chapter 3 magnetic circuit design and analysis using finite

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    CHAPTER 3

    Magnetic Circuit Design and Analysis using Finite Element Method

    3.1 Introduction

    In general, the Finite Element Method (FEM) models a structure as an assemblage of

    small parts (elements). Each element is of simple geometry and therefore is much easier to

    analyze than the actual structure. In essence, a complicated solution is approximated by a model

    that consists of piecewise continuous simple solutions. Elements are called finite to distinguish

    them from differential elements used in calculus. Discretization is accomplished simply by

    sawing the continuum into pieces and then pinning the pieces together again at node points [15].

    FEM is a better solution for electromagnetic circuit design for permanent magnet machines

    [16-23]. Partial Differential Equation (PDE) toolbox of Matlab is used for the design of

    topologies and for getting FEM solution for electromagnetic problems using magnetostatic

    application.

    Design of PMH stepper motor magnetic circuit using equivalent circuit model is difficult

    due to double slotting structure, presence of permanent magnet in the rotor and saturation effects.

    Hybrid stepper motor has a large number of teeth on the stator and rotor surface and a very small

    air gap; the magnetic saturation in the teeth becomes severe while increasing the flux density in

    the airgap. In addition, both radial flux and axial flux are produced because of axially

    magnetized permanent magnet and geometric characteristics [24]. This makes the analysis of

    hybrid stepper motor more difficult using two dimensional (2-D) modeling FEM. Three

    dimensional finite element analysis is one of the solution for nonlinear analysis of axially

    unsymmetrical hybrid stepper motor under this situation [25]. But in order to reduce the

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    computational time involved in the analysis, a 2-D equivalent of the three dimensional (3-D)

    model of the motor was developed and used. In contrast to other methods, the finite element

    method accounts for non-homogeneity of the solution region [26]. PMH stepper motor is

    designed in 2-D for different tooth widths but the design reduces steady state torque and

    increases cogging torque [27].

    This Chapter discusses about the need and fundamental concepts of FEM. Modeling

    aspects of a PMH stepper motor using FEM in 2-D and 3-D, their advantages and disadvantages

    are discussed. Boundary conditions of Neumann and Dirichlet are discussed. Creation of

    different types of mesh and refinement of mesh are discussed. Finally solution by partial

    differential equations (PDE) for the given motor magnetic circuit design using FEM is explained.

    Tooth layer unit (TLU) of PMH stepper motor, which is combination of stator and rotor

    tooth for one tooth pitch, is used for FEM analysis [28, 29]. 2-D Model is used for analysis to get

    magnetic potential and gap permeance using current density of exciting coil in the stator and

    permeability of core materials for stator and rotor [30-34].

    Partial differential equation (PDE) toolbox of Matlab is used to design eight topologies

    of PMH stepper motor [35, 36]. Magnetic potential for all of these eight topologies is evaluated

    using FEM for two core materials at two current densities for two permanent magnetic materials.

    These FEM results are used to obtain the best design which provides best magneto motive force

    (MMF) distribution for better steady-state and dynamic performances of PMH stepper motor.

    3.2 Concepts of Finite Element Method

    3.2.1 A Brief Note on Finite Element Method

    Finite Element Method (FEM) was first developed in 1943 by R. Courant, for application

    of the Ritz method of numerical analysis and minimization of variational calculus to obtain

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    approximate solutions for vibration systems. During early 70s, FEM was limited to expensive

    powerful computers generally owned by the aeronautics, defense and automotive industries.

    Since the price of computers has rapidly decreased with a significant increase in computing

    power, FEM has reached an incredible precision.

    FEM consists of a computer model of a material or design that is stressed or excited and

    analyzed for specific results. It involves dividing a given geometry into a mesh of small

    elements, solving for certain variables at the nodes of these elements, and then interpolating the

    results for the whole region. The size, shape and distribution of the elements determine the

    degree of the accuracy of the results.

    Computational time depends on the number of nodes and elements, and the finer the

    mesh, the longer it takes to solve the problem. Hence, there is a trade-off between accuracy and

    computing time. Generating an optimal mesh is a major topic and requires experience. The mesh

    should be fine enough for good detail with well-shaped elements where information is needed,

    but not too fine, or the analysis requires considerable computer time and memory. This can

    require considerable user intervention, despite FEM software claims of automatic good meshing.

    There are generally two types of analysis that are used in industry: 2-D modelling, and 3-D

    modeling. While 2-D modelling conserves simplicity and allows the analysis to be run on a

    relatively normal computer, it tends to yield less accurate results. On the other hand, 3-D

    modelling produces more accurate results while sacrificing the ability to run on all but the fastest

    computers effectively. Within each of these modelling schemes, the programmer can insert

    numerous functions which may make the system behave linearly or non-linearly. To summarize,

    in the finite element method, complexity of a problem is minimized by dividing the study

    domain into finite elements of simpler geometric shapes and then the partial differential

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    equations related to these elements are solved by the numerical techniques. The finite element

    analysis of a physical event consists of following general steps: [35]

    Representation of the physical event in mathematical model

    Construction of the geometry and its discretization to finite elements

    Assignment sources of excitation (if exist)

    Assignment of boundary conditions

    Derivation and assembling of the element matrix equations

    Solution of the equations for unknown variables

    Post processing or analysis of results obtained

    3.2.2 Basic Principle

    In the finite element method, unknown parameters are determined from minimization of

    energy function of the system. The energy function consists of various physical energies

    associated with a particular event. According to the law of conservation of energy, unless atomic

    energy is involved, the summation of total energies of a device or system is zero. On basis of this

    universal law, the energy function of the finite element model can be minimized to zero. The

    minimum of energy function is found by equating the derivative of the function with respect to

    unknown grid potential to zero i.e. if E is the energy function and A is the unknown grid

    potential, then the unknown potential A is found from the equation = 0. The solutions of

    various differential equations of physical models including electro-magnetic system are obtained

    using this basic principle.

    Since the model in this study has an iron material and is time invariant, the problem can

    be classified as nonlinear magneto-static one. Thus, the energy function E in this case is given by

    eqn (3.1) [35].

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    E = ( H. dBdV $ J. dA' ( dV (3.1) where

    V is the reluctivity in metre/Henrys (m/H) (inverse of permeability),

    H is the field intensity vector in Ampere/metre (A/m),

    B is the flux density vector in tesla (T = Wb/m2),

    J is the current density vector in Ampere/metre2 (A/m2),

    A is the magnetic vector potential in Tesla - metre or (Wb/m) and

    Az is the z-component of magnetic vector potential in Tesla - metre.

    The first term in eqn (3.1) is the energy stored in saturable linear or nonlinear materials,

    and the second term is the input electrical energy. If the permeability is not constant, then the

    stiffness matrix depends on the magnitude of B and J.

    3.3 Tooth Layer Unit of PMH Stepper Motor for FEM Analysis

    Tooth layer unit (TLU) is a rectangle area that has a tooth pitch width and two parallel

    lines behind the teeth of stator and rotor as shown in Fig. 3.1.The factors of the nonlinear

    material and the non-uniform distribution of magnetic field in the teeth of stator and rotor are

    taken full consideration in this computation model. The following are the two basic assumed

    conditions in the computation model of TLU

    1. The lines ab and cd of the TLU in Fig. 3.1 are considered as iso-potential lines.

    2. The magnetic edge effect of stator pole is ignored, which is assumed that the distribution of

    the magnetic field for every tooth pitch width is the same.

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    Fig. 3.1 Tooth layer unit Of PMH stepper motor

    If us and ur are respective scalar quantities of the iso-potential lines ab and cd, the

    magnetic potential difference A is shown in eqn (3.2).

    A = us ur (3.2)

    If () is assumed as the flux in a tooth pitch width per axial u

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