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Calculator of inductance of a straight round wire with frequency effects
 See more: Calculators of inductance. |
* This page is being edited and may be incomplete or incorrect.
 This section is an interactive calculator. |
Inductance of a straight wire or conductor made from non-magnetic material (μr = 1) with round (circular) cross-section can be calculated withe the equation as specified below.
Grover's and Wadell's equations allow accounting for the effect of frequency (skin depth), but in this calculator the permeability of wire is assumed as unity (non-magnetic).
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), and for r ≈ l the calculation errors might be excessive. 3) The surrounding medium is assumed to be non-magnetic with μr ≡ 1. 4) The current is uniformly distributed inside the wire (no skin effect). 5) The equations were converted here to be consistent with SI units.
| 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 | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{d} \right) - 1 + \frac{d}{2⋅l} + \frac{μ_r⋅T_W(x)}{4} \right) $$ | (H) |
| 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 scaling from SI units (1/(Ω·m)) | ||
| (1.1) Grover's coefficient $x$ | $$x = π⋅d⋅100⋅ \sqrt{2⋅μ_r⋅μ_0⋅f⋅ρ} ⋅ k_{SI} $$ | (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) |
| Source: 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 xxxxxxx) | ||
| (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 xxxxxxxx | ||
| (2) [2] Grover, eq. (11), p. 36 | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{d} \right) - 1 + ξ_Z \right) $$ | (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) |
| [2] Grover | ||
| (3.1) [2] Grover, eq. (7), p. 35 highest possible inductance at f = 0 Hz (upper limit) | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{2⋅l}{r} \right) - \frac{3}{4} \right) $$ | (H) |
| (3.2) [2] Grover, simplified eq. (11), p. 36 infinitely high frequency (skin depth s = 0, bottom limit) | $$ L ≈ \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{2⋅l}{r} \right) - 1 \right) $$ | (H) |
| where: $r = d/2$ - radius (m) of the wire | ||