==== Calculator of inductance of a toroidal coil ==== |< 100% 10% 90% >| | {{/calculator/icon_calc.png?60&nolink}} | //[[user/Stan Zurek]], Calculator of inductance of a toroidal coil, Encyclopedia Magnetica//, \\ https://www.e-magnetica.pl/doku.php/calculator/toroidal_coil_inductance, {updated: ~~LASTMOD~~} | | {{/wiki/logo.png?20&nolink}} //See more: [[/Calculators of inductance]]// || Toroidal coil dimensions: D - outer diameter, d - inner diameter, h - thickness, c - round wire diameter, p - average pitch between the turns; the core is not shows in this image but it is assumed to completely fill the inside of the coil [[file/toroidal_coil_dimensions_png|{{/toroidal_coil_dimensions.png}}]] {{page>insert/by_SZ}} [[/Inductance]] of an ideal [[/toroidal coil]] or winding can be calculated from the equations as specified below.
outer diameter D =      

inner diameter d =      

toroid thickness h =      

number of turns N =

relative permeability of the core μr =


(optional) round wire diameter c =      

      

Inductance of toroidal coil:
L1 = [1] eq. (1)

L2 = [2] eq. (2)

L3 = [3] eq. (3), no magnetic core, μr = 1

=== Equations === ^ Approximate inductance of a toroidal coil with a magnetic core ^^^ ^ //Source: [1] [[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884]]// ^^^ | **(1)** \\ //[1], eq. (4.39), p. 138// | $$ L = \frac{μ_r · μ_0 · h · N^2 }{2 · π} · ln \left( \frac{D}{d} \right) $$ | (H) | | where: $μ_r$ - [[/relative permeability]] of the magnetic core (unitless), $μ_0$ - [[/permeability of vacuum]] (H/m), $h$ - thickness of the toroid (m), $N$ - number of turns in the coil (unitless), $D$ - outer diameter of the toroid (m), $d$ - inner diameter of the toroid (m) ||| ^ //Source: [2] [[https://isbnsearch.org/isbn/9781118717790|Marian K. Kazimierczuk, High-Frequency Magnetic Components, Second edition, John Wiley & Sons, Chichester, 2014, ISBN 9781118717790]] // ^^^ | **(2)** \\ //[2], eq. (1.330), p. 48// | $$ L = \frac{μ_r · μ_0 · N^2 · h · (D - d) }{π · (D + d)} = \frac{μ_r · μ_0 · A · N^2 }{ l_c } $$ | (H) | | where other variables as above, and: $A = h · (D - d) / 2 $ - core cross-sectional area (m2), $ l_c = π · (D + d) / 2 $ - core length (m) or [[/magnetic path length]] ||| ^ //Source: [3] [[https://isbnsearch.org/isbn/0876645570|Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]]// ^^^ | **(3)** \\ //[3], eq. (149), p. 170// \\ (valid only for a uniform single layer of turns) | $$ L = \frac{ μ_0 · N }{2 · π} \left( h_G · N · ln \left( \frac{D}{d} \right) - l_t · (G + H) \right) $$ | (H) | | where other symbols as above, and: $G$ - first correction factor for space between turns (unitless), $H$ - second correction factor for space between turns (unitless), $c$ - round wire diameter (m), $p$ - mean pitch between wire centres (m) with the turns assumed to be distributed uniformly in a single layer, and: ||| | //toroid thickness measured between wire centres in the axial direction for a single-layer coils// | $$ h_G = h + c $$ | (m) | | //turn length// | $$ l_t = D - d + 2 · h_G $$ | (m) | | //(first correction factor) \\ [2], Table 38, p. 148, \\ exact function// | $$ G = \frac{5}{4} - ln \left( 2 · \frac{p}{c} \right) $$ | (unitless) | | //(second correction factor) \\ [3], Table 39, p. 150, \\ approximation by S. Zurek (comparison available [[calculator/toroidal_coil_inductance/Grover Table 39 approximation|here]])// | $$ H = 0.33790 - 0.43478 · N^{-0.8} + \frac{0.096876}{N^2} $$ | (unitless) | ↑ {{page>insert/paypal}} {{tag>Calculators Toroid_coil Toroidal_coil Toroid_winding Toroidal_winding Inductance}}