Numerical implementation of the FEM (Electrical Machine) Part 2

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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 densitytmp10924_thumb[2]and its coercive forcetmp10925_thumb[2]With the known material equation

tmp10928_thumb[2]

the reversible permeability

tmp10929_thumb[2]

can be determined. Linear permanent magnets are described by:

tmp10930_thumb[2]

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

Demagnetisation characteristic of a linear permanent magnet.

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

tmp10932_thumb[2]

with

tmp10933_thumb[2]

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

tmp10934_thumb[2]

Applying Green’s formula gives:

tmp10935_thumb[2]

This results in an additional term in the source vectortmp10936_thumb[2]

tmp10937_thumb[2]

Radial magnetisation

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

Radial magnetisation of magnet material within an element.

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

tmp10939_thumb[2]

with

tmp10940_thumb[2]

Binary constraints

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

tmp10941_thumb[2]

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:

tmp10942_thumb[2]tmp10943_thumb[2]

Elimination oftmp10944_thumb[2] using (5.145) gives:

tmp10945_thumb[2]

Symmetrising the matrix

tmp10946_thumb[2]tmp10947_thumb[2]

The values oftmp10948_thumb[2]are calculated with eq.(5.145) after solving the system of equations. Fig. 5.47 shows the matrix assembling.

Assembling of the coefficient matrix.

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:

tmp10952_thumb[2]

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:

tmp10953_thumb[2]

withtmp10954_thumb[2]as 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:

tmp10956_thumb[2]

This yields:

tmp10958_thumb[2]

the Jacobian matrix from iteration step (k) with tmp10959_thumb[2]

tmp10960_thumb[2]

The defect vector can now be evaluated by using:

tmp10961_thumb[2]

The iteration rule to computetmp10962_thumb[2]can be given by:

 

tmp10963_thumb[2]

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

tmp10964_thumb[2]

and afterwards evaluating

tmp10965_thumb[2]

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:

tmp10966_thumb[2]

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 matrixtmp10967_thumb[2]corresponding with the element matrix

tmp10969_thumb[2]

an arbitrary vector E can be defined with the elements:

tmp10971_thumb[2]

to obtain a more convenient formulation for the Jacobian matrix.

tmp10972_thumb[2]

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

tmp10973_thumb[2]

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

tmp10974_thumb[2]

which yields:

 

tmp10975_thumb[2]

Elimination oftmp10976_thumb[2]using (5.125) gives

tmp10977_thumb[2]

and symmetrising the matrix by

tmp10978_thumb[2]yields and symmetrising the matrix by

tmp10979_thumb[2]

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 factortmp10980_thumb[2]

tmp10982_thumb[2]

The idea is to accept the iteratedtmp10984_thumb[2]only if the damping criterion

 

tmp10986_thumb[2]

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

tmp10988_thumb[2]

with the parameterstmp10989_thumb[2]for the damping stepstmp10990_thumb[2]is satisfied or a maximum number of steps is performed without successtmp10991_thumb[2] The parameters are chosen in the range of:

tmp10998_thumb[2]

Typical values aretmp10999_thumb[2]The 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 stepstmp101000_thumb[2]can be implemented to try to recover convergence.