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

Transcription

1 University of California, San Diego UCSD-LPLM D Magnetic Circuit Analysis for a Permanent Magnet Used in Laser Ablation Plume Expansion Experiments Xueren Wang, Mark Tillack and S. S. Harilal December 5, 2002 Revised February 10, 2003 Fusion Division Center for Energy Research University of California, San Diego La Jolla, CA

2 2-D Magnetic Circuit Analysis for a Permanent Magnet Used in Laser Ablation Plume Expansion Experiments I. Introduction Xueren Wang, Mark Tillack and S. S. Harilal December 5, 2002 revised February 10, 2003 The magnetic field distribution from permanent magnets was modeled in order to help in the design of a magnet system for laser plasma expansion studies to be performed in the UCSD Laser Plasma and Laser-Matter Interactions Laboratory [1]. We chose to use rare earth NdFeB permanent magnets due to their low cost and high residual magnetization (>1.3 T). The primary concern with NdFeB is the low demagnetization temperature. However, in our research we do not expect to exceed room temperature in the magnets. The finite element code ANSYS was used for these calculations [2]. ANSYS uses Maxwell s equations as the basis for the magnetic field analysis. The primary degrees of freedom that the finite element solution calculates are either magnetic potentials or flux. Depending on the chosen element type and element option, the degree of freedom may be scalar magnetic potentials, vector magnetic potentials, or edge flux. We examined two primary configurations: (1) two permanent plate magnets held in place with a non-magnetic frame, and (2) permanent magnet pole pieces mounted on magnetic steel to close the magnetic circuit. Parametric studies were performed in order to determine the tradeoffs between 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 and the magnetic field distribution was verified using a Gauss meter. Good agreement between the model and measurement was obtained. II. Design Considerations Our primary design goal was to provide a permanent magnet geometry which produces a magnetic flux density of 0.5 T in the center of a gap with spacing of 2~3 cm and transverse dimensions of 2 cm x 5 cm. The field uniformity in the gap region was a second important consideration. A. Open magnetic circuit We first studied a simple pair of plate magnets mounted in a non-magnetic frame. We considered this to be the cheapest, the simplest configuration which also provides the least potential impact on diagnostic access. The sketch of the open magnetic circuit is shown in Figure 1.

3 B. Closed magnetic circuit Figure 1: Sketch of the opened magnetic circuit (dimensions in cm) Generally, a magnetic circuit closed by magnetic steel can provide a higher magnetic field in the gap as compared with an isolated pair of magnetized plates. The magnetic circuit we analyzed consists of a high permeability steel, two permanent magnetic plates and the air gap where experiments will take place. The magnetization direction of the permanent magnets is assumed to be aligned with the Y-axis. The design sketch of the closed magnetic circuit is shown in Figure 2. Figure 2: Sketch of the closed magnetic circuit (dimensions in cm)

4 III. FEA Modeling A. Open magnetic circuit The FEA model for the opened magnetic circuit has to include the permanent magnets, the air gap between the magnets and the air region surrounding the magnetic circuit to model the effects of infinite boundary (the Biot-Savart Law, B~1/r 2 ). This usually requires a huge number of elements to simulate the magnetic field distribution with acceptable accuracy in the circuit. The element needs for a full 3-D magnetic circuit analysis would probably exceed the maximum allowable element number of To simplify the 3-D magnetic circuit problem and reduce the 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 magnetic circuit The inputs for this case are the material properties of the material regions: the air and the permanent magnets. Each type of material region has certain required material properties. For the air region, ANSYS requests to specify a relative permeability of 1. For the permanent regions, the input requirements are the normal demagnetization B-H curve and components of the coercive force vector. The B-H curve describes the cycling of a magnet in a closed circuit as it is brought to saturation, demagnetized, saturated in the opposite direction, and then demagnetized again under the influence of an external magnetic field [2,3].

5 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 third quadrant, must be translated to the first quadrant in ANSYS magnetic modeling. To do this, a constant shift must be added to all H values. The shift is given by: H c =[(MGXX 2 +MGYY 2 +MGZZ 2 )] 1/2 [2] B (Tesla) B42H (Tesla) 0-1.0E E E E E E+00 H (A/m) Figure 5: Actual demagnetization curve for NdFeB H42 Where H c represents the magnitude of the coercive force, and the coercive force components are used to align the magnetization axis of the magnet with the element coordinate system. In this design, NdFeB 42H permanent magnets were used to generate the required magnetic field of 0.5 Tesla 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 ANSYS demagnetization curve shifted to first quadrant.

6 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 gap between the two magnetic plates was requested to be 2 3 cm to locate the target and allow room for plasma expansion. Generally, there is no flux leakage assumed at the perimeter of the model in an idea closed magnetic circuit, and the magnetic flux is assumed to flow parallel to the surface (A=0, where A is magnetic vector potential). The assumption of no flux leakage out of the magnets and magnetic steel at the perimeter of the model cannot be made because the air gap between the two magnetic plates is too large. Therefore, the FEA model must include the permanent magnets, the permeable steel, the air gap between the plates and the air region surrounding the magnetic circuit to model the effects of the infinite boundary. Like the FEA model of the opened magnetic circuit, a 2-D planar model was used to simplify the full 3-D closed magnetic circuit problem and reduce the huge element number requirement from the ANSYS program. The ANSYS finite element model for the closed magnetic circuit is shown in Figure 7. 2-D element PLANE13 was used in the permanent magnets, the permeable air and the steel material regions. PLANE13 is defined by four nodes with up to four degrees of freedom per node. The element has nonlinear magnetic capability for modeling B-H curves or permanent magnet demagnetization curves. 2-D infinite boundary element INFIN9 was used to simulate the opened boundary of the 2-D planar unbounded field problem. Both the permeable material M54 steel and the permanent magnetic material NdFeB 42H are considered as nonlinear materials in this modeling.

7 The element meshing size for the element PLANE13 is a key factor affecting the accuracy of results. The double-double meshing rule was adopted in this case. The total of elements was considered adequate when comparing the error differences of 0.26% with a previous calculation 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 arrows represent the direction of B flux, and the colors represent the quantities of B. The maximum magnetic flux density at the center of the poles for the opened circuit is about Tesla. The variation of magnetic flux density along the x-axis (in the direction of plume propagation) for the closed magnetic circuit is shown in Figure 9. The maximum magnetic flux density at the center of the poles is about Tesla, and decreases on either side along x. The distance in between the poles is the same as the opened magnetic circuit, but it generates ~2.5 times higher magnetic flux density than the opened magnetic circuit. Figure 10 and 11 show the magnetic flux density contours and flux lines in the closed magnetic circuit. As shown in Figure 8 and 10, flux concentrations are observed on the four corners of circuits. From the 2-D flux line displays, flux discontinuities occur at the interface of the magnetic plates and the magnetic permeable steel because of quite different material permeability. Figure 12 shows the results of parametric studies for the pole distance of 2.0, 1.5 and 1.0 cm.

8 Figure 8: B vector displays of the opened magnetic circuit and partial air region Magnetic Flux Density, Tesla X axis, cm Figure 9: The variation of magnetic flux density along x for a pole distance of 2.5 cm

9 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

10 1 Magnetic Flux Density, Tesla cm pole distance 1.5 cm pole distance 2.0 cm pole distance X axis, cm 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-Meter from F.W. Bell (Model 7030). The Tesla meter makes use of the Hall effect and is capable of simultaneously measuring and displaying different parameters per channel including flux density, frequency, min, max, peak and valley. Figure 13 gives the variation of magnetic flux density along the plume expansion direction. The distance of separation between the magnets is kept at 1.5 cm. The measured magnetic flux density is in good agreement with the estimated values (Figure 12) using the 2D ANSYS finite element model. Figure 13. Magnetic flux density along the plume expansion direction measured using a tesla meter.

11 References [1] [2] ANSYS Release 6.1, ANSYS, Inc., Electromagnetic Field Analysis Guide, April [3] Paul Lorrain, Dale R. Corson and Francois Lorrain, Electromagnetic Fields and Waves (W.H. Freeman and Company, New York, 1988).