==== Calculator of inductance of a straight round wire with frequency effects ====
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| {{/calculator/icon_calc.png?60&nolink}} | //[[user/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: @YEAR@-@MONTH@-@DAY@} |
| {{/wiki/logo.png?20&nolink}} //See more: [[/Calculators of inductance]]// ||
Definition of the dimensions of a **straight round wire**
[[/file/inductance_of_straight_round_wire_png|{{/inductance_of_straight_round_wire.png}}]]
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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/inductance_of_straight_round_magnetic_wire|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 ^^^
^ // 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// ^^^
| **(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 [[/user/Stan_Zurek|S. Zurek]] (this reduces T(x) approximation difference near x = 0 from 6.5 % to below 1 % [[approximation of Grover Tx|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 [[/user/Stan_Zurek|S. Zurek]] and assumption that skin depth is equivalent to tube wall thickness ([[Approximation of Grover zeta|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 [[/user/Stan_Zurek|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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{{tag>Calculators Inductance_of_straight_wire Inductance_of_straight_conductor}}