Editing Inductor (section)

==Inductance formulas==
{{See also|Inductance#Self-inductance of thin wire shapes}}
The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.

{| class="wikitable"
 ! Construction
 ! Formula
 ! Notes
 |-
 ! Cylindrical air-core coil<ref name=Nagaoka>{{cite journal
  | last=Nagaoka | first=Hantaro | author-link=Hantaro Nagaoka
  | title=The Inductance Coefficients of Solenoids
  | url=http://www.g3ynh.info/zdocs/refs/Nagaoka1909.pdf
  | journal=Journal of the College of Science, Imperial University, Tokyo, Japan
  | page=18 | volume=27
  | date=1909-05-06
  | access-date=2011-11-10
  }}</ref>
 | <math>L = \mu_0 K N^2 \frac{A}{\ell} </math>
* ''L'' = inductance in [[Henry (unit)|henries]] (H)
* ''μ<sub>0</sub>'' = [[permeability of free space]] = 4''<math>\pi</math>'' × 10<sup>−7</sup> H/m
* ''K'' = Nagaoka coefficient<ref name=Nagaoka/><ref group=lower-alpha>Nagaoka's coefficient (''K'') is approximately 1 for a coil which is much longer than its diameter and is tightly wound using small gauge wire (so that it approximates a current sheet).</ref>
* ''N'' = number of turns
* ''A'' = area of cross-section of the coil in square metres (m<sup>2</sup>)
* ''ℓ'' = length of coil in metres (m)
 |<math>K\approx 1</math> Calculation of Nagaoka's coefficient (''K'') is complicated; normally it must be looked up from a table.<ref>Kenneth L. Kaiser, ''Electromagnetic Compatibility Handbook'', p. 30.64, CRC Press, 2004 {{ISBN|0849320879}}.</ref>
 |-
 ! rowspan="2"| Straight wire conductor<ref>{{Cite journal |url=http://www.g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf |title=The Self and Mutual Inductances of Linear Conductors |first=Edward B. |last=Rosa |journal=Bulletin of the Bureau of Standards |volume=4 |issue=2 |year=1908 |pages=301–344 |doi=10.6028/bulletin.088|doi-access=free }}</ref>
 | <math>L = \frac{\mu_0}{2\pi}\ \ell \left( A \; - \; B\right) + C</math>,

where:
:<math>\begin{align}
  A &= \ln\left(\frac{\ell}{r} + \sqrt{    \left(\frac{\ell}{r}\right)^2 + 1}\right) \\
  B &= \frac{1}{\frac{r}{\ell} + \sqrt{1 + \left(\frac{r}{\ell}\right)^2}} \\
  C &= \text{Im}\left(\frac{n\rho J_0(nr)}{2\pi\omega\mu rJ_1(nr)}\right)
\end{align}</math>
* ''L'' = inductance
* ''ℓ'' = cylinder length
* ''r'' = cylinder radius
* ''μ''<sub>0</sub> = permeability of free space = 4''<math>\pi</math>''&nbsp;×&nbsp;10<sup>−7</sup>&nbsp;H/m
* ''μ'' = conductor permeability
* ''ρ'' = resistivity
* ''ω'' = angular frequency
* <math>n = \sqrt{-i\frac{\omega\mu}{\rho}} = (-1 + i)\sqrt{\frac{\omega\mu}{2\rho}}</math>
* <math>J_0, J_1</math> are [[Bessel functions of the first kind|Bessel functions]].
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | The term ''C'' gives the ''internal'' inductance of the wire with skin-effect correction (the imaginary part of the internal impedance of the wire). If ω = 0 (DC) then <math>C = \frac{\mu}{8\pi},</math> and as ω approaches ∞, ''C'' approaches 0.<ref>{{cite book|title=Electric transmission lines : distributed constants, theory, and application|year=1951|first1=Hugh Hildreth|last1=Skilling|pages=153–159|publisher=Mcgraw-Hill}}</ref>

The term ''B'' subtracts rather than adds.

 |-
 | <math>L = \frac{\mu_0}{2\pi}\ \ell \left[\ln\left(\frac{4\ell}{d}\right) - 1\right]</math> (when {{nowrap|''d''² ''f'' ≫ 1 mm² MHz}})

<math>L = \frac{\mu_0}{2\pi}\ \ell \left[\ln\left(\frac{4\ell}{d}\right) - \frac{3}{4}\right]</math> (when {{nowrap|''d''² ''f'' ≪ 1 mm² MHz}})
* ''L'' = inductance (nH)<ref>{{Harvnb|Rosa|1908|loc=equation (11a)}}, subst. radius ''ρ'' = d/2 and [[cgs]] units</ref><ref name="Terman straight">{{Harvnb|Terman|1943|pp=48–49}}, convert to natural logarithms and inches to mm.</ref>
* ''ℓ'' = length of conductor (mm)
* ''d'' = diameter of conductor (mm)
* ''f'' = frequency
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | Requires ''ℓ''&nbsp;>&nbsp;100&nbsp;''d''<ref name="Terman adjust">{{Harvtxt|Terman|1943|p=48}} states for ''ℓ''&nbsp;<&nbsp;100 ''d'', include ''d''/2''ℓ'' within the parentheses.</ref>

For relative permeability ''μ''<sub>r</sub>&nbsp;=&nbsp;1 (e.g., [[copper|Cu]] or [[aluminum|Al]]).
 |-
 ! Small loop or very short coil<ref>Burger, O. & Dvorský, M. (2015). ''Magnetic Loop Antenna''. Ostrava, Czech Republic: EDUCA TV o.p.s.</ref>
 | <math>L \approx \frac{\mu_0}{2\pi} N^2 \pi D \left[ \ln\left( \frac{D}{d} \right) + \left(\ln 8 - 2\right) \right]
 + \sqrt{\frac{\mu_0}{2\pi}}\; \frac{N D}{d} \sqrt{\frac{\mu_\text{r}}{2 f \sigma}}
</math>
* ''L'' = inductance in the same units as ''μ''<sub>0</sub>.
* ''D'' = Diameter of the coil (conductor center-to-center)
* ''d'' = diameter of the conductor
* ''N'' = number of turns
* ''f'' = operating frequency (regular ''f'', not ''&omega;'')
* ''&sigma;'' = specific conductivity of the coil conductor
* ''μ''<sub>r</sub> = relative permeability of the conductor 
* Total conductor length <math>\ell_\text{c} \approx N \pi D</math> should be roughly {{frac|1|10}}&nbsp;wavelength or smaller.<ref>Values of <math>\pi D</math> up to {{frac|1|3}}&nbsp;wavelength are feasible antennas, but for windings that long, this formula will be inaccurate.</ref>
* Proximity effects are not included: edge-to-edge gap between turns should be 2×''d'' or larger.
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | Conductor ''μ''<sub>r</sub> should be as close to 1 as possible &ndash; [[copper]] or [[aluminum]] rather than a magnetic or paramagnetic metal.
 |-
 ! Medium or long air-core [[solenoid|cylindrical coil]]<ref>ARRL Handbook, 66th Ed. American Radio Relay League (1989).</ref><ref>{{Cite web|date=2014-07-09|title=Helical coil calculator|url=https://kaizerpowerelectronics.dk/calculators/helical-coil-calculator/|access-date=2020-12-29|website=Kaizer Power Electronics|language=en-US}}</ref>
 | <math>L = \frac{r^2 N^2}{23 r + 25 \ell}</math>
* ''L'' = inductance (μH)
* ''r'' = outer radius of coil (cm)
* ''ℓ'' = length of coil (cm)
* ''N'' = number of turns
 | Requires cylinder length ''ℓ''&nbsp;>&nbsp;0.4&nbsp;''r'': Length must be at least {{frac|1|5}} of the diameter. Not applicable to single-loop antennas or very short, stubby coils.
 |-
 ! Multilayer air-core coil<ref>{{cite journal|last1=Wheeler|first1=H.A.|title=Simple Inductance Formulas for Radio Coils|journal=Proceedings of the Institute of Radio Engineers|date=October 1928|volume=16|issue=10|page=1398|doi=10.1109/JRPROC.1928.221309|s2cid=51638679}}</ref>
 | <math>L = \frac{r^2 N^2}{19 r + 29 \ell + 32 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (cm)
* ''ℓ'' = physical length of coil winding (cm)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (cm)
 |
 |-
 ! rowspan="2"|Flat spiral air-core coil<ref>For the second formula, {{Harvtxt|Terman|1943|p=58}} which cites to {{Harvnb|Wheeler|1928}}.</ref><ref>{{Cite web|url=http://quantum-technologies.iap.uni-bonn.de/de/diplom-theses.html?task=download&file=302&token=fff191dcc4193aae12b9b5b0e9e199c9|title=A Magnetic Elevator for Neutral Atoms into a 2D State-dependent Optical Lattice Experiment|website=Uni-Bonn|access-date=2017-08-15|archive-date=2018-05-23|archive-url=https://web.archive.org/web/20180523063438/http://quantum-technologies.iap.uni-bonn.de/de/diplom-theses.html?task=download&file=302&token=fff191dcc4193aae12b9b5b0e9e199c9|url-status=dead}}</ref><ref>{{Cite web|date=2014-07-10|title=Spiral coil calculator|url=https://kaizerpowerelectronics.dk/calculators/spiral-coil-calculator/|access-date=2020-12-29|website=Kaizer Power Electronics|language=en-US}}</ref>
 | <math>L = \frac{r^2 N^2}{20 r + 28 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (cm)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (cm)
 |
 |-
 | <math>L = \frac{r^2 N^2}{8 r + 11 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (in)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (in)
 | Accurate to within 5&nbsp;percent for ''d''&nbsp;&gt;&nbsp;0.2&nbsp;''r''.<ref name="Terman1943">{{Harvnb|Terman|1943|p=58}}</ref>
 |-
 ! rowspan="2"|Toroidal air-core (circular cross-section)<ref>{{Harvnb|Terman|1943|p=57}}</ref>
 | <math>L = 2\pi N^2 \left(D - \sqrt{D^2 - d^2}\right)</math>
* ''L'' = inductance (nH)
* ''d'' = diameter of coil winding (cm)
* ''N'' = number of turns
* ''D'' = 2 * radius of revolution (cm)
 |
 |-
 | <math>L \approx \pi {d^2 N^2 \over D}</math>
* ''L'' = inductance (nH)
* ''d'' = diameter of coil winding (cm)
* ''N'' = number of turns
* ''D'' = 2 * radius of revolution (cm)
 | Approximation when ''d''&nbsp;&lt;&nbsp;0.1&nbsp;''D''
 |-
 ! Toroidal air-core (rectangular cross-section)<ref name="Terman1943" />
 | <math>L = 2 N^2 h \ln\left({\frac{d_2}{d_1}}\right)</math>
* ''L'' = inductance (nH)
* ''d<sub>1</sub>'' = inside diameter of toroid (cm)
* ''d<sub>2</sub>'' = outside diameter of toroid (cm)
* ''N'' = number of turns
* ''h'' = height of toroid (cm)
|
|}