4]]

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

#### DerivationEdit

$\Phi_{i} = \int_{S_i} \mathbf{B}\cdot\mathbf{da} = \int_{S_i} (\nabla\times\mathbf{A})\cdot\mathbf{da} = \oint_{C_i} \mathbf{A}\cdot\mathbf{ds} = \oint_{C_i} \left(\sum_{j}\frac{\mu_0 I_j}{4\pi} \oint_{C_j} \frac{\mathbf{ds}_j}{|\mathbf{R}|}\right) \cdot \mathbf{ds}_i$

where

$\Phi_i\ \,$ 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 [[5]].

Stokes' theorem [[6]] 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 Edit

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 Edit

The flux $\Phi_i\ \!$ through the ith circuit in a set is given by:

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

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

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