3.2 Physical basis of FEMM calculations
3.2.3 Procedure of FEM utilization
In general, most software for FEM analysis consists of three main parts:
• Preprocessor: preparation and evaluation of the analyzed model
• Solver (processor): assemblage of differential equation system and its solving.
• Postprocessor: analysis of results and calculation of further required parameters
All parts here are described and applied to the FEMM program.
3.2.3.1 Preprocessor
It is a mode or part of the program, where the user creates a model with finite elements. This module contains more sub-modules or other parts. At first, geomet- rical dimensions of model (mm or English units) are defined. It is chosen between 2D and 3D modeling. In this chapter, FEMM program is able to do only 2D analysis.
This is satisfactory in most cases of electrical machines analysis, and calculation time of evaluation is significantly shorter than in 3D. If the user disposes software with 3D, it is required to regard on the manual provided by the software.
Options can be selected between planar x, y coordinates (2D) and axisymmetric system, where it can work with polar coordinates. This system is suitable to solve, e.g., cylindrical coils. In most of the electrical machines, the planar system is used, and also the z-coordinate is selected.
• Drawing mode: This mode is used to draw models, which will be solved. It uses specific points given by their x, y coordinates. These points are connected with
ferromagnetic core. Among such systems are, e.g., electromagnet of contactor and electrical device based on variable reluctance principle (reluctance synchronous machine, switched reluctance machine, see Section 7 and [18–20]). Force calcula- tion is defined by small difference (derivation) of coenergy W´ with respect to small difference (derivation) of position (deflection) x in linear motion orϑin calculation of torque. Thus for force, it is defined as:
F¼dW0 dx ≈ΔW0
Δx (474)
and the torque is defined as:
T¼dW0 dϑ ≈ΔW0
Δϑ : (475)
As it was mentioned above, for calculation of one force or torque value, two FEM calculations are needed, because difference between two coenergies is
required. Setup of this position or deflection step is important. When step is too big, calculated value of force or torque can be inaccurate. Setting of step must be adjusted predictably according to the problem, which is being solved. In linear motion, it can be, e.g.,Δx = 1 mm, in rotating machines single mechanical degree (Δϑ= 1°). Instantaneous value of torque can be calculated as:
T¼∂W0ði,ϑÞ dϑ
i¼const¼ �∂W0ði,ϑÞ dϑ
ψ
¼const:
(476)
3.2.1.3 Lorentz force equation for force and torque calculation
By using Lorentz’s law to calculate force or torque, instantaneous value of torque can be obtained as function of phase induced voltage values and phase current values for three-phase system as sum of products of instantaneous values in all three phases:
T¼ 1
Ω½uiAð Þit Að Þ þt uiBð Þit Bð Þ þt uiCð Þit Cð Þt� (477) where ui(t) are instantaneous induced voltages of three phases, i(t) are instantaneous values of the current, andΩis mechanical angular speed.
3.2.2 Calculation of inductance by means of FEM
The method of steady-state inductance calculation is shown in this chapter.
In electrical machines, it can be calculation of the self-inductance of phase, leakage inductance, armature reaction inductance, or magnetizing inductance. We can use two ways to calculate these:
• Inductance calculation for linear systems from magnetic field energy
• Inductance calculation for nonlinear systems from flux linkage
In linear systems, inductance calculation can be carried out by electromagnetic field energy evaluation, or by coenergy, because these are equal to the linear cases.
It is assumed that electrical machine represents linear system, when operating in
linear area of B-H characteristic. This means that ferromagnetic circuit is not saturated. Calculation is done by equation:
W ¼W0¼1
2LI2)L¼2W
I2 (478)
where electromagnetic field energy is defined as:
W ¼ ð
V
ðB
0
H�dB 2
4
3
5dV (479)
and V is volume, where the electromagnetic field energy is stored.
System is considered as nonlinear, if the machine operates in nonlinear area of B-H characteristic. In nonlinear systems, inductance calculation can be carried out based on flux linkage evaluation, using Stokes theorem and magnetic vector potential:
L¼ψ I ¼
Ð
S∇�A�dS
I ¼∮A�dl
I ¼∮A�JdV
I2 (480)
Utilization of these equations and calculation of all parameters are shown in the next chapters.
3.2.3 Procedure of FEM utilization
In general, most software for FEM analysis consists of three main parts:
• Preprocessor: preparation and evaluation of the analyzed model
• Solver (processor): assemblage of differential equation system and its solving.
• Postprocessor: analysis of results and calculation of further required parameters
All parts here are described and applied to the FEMM program.
3.2.3.1 Preprocessor
It is a mode or part of the program, where the user creates a model with finite elements. This module contains more sub-modules or other parts. At first, geomet- rical dimensions of model (mm or English units) are defined. It is chosen between 2D and 3D modeling. In this chapter, FEMM program is able to do only 2D analysis.
This is satisfactory in most cases of electrical machines analysis, and calculation time of evaluation is significantly shorter than in 3D. If the user disposes software with 3D, it is required to regard on the manual provided by the software.
Options can be selected between planar x, y coordinates (2D) and axisymmetric system, where it can work with polar coordinates. This system is suitable to solve, e.g., cylindrical coils. In most of the electrical machines, the planar system is used, and also the z-coordinate is selected.
• Drawing mode: This mode is used to draw models, which will be solved. It uses specific points given by their x, y coordinates. These points are connected with
lines or curves, aiming to obtain required geometrical shape. In this part of preprocessor, the model can be imported also from CAD programs in dxf format.
• Materials definition: In this mode, used materials of drawn models are defined by specific quantities. Each part of the model must be outlined; thus it is clearly specified which space responds to the given material. In electrical machines materials as: air, insulation, ferromagnetic materials for magnetic circuits of cores, stators and rotors (mostly defined by B-H- characteristic), electric current conductors are mostly made of copper or aluminum materials,
ferromagnetic materials for shafts and permanent magnets, are employed. The definition of these materials is shown in each chapter, related to particular examples. FEMM program offers a library, which contains various materials, and it is possible to define these individually, according to user’s requirements and needs.
• Excitation quantities definition: In this mode, excitation quantities are defined.
These are current density, currents, or voltages, which respond to the
respective parts of model (slot, winding, coil). These values must be calculated directly or obtained for a particular state of the electrical machine (e.g., no load condition).
• Boundaries definition: Analysis and solving of models in FEMM use two important boundary conditions: Dirichlet boundary conditions, which are defined as the constant value of magnetic vector potential A = const. This is used mostly on the surface of electrical machines, where A = 0 is supposed (Figure 42a). Neumann boundary conditions are used for solving symmetrical machines, where we can analyze one quarter only or smaller part which replicates in the model. It is suspected for this condition that normal derivation of magnetic vector potential is zero:∂A/∂n = 0 (Figure 42b).
• Finite element mesh creation: This comprises the FEMM method principle. As titled, the created model is split into too many parts by finite elements. This finite number of elements covers the whole model. The more complicated the shape of the model is, the higher number of elements is required. These elements are mostly triangles, yet some software can use other shapes. By first mesh generating, the program creates a certain number of elements of certain size. This number and size can be set up by the user. It is important that dependence between number and size of elements, calculation speed, and accuracy is not linear. It is not obvious to get more accurate solution by high number of small-size elements. This can be compared to the magnetization curve of ferromagnetic material, where the number of elements is on x-axis and accuracy of solution is on y-axis. This setting is to be done by the user according to the experiences and requirements of a particular model. In each of finite elements, magnetic vector potential A is calculated.
These modes are defined as preprocessor, and we can now continue to the solver part.
3.2.3.2 Solver (processor)
In this part of the program, a system of partial differential equations is created in each node of triangle, i.e. finite element, where vector magnetic potential values are
being solved. Based on the numerical solution of equations utilizing defined itera- tion methods, the program solves this system of partial differential equations.
Result can converge (favorable case), where suspected solution can be obtained, or diverge, where program cannot calculate successful solution. In such case, it is necessary to revise the model and make corrections. As stated above, this part of program will take time according to the selected number and size of finite elements.
3.2.3.3 Postprocessor
In this last part, results of FEMM can be analyzed. The first result is the distribution of magnetic field in the model by means of magnetic field flux line mapping (equipotential lines). This mode allows to calculate other quantities on a particular point, defined line, or curve (by means of space integral calculation). Thus, the following can be obtained: magnetic flux density (normal and tangential compo- nent), magnetic field strength, potential, torques, energies, inductances, losses, etc.
According to this description, FEMM program will be used to analyze and calculate parameters in individual electrical devices and electrical machines. We will start with Section 3 in this chapter, where we calculate the force and inductance of a contactor electromagnet. In Section 4, we will calculate equivalent circuit parameters of single-phase transformer (mostly inductances). In Section 5, we will show calculation of parameters and torques of asynchronous machines. In Section 6, synchronous machines are analyzed, and finally in Section 7, switched reluctance machines are analyzed.
Figure 42.
Boundary conditions: (a) homogenous Dirichlet condition and (b) Neumann condition.
lines or curves, aiming to obtain required geometrical shape. In this part of preprocessor, the model can be imported also from CAD programs in dxf format.
• Materials definition: In this mode, used materials of drawn models are defined by specific quantities. Each part of the model must be outlined; thus it is clearly specified which space responds to the given material. In electrical machines materials as: air, insulation, ferromagnetic materials for magnetic circuits of cores, stators and rotors (mostly defined by B-H- characteristic), electric current conductors are mostly made of copper or aluminum materials,
ferromagnetic materials for shafts and permanent magnets, are employed. The definition of these materials is shown in each chapter, related to particular examples. FEMM program offers a library, which contains various materials, and it is possible to define these individually, according to user’s requirements and needs.
• Excitation quantities definition: In this mode, excitation quantities are defined.
These are current density, currents, or voltages, which respond to the
respective parts of model (slot, winding, coil). These values must be calculated directly or obtained for a particular state of the electrical machine (e.g., no load condition).
• Boundaries definition: Analysis and solving of models in FEMM use two important boundary conditions: Dirichlet boundary conditions, which are defined as the constant value of magnetic vector potential A = const. This is used mostly on the surface of electrical machines, where A = 0 is supposed (Figure 42a). Neumann boundary conditions are used for solving symmetrical machines, where we can analyze one quarter only or smaller part which replicates in the model. It is suspected for this condition that normal derivation of magnetic vector potential is zero:∂A/∂n = 0 (Figure 42b).
• Finite element mesh creation: This comprises the FEMM method principle. As titled, the created model is split into too many parts by finite elements. This finite number of elements covers the whole model. The more complicated the shape of the model is, the higher number of elements is required. These elements are mostly triangles, yet some software can use other shapes. By first mesh generating, the program creates a certain number of elements of certain size. This number and size can be set up by the user. It is important that dependence between number and size of elements, calculation speed, and accuracy is not linear. It is not obvious to get more accurate solution by high number of small-size elements. This can be compared to the magnetization curve of ferromagnetic material, where the number of elements is on x-axis and accuracy of solution is on y-axis. This setting is to be done by the user according to the experiences and requirements of a particular model. In each of finite elements, magnetic vector potential A is calculated.
These modes are defined as preprocessor, and we can now continue to the solver part.
3.2.3.2 Solver (processor)
In this part of the program, a system of partial differential equations is created in each node of triangle, i.e. finite element, where vector magnetic potential values are
being solved. Based on the numerical solution of equations utilizing defined itera- tion methods, the program solves this system of partial differential equations.
Result can converge (favorable case), where suspected solution can be obtained, or diverge, where program cannot calculate successful solution. In such case, it is necessary to revise the model and make corrections. As stated above, this part of program will take time according to the selected number and size of finite elements.
3.2.3.3 Postprocessor
In this last part, results of FEMM can be analyzed. The first result is the distribution of magnetic field in the model by means of magnetic field flux line mapping (equipotential lines). This mode allows to calculate other quantities on a particular point, defined line, or curve (by means of space integral calculation). Thus, the following can be obtained: magnetic flux density (normal and tangential compo- nent), magnetic field strength, potential, torques, energies, inductances, losses, etc.
According to this description, FEMM program will be used to analyze and calculate parameters in individual electrical devices and electrical machines. We will start with Section 3 in this chapter, where we calculate the force and inductance of a contactor electromagnet. In Section 4, we will calculate equivalent circuit parameters of single-phase transformer (mostly inductances). In Section 5, we will show calculation of parameters and torques of asynchronous machines. In Section 6, synchronous machines are analyzed, and finally in Section 7, switched reluctance machines are analyzed.
Figure 42.
Boundary conditions: (a) homogenous Dirichlet condition and (b) Neumann condition.