EB design uses FE analysis

TWI Bulletin, March/April 1990

After gaining a degree in physics. Roger Wise joined the Electron Beam Department at TWI in 1986. He has co-ordinated research on miniaturisation of existing high-power electron beam gun column technology and has also worked on finite element analysis of components for the 150kW non-vacuum electron beam system currently being developed as part of the EUREKA initiative.

Roger Wise looks at some applications for finite element analysis in high power electron beam welding equipment development.

For over a decade, finite element (FE) analysis has been successfully employed at TWI for solving mechanical stress analysis problems. ^[1] Within the last five years a suite of FE programs capable of solving problems governed by Poisson's Equation in two dimensions has found many applications. The programs can model a variety of physical situations ranging from fluid diffusion to electrical eddy currents; this article deals with applications to electrostatic and magnetostatic problems.

In electrostatics, a surface within the problem is generally set at a fixed voltage - the boundary condition - and the program calculates the distribution of voltage equipotentials in the remainder of the problem. Magnetostatic problems are normally defined by a direct current passing in a direction perpendicular to the plane of the problem, but may also involve permanent magnets. The software calculates the distribution of the magnetic vector potential from which other quantities such as field strength and flux density may be derived.

The main application for the software has been in helping to design components for high power electron beam (EB) welding equipment, but it has also been successfully employed in other areas including analysing the voltage stress distribution of high power laser electrode assemblies, and predicting the magnitude of the magnetic field produced by TIG arc oscillation solenoids.

An example of a plot produced using the software is shown in Figure 1, a half section of a possible design for a high power EB gun column. This rotationally-symmetric component has been analysed using the electrostatic solver where the gun is at -300kV (boundary condition) relative to the gun-column wall. The contour lines are voltage stress equipotentials spaced at 20kV intervals.

Any bunching of these lines, especially at the interface between two insulators. indicates a region where high voltage surface breakdown is most likely to occur. By continually modifying and solving the model of the gun column until the high voltage stress can be seen to be spread evenly over the insulating surfaces, the designer can easily minimise the chance of equipment failure without resorting to costly experiments.

The software

The suite of programs currently employed for electromagnetic analysis at TWI is called PE2D, an acronym for Poisson's Equation in 2 Dimensions. ^[2] Although the analysis is limited to two dimensions, both Cartesian and axisymmetric options are available and this is sufficient to model most readily engineered assemblies. Within the two-dimensional limitation, complex geometric shapes can be entered for solution including curved edges which are approximated to arcs of a circle, see Figure 1. Each problem is split into a 'mesh' of triangular elements whose size can be varied depending on the degree of accuracy required at any position in the problem; for a greater accuracy of solution a larger number of smaller elements per unit area are required. Figure 2 (a portion of Figure 1), shows the triangular element distribution forming the mesh. Information concerning the electromagnetic properties of each element (permittivity, permeability, current density and boundary conditions) is also entered, and when the mesh defining the problem is complete the relevant analysis program can be run. For a problem as large as that shown in Figure 1, which has nearly 6000 elements, an experienced operator can generate the full mesh in five or six hours. This problem is exceptionally complicated, and simple jobs typically take one or two hours to define fully.

The magnetostatic solver has the option of non-linear analysis which allows the permeability of ferromagnetic elements to vary as a function of the applied magnetic field. This feature is particularly useful in indicating any areas where problems due to magnetic saturation could arise. A magnetisation curve for each material included in the model is required to achieve a meaningful non-linear solution. Permanent magnets may also be defined, provided that sufficient information concerning the magnitude and direction of their magnetisation is available.

Applications

An example of a non-linear magnetostatic solution is illustrated on the front cover, which shows a model of a focusing lens for a 150kV EB gun column. The lens is fully axisymmetric and the current direction is into the page. The magnetic field is concentrated by the iron shroud surrounding the coil and focuses the electron beam in the region of the gap in the iron work as the beam passes down the centre of the lens ( Figure 3). Although in practice the coil current is variable to allow the electron beam to be focused at any distance, in this example the current is a maximum to assess the peak performance of the lens.

Blue contours (front cover) show lines of constant magnetic vector potential. This is not a straightforward parameter to visualise physically, but it provides a qualitative insight into the integrity of the solution. A more meaningful representation of the solution data may be provided by graphs of physical quantities which can be readily plotted along any line defined within the bounds of the model. Figure 4 shows the modulus of the magnetic flux density (BMOD) along the line AA' shown in Figure 3. Other quantities which may be plotted include components of magnetic flux density, field intensity, force, potential, permeability and (for the eddy currents solver) current density.

The shaded areas in the plot on the front cover indicate areas in the iron shroud of the lens where the permeability has altered because of the applied field of the coil. A colour key is used in the software to identify shaded elements; levels 1 to 9 refer to the following relative permeability levels:

Maximum permeability is 8353.40; minimum is 4991.86.

Although the relative permeability varies through the iron shroud, this model reveals that there is little danger that it will saturate at even the highest coil currents. It is possible to generate an electron beam within the model to test the position and size of the focused beam spot. Although no correction for space-charge is incorporated into the calculation, the results of these tests agree well with empirical formulae and experiment, particularly for electron beams of low current where the space-charge effect is minimal.

In conclusion

This article briefly describes the use of a two-dimension FE computer package to solve electrostatic and magnetostatic problems in EB welding. Many other problems have been analysed at TWl using the software including valve actuator design, non-destructive testing techniques, induction heating and magnetic properties of welds.

Definition of a problem requires great care to ensure that the model is as closely analogous to the physical situation as possible. Members who would like further details of PE2D or its applications are invited to contact Roger Wise at Abington.

References

Bibliography

Zienkiewcz 0 C: The finite element method. 3rd ed McGraw Hill, London 1977.

Hinton E and Owen D R J: An introduction to finite element computations. Pineridge Press Ltd, Swansea 1979.

Chari M V K and Silvester P P: Finite elements in electrical and magnetic field problems. John Wiley and Sons, New York, 1980.