By R. Belmans
advantages 
disadvantages 
relatively fast 
nonlinearities only considered by constant 
factors 

3D fields 
special geometries only 
special boundary conditions have to be assumed 
To demonstrate the strength and shortcomings of this method, on the other hand, two examples are worked out. In the first example a ferromagnetic circuit is calculated. The main limitations in this example are seen in the assumption that the iron circuit is piecewise constant saturated. In contrast to this shortcoming, the geometry of the disctype motor is rather complicated and requires a threedimensional calculation of the field. The assumption of a plane mirror surface can be seen as a very strong boundary condition thus limiting the application range. Therefore, no slots are allowed in this type of model. An air gap winding is recommended in order to be able to compute the air gap field in this machine.
The second example, a high voltage transmission line, is due to the slag of the line, an inherently threedimensional field problem as well, but in this case linear. The assumption of a plane mirror surface, the ground plane below the line, is here the strong limitation. As a consequence, the field can not be calculated in a hilly neighbourhood. Nevertheless, measurements and computations are in good agreement.
Computation of the field quantities of a disctype motor
In this section the basic ideas of the method of point mirroring are given. With a relatively small amount of computation time the field quantities of very complicated geometries can be studied. The method permits threedimensional magnet field calculations. Here, the method is used to calculate the flux distribution of a disc type motor. The rotor of the motor consists of a NdFeB permanent magnet ring. An air gap armature winding is fixed to the stator. Fig. 5.6 shows the construction of the motor.
Fig. 5.6. Construction of the studied disc type motor.
Four poles (2p=4) are axially magnetised on both rotor magnet rings. The used magnet material is NdFeB. In the case of rare earth material, the assumption of a straightlined characteristic of the demagnetisation curve of the magnet material is realistic. In the operating range of the magnet, the magnetisation is almost independent of the demagnetising field strength.
The magnetisation of a volume is the sum of all dipole moments m{ divided by the volume v. Two different computational models, respectively representations for the magnetisation M are possible:
• distributed currents (Fig. 5.7),
• distributed magnetic charge (Fig. 5.8).
The magnetisation is the effect of all elemental currents inside a magnetic medium. The circulating current of one dipole cancels the current of the neighbouring dipole, if the dipoles are parallel and have the same magnitude. If, further, all dipole momentsare uniformly distributed throughout the volume, all volume currentsvanish except the current at the surface, the surface current density(Fig. 5.7). In this model, the distribution of magnetic dipole moments is equivalent to the distribution of currents at the surface of a magnetic medium and within the volume.
With the Maxwell equations the static magnetic field can be expressed as a solenoid field of flux density and the curl of the magnetic field strength:
Fig. 5.7. Magnetic magnetisation M with cunent model.
Under the assumption of no field exciting currents inside the permanent magnets, it can be written:
With vanishingformally the magnetic field strength can be described as a gradient field of a scalar potential function.
Using the demagnetisation characteristic of permanent magnet materialit can be formulated
yielding
Formallyis appropriate to the Poissondifferential equation of the electrostatic field. Analogous to the electric spacecharge density and to the electric surface charge an auxiliary magnetic quantity can be defined.
With known inner magnetisation M of the magnet, the divergence of the magnetisation can be identified as an auxiliary magnetic surface charge
Fig. 5.8. Permanent magnet shape with auxiliary magnetic surface charge.
As shown in Fie. 5.8 for the upper pole surface, this yields
and for the lower pole surface
In this consideration, the laws of electrostatic field can be used to evaluate the scalar potentialand magnetic field strength H(P) in a point P. Integration is performed over the surface of the north AN and the south pole surface As
To determine the flux density of the ringformed permanent rotor of the mentioned disctype motor, the ring is subdivided into trapezoidal single magnet elements (Fig. 5.9).
Fig. 5.9. Permanent magnet ring.
The superposition of elementary and simple magnet shapes is performed in order to form a complicatedshape permanent magnet. Here, the superposition of the field components of cubed and triangle magnet elements result in the required trapezoidal magnet as indicated in Fig. 5.10. The magnetic field excited by the permanent magnet ring can now be calculated at every point P outside the magnet volume.
To consider the ferromagnetic back iron inside the real machine, the magnetic surface charges have to be minrored at the boundaries of the air gap. The laws governing the electrostatic field can be used here. A mirror interval of 4 to 5 steps is sufficient to obtain an acceptable accuracy (Walkhoff "’).
Fig. 5.10. Superposition of cube and triangle to obtain a trapezoidal shaped magnet.
With these considerations, the auxiliary configuration for the calculation of the air gap field of the discshaped motor, using the point mirroring method, can be constructed (Fig. 5.11).
Fig. 5.11. Auxiliary configuration and coordinate system.
Mirroring the mentioned surface charges is done at the material boundaries having permeabilities(Fig 5.11). Saturation is considered by the constant factors.
The zcomponent of the resulting flux density distribution of a pole pitch of the mentioned disctype motor can be taken from Fig. 5.12.
Fig. 5.12. Threedimensional air gap flux density distribution.
Computation of the fields below AC high voltage lines
Overhead transmission lines generate in their vicinity electric and magnetic fields. The source of the magnetic field is the current in the phase conductors. The electric field is caused by the high potential at the phase conductors.
The problem specifies small diameter conductors above a large flat conducting ground plane. The phase conductors are at a timedependent specified electrical potential and cany a timedependent current. Due to the slag of the phase conductors, the field problem turns out to be threedimensional. Only symmetric threephase voltage and current systems are considered. The ground below the transmission line is a uniform plane.
The field problem may be considered as quasi static. Therefore, the solution can be determined by static techniques. With respect to the slag of the phase conductors, infinitesimally thin, segmented filaments approximate the geometry of a single conductor. Due to the symmetry between two poles, one half of the arrangement is drawn in Fig. 5.13 only. The value of s indicates the slag, / is the distance between the two high voltage poles, the span field length.
Fig. 5.13. Geometric modelling of a conductor.
Electric field
The electric field is computed by mirroring single line charges at the assumed to be ideal conducting ground plane below the phase conductors. Each infinitesimally thin filament segment represents in this case a linecharge. A constant linecharge at any position in the original coordinate system (x, y, z) is drawn in Fig. 5.14. To evaluate the field quantities of the linecharge, this coordinate system has to be transformed into a systemThis transformation is performed in two steps. The first step consists of a parallel shift of the origin into the starting point of the linecharge. In a second step a rotation of this temporary coordinate systemaround theaxis is carried out in such a way that the line charge lies in the plane. The last rotation in this step is around theso that the linecharge lies in the x°axis. In this coordinate system the potential cp of the linecharge in the pointis given by:
Fig. 5.14. Coordinate transformation of an infinitesimally thin filament segment.
To evaluate the field quantities with respect to this boundary condition, the linecharge has to be mirrored with respect to the plane xy. Superposition of linecharge q and mirrorcharge q, indicated in Fig. 5.14, gives the potential tp at the point ?(x,y,z) inside the global coordinate system.
To consider the slag of the conductors, a quadratic approximation is used. Referring to Fig. 5.13, it can be written as
With the known complex potentialsof the i conductors and transforming eq. (5.15) to compute the coefficient matrix A, a linear set of equations can be formulated.
The solution determines the chargeof each element of the conductors. With these values the components of the electrostatic field strength in the point P(x, y, z) can be computed.
Fig. 5.15. Convergence of the simulation with respect to the slag of the hightension line.
To illustrate the convergence behaviour of the method, attention should be paid to Fig. 5.15. With an increasing number of infinitesimally thin filament segments for one half of the span field, the electrostatic field strength converges to the correct value. Calculations with 5…7 polygon elements deliver results with a reasonable accuracy at acceptable computational costs.