Download - CHAPTER 3 Magnetic Circuit Design and Analysis using Finite

24

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

25

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

26

approximate solutions for vibration systems. During early 70’s, 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

27

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 δδ = 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].

28

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.

29

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 Ф (α) is assumed as the flux in a tooth pitch width per axial unit length of iron core and

α is the relative position angle between stator and rotor, then the specific magnetic conductance

G of TLU is shown in eqn (3.3).

G = Фα (3.3)

Apparently, G is related to the saturation of iron core and is changed with A. The relative

position angle α can be obtained by the numerical computation on the magnetic field of TLU.

The lines ac and bd are the periodic boundary lines because the distribution of the magnetic field

is same for every tooth pitch width. The magnetic field in TLU is irrational field and the

magnetic equations for the field are given in the rectangular coordinates as shown in eqn (3.4).

30

)***+ ,

,- .µ /0/12 3 ,

,4 .µ /0/52 6 0

89 6 08: 6 08 x, y9 6 8 x 3 T?, y:@AAAB (3.4)

where φ is the scalar quantity, µ is the magnetic permeability and TP is the tooth pitch. For a

certain position angle α and a magnetic potential difference A, the distribution of the magnetic

field of TLU can be calculated by the 2-D finite element analysis in x, y directions. The flux per

axial length of TLU is as shown in eqn (3.5).

φα, A 6 ∑ B DmFGGGG (3.5)

Here the nodes j and m are on the border ab as shown in Fig. 3.1 HIFGGGGG is the length of

unit e from node j to m and Be is the flux density.

3.4 The Partial Differential Equation (PDE) Toolbox of Matlab for FEM Analysis The Partial Differential Equation (PDE) Toolbox provides a powerful and flexible environment

for the study and solution of partial differential equations in two space dimensions and time. The

equations are discretized by the Finite Element Method (FEM). The objectives of the PDE

Toolbox are mentioned below [36]

• Define a PDE problem, i.e., define 2-D regions, boundary conditions and PDE coefficients.

• Numerically solve the PDE problem, i.e., generate unstructured meshes, discretize the equation

and produce an approximation to the solution.

• Visualize the results.

This invokes the graphical user interface (GUI), which is a self-contained graphical

environment for PDE solving. Advanced applications are also possible by downloading the

domain geometry, boundary conditions and mesh description to the MATLAB workspace. From

the command line, (or M-files) functions are called from the toolbox to execute the works, like

31

generation of meshes, discretization of problem, performing interpolation, plotting data on

unstructured grids etc.

The basic equation of the PDE Toolbox is shown in eqn (3.6) as

$∇ . c∇u 3 au 6 f Ω (3.6)

which is referred as the elliptic equation, regardless of whether its coefficients and boundary

conditions make the PDE problem elliptic in the mathematical sense. Analogously, the terms

parabolic equation and hyperbolic equation are used for equations with spatial operators like the

one above and first and second order time derivatives respectively. In eqn (3.6), Ω is a bounded

domain in the plane: c, a, f, and the unknown u are scalar, complex valued functions defined on

Ω. ’c’ can be a 2-by-2 matrix function on Ω. The toolbox can also handle the parabolic PDE,

hyperbolic PDE and the eigen value problem shown in eqns (3.7), (3.8) and (3.9) [36].

d /N/ $ ∇. c∇u 3 au 6 f (3.7)

d /ON/PO $ ∇. c∇u 3 au 6 f (3.8)

$∇. c∇u 3 au 6 λ du (3.9)

where d is a complex valued function on Ω, and λ is an unknown eigenvalue. For the parabolic

and hyperbolic PDE the coefficients c, a, f, and d can depend on time. A nonlinear solver is

available for the nonlinear elliptic PDE shown in eqn (3.10)

$∇. c u ∇u 3 a u u 6 fu (3.10)

where c, a, and f are functions of the unknown solution u.

All solvers can handle the system case .using eqns (3.11) and (3.12) with systems of

arbitrary dimension from the command line. For the elliptic problem, an adaptive mesh

refinement algorithm is implemented. It can also be used in conjunction with the nonlinear

solver. In addition, a fast solver for Poisson’s equation on a rectangular grid is available

32

$∇. cQQ∇uQ 3 ∇. cQ∇u 3 aQQuQ 3 aQu 6 fQ (3.11)

$∇. cQ∇uQ 3 ∇. c∇u 3 aQuQ 3 au 6 f (3.12)

The following boundary conditions are defined for scalar u shown in eqns (3.13) and (3.14).

• Dirichlet boundary condition:

h. u = r, on the boundary ∂Ω (3.13)

• Generalized Neumann boundary condition:

ñ. c∇u + qu = g, on the boundary ∂Ω (3.14)

where g, q, h, and r are complex valued functions, ñ is the outward unit normal defined on ∂Ω .

(The eigenvalue problem is a homogeneous problem, i.e., g = 0, r = 0.) In the nonlinear case, the

coefficients, g, q, h, and r can depend on u, and for the hyperbolic and parabolic PDE, the

coefficients can depend on time. For the two-dimensional system case, Dirichlet boundary

condition is shown in eqns (3.15) and (3.16).

h11u1 + h12u2 = r1 (3.15)

h21u1 + h22u2 = r2 (3.16)

The generalized Neumann boundary condition is shown in eqns (3.17) and (3.18).

$ñ. cQQ∇uQ 3 ñ. cQ∇u 3 qQQuQ 3 qQu 6 gQ (3.17)

$ñ. cQ∇uQ 3 ñ. c∇u 3 qQuQ 3 qu 6 g (3.18)

The mixed boundary condition is shown in eqns (3.19) and (3.20).

$ñ. cQQ∇uQ 3 ñ. cQ∇u 3 qQQuQ 3 qQu 6 gQ 3 hQQµ (3.19)

$ñ. cQ∇uQ 3 ñ. c∇u 3 qQuQ 3 qu 6 g 3 hQµ (3.20)

where µ is computed such that the Dirichlet boundary condition is satisfied. Dirichlet boundary

conditions are also called essential boundary conditions and Neumann boundary conditions are

also called natural boundary conditions [36].

33

The process of defining a problem and solving it is reflected in the design of the GUI. A

number of data structures define different aspects of the problem, and the various processing

stages produce new data structures out of old ones. Fig. 3.2 shows flow chart of this process. The

rectangles are functions, and ellipses are data represented by matrices or M-files. Arrows

indicate data necessary for the functions. Here geometry parameters of the problem are fed in

matrices and DECSG (Description of Constructive Solid Geometry) constructs solid geometry of

the given problem. These matrices are decomposed according to different structures using

Decomposed Geometry Matrix and saved as M-file. Mesh is generated for the designed

geometry using Intimesh and mesh diagram is plotted using Mesh Data. If this mesh is not giving

satisfied PDE results, it is refined using Refine mesh. Boundary conditions are fed for required

analysis like one pole pitch in Boundary Condition Matrix and saved as M-file. PDE coefficients

are fed according to the given problem like permeance, current density for Magnetostatic

problem in Coefficient Matrix and saves as M-file. PDE solution data is executed like magnetic

potential for Magnetostatic problem using Assemble PDE and PDE plot is obtained in 2-D.

Magnetostatics application of PDE toolbox is used for magnetic circuit design of any

machine. The “statics” implies that the time rate of change is slow, so it is started with

Maxwell’s equations for steady cases are shown in eqns (3.21) and (3.22) [36].

∇ × H = J (3.21)

∇ ⋅ B = 0 (3.22)

34

Fig. 3.2 Flow chart about the process for the FEM solution of a problem using PDE toolbox

35

The relationship between B and Η is given in eqn (3.23).

B = µH (3.23)

where B is the magnetic flux density, H is the magnetic field intensity, and µ is the permeability

of the magnetic material.

Since ∇ × B = 0, there exists a magnetic vector potential A such that

B = ∇ × A (3.24)

and

∇ × ( QV× A) = J (3.25)

The plane case assumes that the current flows are parallel to the z-axis, so only the z component

of A is present as

A = (0, 0, A), J = (0, 0, J) (3.26)

and the eqn (3.25) can be simplified to the scalar elliptic PDE as shown in eqn (3.27)

$∇ . .QV∇u2 = J (3.27)

where J is equal to J (x, y)

For the 2-D case, the magnetic flux density B is computed as shown in eqn (3.28).

B = .//5 , $ /

/1 , 02 (3.28)

and the magnetic field H is expressed from eqn (3.23) as

H = QV B

The interface condition across sub-domain borders between regions of different material

properties is that H × ñ be continuous. This implies the continuity of magnetic field QV

//ñ ) and

does not require special treatment since the variation formulation of the PDE problem is used. In

36

ferromagnetic materials, µ is usually dependent on the field strength |B| = |∇A|, so the nonlinear

solver is needed.

Dirichlet boundary condition specifies the value of the magnetostatic potential A on the

boundary. Neumann condition specifies the value of the normal component of ñ (QV∇A ) on the

boundary. This is equivalent to specifying the tangential value of the magnetic field H on the

boundary. B and H can be plotted as vector fields.

3.5 Design of Magnetic Circuit for Different Topologies for FEM Analysis

3.5.1 Geometry Design Using Graphical User Interface (GUI) of PDE Toolbox

A practical 1.80 step angle four phase bipolar PMH stepper motor is chosen for design,

having 4 poles in the stator and 2 sections in the rotor with 50 teeth on each disk with AlNiCo5

magnet radially magnetized. The main structural parameters of the PMH stepper motor required

for Topology design are given in Table 3.1. Geometry of PMH motor is designed using PDE

toolbox GUI for eight topologies. Airgap length, tooth width and tooth pitch are designed in such

a way that ratios of tooth width to tooth pitch is 0.75 and tooth pitch to airgap length is 20.

Topologies are designed considering tooth pitch as 1.86 mm, tooth width as 1.42 mm and slot

width as 1.32 mm. The eight topologies designed are given below

1. Non-uniform air-gap of 0.137 mm length with extra teeth on stator.

2. Non-uniform air-gap of 0.137 mm length without extra teeth on stator.

3. Non-uniform air-gap of 0.93 mm length with extra teeth on stator.

4. Non-uniform air-gap of 0.93 mm length without extra teeth on stator.

5. Uniform air-gap of 0.137 mm length with extra teeth on stator.

6. Uniform air-gap of 0.137 mm length without extra teeth on stator.

7. Uniform air-gap of 0.93 mm length with extra teeth on stator.

37

8. Uniform air-gap of 0.93 mm length without extra teeth on stator

In the window of GUI of PDE toolbox, different geometry shapes are available for

design. Circles shapes are created for rotor, stator and permanent magnet cores with their given

diameter values shown in Table 3.1.

Table 3.1 Structural parameters of PMH stepper motor for Topology design Stator poles Tooth per stator

pole Outer diameter

of stator Inner diameter

of stator Outer diameter of

stator shell

4 10 10.108 cm 5.936 cm 10.652 cm

Number of rotor teeth

Number of turns per phase

Section length of rotor

Outer diameter of rotor

Inner diameter of rotor

50 102 10.26 cm 4.2 cm 1.74 cm

3.5.1.1 Rotor Design

Rotor design is same for all Topologies. The circumference of outer circle for rotor is

divided with 100 as there are 50 teeth and 50 slots per rotor disk and is given as

Circumference of outer circle of rotor = П × D = 13.195 cm

Width of each rotor tooth = П × X Q = 0.132 cm

Rectangles are created with 0.132 cm side. Each rectangle is displaced with 7.20 (360 ÷ 50). The

geometry is designed for one stator pole pitch as it is a symmetric design.

3.5.1.2 Geometry Design

Stator poles are designed according to the required airgap. Stator core is designed with

circles with its inner and outer diameters using the data given in Table 3.1. Number of turns per

phase is calculated as shown below:

Rated voltage =12 V

Rated current = 1 A

38

Resistance = 12 Ω

SWG of copper winding = 36

Specific resistance = 1.68 × 10-8 Ω-m

Cross section area of 36 SWG wire = 0.0293 mm2

Length of single turn from geometry = 205.68 mm

No of turns = 9 × Y 9Z9 9 × [ \ N = 102

3.5.1.2.1 Geometry Design of Non-uniform Airgap Topologies

K.R.Rajgopal, Bhim Singh and B.P.Singh [37] reported the design of non-uniform airgap using

equivalent circuit model. In this thesis, an attempt is made to design non-uniform Topologies

using FEM. Stator poles are created with two rectangles (R1, R2 for pole 1 and R3, R4 for pole

2), without pole arc; and made union with stator inner core circle C3. Ten stator pole teeth are

created as squares and intersected from stator pole end rectangles (R1, R3 for pole one and pole

two respectively). One more rectangle is created and made union with pole shoe rectangle for

current coil design (R5, R6 for pole one and pole two respectively). Fig. 3.3 shows geometry

design diagram for a pair of poles for Topology 1 without stator core. While designing, airgap

length is varied with the height of rectangles (R1, R3) drawn for pole shoes. The geometry

portion for FEM solution of one pole pitch for Topologies with extra teeth (R7) is calculated

using eqn (3.29); and the detailed procedure is given in APPENDIX-B.

]^R1 3 R2 3 R3 3 R4d 3 C1 3 C3 3 C4 3 R5 3 R6 3 R7 $ ^SQ1 3 SQ2 3 SQ3 3SQ4 3 SQ5 3 SQ6 3 SQ7 3 SQ8 3 SQ9 3 SQ10 3 SQ11 3 SQ12 3 SQ13 3 SQ14 3 SQ15 3SQ16 3 SQ17 3 SQ18d 3 ^C2 – ^SQ19 3 SQ20 3 SQ21 3 SQ22 3 SQ23 3 SQ24 3 SQ25 3

SQ26 3 SQ27 3 SQ28 3 SQ29 3 SQ30 3 SQ31 3 SQ32 3 SQ33 3 SQ34 3 SQ35 3 SQ36 3 SQ37 3 SQ38 3 SQ39 3 SQ40 3 SQ41ddn SQ42 (3.29)

39

` Similarly, for Topologies without extra teeth between stator poles can be obtained by

neglecting R7 in eqn (3.29) as given in eqn (3.30).

]^R1 3 R2 3 R3 3 R4d 3 C1 3 C3 3 C4 3 R5 3 R6 $ ^SQ1 3 SQ2 3 SQ3 3 SQ4 3SQ5 3 SQ6 3 SQ7 3 SQ8 3 SQ9 3 SQ10 3 SQ11 3 SQ12 3 SQ13 3 SQ14 3 SQ15 3 SQ16 3SQ17 3 SQ18d 3 ^C2 – ^SQ19 3 SQ20 3 SQ21 3 SQ22 3 SQ23 3 SQ24 3 SQ25 3 SQ26 3

SQ27 3 SQ28 3 SQ29 3 SQ30 3 SQ31 3 SQ32 3 SQ33 3 SQ34 3 SQ35 3 SQ36 3 SQ37 3 SQ38 3 SQ39 3 SQ40 3 SQ41ddn SQ42 (3.30)

where R1, R2 are rectangles, designed for stator pole 1 and R5 is rectangle designed for current

coil on pole 1. R3, R4 are rectangles designed for stator pole 2 and R6 is rectangle designed for

current coil on pole 2. SQ1 to SQ18 are squares, designed as teeth on stator poles. SQ19 to SQ41

are squares, created equally to provide 23 rotor teeth. SQ42 is square to provide the required

boundary for one pole pitch geometry for FEM analysis. R7 is the extra teeth on stator for

smooth performance of the motor. C1 is a circle created for permanent magnet. C2 is a circle

created as outer rotor circle. C3 and C4 are circles created for stator core as inner and outer

circles respectively. Executing these eqns (3.29) and (3.30); design Topologies are developed

corresponding to non – uniform airgap as shown from Fig. 3.4, to Fig. 3.7 for Topology 1 to

Topology 4 respectively. The scales for x and y axes are in decimetres.

40

Fig. 3.3 Geometry design of Topology 1 for a pair of poles

Fig. 3.4 Geometry design of Topology 1 for one pole pitch

41



42


3.5.1.2.2 Geometry Design of Uniform Airgap Topologies

Uniform airgap is obtained by designing stator poles with pole arc. For this design

process, two rectangles are in union with each other (R1, R2 for pole one and R3, R4 for pole

two). These rectangles are intersected by a circle (C3) whose diameter is equal to the outer rotor

circle (C2) diameter plus twice the airgap length and made union with stator inner core circle

(C5). Ten stator pole teeth are created as squares and intersected from stator pole end rectangle.

One more rectangle is created and made union with pole shoe rectangle for current coil design.

Fig. 3.8 shows the obtained design for a pair of poles for the Topology 7 without stator core.

Remaining design procedure is similar to non-uniform airgap Topologies. Equations (3.28) and

(3.29) are used to get the geometry for one pole pitch. Fig.3.9, Fig.3.10 and Fig.3.11 show

designed diagrams of Topology 5, Topology 6 and Topology 8 respectively for one pole pitch.

As Topology 7 design diagram is shown for pair of poles it is not shown again for one pole pitch.

43

Fig. 3.8 Geometry design of Topology 7 for a pair of poles


44



45

3.5.2 Magnetic Potential analysis using FEM

Once geometry for eight Topologies is designed, boundary conditions are specified for

getting vector fields of magnetic potential and flux densities. Dirichlet boundary condition is

considered as (1, 0) and Neumann boundary condition is considered as (0, 0) for 2-D analysis.

Once the boundary conditions are given, programme could be executed to verify correct

boundary conditions. After verifying boundary conditions, permeability and current density of

different parts like stator core, rotor core, stator current coil and permanent magnet are calculated

using eqns (3.31) and (3.32) respectively. Permeability (µ) is one and current density (J) is zero

for airgap.

µ 6 µopqQrst∇Ot + µu (3.31)

where µMax, µMin are maximum permeability and minimum permeability respectively of core

material used for stator and rotor. C is coercive force of the core material. ∇A is equivalent to

(– µJ [38]. Two core materials Iron (99.8%) and Iron (99.95%) are considered for analysis.

Permeability (µ) is calculated for Iron (99.8%), Iron (99.95%) using (3.30) and obtained as

5,150 H/m and 2,10,000 H/m respectively. Standard wire gauge (SWG) 36 conductors are

considered for current coil design. Analysis is considered for 0.5 A and 1 A whose current

densities (Jc) are calculated using eqn (3.32). Current densities of current coil with 36 SWG

conductors for 0.5 A, 1 A are 170648 A/m2 and 341296 A/m2 respectively. Current density (Jc)

in core materials is equivalent to zero. Two permanent magnetic materials Neodymium Iron

Boron (NdFeB) and Samarium Cobalt (Sm2Co17) are used for analysis and the data is given in

APPENDIX-C. Current density for permanent magnet ( Jpm) is calculated using eqn (3.33) [39].

Jc = sN

9N9 A/m2 (3.32)

Jpm = Bs - µ0Hs A/m2 (3.33)

46

where Bs, Hs are saturated flux density and field strength of permanent magnet respectively . µ0

is permeability of air.

Designed Topology is executed for mesh generation when boundary conditions are

properly mentioned. PDE coefficients (permeability and current density) for all parts of the

Topology are mentioned. Solutions for magnetic potential and flux density vectors for these eight

Topologies for two core materials at two current densities for two permanent magnetic materials

are obtained through nonlinear solution.

3.5.2.1 Magnetic Potential Analysis of Topology 1 using FEM

Fig. 3.12 shows mesh diagram with Iron (99.8%) core at the current density of

170648 A/m2 for NdFeB as permanent magnet. Fig. 3.13 shows magnetic potential and flux

density vectors diagram with Iron (99.8%) core at the current density of 170648 A/m2 for NdFeB

as permanent magnet. Fig. 3.14 shows mesh diagram with Iron (99.95%) core at the current

density of 170648 A/m2 for NdFeB as permanent magnet. Fig. 3.15 shows magnetic potential and

flux density vectors diagram with Iron (99.95%) core at the current density of 170648 A/m2 for

NdFeB as permanent magnet. Fig. 3.16 shows mesh diagram with Iron (99.8%) core at current


flux density vectors diagram with Iron (99.8%) core at current density of 341296 A/m2 for



flux density vectors diagram with Iron (99.95%) core at current density of 341296 A/m2 for


density of 170648 A/m2 for Sm2Co17 as permanent magnet.

47

Fig. 3.21 shows magnetic potential and flux density vectors diagram with Iron (99.8%)

core at current density of 170648 A/m2 for Sm2Co17 as permanent magnet. Fig. 3.22 shows mesh

diagram with Iron (99.95%) core at current density of 170648 A/m2 for Sm2Co17 as permanent

magnet. Fig. 3.23 shows magnetic potential and flux density vectors diagram with Iron (99.95%)








diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet.

Fig. 3.12 Mesh diagram of Topology 1 with Iron (99.8%) core at current density of 170648 A/m2 for NdFeB as permanent magnet

48

Fig. 3.13 Magnetic potential and flux density vectors diagram of Topology 1 with Iron

(99.8%) core at current density of 170648 A/m2 for NdFeB as permanent magnet

Fig. 3.14 Mesh diagram of Topology 1 with Iron (99.95%) core at current density of

170648 A/m2 for NdFeB as permanent magnet

49





50





51




170648 A/m2 for Sm2Co17 as permanent magnet

52


(99.8%) core at current density of 170648 A/m2 for Sm2Co17 as permanent magnet



53





54





55



Fig. 3.28 Mesh diagram of Topology 1 with Iron (99.8%) core without current density for

NdFeB as permanent magnet

56

Fig. 3.29 shows magnetic potential and flux density vectors diagram with Iron (99.8%)

core without current density for NdFeB as permanent magnet. Fig. 3.30 shows mesh diagram

with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig. 3.31

shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without

current density for NdFeB as permanent magnet. Fig. 3.32 shows mesh diagram with Iron

(99.8%) core without current density for Sm2Co17 as permanent magnet. Fig. 3.33 shows

magnetic potential and flux density vectors diagram with Iron (99.8%) core without current

density for Sm2Co17 as permanent magnet. Fig. 3.34 shows mesh diagram with Iron (99.95%)

core without current density for Sm2Co17 as permanent magnet. Fig. 3.35 shows magnetic

potential and flux density vectors diagram with Iron (99.95%) core without current density for

NdFeB as permanent magnet.


(99.8%) core without current density for NdFeB as permanent magnet

57





58


Sm2Co17 as permanent magnet


(99.8%) core without current density for Sm2Co17 as permanent magnet

59


Sm2Co17 as permanent magnet


(99.95%) core without current density for Sm2Co17 as permanent magnet

60

Magnetic potential is obtained when the rotor teeth are totally aligned with stator teeth.

This is indicated with arrow in Fig. 3.35. MMF is the product of magnetic potential from FEM

results and section length of the motor shown in Table 3.1. Table 3.2 shows MMF values of

Topology 1 for two different core materials at two different current densities and for two

different permanent magnet materials.

Table 3.2 MMF values of Topology 1 for two different core materials at two different current densities and for two different permanent magnetic materials

Topology 1

Core Material

Permanent Magnet (PM)

MMF due to PM

(Wb/H )

Current density (A/m2)

MMF due to excitation

(AT)

Non-uniform narrow airgap

(0.137 mm) with extra stator teeth

between pair of poles

Iron (99.8%)

NdFeB

1.539 x10-4

170648

18.312

341296

38.4925

Sm2Co17

1.539 x10-4

170648

18.312

341296

38.4925

Iron (99.95%)

NdFeB

4.578x10-4

170648

61.588

341296

84.779

Sm2Co17

4.578x10-4

170648

40.588

341296

84.779

61

Change in the magnitude of MMF is not found with change of permanent magnet from

NdFeB to Sm2Co17, because of the small size of permanent magnet. Non-uniform airgap results

weak magnetic potential between stator and rotor. MMF is two times more in Iron (99.95%) than

Iron (99.8%).


Fig. 3.36 shows mesh diagram with Iron (99.8%) core at current density of 170648 A/m2

for NdFeB as permanent magnet. Fig. 3.37 shows magnetic potential and flux density vectors

diagram with Iron (99.8%) core at current density of 170648 A/m2 for NdFeB as permanent

magnet. Fig. 3.38 shows mesh diagram with Iron (99.95%) core at current density of


density vectors diagram with Iron (99.95%) core at current density of 170648 A/m2 for NdFeB as

permanent magnet.



62





63









permanent magnet. Fig. 3.44 shows mesh diagram with Iron (99.8%) core without current

density for NdFeB as permanent magnet. Fig. 3.45 shows magnetic potential and flux density

vectors diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet.

Fig. 3.46, Fig. 3.47 show mesh diagram, magnetic potential and flux density vectors diagram

with Iron (99.95%) core without current density for NdFeB as permanent magnet.

64

Fig. 3.40 Mesh diagram of Topology 2 with Iron (99.8%) core at current density of 341296 A/m2 for NdFeB as permanent magnet

Fig. 3.41 Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.8%) core at current density of 341296 A/m2 for NdFeB as permanent magnet

65


341296 A/m2 for NdFeB permanent magnet


66



Fig. 3.45 Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.8%) core without current density for NdFeB as permanent magnet

67

Fig. 3.46 Mesh diagram of Topology 2 with Iron (99.95%) core without current density for NdFeB as permanent magnet



Table 3.3 shows MMF values of Topology 2 for two different core materials and at two

different current densities.

68

Table 3.3 MMF values of Topology 2 for two different core materials and at two different current densities

Topology 2

Core Material


MMF due to PM

(Wb/H )



(AT)

Non-uniform narrow airgap

(0.137 mm) without extra stator teeth


Iron (99.8%)

NdFeB

1.232x10-4

170648

13.397

341296

30.794

Iron (99.95)%

NdFeB

3.849 x10-4

170648

36.491

341296

79.382

Non-uniform airgap without extra teeth between stator poles reduces magnetic potential

between stator and rotor. MMF is more than two times in Iron (99.95%) when compared to

Iron (99.8%).








permanent magnet. Fig. 3.52 shows mesh diagram with Iron (99.8%) core at current density of



permanent magnet.

69




(99.8%) core with current density of 170648 A/m2 for NdFeB as permanent magnet

70

Fig. 3.50 Mesh diagram of Topology 3 with Iron (99.95%) core with current density of



71




72




magnet. Fig. 3.56 shows mesh diagram with Iron (99.8%) core without current density for

NdFeB as permanent magnet. Fig. 3.57 shows magnetic potential and flux density vectors

diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig.

3.58 shows mesh diagram with Iron (99.95%) core without current density for NdFeB as

permanent magnet. Fig. 3.59 shows magnetic potential and flux density vectors diagram with

Iron (99.95%) core without current density for NdFeB as permanent magnet.



73





74




75






Topology 3

Core Material


MMF due to PM

(Wb/H )



(AT)

Non-uniform large airgap

(0.93 mm) with extra stator teeth between pair of

poles

Iron (99.8%)

NdFeB

4.492x10 -5

170648

3.246

341296

8.115

Iron (99. 95%)

NdFeB

5.615x10 -5

170648

5.692

341296

13.984

When airgap length is increased from 0.137 mm to 0.93 mm magnetic potential between

stator and rotor is drastically reduced. MMF is 40% more in Iron (99.95%) than Iron (99.8%).

76





magnet. Fig. 3.62 shows mesh diagram with Iron (99.95%) core at current density of 170648

A/m2 for NdFeB as permanent magnet. Fig. 3.63 shows magnetic potential and flux density

vectors diagram with Iron (99.95%) core at current density of 170648 A/m2 for NdFeB as




permanent magnet.



77




78




341296 A/m2for NdFeB as permanent magnet

79



Fig. 3.66 shows mesh diagram with Iron (99.95%) core at current density of






Fig. 3.70 shows mesh diagram with Iron (99.95%) core without current density for NdFeB as



80





81





82



83




Topology 4

Core

Material


MMF due to PM

(Wb/H )



(AT)

Non-uniform large airgap (0.93 mm)

without extra stator teeth


Iron (99.8%)

NdFeB

4.251x10-5

170648

2.434

341296

5.492

Iron (99. 95%)

NdFeB

5.115x10-5

170648

4.115

341296

11.23

Magnetic potential between stator and rotor is more reduced for this Topology than

Topology 3. MMF is more than two times in Iron (99.95%) when compared to Iron (99.8%).










84


permanent magnet.





85





86





87

Fig. 3.78 shows mesh diagram with Iron (99.95%) core at current density of 341296











88





89





90





Topology 5

Core Material


MMF due to PM

(Wb/H )



(AT)

Uniform narrow airgap (0.137 mm) with extra

stator teeth between pair of poles

Iron (99.8%)

NdFeB

2.763x10-4

170648

20.794

341296

48.493

Iron (99. 95%)

NdFeB

5.883x10-4

170648

45.779

341296

99.863

91

Uniform low airgap (0.137 mm) with extra teeth between stator poles gives excellent

magnetic potential between stator and rotor. MMF is nearly twice in Iron (99.95%) than

Iron (99.8%).











permanent magnet.



92





93





94












95





96





97





98




Topology 6

Core Material


MMF due to PM

(Wb/H )



(AT)

Uniform narrow airgap (0.137 mm) without

extra stator teeth between pair of poles

Iron (99.8%)

NdFeB

2.164x10-4

170648

16.635

341296

35.794

Iron (99. 95%)

NdFeB

4.240x10-4

170648

40.382

341296

85.434

Uniform low airgap (0.137 mm) without extra teeth between stator poles gives excellent

magnetic potential between stator and rotor when compared to non – uniform airgap Topologies

but less than Topology 5. MMF is nearly 60% more in Iron (99.95%) than Iron (99.8%).









99



permanent magnet.





100





101





102




magnet. Fig. 3.104 shows mesh diagram with Iron (99.8%) core without current density for

NdFeB as permanent magnet. Fig. 3.105 shows magnetic potential and flux density vectors

diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig.3.

106 shows mesh diagram with Iron (99.95%) core without current density for NdFeB as





103





104



Fig. 3.106 Mesh diagram of Topology 7 with Iron (99.95%) core without current density

for NdFeB as permanent magnet

105





Topology 7

Core Material


MMF due to PM

(Wb/H )



(AT)

Uniform large airgap (0.19 mm) with extra stator teeth between

pair of poles

Iron (99.8%)

NdFeB

5.575x10-5

170648

04.689

341296

09.738

Iron (99. 95%)

NdFeB

6.928x10-5

170648

06.858

341296

15.230

106

For airgap length is 0.93 mm magnetic potential between stator and rotor is reduced when

compared to Topology 5, Topology 6, Topology 1 and Topology 2. But it has better interaction

than Topology 3 and Topology 4. MMF is 35% more in Iron (99.95%) than Iron (99.8%).











permanent magnet.



107





108





109



Fig. 3.114 shows mesh diagram with Iron (99.95%) core at current density of 341296









110





111





112

Fig. 3.118 Mesh diagram of Topology 8 with Iron (99.95%) core without current density

for NdFeB as permanent magnet


113




Topology 8

Core Material


MMF due to PM,

(Wb/H )


MMF due to excitation,

AT

Uniform large airgap (0.19 mm) without extra stator teeth


Iron (99.8%)

NdFeB

3.313x10-5

170648

02.989

341296

07.439

Iron (99. 95%)

NdFeB

6.700x10-5

170648

05.315

341296

12.607

Magnetic potential between stator and rotor is less than Topology 7. MMF is nearly 40%

more in Iron (99.95%) than Iron (99.8%).

3.6 Summary

Magnetic circuit analysis is carried out for different Topologies using FEM and from the

results it is observed that MMF distribution of PMH stepper motor is found to be uniform with

uniform airgap Topologies (Topology 5, Topology 6, Topology 7 and Topology 8). More MMF

interaction between stator and rotor is found for low airgap Topologies (0.137 mm). Leakage

flux is minimized using extra teeth on stator (Topology 1 and Topology 5). There is not much

difference found in MMF distribution for two different permanent magnetic materials (NdFeB

and Sm2Co17). More MMF interaction observed for Iron (99.95%) core material. This analysis

precisely predicts steady-state and dynamic response of PMH stepper motor.

Download - CHAPTER 3 Magnetic Circuit Design and Analysis using Finite

Top Related