By R. Belmans
Field computation and numerical techniques
We consider that a variational principle or a boundary problem can describe a given physical-technical problem. Thus, this field problem is given by a differential equation. The problem is now to find a feasible solution of this differential equation.
Fig. 5.1 shows the various possibilities for solving general field problems. The methods applicable for use can be divided into two general classes, analytical and numerical methods. Methods which are based on simplified analytical models are called semi-numerical.
Fig. 5.1. General field analysis.
When compared to numerical techniques, analytical methods have the opportunity to deliver the exact solution of the differential equation. Those approaches, separation of the variables and Laplace transformations or other methods, can be applied to geometrically simple problem formulations only. Analytical approximations are suitable where the problem itself is very well known so that it is possible to apply appropriate simplifications. The entire above-mentioned reasons limit the general application range for analytical methods. They are very suitable for specific problems. If an analytical approach is applicable, the solution is obtained in a rather short time; this is another great advantage of the analytical techniques when compared to the numerical methods.
Numerical integration, Runge-Kutta, Euler and other techniques can obtain direct solutions from systems of partial differential equations. The finite difference method computes the solution by applying Taylor series to approximate the field quantities in points of a mesh grid covering the domain of interest. The finite element approach belongs to the discrete methods and will be the topic of the following topics. The discrete element methods have the ability to be used in general applications. Therefore, different types of problems can be solved by the same method. They can be solved employing the same numerical structure.
To solve a technical field problem numerically, an appropriate method has to be chosen. The most important methods are listed here:
• finite element method (FEM)
• finite difference method (FDM)
• boundary element method (BEM)
• magnetic equivalent circuit (MEC)
• point mirroring method (PMM).
Table 5.1. Numerical field computation methods.
Today, the FEM is a well known method and is commonly used for electromagnetic field problems. The two most popular methods for deriving the finite element equations are the variational and the Galerkin approach, being a special case of the method of weighted residuals. For a two-dimensional analysis, the domain of interest is discretised into a number of simple triangular or rectangular elements, the finite elements, with homogenous properties. For three-dimensional problems, tetrahedra or other simple volume elements are used. The potential function is approximated in those finite elements by simple shape functions, mainly linear or quadratic. This results in a large linear system of equations. Saturation effects can be considered easily. Using triangular elements for two-dimensional field problems and tetrahedra in three dimensions, a very good approximation of the geometry is obtained. The FEM is the most flexible method when compared to all the other techniques listed in Table 5.1.
Historically the FDM is the oldest method. Here, the domain of interest is discretised by a grid with discrete points. The differential equation of the particular field problem is locally transferred into a difference equation. This leads to a linear system of equations to be solved. The solution at the grid points approximates the field.
Due to the discretisation of the domain by a grid, in xy or polar coordinates, this method is in some cases troublesome in accurately approximating the geometry (Table 5.1). A local grid refinement to increase the solution accuracy can not be obtained in an efficient way. Non-linearities can easily be implemented using Newton iteration schemes. A three dimensional FDM is possible under the same restrictions as mentioned before. The FDM lost its importance and is nowadays used for problems in the time domain only and is still popular in fluid dynamics.
Using a particular approach at the boundaries of a field domain including the solution of the fundamental system (Green functions) of the partial differential equation represents the basic idea of the BEM. Using this connection, only the boundaries of the region of interest are discretised. Due to this discretisation the geometry of a domain can be approximated very accurately. The coefficient matrix of the BEM is, in contrast to the FDM and FEM, completely filled, non-symmetric and not positive definite. Therefore, special solvers have to be used for the resulting system of equations. Non-linearities are very difficult to account for.
The equivalence of the steady state electrical flow field and electromagnetic field is exploited in the MEC. This method recommends a principal knowledge of the field distribution. Here, the field domain is discretised by lumped parameter elements, representing the reluctances and sources of the problem. To obtain a potential at the nodes of this network, a rather small linear system of equations has to be solved. Therefore, the MEC is fast. The low level of discretisation, when compared to the FEM, results in an acceptable accuracy for field quantities. The computation of forces in electrical machines is troublesome, as a derivative of the reluctivity is required. Non-linearities can be implemented easily.
The PMM exploits the analogue formulations of the magnetostatic and electrostatic field problems. The PMM has its origin in analytical field calculations. Permanent magnets are considered by magnetic surface charges and are mirrored at the boundaries of the region of interest following the rules of the electrostatic field. This method is very fast, but restricted to very special geometries. Saturation can be considered by constant permeabilities only. An advantage using this method is the fact that special problems (3D high voltage transmission lines, 3D permanent magnet constructions) can be solved relatively easily and fast, whereas the FEM or other discretisation methods require huge efforts to define the problem, at high computational costs.
The advantages and disadvantages of the above-mentioned methods are collected in Table 5.1. In the following section examples are given for the MEC and PMM to demonstrate their strength in selected problem classes of the design of electromagnetic devices.
advantages |
disadvantages |
fast |
simple geometries only |
easy to implement |
flux paths must be known to build up the model |
non-linearities possible |
forcc computations are troublesome |
Computation of field quantities of an electromagnetic actuator
For the optimisation of magnetic circuits by numerical methods, fast field computation algorithms are recommended. If the field problem is not too complicated, the MEC can be employed. Here, the computation of an actuator with permanent magnets is discussed.
Using the formal equivalence of the electric flow field, the magnetic field components of an electromagnetic device can be obtained using a magnetic equivalent circuit. With the rules of the circuit theory, this mode] of the electromagnetic circuit is solved. Compared to the finite element method, this approach offers the ability to obtain accurate results with low computational costs. Inherent non-linearity due to the characteristic of the ferromagnetic parts in the magnetic circuit is implemented. Non-linearities caused by the relative displacement between moving parts are implemented as well. Results obtained by simulations are compared with measurements on a small permanent magnet-excited actuator (Fig. 5.2).
Fig. 5.2. Geometry and flux plot of the actuator computed by FEM.
Fig. 5.3. Complete magnetic equivalent circuit.
A two-pole diametrically magnetised permanent magnet rotor ring is centred in the stator bore. The two-phase armature winding is arranged in closed stator slots.
The equivalent magnetic circuit (Fig. 5.3) consists of magnetic resistors, defined by flux tubes, and flux and/or mmf sources. The solution of this field problem is the analysis of a non-linear network.
S(x) is the area perpendicular to the direction of the fluxat the position x whereare the magnetic potentials at both ends of the flux tube. The differencecorresponds to the magnetic voltage drop along the flux path. The magnetic resistorfor the equivalent magnetic circuit is
Since in iron parts of the electromagnetic device the permeability |i(x) is a function of the flux density, the field problem is non-linear. Permanent magnet material with its demagnetisation characteristic is included in the equivalent magnetic circuit as well. An evaluation of Ampere’s law leads to the mmf sources modelling the windings in an electromagnetic device.
To solve the network problem, a node-based method is used, enabling, when compared with branch-oriented algorithms, a more easy assembly of the node permeance matrix. The solution of three dimensional networks is possible as well. As the result of this network circuit analysis, the potentials at the nodes are obtained. From the known node potentials, the interesting field quantities can be derived. The advantages of the method used are:
• direct assembling of the system of equations
• diagonal dominant coefficient matrix
• sparse system
• no restrictions to planar graphs.
Fig. 5.4. a) Network and b) directed graphs.
The clearest and most flexible way to describe the structure of a network is by setting up a graph or mathematical topological matrices. The topology describes the properties of the network concerning its structure without considering the properties of the network elements. The structure is modelled by a directed graph (Fig. 5.4b). The incidence matrix A of the directed graph describes the topology of the network. Its columns indicate the branch number and its’rows the node number. The elements of
are
The subscript k denotes the number of nodes and z the number of branches.
After introducing the vectors,modelling the flux andgiving the mmf sources in the network, and with the diagonal matrix D representing the branch permeances, the complete system of equations is
In (5.4) the vectorcontains the required node potentials for further consideration. The system (5.4) can be solved either by direct or by iterative methods.
The network of a magnetic equivalent circuit consists mainly of non-linear elements where the permeance of a flux tube depends via the permeability p on the flux density. Therefore, the flux density B as a function of field strength H of such elements must be given and incorporated into the solution process. The Newton algorithm obtains the iterative solution. The iteration instruction for iteration step (k+1) is
whererepresents the solution vector containing the node potentials from iteration stepis the Jacobi matrix andthe fundamental system. To assemble the Jacobi matrix with the term the given non-linear material characteristicis evaluated by cubic spline interpolations. This enables the use of the term directly because the derivative is already included in the interpolation algorithm and thus available without additional numerical expenses.
When computing the magnetic field quantities mainly the generated forces and/or torques of an electromagnetic energy converter is of interest. The electromagnetic torque of the actuator is calculated using the energy principles according to virtual work.
Computations with increasing winding currents are performed to verify the accuracy of the magnetic equivalent circuit model at different saturation levels inside the iron parts. Fig. 5.5 shows a very good agreement between the methods. For the computed and measured torque versus position, a good agreement is found as well.
The computation time to solve the non-linear magnetic equivalent circuit with 210 elements (Fig. 5.3) is of the order of seconds.
Fig. 5.5. Air gap flux density with different winding currents.