Calculator of inductance of a toroidal coil
 | Stan Zurek, Calculator of inductance of a toroidal coil, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/toroidal_coil_inductance, {updated: 2025/04/05 21:22} |
 See more: Calculators of inductance |
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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
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Inductance of an ideal toroidal coil or winding can be calculated from the equations as specified below.
Equations
| Approximate inductance of a toroidal coil with a magnetic core | ||
|---|---|---|
| Source: [1] 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] 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] 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 here) | $$ H = 0.33790 - 0.43478 · N^{-0.8} + \frac{0.096876}{N^2} $$ | (unitless) |
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