Conclusiones

The finite element method is based on the concept of dividing the original problem domain into a group of sub-domains, the ‘elements’, and applying a numerical for-mulation based on interpolation theory to the elements. A numerical solution is then sought with respect to some optimal criterion.

The finite element method was first applied to solve structural analysis problems as early as the 1950s, also the method was applied to heat transfer and fluid flow problems. In 1970, an article by Silvester and Chari entitled ‘Finite element solution of saturable magnetic field problems’, in which they proposed a formulation capable of dealing with complex geometry and the problem of magnetic non-linearity, signalled the beginning of a new era in the field of applied electromagnetism [9.1]. They made major contributions to the development of the method which is so widely applied today in electrical engineering.

The finite element method is in fact a numerical technique for solving large-scale, complex problems, using a simple and flexible data structure. In order to discretise a problem region, it is necessary to choose elements of a given shape. Several element shapes are in use and the principle ones are: the triangle, the quadrilateral, and curvilinear shapes [9.2]. Elements are defined in terms of their shape and the order of polynomial interpolation of the trial function ( the function that describes the variation of the solution parameter within an element in terms of the element’s nodal values).

Figure 9.1 illustrates the formation of a net and the effect of varying the mesh fineness.

0.035

Figure 9.1 Effect of varying fineness of mesh

Finite element design methods – availability of suitable data 101 9.2.1 Finite element formulations

Instead of expressing the laws of electromagnetism in terms of electric and mag-netic fields, it turns out that it is often more convenient to express the theory in terms of potentials which are related to the field equations by curl (∇ × ) or gradient (∇ · ).

In any 3-dimension, the vector ¯b has the property that

∇ · (∇ × ¯b) = 0 (9.1)

and the Maxwell equations govern the fields in electromagnetic devices are

∇ · ¯B = 0 (9.2)

∇ × ¯H = ¯J (9.3)

where ¯B is the magnetic flux density, ¯H is the magnetic field intensity and ¯J is the current density.

Therefore, the magnetic vector potential ¯A, is defined as

∇ × ¯A = ¯B (9.4)

By the analogy to the electrostatic field in which the electric field ¯E is related to the applied voltage V as ( ¯E = −∇V ), the magnetic scalar potential ψ is defined as

H = −∇ψ¯ (9.5)

Therefore the Maxwell equations could be formulated in terms of the magnetic poten-tials ¯A and ψ, and finding a solution in terms of such potentials allows the calculation of ¯H and ¯B.

Computer programs are used to solve the FEM formulated equations. Although the available computer power has greatly increased, computer power still limits large 3D models. Therefore the FEM formulations are usually optimised towards faster solutions. One common strategy is to split the problem regions up into non-conducting and conducting volumes and use the optimum field variable in each. Often non-conducting regions are modelled using magnetic scalar potentials, this is because scalar variables are very economical as only one variable is required per node, in contrast with vector potential variables which require three.

Various magnetic scalar potentials are in common use, the total scalar ψ is optimal in regions which contain no current and the reduced scalar φ allows current to be introduced into magnetic scalar potential regions. The total scalar ψ is defined as in equation (9.5):HT = −∇ψ, and the reduced scalar [9.3] φ is defined as HT =

−∇φ + HS. HereHT is the total magnetic field intensity andHSis the field defined as∇ × HS = JS, whereJS is the source current density.

The basic method has been extended to allow voltage forced conditions [9.4] and to automatically produce cuts for solving multiply connected problems [9.5]. Both scalars give rise to a Laplacian type equation which has to be solved:

∇ · μ ∇ψ = 0 (9.6)

Eddy current regions must always be described by a vector variable. The most common in use today is the magnetic vector potentialA. This is defined as ∇ × A = B:

∇ × A fourth equation may be obtained by reiterating the requirement that∇ · J = 0, already implicit in (9.7):

In the above V is the electric scalar potential and is not always required. TheA and ψ variables may be conveniently coupled [9.6] at common interfaces throughout the problem. It should be noted that many other formulations exist. One of the most important is theT – method [9.7]. Here a vector variable T , defined in conducting regions, is coupled to a magnetic scalar defined in non-conducting regions.

One component of the magnetic vector potentialA is by far the most common variable in use for 2D problems. This gives rise to a Poisson-type equation as follows:

∇ ×

When a coil is wound on a core of ferromagnetic material the resulting field is greatly magnified and may be described by the relationshipB = μ0(H + M), where H is the field as derived from Ampère’s law and μ0the permeability of free space.M is the extra magnetisation arising from ferromagnetism.

The machine designer is interested inB rather than M, and because B may be composed of a complex mix of vectors which are a function ofH and directional material properties, it is hard to quantify. Particularly B shows hysteresis and is direction and time dependent.

The relationship betweenB and H may be approximately realised in the type of curve shown in Figure 9.2. The steep portion of the curve fades into a region of shallower slope leading to saturation.

9.2.3 Magnetisation curves for use in finite element schemes

Usually in finite element schemes only the mean magnetisation curve is used. This means that such effects as hysteresis cannot be described and the B–H curve becomes single valued and monotonic. The equations will be non-linear if the permeability of the iron cannot be considered constant over the range of operation of the device.

These non-linear equations are solved numerically using a standard technique such as simple iteration or Newton–Raphson.

The mean magnetisation curve would normally be calculated from a B–H curve, measured in a standard way. The B–H curve itself is rarely in fact used directly. In magnetic scalar equations such as (9.1), the magnetic scalar is defined asHT = −∇ψ,

Finite element design methods – availability of suitable data 103

0 1000 2000 3000 4000

Flux density B (T)

Figure 9.2 B–H characteristic

and the magnetisation information appears as μ in the governing equation. Since the term μ is initially unknown in a non-linear problem, an iterative solution is required. This involves repeated solution of (9.1) and the value of μ is updated at each iteration.

The procedure is to findH from H = −∇ψ and to find μ from a graph of μ versus H². The Newton–Raphson method requires the slope of the graph of μ versus H².

In a similar way, if we are dealing with a governing equation describing the magnetic vector potential, as in (9.9) or (9.7), since the magnetic vector potential is described as∇ × A = B and the magnetisation information appears as 1/μ in the governing equation, a graph of 1/μ versus B²is required. In addition to this, a graph of the slope of 1/μ versus B²is also required if the Newton–Raphson method is used.

The smoothness of the various graphs is important for the convergence of the various non-linear solution methods. It can affect the speed of convergence and in extreme cases there will be no convergence at all.

To this end, the magnetisation details would normally be stored in the computer in such a way as to ensure smooth curves. Various curve fitting methods have been used to represent the measured data, including cubic splines (probably the most common method), cubic Hermite polynomials or exponential functions. In addition to this, a good finite element package would allow the user to easily create (or automatically create) curves of μ versus H² and 1/μ versus B²and their respective derivatives from the measured B–H data.

A good introduction to basic material briefly described here may be obtained from [9.8] or [9.9]. More detail and some practical examples may be found in [9.10].

9.2.4 Hysteresis and iron loss

Some new developments involve the inclusion of hysteresis effects into finite element models; here various different representations have been used including Preisach models [9.11], [9.12] and graphical methods [9.13]. Semi-analytic methods have also been developed [9.14].

Iron loss can be determined in an approximate way by performing a time transient analysis, Fourier analysing the resultingB field in each element and calculating the

Magnetic intensity H (kA/m)

Figure 9.3 Permanent magnet: second quadrant characteristic

loss from manufacturer’s iron loss curves. Usually these curves are of loss versus peakB and frequency, so interpolation would be required [9.15]. Permanent magnets are used in the second quadrant of the B–H plane, see Figure 9.3.

A permanent magnet will operate along the line AB in use (as part of a machine) and may exhibit minor recoil loops. μr, the relative permeability which the line expresses, is called the relative recoil permeability.