Editing Inductance (section)

==Calculating self inductance==
In the most general case, inductance can be calculated from Maxwell's equations. Many important cases can be solved using simplifications. Where high frequency currents are considered, with [[skin effect]], the surface current densities and magnetic field may be obtained by solving the [[Laplace equation]]. Where the conductors are thin wires, self-inductance still depends on the wire radius and the distribution of the current in the wire. This current distribution is approximately constant (on the surface or in the volume of the wire) for a wire radius much smaller than other length scales.

===Straight single wire===
As a practical matter, longer wires have more inductance, and thicker wires have less, analogous to their electrical resistance (although the relationships are not linear, and are different in kind from the relationships that length and diameter bear to resistance).

Separating the wire from the other parts of the circuit introduces some unavoidable error in any formulas' results. These inductances are often referred to as “partial inductances”, in part to encourage consideration of the other contributions to whole-circuit inductance which are omitted.

====Practical formulas====
For derivation of the formulas below, see Rosa (1908).<ref name=Rosa1908>{{cite journal |first=E.B. |last=Rosa |title=The self and mutual inductances of linear conductors |journal=Bulletin of the Bureau of Standards |volume=4 |issue=2 |year=1908 |page=301 ff |publisher=[[U.S. Bureau of Standards]]|doi=10.6028/bulletin.088 |doi-access=free }}</ref>
The total low frequency inductance (interior plus exterior) of a straight wire is:

<math display=block>L_\text{DC} = 200\text{ }\tfrac{\text{nH}}{\text{m}}\, \ell \left[\ln\left(\frac{\,2\,\ell\,}{r}\right) - 0.75 \right]</math>

where 
* <math>L_\text{DC}</math> is the "low-frequency" or DC inductance in nanohenry (nH or 10<sup>&minus;9</sup>H),
* <math>\ell</math> is the length of the wire in meters,
* <math>r</math> is the radius of the wire in meters (hence a very small decimal number),
* the constant <math>200\text{ }\tfrac{\text{nH}}{\text{m}}</math> is the [[Vacuum permeability|permeability of free space]], commonly called <math>\mu_\text{o}</math>, divided by <math>2 \pi</math>; in the absence of magnetically reactive insulation the value 200 is exact when using the classical definition of ''μ''<sub>0</sub> = {{val|4|end=π|e=-7|u=H/m}}, and correct to 7 decimal places when using the [[2019 revision of the SI|2019-redefined SI value]] of ''μ''<sub>0</sub> = {{val|1.25663706212|(19)|e=-6|u=[[Henry (unit)|H]]/m}}.

The constant 0.75 is just one parameter value among several; different frequency ranges, different shapes, or extremely long wire lengths require a slightly different constant ([[#current_distribution_parameter_Y|see below]]). This result is based on the assumption that the radius <math>r</math> is much less than the length {{nowrap|<math>\ell</math>,}} which is the common case for wires and rods. Disks or thick cylinders have slightly different formulas.

For sufficiently high frequencies skin effects cause the interior currents to vanish, leaving only the currents on the surface of the conductor; the inductance for alternating current, <math>L_\text{AC}</math> is then given by a very similar formula:

<math display=block>L_\text{AC} = 200\text{ }\tfrac{\text{nH}}{\text{m}}\, \ell \left[\ln\left(\frac{\,2\,\ell\,}{r}\right) - 1 \right]</math>
where the variables <math>\ell</math> and <math>r</math> are the same as above; note the changed constant term now 1, from 0.75 above.

For example, a single conductor of a lamp cord {{val|10|u=m}} long, made of 18&nbsp;[[American wire gauge|AWG]] ({{val|1.024|u=mm}}) wire, would have a low frequency inductance of about {{val|19.67|u=µH}}, at k=0.75, if stretched out straight.

===Wire loop===
Formally, the self-inductance of a wire loop would be given by the above equation with <math>\ m = n\ .</math> However, here <math>\ 1/\left|\mathbf{x} - \mathbf{x}'\right|\ </math> becomes infinite, leading to a logarithmically divergent integral.{{efn|
The integral is called "logarithmically divergent" because <math>\ \int \frac{1}{x}\ \mathrm{d}x = \ln(x)\ </math> for {{nowrap|<math>\ x > 0\ </math>,}} hence it approaches infinity like a logarithm whose argument approaches infinity.
}}
This necessitates taking the finite wire radius <math>\ a\ </math> and the distribution of the current in the wire into account. There remains the contribution from the integral over all points and a correction term,<ref name=den12>
{{cite journal
 | first = R. | last = Dengler
 | year = 2016
 | title = Self inductance of a wire loop as a curve integral
 | journal = Advanced Electromagnetics
 | volume = 5 | issue = 1 | pages = 1–8
 | bibcode=  2016AdEl....5....1D  | s2cid = 53583557
 | doi= 10.7716/aem.v5i1.331  | arxiv = 1204.1486
}}
</ref>

: <math display=block> L = \frac{\mu_0}{4\pi} \left[\ \ell\ Y + \oint_{C}\oint_{C'} \frac{\mathrm{d}\mathbf{x}\cdot \mathrm{d}\mathbf{x}'}{\ \left|\mathbf{x} - \mathbf{x}'\right|\ }\ \right] + \mathcal{O}_\mathsf{bend} \quad \text{ for } \; \left|\mathbf{s} - \mathbf{s}'\right| > \tfrac{1}{2}a\ </math>

where
: <math>\ \mathbf s\ </math> and <math>\ \mathbf{s}'\ </math> are distances along the curves <math>\ C\ </math> and <math>\ C'\ </math> respectively
: <math>\ a\ </math> is the radius of the wire 
: <math>\ \ell\ </math> is the length of the wire
: <math>\ Y\ </math> is a constant that depends on the distribution of the current in the wire:
:: <math>\ Y = 0\ </math> when the current flows on the surface of the wire (total [[skin effect]]),
:: <math display="inline">\ Y = \tfrac{1}{2}\ </math> when the current is evenly over the cross-section of the wire. 
: <math>\ \mathcal{O}_\mathsf{bend}\ </math> is an error term whose size depends on the curve of the loop:
:: <math>\ \mathcal{O}_\mathsf{bend} = \mathcal{O}(\mu_0 a)\ </math> when the loop has sharp corners, and
:: <math display="inline">\ \mathcal{O}_\mathsf{bend} = \mathcal{O}\mathord\left( {\mu_0 a^2}/{\ell} \right)\ </math> when it is a smooth curve.
::  Both are small when the wire is long compared to its radius.

===Solenoid===
A [[solenoid]] is a long, thin coil; i.e., a coil whose length is much greater than its diameter. Under these conditions, and without any magnetic material used, the [[magnetic field|magnetic flux density]] <math>B</math> within the coil is practically constant and is given by
<math display=block>B = \frac{\mu_0\, N\, i}{\ell}</math>

where <math>\mu_0</math> is the [[magnetic constant]], <math>N</math> the number of turns, <math>i</math> the current and <math>l</math> the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density <math>B</math> by the cross-section area {{nowrap|<math>A</math>:}}
<math display=block>\Phi = \frac{\mu_0\, N\, i\, A}{\ell},</math>

When this is combined with the definition of inductance {{nowrap|<math>L = \frac{N\, \Phi}{i}</math>,}} it follows that the inductance of a solenoid is given by:
<math display=block>L = \frac{\mu_0\, N^2\, A}{\ell}.</math>

Therefore, for air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.

===Coaxial cable===
Let the inner conductor have radius <math>r_i</math> and [[Permeability (electromagnetism)|permeability]] {{nowrap|<math>\mu_i</math>,}} let the dielectric between the inner and outer conductor have permeability {{nowrap|<math>\mu_d</math>,}} and let the outer conductor have inner radius {{nowrap|<math>r_{o1}</math>,}} outer radius {{nowrap|<math>r_{o2}</math>,}} and permeability {{nowrap|<math>\mu_0</math>.}} However, for a typical coaxial line application, we are interested in passing (non-DC) signals at frequencies for which the resistive [[skin effect]] cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate

<math display=block>L' = \frac{\text{d}L}{\text{d}\ell} \approx \frac{\mu_d}{2 \pi} \ln \frac{r_{o1}}{r_i}</math>

===Multilayer coils===
Most practical air-core inductors are multilayer cylindrical coils with square cross-sections to minimize average distance between turns (circular cross -sections would be better but harder to form).

===Magnetic cores===
Many inductors include a [[magnetic core]] at the center of or partly surrounding the winding. Over a large enough range these exhibit a nonlinear permeability with effects such as [[saturation (magnetic)|magnetic saturation]]. Saturation makes the resulting inductance a function of the applied current.

The secant or large-signal inductance is used in flux calculations. It is defined as:

<math display=block>L_s(i) \mathrel\overset{\underset{\mathrm{def}}{}}{=} \frac{N\ \Phi}{i} = \frac{\Lambda}{i}</math>

The differential or small-signal inductance, on the other hand, is used in calculating voltage. It is defined as:

<math display=block>L_d(i) \mathrel\overset{\underset{\mathrm{def}}{}}{=} \frac{\text{d}(N \Phi)}{\text{d}i} = \frac{\text{d}\Lambda}{\text{d}i}</math>

The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and the [[chain rule]] of calculus.

<math display=block>v(t) = \frac{\text{d}\Lambda}{\text{d}t} = \frac{\text{d}\Lambda}{\text{d}i}\frac{\text{d}i}{\text{d}t} = L_d(i)\frac{\text{d}i}{\text{d}t}</math>

Similar definitions may be derived for nonlinear mutual inductance.