Magnetic Circuit Design and Analysis using Finite Element Method

Transcription

1 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 24

2 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 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 25

3 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 26

4 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 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 δ δ = 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]. 27

5 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/m 2 ), J is the current density vector in Ampere/metre 2 (A/m 2 ), A is the magnetic vector potential in Tesla - metre or (Wb/m) and A z 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. 28

6 Fig. 3.1 Tooth layer unit Of PMH stepper motor If u s and u r are respective scalar quantities of the iso-potential lines ab and cd, the magnetic potential difference A is shown in eqn (3.2). A = u s u r (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). 29

7 +,,-.µ/023, /1,4.µ/0260 /5 B * 8 * 9 60 A * 8 : 60 ) 8 x,y 9 68 x3t?,y AA (3.4) where φ is the scalar quantity, µ is the magnetic permeability and T P 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,a6 B Dm FGGGG (3.5) Here the nodes j and m are on the border ab as shown in Fig. 3.1 HI FGGGGG is the length of unit e from node j to m and B e 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 30

8 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 u3 au6f 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 / $. c u3 au6f (3.7) d /O N /PO$. c u3 au6f (3.8) $. c u3 au6 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 u3 a u u6 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 31

9 $. c QQ u Q 3. c Q u 3 a QQ u Q 3a Q u 6 f Q (3.11) $. c Q u Q 3. c u 3 a Q u Q 3a u 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). h 11 u 1 + h 12 u 2 = r 1 (3.15) h 21 u 1 + h 22 u 2 = r 2 (3.16) The generalized Neumann boundary condition is shown in eqns (3.17) and (3.18). $ñ. c QQ u Q 3ñ. c Q u 3 q QQ u Q 3q Q u 6 g Q (3.17) $ñ. c Q u Q 3ñ. c u 3 q Q u Q 3q u 6 g (3.18) The mixed boundary condition is shown in eqns (3.19) and (3.20). $ñ. c QQ u Q 3ñ. c Q u 3 q QQ u Q 3q Q u 6 g Q 3h QQ µ (3.19) $ñ. c Q u Q 3ñ. c u 3 q Q u Q 3q u 6 g 3h Q µ (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]. 32

10 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) 33

11 Fig. 3.2 Flow chart about the process for the FEM solution of a problem using PDE toolbox 34

12 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 and B = A (3.24) ( Q A) = J (3.25) V 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) where J is equal to J (x, y) $.. Q u2 = J (3.27) V For the 2-D case, the magnetic flux density B is computed as shown in eqn (3.28). B =. / /5,$/ /1 and the magnetic field H is expressed from eqn (3.23) as,02 (3.28) H = Q V 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 Q V / /ñ ) and does not require special treatment since the variation formulation of the PDE problem is used. In 35

13 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 ( Q A ) on the V 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 Geometry Design Using Graphical User Interface (GUI) of PDE Toolbox A practical 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 AlNiCo 5 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 mm length with extra teeth on stator. 2. Non-uniform air-gap of 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 mm length with extra teeth on stator. 6. Uniform air-gap of mm length without extra teeth on stator. 7. Uniform air-gap of 0.93 mm length with extra teeth on stator. 36

14 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. Stator poles Table 3.1 Structural parameters of PMH stepper motor for Topology design Tooth per stator Outer diameter Inner diameter Outer diameter of pole of stator of stator stator shell cm cm cm Number of rotor teeth Number of turns per phase Section length of rotor Outer diameter of rotor Inner diameter of rotor cm 4.2 cm 1.74 cm 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 = cm Width of each rotor tooth = П X Q = cm Rectangles are created with cm side. Each rectangle is displaced with (360 50). The geometry is designed for one stator pole pitch as it is a symmetric design 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 37

15 Resistance = 12 Ω SWG of copper winding = 36 Specific resistance = Ω-m Cross section area of 36 SWG wire = mm 2 Length of single turn from geometry = mm No of turns = 9 Y9Z9 9 [ \ N = 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 R4d3C13 C3 3 C4 3R53 R63R7 $ ^SQ13SQ23SQ33 SQ43SQ53SQ63SQ73SQ83SQ93SQ103SQ113SQ123SQ133SQ143 SQ153 SQ163SQ173SQ18d3^C2 ^SQ193SQ203SQ213SQ223SQ233SQ243SQ253 SQ263 SQ273 SQ283 SQ293 SQ303 SQ313 SQ323 SQ333 SQ343 SQ353 SQ363 SQ373 SQ383SQ393 SQ40 3 SQ41ddn SQ42 (3.29) 38

16 ` 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 R4d3C13 C3 3 C4 3 R5 3 R6$ ^SQ13SQ23SQ33SQ43 SQ53SQ63SQ73SQ83SQ93SQ103SQ113SQ123SQ133SQ143 SQ153SQ163 SQ173SQ18d3^C2 ^SQ193SQ203SQ213SQ223SQ233SQ243SQ253 SQ263 SQ273 SQ283 SQ293 SQ303 SQ313 SQ323 SQ333 SQ343 SQ353 SQ363 SQ373 SQ383SQ393 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. 39

17 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 40

18 Fig. 3.5 Geometry design of Topology 2 for one pole pitch Fig. 3.6 Geometry design of Topology 3 for one pole pitch 41

19 Fig. 3.7 Geometry design of Topology 4 for one pole pitch 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. 42

20 Fig. 3.8 Geometry design of Topology 7 for a pair of poles Fig. 3.9 Geometry design of Topology 5 for one pole pitch 43

21 Fig Geometry design of Topology 6 for one pole pitch Fig Geometry design of Topology 8 for one pole pitch 44

22 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. µ6 µ opq Qrst O t + µ u (3.31) where µ Max, µ Min 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 (J c ) are calculated using eqn (3.32). Current densities of current coil with 36 SWG conductors for 0.5 A, 1 A are A/m 2 and A/m 2 respectively. Current density (J c ) in core materials is equivalent to zero. Two permanent magnetic materials Neodymium Iron Boron (NdFeB) and Samarium Cobalt (Sm 2 Co 17 ) are used for analysis and the data is given in APPENDIX-C. Current density for permanent magnet ( J pm ) is calculated using eqn (3.33) [39]. J c = sn 9N9 A/m 2 (3.32) J pm = B s - µ 0 H s A/m 2 (3.33) 45

23 where B s, H s are saturated flux density and field strength of permanent magnet respectively. µ 0 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 Magnetic Potential Analysis of Topology 1 using FEM Fig shows mesh diagram with Iron (99.8%) core at the current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at the current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at the current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at the current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. 46

24 Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig Mesh diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 47

25 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 48

26 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 49

27 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 50

28 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet 51

29 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet 52

30 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet 53

31 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet 54

32 Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core at current density of A/m 2 for Sm 2 Co 17 as permanent magnet Fig Mesh diagram of Topology 1 with Iron (99.8%) core without current density for NdFeB as permanent magnet 55

33 Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for Sm 2 Co 17 as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core without current density for Sm 2 Co 17 as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core without current density for Sm 2 Co 17 as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core without current density for NdFeB as permanent magnet 56

34 Fig Mesh diagram of Topology 1 with Iron (99.95%) core without current density for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core without current density for NdFeB as permanent magnet 57

35 Fig Mesh diagram of Topology 1 with Iron (99.8%) core without current density for Sm 2 Co 17 as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.8%) core without current density for Sm 2 Co 17 as permanent magnet 58

36 Fig Mesh diagram of Topology 1 with Iron (99.95%) core without current density for Sm 2 Co 17 as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 1 with Iron (99.95%) core without current density for Sm 2 Co 17 as permanent magnet 59

37 Magnetic potential is obtained when the rotor teeth are totally aligned with stator teeth. This is indicated with arrow in Fig 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 Core Material Topology 1 Permanent Magnet (PM) MMF due to PM (Wb/H ) Current density (A/m 2 ) MMF due to excitation (AT) NdFeB x Non-uniform narrow airgap (0.137 mm) with extra stator teeth between pair of poles Iron (99.8%) Iron (99.95%) Sm 2 Co x NdFeB 4.578x Sm 2 Co x

38 Change in the magnitude of MMF is not found with change of permanent magnet from NdFeB to Sm 2 Co 17, 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%) Magnetic Potential Analysis of Topology 2 using FEM Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig Mesh diagram of Topology 2 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 61

39 Fig Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 2 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 62

40 Fig Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig 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 show mesh diagram, magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. 63

41 Fig Mesh diagram of Topology 2 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 64

42 Fig Mesh diagram of Topology 2 with Iron (99.95%) core at current density of A/m 2 for NdFeB permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 65

43 Fig Mesh diagram of Topology 2 with Iron (99.8%) core without current density for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 2 with Iron (99.8%) core without current density for NdFeB as permanent magnet 66

44 Fig Mesh diagram of Topology 2 with Iron (99.95%) core without current density for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors 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. 67

45 Table 3.3 MMF values of Topology 2 for two different core materials and at two different current densities Core Topology 2 Material Permanent Magnet (PM) MMF due to PM (Wb/H ) Current density (A/m 2 ) MMF due to excitation (AT) Non-uniform narrow airgap (0.137 mm) without extra stator teeth between pair of poles Iron (99.8%) Iron (99.95)% NdFeB x 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%) Magnetic Potential Analysis of Topology 3 using FEM Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. 68

46 Fig Mesh diagram of Topology 3 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.8%) core with current density of A/m 2 for NdFeB as permanent magnet 69

47 Fig Mesh diagram of Topology 3 with Iron (99.95%) core with current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 70

48 Fig Mesh diagram of Topology 3 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 71

49 Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig Mesh diagram of Topology 3 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 72

50 Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 3 with Iron (99.8%) core without current density for NdFeB as permanent magnet 73

51 Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.8%) core without current density for NdFeB as permanent magnet Fig Mesh diagram of Topology 3 with Iron (99.95%) core without current density for NdFeB as permanent magnet 74

52 Fig Magnetic potential and flux density vectors diagram of Topology 3 with Iron (99.95%) core without current density for NdFeB as permanent magnet Table 3.4 shows MMF values of Topology 3 for two different core materials and at two different current densities. Table 3.4 MMF values of Topology 3 for two different core materials and at two different current densities Topology 3 Core Material Permanent Magnet (PM) MMF due to PM (Wb/H ) Current density (A/m 2 ) MMF due to excitation (AT) Non-uniform large airgap (0.93 mm) with extra stator teeth between pair of poles Iron (99.8%) Iron (99. 95%) NdFeB 5.615x When airgap length is increased from 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%). 75

53 Magnetic Potential Analysis of Topology 4 using FEM Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig Mesh diagram of Topology 4 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 76

54 Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 4 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 77

55 Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Mesh diagram of Topology 4 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 78

56 Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. 79

57 Fig Mesh diagram of Topology 4 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 80

58 Fig Mesh diagram of Topology 4 with Iron (99.8%) core without current density for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.8%) core without current density for NdFeB as permanent magnet 81

59 Fig Mesh diagram of Topology 4 with Iron (99.95%) core without current density for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 4 with Iron (99.95%) core without current density for NdFeB as permanent magnet 82

60 Table 3.5 shows MMF values of Topology 4 for two different core materials and at two different current densities. Table 3.5 MMF values of Topology 4 for two different core materials and at two different current densities Topology 4 Core Material Permanent Magnet (PM) MMF due to PM (Wb/H ) Current density (A/m 2 ) MMF due to excitation (AT) Non-uniform large airgap (0.93 mm) without extra stator teeth between pair of poles Iron (99.8%) Iron (99. 95%) NdFeB 5.115x 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%) Magnetic Potential Analysis of Topology 5 using FEM Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux 83

61 density vectors diagram with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig Mesh diagram of Topology 5 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 5 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 84

62 Fig Mesh diagram of Topology 5 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 5 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 85

63 Fig Mesh diagram of Topology 5 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet Fig Magnetic potential and flux density vectors diagram of Topology 5 with Iron (99.8%) core at current density of A/m 2 for NdFeB as permanent magnet 86

64 Fig shows mesh diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.8%) core without current density for NdFeB as permanent magnet. Fig shows mesh diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig shows magnetic potential and flux density vectors diagram with Iron (99.95%) core without current density for NdFeB as permanent magnet. Fig Mesh diagram of Topology 5 with Iron (99.95%) core at current density of A/m 2 for NdFeB as permanent magnet 87