Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

NoahH/ CreateCoil

Create simple, discrete coils using current-carrying loops, saddles or ellipses to generate magnetic fields

Contributed by: Noah Hardwicke, Peter J. Hobson, Michael Packer

Create simple coils made from cylindrical loop, saddle, or ellipse-based primitives, optimised to generate a target magnetic field harmonic. Visualise and examine the coils and the magnetic fields they generate.

CreateCoil implements the theoretical model derived in Designing optimal loop, saddle, and ellipse-based magnetic coils by spherical harmonic mapping.

We will soon be adding the ability to export coil data and field data, so check back for updates!

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["NoahH/CreateCoil"]


To load the code after installation, evaluate this code:
Needs["NoahH`CreateCoil`"]

Details

Coils are built from loop, saddle, or ellipse-based primitives. They are optimised to generate a desired magnetic field harmonic of order n and degree m, denoted Bn,m.
Loop primitives are pairs of axially-separated loops, which generate zonal field harmonics of total azimuthal symmetry (Bn,0). The axially-separated pairs carry the same current if n is odd and opposite current if n is even.
Saddle and ellipse-based primitives both generate tesseral field harmonics. They contain a number of axially-separated and azimuthally-periodic saddles or ellipses, whose periodicity matches that of the generated field harmonic. The axially separated sets carry the same current if n+m is odd and opposite current if n+m is even.

Examples

Basic Examples (2) 

Compare the uniform axial Bz field (encoded as the B1,0 field harmonic) generated by a Helmholtz pair, and by three optimised pairs of loops covering the same axial extent:

In[1]:=
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Out[1]=
In[2]:=
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List[355.0710144042969`, 125.46986389160156`], List[219.`, 46.90887451171875`], List[219.`, 204.0298843383789`], List[355.0710144042969`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6901960784313725`, 0.5882352941176471`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[200.`, 394.0606994628906`], List[336.0710144042969`, 315.4997024536133`], List[200.`, 236.93968200683594`], List[63.928985595703125`, 315.4997024536133`], List[200.`, 394.0606994628906`]]]]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, 15]]], StyleBox[RowBox[List["LoopFieldPlot", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, 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Out[3]=

Optimise a saddle coil and an ellipse coil to generate the uniform transverse Bx field (encoded as the B1,1 field harmonic):

In[4]:=
{saddle, ellipse} = {First /@ InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], 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Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["NoahH/CreateCoil", "NoahH`CreateCoil`FindSaddleCoil"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{1, -2, 2}, {1, 1}, 2, {.01, 3}], First@InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], 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Out[4]=

Visualise the coils as 2D schematics:

In[5]:=
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Out[5]=

Visualise the coils as 3D plots. Colour each ellipse primitive differently to distinguish them more easily:

In[6]:=
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Out[6]=

Plot the Bx field component generated by each coil in the xz-plane:

In[7]:=
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Out[7]=

Scope (3) 

The search domain can be constrained to account for physical constraints. Optimise a loop coil of three pairs to generate the B2,0 field harmonic, with axial separations constrained between 0.5 and 1 times the coil radius:

In[8]:=
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Out[8]=
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Out[9]=

When the mouse hovers over a wire in a 2D or 3D coil plot, all other wires belonging to the same coil primitive are highlighted, and a tooltip shows the parameters describing that primitive:

By default, FindLoopCoil, FindSaddleCoil, and FindEllipseCoil null one fewer leading-order error field harmonics than there are coil parameters, meaning that solutions lie on a 1D contour embedded in a solution space whose dimensionality is equal to the number of coil parameters. Points are found on the solution contour by searching over a coarse mesh of search seeds. These points are interpolated to produce a fine mesh of search seeds, which yield the final set of solutions:

In[10]:=
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Out[11]=

The solutions are then ranked by the ratio of the desired-to-leading-order error field harmonic magnitudes.

Specifying fewer field harmonics to be nulled results in a solution contour of higher dimensionality. Use three saddle primitives to null only one field harmonic, and the solution contour is a 2D surface embedded in a 3D space:

In[12]:=
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{292., 0}}, {0, 255}, ColorFunction->RGBColor, ImageResolution->{144., 144.}, RasterInterpolation->"High"], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->Automatic, ImageSizeRaw->{292., 25.}, PlotRange->{{0, 292.}, {0, 25.}}]\)
Out[13]=

Although the interpolation algorithm works for arbitrary solution contour and search space dimensions, it is currently only optimised for a 1D solution contour (the default and most useful case).

Publisher

Noah Hardwicke

Compatibility

Wolfram Language Version 13.0

Version History

  • 1.1.2 – 25 September 2023
  • 1.1.1 – 19 September 2023
  • 1.1 – 19 September 2023
  • 1.0.2 – 28 June 2023
  • 1.0.1 – 28 June 2023
  • 1.0.0 – 28 June 2023

License Information

Creative Commons Attribution Non Commercial 3.0 Unported

Paclet Source

Source Metadata