Editing Inductance (section)

==Self-inductance of thin wire shapes==
{{See also|Inductor#Inductance formulas}}
The table below lists formulas for the self-inductance of various simple shapes made of thin cylindrical conductors (wires). In general these are only accurate if the wire radius <math>a</math> is much smaller than the dimensions of the shape, and if no ferromagnetic materials are nearby (no [[magnetic core]]).

{| class="wikitable"
|+ Self-inductance of thin wire shapes
! scope="col" | Type
! scope="col" | Inductance
! scope="col" | Explanation of symbols
|-
! scope="row" | Single layer<br/>solenoid
|
Wheeler's approximation formula for current-sheet model air-core coil:<ref>{{cite journal |doi=10.1109/JRPROC.1942.232015|title=Formulas for the Skin Effect |year=1942 |last1=Wheeler |first1=H.A. |journal=Proceedings of the IRE |volume=30 |issue=9 |pages=412–424 |s2cid=51630416 }}</ref><ref>{{cite journal |doi=10.1109/JRPROC.1928.221309|title=Simple Inductance Formulas for Radio Coils |year=1928 |last1=Wheeler |first1=H.A. |journal=Proceedings of the IRE |volume=16 |issue=10 |pages=1398–1400 |s2cid=51638679 }}</ref>

<math>\mathcal{L} = \frac{N^2 D^2}{18D + 40\ell}</math> (inches) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math>\mathcal{L} = \frac{N^2 D^2}{45.72D + 101.6\ell}</math> (cm)

This formula gives an error no more than 1% {{nowrap|when <math>\ell > 0.4\, D ~.</math>}}
| {{plainlist|
* <math>\mathcal{L}</math>: inductance in μH ({{10^|−6}} henries)
* <math>N</math>: number of turns
* <math>D</math>: diameter in (inches) (cm)
* <math>\ell</math>: length in (inches) (cm)
}}
|-
! scope="row | Coaxial <br/>cable (HF)
| <math>\mathcal{L} = \frac{\mu_0}{2\pi} \ell \ln\left(\frac{b}{a}\right)</math>
| {{ubl
 | <math>b</math>: Outer conductor's inside radius
 | <math>a</math>: Inner conductor's radius
 | <math>\ell</math>: Length
 | <math>\mu_0 </math>: see table footnote.
 }}
|-
! scope="row | Circular loop<ref>{{cite book |last=Elliott |first=R.S. |title=Electromagnetics |publisher=IEEE Press |year=1993 |location=New York}} Note: The published constant {{frac|−3|2}} in the result for a uniform current distribution is wrong.</ref>
| <math>\mathcal{L} = \mu_0\ r\ \left[\ln\left(\frac{8 r}{a}\right) - 2 + \tfrac{1}{4}Y + \mathcal{O} \left(\frac{a^2}{r^2}\right)\right]</math>
| {{ubl
 | <math>r</math>: Loop radius
 | <math>a</math>: Wire radius
 | <math>\mu_0, Y</math>: see table footnotes.
 }}
|-
! scope="row | Rectangle from <br/>round wire<ref>{{cite book |first=Frederick W. |last=Grover |title=Inductance Calculations: Working formulas and tables |publisher=Dover Publications, Inc. |location=New York |year=1946}}</ref> 
|
<math>\begin{align}
  \mathcal{L} = \frac{\mu_0}{\pi}\ \biggl[\ &\ell_1\ln\left(\frac{2\ell_1}{a}\right) + \ell_2\ \ln\left(\frac{2\ell_2}{a}\right) + 2\sqrt{\ell_1^2 + \ell_2^2\ } \\
            &- \ell_1\ \sinh^{-1}\left(\frac{\ell_1}{\ell_2}\right) - \ell_2 \sinh^{-1}\left(\frac{\ell_2}{\ell_1}\right) \\
            &- \left(2 - \tfrac{1}{4}Y\ \right)\left(\ell_1 + \ell_2\right)\ \biggr]
\end{align}</math>
| {{ubl
 | <math>\ell_1, \ell_2</math>: Side lengths
 | {{nowrap|<math>\ \ell_1 \gg a, \ell_2 \gg a\ </math>}}
 | <math>a</math>: Wire radius
 | <math>\mu_0, Y</math>: see table footnotes.
 }}
|-
! scope="row | Pair of parallel<br/> wires
| <math>\mathcal{L} = \frac{\ \mu_0 }{\pi}\ \ell\ \left[ \ln\left(\frac{s}{a}\right) + \tfrac{1}{4}Y \right] </math>
| {{ubl
 | <math>a</math>: Wire radius
 | <math>s</math>: Separation distance, {{nowrap|<math>s \ge 2a</math>}}
 | <math>\ell</math>: Length of pair
 | <math>\mu_0, Y</math>: see table footnotes.
 }}
|-
! scope="row | Pair of parallel<br/> wires (HF)
|
<math>\begin{align}
  \mathcal{L} &= \frac{\mu_0}{\pi}\ \ell\ \cosh^{-1}\left(\frac{s}{2a}\right) \\
              &= \frac{\mu_0}{\pi}\ \ell\ \ln\left(\frac{s}{2a} + \sqrt{\frac{s^2}{4a^2} - 1}\right) \\
              &\approx \frac{\mu_0}{\pi}\ \ell\ \ln\left(\frac{s}{a}\right)
\end{align}</math>
| {{ubl
 | <math>a</math>: Wire radius
 | <math>s</math>: Separation distance, {{nowrap|<math>s \ge 2a</math>}}
 | <math>\ell</math>: Length (each) of pair
 | <math>\mu_0</math>: see table footnote.
 }}
|}

{{anchor|current_distribution_parameter_Y}}<math>Y</math> is an approximately constant value between 0 and 1 that depends on the distribution of the current in the wire: {{nowrap|<math>Y = 0</math>}} when the current flows only on the surface of the wire (complete [[skin effect]]), {{nowrap|<math>Y = 1</math>}} when the current is evenly spread over the cross-section of the wire ([[direct current]]). For round wires, Rosa (1908) gives a formula equivalent to:<ref name=Rosa1908/>

<math display=block>Y \approx \frac{1}{\, 1 + a\ \sqrt{\tfrac{1}{8}\mu\sigma\omega \,} \,}</math>

where

{{plainlist|indent=1|1=
* <math>\omega = 2\pi f</math> is the angular frequency, in radians per second;
* <math>\mu = \mu_0\,\mu_\text{r}</math> is the net [[magnetic permeability]] of the wire;
* <math>\sigma</math> is the wire's specific conductivity; and
* <math>a</math> is the wire radius.
}}

<math>\mathcal{O}(x)</math> is represents small term(s) that have been dropped from the formula, to make it simpler. Read the term <math>{}+ \mathcal{O}(x)</math> as "plus small corrections that vary on the order of {{nowrap|<math>x</math>"}} (see [[big O notation]]).