Calculator of inductance of a straight round wire with frequency effects
 | Stan Zurek, Calculator of inductance of a straight round wire with frequency effects, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/inductance_of_straight_round_wire_vs_frequency, {accessed: 2025-03-13} |
 See more: Calculators of inductance |
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Inductance of a straight wire or conductor made from non-magnetic material (μr = 1) with round (circular) cross-section can be calculated with the the equation as specified below.
Grover's and Wadell's equations allow accounting for magnetic permeability of the wire (μr > 1), but in this calculator its permeability is assumed as unity (non-magnetic). For inductance of round magnetic wire see: Calculator of inductance of a straight round magnetic wire.
Note: Several assumptions are made for all these equations: 1) The return path is not considered so the total inductance of the complete circuit can be significantly different. 2) The length of the wire is assumed to be significantly longer than it radius (r « l), otherwise the errors might be excessive. 3) The surrounding medium is assumed to be non-magnetic (μr = 1). 4) Unless stated otherwise, the current is uniformly distributed inside the wire (no skin effect). 5) The equations were converted here to be consistent with SI units.
Equations
| Inductance of a straight round wire with frequency effects | ||
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| Sources: [1] B.C. Wadell, Transmission Line Design Handbook, Artech House, Norwood, 1991, ISBN 0890064369, and [2] F.W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570 |
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| (1) [1] Wadell, eq. (6.2.1.1), p. 380 | $$ L_{AC} = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{d} \right) - 1 + \frac{d}{2⋅l} + \frac{μ_r⋅T_W(x)}{4} \right) $$ | (H) |
| (1.1) Grover's coefficient $x$ | $$x = π⋅d⋅100⋅k_{SI}⋅ \sqrt{2⋅μ_r⋅μ_0⋅f⋅ρ} $$ | (unitless) |
| (1.2) Grover's function $T(x)$, with Wadell's approximation | $$T_W(x) = \sqrt{\frac{0.873011 + 0.00186128 ⋅ x}{1 - 0.279381⋅x + 0.127964⋅x^2}} $$ | (unitless) |
| where: $μ_0$ - magnetic permeability of vacuum (H/m), $l$ - wire length (m), $d$ - wire diameter (m), $μ_r$ - relative magnetic permeability of the wire (unitless), $x$ - Grover's coefficient (unitless), $T_W(x)$ - Wadell's approximation of Grover's function T(x), $k_{SI}$ - factor (1/(Ω·m)) scaling from SI units to unitless, $f$ - frequency (Hz), $ρ$ - resistivity of wire (Ω·m) | ||
| Source: [1] Wadell's function TW(x) with minor addition by S. Zurek (this reduces T(x) approximation difference near x = 0 from 6.5 % to below 1 % see more) | ||
| (1.3) Wadell's approximation of $T(x)$ with Zurek's correction | $$T_{W,Z}(x) = \sqrt{\frac{0.873011 + 0.00186128 ⋅ x}{1 - 0.279381⋅x + 0.127964⋅x^2}} + \frac{0.06}{(x+1)^6} $$ | (unitless) |
| Source: [2] Grover's eq. (11) for tubular conductor, with polynomial approximation of the function ln(ξ) by S. Zurek and assumption that skin depth is equivalent to tube wall thickness (see more) | ||
| (2) [2] Grover, eq. (11), p. 36 | $$ L_{AC} = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{d} \right) - 1 + ξ_Z \right) $$ | (H) |
| where: $ξ_Z$ - Grover's function from Table 4 (unitless) approximated by S. Zurek with a polynomial function, other variables as above | ||
| (2.1) Grover's Table 4 with Zurek's approximation by polynomials | $$ ξ_Z = 0.1705⋅z^3 - 0.3979⋅z^2 - 0.0214⋅z + 0.25 $$ | (unitless) |
| (2.2) ratio $z$ of diameters | $$ z = d_{s}/d $$ | (unitless) |
| (2.3) skin depth diameter $d_s$ | $ d_s = d - 2⋅s $ (for d > 2⋅s, otherwise $ d_s = 0 $ ) | (m) |
| (2.4) skin depth $s$ | $$ s = \sqrt{\frac{2⋅ρ}{π⋅f⋅μ_r⋅μ_0}} $$ | (m) |
| Source: [2] Grover | ||
| (3.1) [2] Grover, eq. (7), p. 35 highest possible inductance at f = 0 Hz (upper limit) | $$ L_{DC} = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{2⋅l}{r} \right) - \frac{3}{4} \right) $$ | (H) |
| (3.2) [2] Grover, simplified eq. (11), p. 36 infinitely high frequency (skin depth s = 0, lower limit) | $$ L_{AC,HF} ≈ \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{2⋅l}{r} \right) - 1 \right) $$ | (H) |
| where: $r = d/2$ - radius (m) of the wire | ||
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