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Input files
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LoopInductRF.MIN, LoopInductRF.PIN, LoopInductRF.png, LInductNorm.SCR, LoopInductRF.SCR, permag_flux.cfg
LoopInductRF.zip
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Description
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The example is the second in a series illustrating techniques for self and mutual inductance calculations with PerMag, Nelson and Magnum. This 2D PerMag calculation addresses the self-inductance of a simple current loop at high frequency. An analysis configuration (permag_flux.cfg) file was created for calculations of magnetic flux (Phi) through surfaces and enclosed current using the line integral function.
The previous example described calculations of the inductance of a circular current loop with a uniform distribution of current over the loop cross section. The example approximated DC excitation of a solid circular rod or a multiturn toroidal coil. This example covers high frequency excitation of a solid circular rod. Here, the skin depth is small compared to the rod radius. It would be challenging and inefficient to represent a thin surface current layer by finite-element methods. Instead, we represent the rod as a region with a fixed stream-function,
Psi = r*ATheta
where ATheta is the vector potential. The condition that derivatives of Psi inside the rod equal zero implies that the magnetic flux density equals zero. In other words, the rod is an ideal flux excluder. The change in B between the inside and outside of the rod corresponds to the surface current density necessary for field exclusion. As in the previous example, The rod has diameter d = 2.0 cm and the loop diameter is D = 20.0 cm. The solution volume has large dimensions (50 cm) and a flux-excluding boundary to approximate infinite space.
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Results
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The solution boundary has Psi = 0.0. Initially, we do not know the correspondence between current and the value of Psi assigned the rod. The following procedure is used:
- Set values for the rod elements and nodes initially to the fixed condition Psi = 1.0 and find a solution.
- Take an Ampere's law integral around the rod to find the enclosed current, Ic.
- To assign a current of 1.0 A, change the fixed stream function value to Psi = 1.0/Ic and recalcuate the solution.
The configuration file permag_flux.cfg includes the entry
IEnclosed = &Bxz $IMu0 *;&Byr $IMu0 *
in the SURFACE section. The script LInductNorm.SCR specifies integrals along four lines that enclose the rod:
INPUT LoopInductRF.POU
OUTPUT LInductNorm.DAT
LINEINT -10.0 0.0 10.0 0.0
LINEINT 10.0 0.0 10.0 20.0
LINEINT 10.0 20.0 -10.0 20.0
LINEINT -10.0 20.0 -10.0 0.0
ENDFILE
Adding parallel components of the quantity IEnclosed as listed in LInductNorm.DAT gives a total current of 21.253 MA. The statement assigning a fixed vector potential value of the rod is changed to
VecPot(2) = 4.7051E-8
and the calculation repeated. The analysis script
INPUT LoopInductRF.POU
OUTPUT LoopInductRF.DAT
NSCAN 200
VOLUMEINT
LINEINT 0.0 0.0 0.0 9.0
LINEINT 50.0 10.0 1.0 10.0
LINEINT -50.0 10.0 -1.0 10.0
ENDFILE
applied to the normalized solution gives the following results. The volume integral of field energy is U = 1.4680E-7 J, impying an inductance L = 2.9360E-7 H. The flux integrals give the following values of inductance, L = 2.9467E-7, 2.9427E-7 and 2.9433E-7. The results are consistent with the analytic predictions and those of the previous example. The figure below shows lines of magnetic flux density near the flux-excluding rod.
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