**By R. Belmans**

## Permanent magnet material

**It is assumed that the** demagnetisation characteristic in the second quadrant of the magnetisation curve of the considered magnet material is linear (Fig. 5.45). This assumption is realistic for most of the modern rare earth magnets such as the grades of NdFeB, SmCo or the Strontium or Barium Ferrites. The permanent magnet is characterised by its remanence flux densityand its coercive forceWith the known material equation

the reversible permeability

can be determined. Linear permanent magnets are described by:

This results in an extra exciting term in the differential equation and in the functional.

**Fig. 5.45. Demagnetisation characteristic of a linear permanent magnet.**

with

This functional is valid for magnetostatic problems where a magnetic field is imposed by currents and permanent magnets.

### Functional within an element

**The third term in the** functional within an element becomes

**Applying Green’s formula gives:**

This results in an additional term in the source vector

### Radial magnetisation

**The direction of the magnetisation for** a radial magnetised permanent magnet must be calculated for each element separately (Fig. 5.46).

**Fig. 5.46. Radial magnetisation of magnet material within an element.**

with

## Binary constraints

Binary constraints enforce a relation between values of the magnetic vector potential at a first boundary and values at a second boundary.

Nodes belonging to the first boundary are indicated as binary constrained (index 1), while nodes belonging to the second boundary are seen as free nodes (index 2), yielding the system of linear equations in matrix-vector notation:

**Elimination of**** using (5.145) gives:**

Symmetrising the matrix

The values ofare calculated with eq.(5.145) after solving the system of equations. Fig. 5.47 shows the matrix assembling.

**Fig. 5.47. Assembling of the coefficient matrix.**

## Non-linear materials

**To implement the ferromagnetic properties of iron,** the non-linear characteristics have to be considered. This results in a system of nonlinear equations and this system can not be solved in a closed form. A numerical iteration scheme has to be used to obtain a field solution with the presence of ferromagnetic material. To obtain a solution of such a non-linear system, the Newton iteration can be used.

### Newton iteration

**The Newton iteration is a fast approach.** Applying appropriate start solutions, its rate of convergence is locally of quadratic order. The numerical solution of a non-linear system is transferred to a series of linear solutions, i.e. in every iteration step a linear system of equation is solved. Various modified Newton methods are in common use.

**It is assumed to have the known system of equations:**

with K the coefficient matrix, A the vector potential and R the right hand side; we call this the fundamental system F(A) or residual. When compared to the set of linear equations, in the non-linear case the coefficients of the system matrix are dependent on the solution vector A, The difference between a solution in iteration step (k) and (k+1) is:

withas the defect vector. The defect is used as a stopping criterion for the iterations.

**The fundamental system is expressed by a Taylor series neglecting the second order and higher terms:**

This yields:

the Jacobian matrix from iteration step (k) with

**The defect vector can now be evaluated by using:**

The iteration rule to computecan be given by:

For practical considerations the formulation of the iteration in this way is not very useful. But solving the system

and afterwards evaluating

results in the overall solution of the non-linear problem. If the start solution is not chosen close to the exact solution of the system, the iteration may oscillate, diverge and fail.

### Jacobian matrix and final system of equations

**The Jacobian can be assembled by applying:**

The first term corresponds to the linear equations, while the second term exists only in the presence of a non-linear material. By using an arbitrary matrixcorresponding with the element matrix

an arbitrary vector E can be defined with the elements:

to obtain a more convenient formulation for the Jacobian matrix.

**The system of equations to obtain the defect can be written by:**

**The source vector S for the calculation of the residual is introduced as:**

which yields:

**Elimination of****using (5.125) gives**

and symmetrising the matrix by

yields and symmetrising the matrix by

**Damped Newton iteration** For the Newton method, an appropriate step length of the algorithm has to be chosen to save the iteration from divergence. The damped Newton iteration is a variant of the original method. The defect vector, Newton correction, is damped by a factor

The idea is to accept the iteratedonly if the damping criterion

is satisfied to ensure convergence. If this condition is not satisfied, damping steps must be performed until

with the parametersfor the damping stepsis satisfied or a maximum number of steps is performed without success The parameters are chosen in the range of:

Typical values areThe maximum number of steps is typically between 5 and 10.

This scheme represents an adaptive damping factor for the Newton iteration. If further problems occur, and the solution already diverges; special algorithms such as gradient stepscan be implemented to try to recover convergence.