# 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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