==== Calculator of inductance of straight round strands distributed on a circle ====
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| {{/calculator/icon_calc.png?60&nolink}} | //[[user/Stan Zurek]], Calculator of inductance of straight round strands distributed on a circle, Encyclopedia Magnetica//, \\ https://www.e-magnetica.pl/doku.php/calculator/inductance_of_straight_round_strands_on_circle, {accessed: @YEAR@-@MONTH@-@DAY@} |
| {{/wiki/logo.png?20&nolink}} //See more: [[/Calculators of inductance]]// ||
Definition of multiple round [[/strands]] distributed on a circle (example shown with 4 wires), all connected electrically in [[/parallel connection|parallel]], without a return current path
[[/file/inductance_of_straight_strands_on_circle_png|{{/inductance_of_straight_strands_on_circle.png}}]]
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[[/Inductance]] of a number //N// of identical parallel [[/strand|strands]], [[/straight wire|straight wires]] or conductors, connected in parallel and distributed uniformly on a perimeter of a circle (without current return path) can be calculated with the equations as below.
==== Equations ====
//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 other dimensions (a,d,D << l), otherwise the calculation errors might be excessive. 3) The medium and the wire are assumed to be non-magnetic with μr = 1. 3) The current is uniformly distributed inside the wire (no [[/skin effect]]). 4) The equations were re-arranged and converted here to be consistent with [[/SI units]], and making use of diameter rather than radius. 5) The circle diameter D is such such that it coincides with the centres of the wires. 6) The calculations may fail for large N (e.g. N > 150), due to limitations of raising to such a large power value.//
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^ Inductance of a several round wires or conductors distributed uniformly on a circle ^^^^
| //[1] Source: [[https://isbnsearch.org/isbn/0876645570|F.W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]]// ||||
| for N wires \\ (uniformly spaced, regular polygon) | **(1)** \\ //[1], full eq. (14), p. 37// | $$ L = \frac{μ_0 ⋅ l}{2⋅π} ⋅ \left( ln \left( \frac{4 ⋅ l}{\sqrt[N]{N ⋅ d ⋅ D^{N-1} }} \right) - \frac{4⋅N-1}{4⋅N} \right) $$ | (H) |
| for N = 3 wires \\ (equilateral triangle) | **(2)** \\ //[1], full eq. (13), p. 37// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{\sqrt[3]{3⋅d ⋅ D^2}} \right) - \frac{11}{12} \right) $$ | (H) |
| for 2 wires \\ (one on each side of the circle) | **(3)** \\ //[1], full eq. (12), p. 37 // | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{\sqrt{2 ⋅ d ⋅ D}} \right) - \frac{7}{8} \right) $$ | (H) |
| for N = 1 wire \\ (single wire, for comparison only) | **(4)** \\ //[1], full eq. (7), p. 35// | $$ L = \frac{μ_0 ⋅ l}{2⋅π} ⋅ \left( ln \left( \frac{4 ⋅ l}{d} \right) - \frac{3}{4} \right) $$ | (H) |
| where: $μ_0$ - [[/permeability of vacuum]] (H/m), $l$ - wire length (m), $d$ - diameter (m) of the wire, $D$ - diameter (m) of the circle. ||||
| //These equations are based on [1] but were rearranged by S. Zurek to make use of diameter rather than radius, and for SI units. For N = 3 the the length of the side of the triangle (as used in [1]) was calculated as $a = D ⋅ sin(π/3) = \sqrt{3} ⋅ D /2 $ and used for simplifying and unifying the equation, so that only the D input variable is needed. The conversion process is listed [[/calculator/inductance_of_straight_round_strands_on_circle/SI_conversion|here]].// ||||
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{{tag>Calculators Inductance_of_straight_strands_distributed_on_circle Inductance_of_round_strands Inductance_of_straight_conductors_on_a_circle Inductance_of_pair_of_conductors Inductance_of_wires_on_triangle Inductance_wires_on_square Inductance_of_wires_on_regular_polygon}}