**By R. Belmans**

**In this section the** most important material models such as permanent magnet and non-linear ferromagnetic materials are introduced.

## Non-linear material

**Most of the electromagnetic field problems** are inherently non-linear. Having accurate numerical techniques able to handle non-linearities strengthens the trend to have problems out of this class because material costs and a desired minimisation of the devices force the exploitation of material to its limits. As a consequence mainly highly saturated magnetic circuits have to be analysed.

**Fig. 5.35. Hysteresis loops and magnetisation characteristic.**

**The analysis of magnetic devices requires** knowledge of the physical properties of the materials used. Slowly magnetising a piece of ferromagnetic material to a value Hi and then reducing the field until it reaches -Hi , and repeating the process until the characteristic remains the same, yields a hysteresis loop (Fig. 5.35). If the field is increased to a value H2 and the process is repeated, and then to H3, etc., a family of nested hysteresis loops is obtained as shown in (Fig. 5.35). The connected tips of the hysteresis characteristics represent the normal magnetisation characteristic. This curve is most commonly used in the finite element analysis to represent the non-linear properties of ferromagnetic materials. Using soft magnetic materials, the curve is narrow and thus an acceptable approximation of the real behaviour of the material. Here, only non-hysteretic materials are discussed. Manufacturers having different grades normally deliver the material characteristics by B-H curves (Fig. 5.36).

**Fig. 5.36. Magnetisation characteristic in BH-form.**

The basic principles of the linear finite element analysis carry over to non-linear problems almost without modification. As always, a stationary functional is constructed and discretised over finite elements. As might be expected, the equations resulting from non-linear problems are non-linear as well. They can be solved by several different methods. Simple iterative methods are not always stable and can take a long time to converge. A more common approach is to use Newton iterations.

**In order to extend the linear finite** element procedure to include nonlinear material properties, a mathematical model describing the magnetic properties of the material is recommended. Therefore, computer readable files of material properties must be maintained ready for use. Numerous ways of modelling magnetic property curves have been tried out:

• reluctivity as a function of flux density squared,

• field as a function of flux density,

• permeability as a function of field squared,

• permeability as a function of flux density,

**The Newton method is the** reason for using squared values of the independent variables B or H, rather than magnitudes. The variables are usually derived from potentials in vector component form, so that finding the magnitude involves first finding the squares of the components and then extracting the square root of their sum. Using the Newton method the Jacobian matrix can be evaluated in the form:

**Therefore**, the classical B-H characteristic is not the best choice to introduce ferromagnetic material properties in finite element software. In the most common program designs therepresentation is employed.

**Fig. 5.37. Magnetisation characteristic in****form.**

The rangeof thematerial characteristic is subdivided into segments. The data samplesare tabulated in an ASCII file readable by the finite element program. For the points between the given data an interpolation scheme has to be chosen. In the simplest case a linear approximation could be chosen. The ferromagnetic characteristic would then be represented by a polygon. This approximation would be discontinuous in the points of the given data sample. However, the Newton iteration recommends a continuous function. Therefore, the values between the single data sample are usually approximated by cubic spline-interpolating polynomials. For all points beyond the range of available data the curve can be extrapolated linearly.

### Cubic spline interpolation

**Splines are curves to approximate functions**. Cubic splines in particular have received much attention in numerical analysis in the past, replacing other polynomial and exponential approximations. The main reason for the interest in splines is that they result in a simple formulation. They interpolate exactly at the given data points and have a continuous first and second order derivative. The continuous first derivative makes the method suitable for the Newton iteration.

Assume a functionto be interpolated by cubic splines in the interval [a, b].

Given a tabulated functionwith x arranged in monotone way.

**The spline interpolating polynomial is called S(x) with properties:**

• S(x) is two times continuous by differentiable in [a, b].

• S(x) is in every intervalgiven by a cubic polynomial.

• As a boundary condition it is taken thatin order to obtain a so called natural spline.

Using the approach

the coefficients can be calculated by applying:

**The first derivative with respect to x of the interpolating function is:**

while the second is:

With this, the requirement of continuity of the second derivative over the boundaries of the intervalis satisfied.

Because of a required continuous derivative of first order of the interpolating spline function, the values of dS/dx at the pointformust be equal. Employingfor both intervals yields:

This equation can be evaluated withfor every interval.

This results inlinear independent equations for the n unknownat the given data samples. Using the conditions of a natural spline yields two additional constraints. This is a symmetric tri-diagonal system of equations and is easy to solve. With the derivatives known, the coefficients of the interpolating spline function are now determined.

**A great advantage of using cubic splines** is the fact that this linear system of equations has to be solved only once to obtain the values of the second derivative. Therefore, space for the solution of dimension n has to be allocated in the memory of the computer only. For the finite element method this means that this system of equation has to be solved only once independently of the number of finite elements used in the model of the magnetic circuit. For the practical use of the method a field of 20-25 data samples is sufficient to represent a non-linear ferromagnetic material characteristic.

**The non-linear material characteristics** are normally delivered by the manufacturer in a B=B(H) form. The data can be given in a tabular or a graphical form. This implies that the user of finite element softwaretakes the values and rearranges them into the appropriate representation. Converting the given data sample in this way may lead to numerical difficulties. To ensure stable convergence and the highest possible computational speed of the Newton iteration, particular attention has to be paid to the numerical representation of the data samples. The curve representation has to fulfil some numerical requirements. The functionhas to be monotonic. If the characteristic is not monotonic, the derivative changes sign, eventually leading to slow or even to non-convergence of the iteration scheme. A possible way to overcome this problem lies in the optimisation of the given data samples. Here, a numerical optimisation algorithm can change the values of the given points to ensure a monotonic behaviour of the characteristic. After optimisation a technically pure curve is constructed.

## Permanent magnets

**The development of high energy** permanent magnet materials such as SmCo and NdFeB grades has led to increased interest in the use of permanent magnet material in electrical machines and actuators. The representation of these hard magnetic materials is difficult and the subject of ongoing research. As mentioned in the last section, ferromagnetic materials are characterised by a narrow hysteresis loop. In contrast, hard magnetic materials such as permanent magnets exhibit wide loops. It is often acceptable to consider the magnetic characteristic of a permanent magnet by a straight line in the second quadrant of the hysteresis loop. This is not a limitation of the finite element method. During the design of permanent magnet excited devices, particular attention must be paid to the operating temperature of the magnets.

The intersection of the hysteresis loop with the ordinate is called the residual or remanence flux density BR. The intersection of the abscissa and the loop is called the coercive force He-

**There are two possibilities allowing the modelling of a permanent magnet material:**

**• magnetisation model**

**• current sheet approach.**

**Although these two methods** have a different starting point, they both result in the same set of equations. Assuming a straight line as the characteristic of the permanent magnet material (Fig. 5.38), there are only two parameters required to define the characteristic:

**• the y-axis intercept BR.**

**Fig. 5.38. Definition of permanent magnet material.**

In Table 5.5 the most important permanent magnet properties are collected. During a design of a device excited by permanent magnets, particular attention has to be paid to the temperature dependence of the grade used.

**Fig. 5.39. Demagnetisation characteristics for different permanent magnet material at room temperature 20° C.**

**Table 5.5. Properties of different permanent magnet material at room temperature.**

Typical demagnetisation characteristics of different grades are shown in Fig. 5.39.

### Magnetic vector model

The demagnetisation characteristic is defined by

whereis the magnetic susceptibility, M the magnetisation vector and H the field strength at the operating point AP. In terms of the remanent flux density

The incremental permeability, the slope of the demagnetisation characteristic, is

is a very small positive number so that the apparent permeability of the magnet is only slightly larger than that of the free space (Table 5.5).

The reluctivity is defined as

applying to the demagnetisation characteristic, yields

Using the Maxwell equation for a magneto static problem

yields

The second term, the magnetic vector, on the right-hand side represents a source term and can be identified as an equivalent magnetic current.

### Current sheet approach

**Using an equivalent current sheet** representing the permanent magnet material is an easy way to introduce the material properties in a finite element program. In its original form it is not easy to apply permanent magnets with an odd shape. However, if the model is extended, an arbitrary shaped magnet can be described. In the following section a linear demagnetisation characteristic of the material is assumed.

**Fig. 5.40. Idealised magnetic core excited a) by a permanent magnet and b) by a current sheet.**

Taking the permeability of the iron core in Fig. 5.40 to be infinite, Ampere’s law yields

Now, for a uniform B, it can be written (H is negative)

The intersection between the air gap characteristic, the load line, and the demagnetisation curve represents the point AP of the magnet material used (Fig. 5.38) with

The permanent magnet in Fig. 5.38 can now be represented as a current sheet with the total ampere-turnsand a material of equivalent permeability equivalent permeability

Again assuming an infinite permeability of the iron parts of the magnetic core,

This yields

All magnetic quantities outside the magnet remain the same as in the case of the magnetic vector, but are shifted to the first quadrant of the magnetisation characteristic.

**This method is easy to implement** for rectangular magnets with a magnetisation parallel to two sides of the rectangle (Fig. 5.41).

**Fig. 5.41. Triangular finite element with magnetisation.**

**These ideas now can be transferred** to permanent magnets with an arbitrary shape. Therefore, current sheets are assumed on all sides of the finite element. After some elementary trigonometrically manipulations it can be written

with

In a similar way theare calculated. With the equivalent magnetisation vector

the currents are

**The other edge currents** can be calculated in a analogue way Ij. This procedure applied on an element by element base enables the construction of arbitrary shaped permanent magnet material.