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Self-Inductance of a Single-Layer Rectangular Cross-Section Coil

The self­inductance of a single-layer rectangular cross-section coil is a property that describes the electromotive force caused by the change in current in the coil.

The self\[Hyphen]inductance increases quadratically with the number of coils. The coil cross-section edge lengths have cubic effects on the self-inductance. Increasing coil length generally diminishes self-inductance.

Formula

QuantityVariable["L", "MagneticInductance"] == (Quantity[1/(3*Pi), "MagneticConstant"]*(-QuantityVariable["a", "Length"]^3 - QuantityVariable["b", "Length"]^3 + 3*Pi*QuantityVariable["a", "Length"]*QuantityVariable["b", "Length"]*QuantityVariable["l", "Length"] - 3*(ArcSinh[QuantityVariable["b", "Length"]/QuantityVariable["a", "Length"]]*QuantityVariable["a", "Length"]^2*QuantityVariable["b", "Length"] - ArcSinh[QuantityVariable["b", "Length"]/Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["l", "Length"]^2]]*QuantityVariable["a", "Length"]^2*QuantityVariable["b", "Length"] + ArcSinh[QuantityVariable["a", "Length"]/QuantityVariable["b", "Length"]]*QuantityVariable["a", "Length"]*QuantityVariable["b", "Length"]^2 - ArcSinh[QuantityVariable["a", "Length"]/Sqrt[QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2]]*QuantityVariable["a", "Length"]*QuantityVariable["b", "Length"]^2 + 2*ArcTan[(QuantityVariable["a", "Length"]*QuantityVariable["b", "Length"])/(QuantityVariable["l", "Length"]*Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2])]*QuantityVariable["a", "Length"]*QuantityVariable["b", "Length"]*QuantityVariable["l", "Length"] - ArcSinh[QuantityVariable["a", "Length"]/QuantityVariable["l", "Length"]]*QuantityVariable["a", "Length"]*QuantityVariable["l", "Length"]^2 + ArcSinh[QuantityVariable["a", "Length"]/Sqrt[QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2]]*QuantityVariable["a", "Length"]*QuantityVariable["l", "Length"]^2 - ArcSinh[QuantityVariable["b", "Length"]/QuantityVariable["l", "Length"]]*QuantityVariable["b", "Length"]*QuantityVariable["l", "Length"]^2 + ArcSinh[QuantityVariable["b", "Length"]/Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["l", "Length"]^2]]*QuantityVariable["b", "Length"]*QuantityVariable["l", "Length"]^2) + QuantityVariable["a", "Length"]^2*(Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2] + Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["l", "Length"]^2] - Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2]) + QuantityVariable["b", "Length"]^2*(Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2] + Sqrt[QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2] - Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2]) + 2*QuantityVariable["l", "Length"]^2*(QuantityVariable["l", "Length"] - Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["l", "Length"]^2] - Sqrt[QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2] + Sqrt[QuantityVariable["a", "Length"]^2 + QuantityVariable["b", "Length"]^2 + QuantityVariable["l", "Length"]^2]))*QuantityVariable["N", "Unitless"]^2)/QuantityVariable["l", "Length"]^2

symbol description physical quantity
L self­inductance "MagneticInductance"
l coil length "Length"
a coil cross-section edge length 1 "Length"
b coil cross-section edge length 2 "Length"
N coil turns "Unitless"

Forms

Examples

Get the resource:

In[1]:=
ResourceObject["Self-Inductance of a Single-Layer Rectangular \ Cross-Section Coil"]
Out[1]=

Get the formula:

In[2]:=
FormulaData[ ResourceObject[ "Self-Inductance of a Single-Layer Rectangular Cross-Section Coil"]]
Out[2]=

Use some values:

In[3]:=
FormulaData[ ResourceObject[ "Self-Inductance of a Single-Layer Rectangular Cross-Section \ Coil"], {QuantityVariable["N","Unitless"] -> 10, QuantityVariable["b","Length"] -> Quantity[1.5`, "Centimeters"], QuantityVariable["l","Length"] -> Quantity[2.`, "Centimeters"], QuantityVariable["L","MagneticInductance"] -> None}]
Out[3]=

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