University of California, San Diego UCSD-LPLM-02-04

Fusion DivisionCenter for Energy Research

University of California, San DiegoLa Jolla, CA 92093-0417

2-D Magnetic Circuit Analysis for a PermanentMagnet Used in Laser AblationPlume Expansion Experiments

Xueren Wang, Mark Tillack and S. S. Harilal

December 5, 2002Revised February 10, 2003

2-D Magnetic Circuit Analysis for a Permanent MagnetUsed in Laser Ablation Plume Expansion Experiments

Xueren Wang, Mark Tillack and S. S. Harilal

December 5, 2002revised February 10, 2003

I. Introduction

The magnetic field distribution from permanent magnets was modeled in order to help in thedesign of a magnet system for laser plasma expansion studies to be performed in the UCSDLaser Plasma and Laser-Matter Interactions Laboratory [1]. We chose to use rare earth NdFeBpermanent magnets due to their low cost and high residual magnetization (>1.3 T). The primaryconcern with NdFeB is the low demagnetization temperature. However, in our research we donot expect to exceed room temperature in the magnets.

The finite element code ANSYS was used for these calculations [2]. ANSYS uses Maxwell’sequations as the basis for the magnetic field analysis. The primary degrees of freedom that thefinite element solution calculates are either magnetic potentials or flux. Depending on thechosen element type and element option, the degree of freedom may be scalar magneticpotentials, vector magnetic potentials, or edge flux.

We examined two primary configurations: (1) two permanent plate magnets held in place with anon-magnetic frame, and (2) permanent magnet pole pieces mounted on magnetic steel to closethe magnetic circuit. Parametric studies were performed in order to determine the tradeoffsbetween gap dimensions and achievable magnetic field strength and uniformity.

Based on the analysis, a design with an iron core was adopted. The magnet was fabricated andthe magnetic field distribution was verified using a Gauss meter. Good agreement between themodel and measurement was obtained.

II. Design Considerations

Our primary design goal was to provide a permanent magnet geometry which produces amagnetic flux density of 0.5 T in the center of a gap with spacing of 2~3 cm and transversedimensions of 2 cm x 5 cm. The field uniformity in the gap region was a second importantconsideration.

A. Open magnetic circuit

We first studied a simple pair of plate magnets mounted in a non-magnetic frame. Weconsidered this to be the cheapest, the simplest configuration which also provides the leastpotential impact on diagnostic access. The sketch of the open magnetic circuit is shown inFigure 1.

Figure 1: Sketch of the opened magnetic circuit (dimensions in cm)

B. Closed magnetic circuit

Generally, a magnetic circuit closed by magnetic steel can provide a higher magnetic field in thegap as compared with an isolated pair of magnetized plates. The magnetic circuit we analyzedconsists of a high permeability steel, two permanent magnetic plates and the air gap whereexperiments will take place. The magnetization direction of the permanent magnets is assumedto be aligned with the Y-axis. The design sketch of the closed magnetic circuit is shown inFigure 2.

Figure 2: Sketch of the closed magnetic circuit (dimensions in cm)

III. FEA Modeling

A. Open magnetic circuit

The FEA model for the opened magnetic circuit has to include the permanent magnets, the airgap between the magnets and the air region surrounding the magnetic circuit to model the effectsof infinite boundary (the Biot-Savart Law, B~1/r2). This usually requires a huge number ofelements to simulate the magnetic field distribution with acceptable accuracy in the circuit. Theelement needs for a full 3-D magnetic circuit analysis would probably exceed the maximumallowable element number of 128000. To simplify the 3-D magnetic circuit problem and reducethe huge element number requirement from the ANSYS program, a 2-D planar model was used.The ANSYS finite element model for the opened magnetic circuit is shown in Figure 3.

Figure 3: 2-D ANSYS finite element model for the opened magneticcircuit

The inputs for this case are the material properties of the material regions: the air and thepermanent magnets. Each type of material region has certain required material properties. Forthe air region, ANSYS requests to specify a relative permeability of 1. For the permanentregions, the input requirements are the normal demagnetization B-H curve and components ofthe coercive force vector. The B-H curve describes the cycling of a magnet in a closed circuit asit is brought to saturation, demagnetized, saturated in the opposite direction, and thendemagnetized again under the influence of an external magnetic field [2,3].

Figure 4: The magnetization B-H curve for a permanent magnet

As shown in above figure, the demagnetization B-H curve, which is normally lies in the thirdquadrant, must be translated to the first quadrant in ANSYS magnetic modeling. To do this, aconstant shift must be added to all H values. The shift is given by:

Hc=[(MGXX2+MGYY2+MGZZ2)]1/2 [2]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1.0E+06 -8.0E+05 -6.0E+05 -4.0E+05 -2.0E+05 0.0E+00

H (A/m)

B (

Tes

la) B42H (Tesla)

Figure 5: Actual demagnetization curve for NdFeB H42

Where Hc represents the magnitude of the coercive force, and the coercive force components areused to align the magnetization axis of the magnet with the element coordinate system. In thisdesign, NdFeB 42H permanent magnets were used to generate the required magnetic field of 0.5Tesla at the center of the air gap. The residual induction of the NdFeB 42H permanent magnets,Br, is about 1.33 Tesla. Figures 5 and 6 show the actual demagnetization curve and the ANSYSdemagnetization curve shifted to first quadrant.

Figure 6: ANSYS demagnetization curve shifted to the first quadrant

B. Closed magnetic circuit

The permanent magnets, NdFeB 42H, were used in the closed magnetic circuit. The air gapbetween the two magnetic plates was requested to be 2–3 cm to locate the target and allow roomfor plasma expansion. Generally, there is no flux leakage assumed at the perimeter of the modelin an idea closed magnetic circuit, and the magnetic flux is assumed to flow parallel to thesurface (A=0, where A is magnetic vector potential). The assumption of no flux leakage out ofthe magnets and magnetic steel at the perimeter of the model cannot be made because the air gapbetween the two magnetic plates is too large. Therefore, the FEA model must include thepermanent magnets, the permeable steel, the air gap between the plates and the air regionsurrounding the magnetic circuit to model the effects of the infinite boundary. Like the FEAmodel of the opened magnetic circuit, a 2-D planar model was used to simplify the full 3-Dclosed magnetic circuit problem and reduce the huge element number requirement from theANSYS program. The ANSYS finite element model for the closed magnetic circuit is shown inFigure 7.

2-D element PLANE13 was used in the permanent magnets, the permeable air and the steelmaterial regions. PLANE13 is defined by four nodes with up to four degrees of freedom pernode. The element has nonlinear magnetic capability for modeling B-H curves or permanentmagnet demagnetization curves. 2-D infinite boundary element INFIN9 was used to simulate theopened boundary of the 2-D planar unbounded field problem. Both the permeable material M54steel and the permanent magnetic material NdFeB 42H are considered as nonlinear materials inthis modeling.

The element meshing size for the element PLANE13 is a key factor affecting the accuracy ofresults. The double-double meshing rule was adopted in this case. The total of 108500 elementswas considered adequate when comparing the error differences of 0.26% with a previouscalculation using half the number of elements.

Figure 7: 2-D ANSYS finite element model for the closed magnetic circuit

IV. Summary of Modeling Results

Figure 8 shows the B vector for the opened magnetic circuit. In this vector displays, the arrowsrepresent the direction of B flux, and the colors represent the quantities of B. The maximummagnetic flux density at the center of the poles for the opened circuit is about 0.235 Tesla. Thevariation of magnetic flux density along the x-axis (in the direction of plume propagation) for theclosed magnetic circuit is shown in Figure 9. The maximum magnetic flux density at the centerof the poles is about 0.574 Tesla, and decreases on either side along x. The distance in betweenthe poles is the same as the opened magnetic circuit, but it generates ~2.5 times higher magneticflux density than the opened magnetic circuit. Figure 10 and 11 show the magnetic flux densitycontours and flux lines in the closed magnetic circuit. As shown in Figure 8 and 10, fluxconcentrations are observed on the four corners of circuits. From the 2-D flux line displays, fluxdiscontinuities occur at the interface of the magnetic plates and the magnetic permeable steelbecause of quite different material permeability. Figure 12 shows the results of parametricstudies for the pole distance of 2.0, 1.5 and 1.0 cm.

Figure 8: B vector displays of the opened magnetic circuit and partial air region

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

-2.50

-2.00

-1.50

-1.00

-0.50 0.0

00.5

01.0

01.5

02.0

02.5

0

X axis, cm

Mag

neti

c F

lux

Den

sity

, Tes

la

Figure 9: The variation of magnetic flux density along x for a pole distance of 2.5 cm

Figure 10: The contour of magnetic flux density in the circuit and partial air region

Figure 11: 2-D flux lines in the magnetic circuit

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

X axis, cm

Mag

neti

c F

lux

Den

sity

, Tes

la1 cm pole distance

1.5 cm pole distance

2.0 cm pole distance

Figure 12: The variations of magnetic flux density along the X-axis for different pole distance

IV. Magnetic flux density measurements using a Gauss/Tesla meter

Magnetic flux measurements were made using a three-channel Advanced Gauss/Tesla-Meterfrom F.W. Bell (Model 7030). The Tesla meter makes use of the Hall effect and is capable ofsimultaneously measuring and displaying different parameters per channel including fluxdensity, frequency, min, max, peak and valley. Figure 13 gives the variation of magnetic fluxdensity along the plume expansion direction. The distance of separation between the magnets iskept at 1.5 cm. The measured magnetic flux density is in good agreement with the estimatedvalues (Figure 12) using the 2D ANSYS finite element model.

Figure 13. Magnetic flux density along the plume expansion direction measured using a teslameter.

References

[1] http://aries.ucsd.edu/LASERLAB

[2] ANSYS Release 6.1, ANSYS, Inc., Electromagnetic Field Analysis Guide, April 2002.

[3] Paul Lorrain, Dale R. Corson and Francois Lorrain, Electromagnetic Fields and Waves(W.H. Freeman and Company, New York, 1988).