3-D Equivalent Magnetic Circuit Network Method for Precise and Fast Analysis of PM-Assisted Claw-Pole Synchronous Motor

Transcription

1 3-D Equivalent Magnetic Circuit Networ Method for Precise and Fast Analysis of PM-Assisted Claw-Pole Synchronous Motor Jae-Han Sim Automotive Engineering Hanyang University Seoul, Republic of Korea Dong-Gyun Ahn Automotive Engineering Hanyang University Seoul, Republic of Korea Doo-Young Kim Automotive Engineering Hanyang University Seoul, Republic of Korea Jung-Pyo Hong Senior Member, IEEE Automotive Engineering Hanyang University Seoul, Republic of Korea Abstract PM-assisted claw-pole synchronous motor (CPSM has coonly been used in a variety of industrial applications, thans to its robust structure and high energy density. However, such motor has an axially asyetric rotor configuration, which induces the corresponding 3-D magnetic field distribution. Thus, this paper proposes an infinitesimal hexahedron-element-based 3-D equivalent magnetic circuit networ (EMCN method in order to estimate the performance of the PM-assisted CPSM. The hexahedron element is considered as an optimum unit element in describing the configuration of the PM-assisted CPSM in detail. That eventually maes it possible to evaluate the specific 3-D magnetic field distribution from which the flux linage, the bac electro-motive force, and the torque are also being calculated. The results are compared with data and/or experimental data to validate their accuracy. Finally, it is proved that the scalar-potential-based method requires less computing time than the vector-potential-based. Index Terms 3-D equivalent magnetic circuit networ method, claw-pole synchronous motor, infinitesimal hexahedron element, magnetic scalar potential, permanent magnets. I. INTRODUCTION Claw-pole synchronous motor (CPSM is one of the most fascinating alternatives to permanent magnet synchronous motor (PMSM with embedded or surface-mounted rare-earth materials. Compared with the PMSM, the CPSM features the more robust structure, the lower manufacturing cost, and the higher productivity [1]-[3]. Its operating principle is similar to that of wound field synchronous motor [4]. That is to say, it has a DC-current-fed field coil in rotor and an AC-current-fed armature winding in stator [5]. Furthermore, such motor can produce an enhanced energy density by applying the rotor overhang structure and supplementing the permanent magnets (PMs between the rotor poles. Thans to those advantages, the PM-assisted CPSM has the potential industrial applications such as electrical water pump, integrated starter generator, and hybrid vehicle traction drives [6], [7]. However, the PM-assisted CPSM has an asyetric rotor structure along the longitudinal axis. Consequently, a 3-D magnetic field distribution, especially in the air-gap, is induced and thus the corresponding numerical approach is needed [3], [8]. Most academic papers deal with the 3-D finite element analysis (FEA to predict the performance of the PM-assisted CPSM [9]-[12]. The provides the precise simulation results, but requires the massive computational wor from the geometry and mesh modeling to the time-consuming solving process. Such numerical method is eventually not efficient in the iterative initial and optimiation design procedures. The lumped parameter magnetic circuit model has also been suggested as another numerical approach to evaluate or design a wide variety of electric machines. Chen et al. [13], [14] and Zhu et al. [] deal with and focus on the flux-switching PM motors. Gorginpour et al. [16] and Tavana et al. [17] discuss the induction motors. The PM-assisted or the basic CPSMs are dealt with in [3] and [18]-[2]. Particularly, the conventional equivalent magnetic circuit (EMC of [3] is based upon the probable magnetic flux paths method and assumes that the magnetic flux flows in the radial direction only at the air-gap. However, the probable magnetic flux paths can vary with the saturation degree and the corresponding unexpected magnetic flux paths and thus induces the error. Such approach also has a difficulty in representing a specific waveform of the radial magnetic flux density at the one rotor position and does not reflect the rotor relative motion to the stator. Hence, the conventional approach needs to assume that the radial flux density and the corresponding flux linage are sinusoidal to calculate the fundamental bac electro-motive force (BEMF although they are actually nonsinusoidal. Moreover, the electromagnetic torque cannot be calculated without the polar magnetic flux density at the air-gap. Thus, the conventional EMC eventually just provide the approximate estimate of the magnetic field distribution and the fundamental BEMF. This paper proposes an infinitesimal hexahedron-elementbased 3-D equivalent magnetic circuit networ (EMCN method to supplement the limitations of the conventional approach. The proposed method is able to evaluate the fundamental, the harmonics, and the time-varying magnetic field distribution of the PM-assisted CPSM with the rotor overhang structure. The approach does not need to predict or assume the magnetic flux paths thans to the formulation of the 3-D nonlinear networ and thus can be applied to the machine regardless of the load conditions. The method also enhances its accuracy employing

2 the infinitesimal hexahedron elements and reflects the timevarying rotor position considering the time-varying connectivity between the elements. Finally, it reduces its solving time by applying the magnetic scalar potential (MSP approach compared to employing magnetic vector potential (MVP approach. The reason is related to the decreased number of the unnown parameters in the MSP approach. To begin with, the comparison between the MSP and MVP approaches is described in Section II, A. The formulation procedure of the is supplemented in detail in other subsections of Section II. It is related to the element equation, the magneto-motive force and the permeance calculation, the system matrix organiation, and the boundary conditions application. The formulation eventually enabled to calculate the specific magnetic field or flux density distribution, the flux linage, the BEMF, the electromagnetic torque, the loss evaluation, and the PM demagnetiation. The results were compared with data and/or experimental data to validate its accuracy. Furthermore, it was proven that the proposed method palliates more computational burden than 3- D FEA. Even though it shows its high accuracy and shortened solving time, the inherent limitations are also explained. II. 3-D EQUIVALENT MAGNETIC CIRCUIT NETWORK (EMCN METHOD A. MSP Approach versus MVP Approach For 2-D magnetic field problem, the MSP approach is not efficient compared to the MVP approach in terms of solving time. It is because the unnown parameters of the MSP and MVP approaches are exactly same. However, the MSP approach is effective for the 3-D magnetic field problem because the number of unnowns is reduced by third owing to the fact that in this approach only one unnown needs to be treated for each node. On the other hand, the MVP approach need to treat three unnowns for each node. The number of unnowns determines the sie of the system matrix and provides the rough estimate of the solving time. Fig. 1 shows the aforementioned number of unnowns for each approach in the 2-D and 3-D problems where u is the unnown MSP, A is the unnown MVP. The superscripts ρ, φ, and denote the radial, the polar, and the horiontal directions. The subscripts and denote the corresponding node positioning indexes, respectively, which play a role in formulating the system matrix. Fig. 1. Comparison of the unnown parameters between MSP and MVP approaches; (a 2-D problem and (b 3-D problem. Fig. 2. Infinitesimal hexahedron element in cylindrical coordinate system B. Element Equation The electric machines, such as the PM-assisted CPSM, are best described in the cylindrical coordinate system and thus the unit element becomes a hexahedron as expressed in Fig. 2. The hexahedron element has an infinitesimal radial distance Δρ, an infinitesimal angular coordinate Δφ, and an infinitesimal height Δ. Each element is awarded a constant or a nonlinear relative permeability according to its assigned material. The nonlinear relative permeability is converged through the chord method in this wor. The constant or converged value is being used so as to calculate the permeance and the magneto-motive force in Subsection C. In addition, it is assumed that all the nodes are positioned at the centers of individual hexahedron elements. In this case, an arbitrary node is connected with 6 adjacent nodes at most and the corresponding circuit diagram is described in Fig. 3. The element equation can be represented as in (1. It is based on the aforementioned connection relationship and the Gauss s law, which states that the net outward magnetic flux from a node is equal to ero [21]-[23]. where (1 i1, j, i, j1, 1 u u F P i1, j, i1, j, i1, j, i1, j, u u F P i, j, i, j, i1, j, i, j, i, j, u u F P i, j1, i, j1, i, j1, i, j1, u u F P i, j, i, j, i, j1, i, j, i, j, u u F u u F P i, j, i, j, i, j, 1 i, j, i, j, P In the preceding equations, Φ is the magnetic flux, F is the magneto-motive force, and P is the permeance, i.e. the reciprocal of magnetic reluctance or magnetic resistance. C. Permeance and Magneto-Motive Force The permeance between 2 adjacent nodes can be defined by the absolute permeability, the distance, and the sectional-area. The absolute permeability can be defined as a product of the vacuum permeability μ and the relative permeability μ r. As m- (2

3 + Direction ui, j 1, - Direction Fig. 3. Circuit diagram of an arbitrary node (i,j,. It has 6 branches and 6 adjacent nodes at the most (the red arrows express the direction of the magnetic flux between the 2 adjacent nodes entioned before, the nonlinear relative permeability is only applied in the rotor and stator core elements because of their nonlinearity. It is based on their nonlinear magnetiing curves, i.e. their BH curves. If the applied material is not the core material, for instance for air, copper, and PM elements, the relative permeability has a constant value: 1. in the case of air and copper elements and in the case of PM elements. The values does not change according to the load conditions. The distance denotes the length of a magnetic flux path between the 2 adjacent nodes. Moreover, the sectional-area is perpendicular to the magnetic flux path and its value depends on the hexahedron element sie. Equations (3-(5 express the radial permeance P ρ, the polar permeance P φ, and the axial permeance P, respectively, where ρ is the radial distance from the origin to the infinitesimal hexahedron element. P P P u 1 F i, j, P i, j, F u P i, j, P i1, j, F i1, j, u P 1 i 1, j, u i, j1, - Direction + Direction 1 r ln 1 r ln r 2 Moreover, there are three types of excitation sources in the PM-assisted CPSM: DC-current-fed field coil, AC-current-fed armature winding, and PMs. To begin with, the magnetomotive force produced by the field coil, F Field_Coil, is defined as a product the field coil turns N Field_Coil and its direct current I DC in (6. It gives rise to magnetic fields in the rotor yoe elements. Liewise, the magneto-motive force produced by the armature coil, F Armature_Coil, which constitutes the armature winding, can be defined as a product of the armature coil turns N Armature_Coil and its alternating current I AC as in (7. That magneto-motive force also causes magnetic fields in the stator teeth elements which belong to within the armature coil pitch as in Fig. 5. F u ui 1, j, F P 1 P i, j1, F i, j1, + Direction - Direction (3 (4 (5 F N I (6 Field _C oil Field _Coil DC F N I (7 Armature _C oil Armature _Coil AC Moreover, the NdFeBs are supplemented between the rotor positive and negative poles to decrease the saturation level and improve the energy density of the object model. The magnetomotive force produced by a PM is shown in (8. F PM L B L P PM PM PM r PM S PM rec PM PM rec PM where P PM is the PM permeance, Φ PM is the PM magnetic flux, L PM is the PM length along the magnetiation direction, S PM is the PM sectional-area that is orthogonal to the magnetiation direction, B r is the PM residual induction, and (μ rec PM is the PM recoil permeability positioned between the rotor poles. D. System Matrix and Boundary Conditions The element equations in (1 are constituted in all the nodes to organie a system matrix in (9. The system matrix consists of the permeance coefficient matrix P, the MSP column vector u, and the driving column vector F. (8 P u F (9 Assuming that the entire analysis region is divided by n i, n j, and n elements in the radial, the polar, and the axial directions, the total element number is same as their multiplication n = n i *n j *n. The n i, n j, and n are the variable numbers, which allow the flexible-sied mesh generation. In addition, all the nodes were listed from (1,1,1, (2,1,1, (3,1,1 to (n i,n j,n in sequence. In this case, the order of an arbitrary node (i,j,, n(i,j,, could be expressed as in (1. nij,, ( 1 nn( j1 n i (1 i j j It should be noted that the element equation of the arbitrary node (i,j, belongs to n(i,j,th row of the system matrix. Equation (11-(19 show that the corresponding u, F, and P belong to the n(i,j,th row. u,, u nij,, (11 i j,, i1, j, i1, j, Fi, j1, Pi, j1, F P F 1 P 1 F P F nij F P F P (12 In particular, the permeance coefficient matrix P described in (13-(19 becomes a square matrix that has the syetric components with regard to the main diagonal components: i.e. P[1,2] = P[2,1] and P[3,1] = P[1,3], etc. Such syetric matrix P especially helps reducing the computational burden.

4 Fig. 4. Permeance coefficient matrix P (3 by 3 by 3 elements in the radial, the polar, and the axial directions Fig. 4 illustrates an example of a 27 by 27 P matrix where the number 27 means the total element number. In addition, the n i, n j, and n were selected as 3. In any cases of the combinations, the syetry of P matrix is maintained. P ni (, j,, ni (, j, Pi1, j, P Pi, j1, P P 1 P ni j ni j P 1,, (13 P (,,, ( 1,, i j (14 ni j ni j P,, P (,,, ( 1,, i j (, 1, P ni (, j,, ni (, j 1, P i j (16 ni j ni j P,, P (,,, (, 1, i j (17 (,,, (,, 1,, 1 P ni j ni j P (18 i j (,,, (,, 1,, P ni j ni j P i j (19 In (13-(19, the former term and the latter term denote the row and the column, respectively. Consequently, the system matrix consists of n equations and n unnown MSPs. It needs a boundary condition that is related to the determination of a ground potential. The u(1,1,1 was set as ero in this wor. It is not important that any ground potential is ero because the magnetic potential difference between the 2 adjacent nodes is of concern. Additionally, the periodic boundary conditions are explained in (2 and (21. The equations deal with the periodicity of the electric machines in the polar direction. The model is generated considering its periodicity. j (2 1 j 1 n j j 1 1 (21 jn j With the boundary conditions above, the MSPs at all the nodes can be obtained. The nown MSPs are being used to ca- Magnetic Flux Density [T] Magnetic Field Intensity [A/m] Fig. 5. Nonlinear magnetiing curve of the core material and its chord method for convergence lculate the magnetic flux density distribution, the flux linage, the BEMF, and the electromagnetic torque. E. Nonlinear Analysis Fig. 5 describes the nonlinear magnetiing curve of the core material and its chord method for convergence. The magnitude of the magnetic flux density at the arbitrary node or element (i,j, for the th iteration process, B ( i,j,, is calculated based on its existing absolute permeability μ ( i,j,. The H ( i,j, can be calculated by dividing the B ( i,j, into μ ( i,j, as shown in Fig. 5. Afterwards, the μ ( i,j, is renewed as μ (+1 i,j, using H ( i,j,. If ε ( tolerance, save all the calculated parameters where ε ( is (μ (+1 - μ ( /μ ( *1%. If ε ( > tolerance, the above nonlinear analysis is performed again and is also renewed as +1. The arrows show the sequence of the aforementioned description. F. Flux Linage and Bac Electro-Motive Force Equation (2 expresses the generalied representation of the magnetic flux flowing between the 2 adjacent nodes. Based on (2, the radial magnetic flux flowing through each stator tooth can be calculated. The polar and axial magnetic fluxes are not considered since such components do not contribute to generate the electromagnetic torque and the output power. Fig. 6 illustrates the 1-phase armature winding layout and the radial magnetic flux distribution of the PM-assisted CPSM. The object model has a combination of 8-pole, 48-slot, and 5- coil pitch whose 3-phase winding layout is described. In this case, the flux linage of the phase winding, Ψ ph, is able to be expressed as in (22 where the radial magnetic fluxes through each stator tooth are denoted as from Φ ρ 1 to Φ ρ 12. ph NArmature _ Coil where ( 2 B ( 1 B ( B i, j, Slope : ( 2 H ( 2 ( 1 H ( H i, j, Slope : ( ( 1 Slope : ( i, j, ( ( ( (22 4,5,6,7,8, 5,6,7,8,9 1,11,12,1, 2, 11,12,1, 2,3 1% Furthermore, the can reflect the rotating motion of the PM-assisted CPSM. As a result, the flux linages of the phase windings in (22 can be computed according to the rotor

5 Fig. 6. Armature winding layout and radial magnetic flux distribution of the PM-assisted CPSM with a combination of 8-pole, 48-slot, and 5-coil pitch (one phase winding of the quarter model is illustrated only Load Line Slope of Load Line : Permeance Coefficient Knee Point (Temperature : T 1 < T 2 Not Demagnetied Demagnetied ( T1 B r ( T2 B r ( T1 B m ( T1 B nee ( T2 B nee ( T2 B m Magnetic Flux Density [T] 1 Magnetic Flux Density [T] 4 Transient Analysis Fig. 7. Time [sec.] Core loss calculation procedure for the PM-assisted CPSM position. The phase BEMF E ph in (23 can also be given depending on the rotor position. The E ph is same as the rate of change of the Ψ ph by the Faraday s law of induction. The negative sign in (23 indicates that this voltage opposes the original applied voltage based on the Len s law. E 2 Magnitude Suation of All Harmonic Core Losses at One Element i i P Pv ( Bv, vf v1 ph d d N dt ph Armature _ Coil dt (23 G. Electromagnetic Torque The radial, the polar, and the axial magnetic flux densities, B ρ, B φ, and B, are calculated from dividing the individual magnetic fluxes by their orthogonal sectional-area. The polar magnetic pressure P φ and the electromagnetic torque T φ are obtained and expressed based on the Maxwell s Stress Tensor in (24 and (25, respectively [24]. P Harmonic Analysis B B (24 2 L r 2 Harmonic Order Core Loss Profile T B B d d (25 where ρ is the radial distance from the origin to the air-gap, and L r is the rotor axial length. The P φ and the T φ can also be given according to the rotor position. H. Core Loss Evaluation Fig. 7 represents the procedure to calculate the core loss. The core loss is computed considering the fundamental and the harmonic components of the magnetic flux density according 5 3 Core Loss [W/g] 1H H 2H 4H 5H 6H 8H 1H 2H Magnetic Flux Density [T] Suation of All Harmonic Core Losses at All Elements m i Pc P i1 ( T 1 ( T 1 ( T2 ( T2 H c H m H m H c Magnetic Field Intensity [A/m] Fig. 8. PM demagnetiation curves at the operating temperature T1 and T2. to the magnitude and phase of the applied current. The following describes the procedure step by step [25]. Step 1 The magnetic flux density of each element during one electrical angle period is calculated as the rotor is moved under a load condition. Step 2 The Fourier transform of the magnetic flux density is performed in order to find the magnitude of the fundamental and the harmonic components. Step 3 From the core loss data of the material, the core loss corresponding to the magnitude and the frequency of individual harmonics is calculated. Step 4 The sum of the core losses due to all the harmonic components for each element is calculated. Step 5 The core losses of all the elements are added to obtain the total core loss of the electric machine. I. PM Demagnetiation The proposed method also provides the internal magnetic flux density and the corresponding magnetic field intensity of the PM. Such information enables for the engineers to mae a judgement whether its irreversible demagnetiation is occurred or not. If the operating point of the PM B m B nee, the irreversible demagnetiation is not occurred. If B m < B nee, the demagnetiation is occurred. In the case of Fig. 8, the former corresponds to the temperature T 1 and the latter to the temperature T 2. The judgement can be performed for all the PM elements considering the load conditions. J. Utiliation and Inherent Limitation The proposed method can generate a variety of topologies even though the geometrical parameters change as in Fig. 9 and decrease its solving time with the reduced unnowns. It can be utilied in order to find the influence of the geometrical parameters on the performance precisely and swiftly. In contrast, there exist some inherent limitations that the hexahedron elements show. According to Fig. 9, the hexahedron elements do not provide the ability to fan in or out the meshing as opposed to tetrahedron elements coonly used in. This limited meshing adjustability results in excessive number

6 Stator Outer Radius Stator Tooth Width Rotor Outer Radius Rotor Yoe Radius Shaft Radius Rotor Slot Depth Rotor Root Embrace Rotor Tip Embrace Stator Yoe Thicness Stator Axial Length Rotor Tip Thicness Rotor Pole Length of elements in less critical regions (shaft of the solution space and non-optimal mesh in high energy regions (air-gap. The proposed method requisitely needs that the facets between adjacent elements are at the same sie. Such premise condition eventually limits its flexibility. Rotor Axial Length Fig. 9. Top view and front view of 6-pole and 9-slot PM-assisted CPSM models having different pole shapes Rotor Shoe Thicness Air-Gap Length Rotor Root Thicness III. PM-ASSISTED CLAW-POLE SYNCHRONOUS MOTOR Table I shows a suariation of the design specification of the PM-assisted CPSM prototype which an automotive electronic components company developed and manufactured. As mentioned earlier, the prototype consists of 8-pole, 48-slot, and 5-coil pitch. The N42 series are supplemented between the rotor poles. Such PMs formulate a closed magnetic flux path in the rotor only. Their consequent magnetiation directions are opposite to the main flux generated by a field coil. Their existence contributes to decrease the rotor saturation level and improve the power and torque densities. Additionally, the rotor axial length is higher than the stator axial length. It is expected that the prototype can produce an improved power density through the overhang structure. Fig. 1 shows all the design variables to define a specific topology of the PM-assisted CPSM in the. Once the specific topology is determined, the requires the field coil, the armature winding, and the individual material properties. The field coil has 136 turns and the armature winding consists of 4 turns per coil. The rotor, the stator, and the shaft are made up of S2C, 35A23, and SC. Figs. 11 and 12 indicate the specific topology of the prototype produced by the JMAG Designer ( software and the method. The Figs. prove that the prototype can be drawn in detail through the method where the air region that encompasses the whole model is spacious enough to reflect the influence of the axial leaage fluxes. Fig. 1. Design variables in rotor and stator in order to determine the specific topology of the PM-assisted CPSM TABLE I DESIGN SPECIFICATION OF THE PROTOTYPE Division Value Unit Pole Number / Slot Number / Coil Pitch 8 / 48 / 5 - Rotor / Stator Axial Length 67 / 48 Rotor / Stator Outer Radius 48.5 / 7.5 Shaft / Rotor Yoe Radius 9 / 33 Rotor Tip Embrace / Thicness / 7 Rotor Root Embrace / Thicness 37 / 9.5 Rotor Slot Depth / Pole Length 9.5 / 37.4 Rotor Shoe Thicness / Air-Gap Length 14.8 /.5 Stator Tooth Width / Yoe Thicness 3.6 / 9 Stator Slot Opening / Tooth Tip Thicness 1 /.5 PM width / PM length / PM height 7 / 11 / 35.7 Rotor / Stator Core Material S2C / 35A23 - Shaft / PM Material SC / N42H - IV. VERIFICATION A. FEA Verification The verification was accomplished in order to compare the flux linage waveforms, the electromagnetic torque waveforms, and the magnetic flux density distributions at the air-gap. Such verification was done under the no-load and

7 Fig. 11. Configuration of PM-assisted CPSM prototype by JMAG Designer (one-eighth rotor and one-sixteenth stator are described Flux Linage [mwb] Field Current : 1 A dc, Armature Current : A rms Field Current : 1 A dc, Armature Current : 1 A rms (3 o Electrical Angle [ o ] Fig. 13. Comparison of the flux linage waveforms (Case 1 and Case 2 3 Flux Linage [mwb] Field Current : 16 A dc, Armature Current : A rms Field Current : 16 A dc, Armature Current : 213 A rms (3 o Electrical Angle [ o ] Fig. 14. Comparison of the flux linage waveforms (Case 3 and Case 4 Fig. 12. Configuration of PM-assisted CPSM prototype by (one-eighth rotor and one-sixteenth stator are described the load conditions. The no-load condition means when only the field current I DC is fed and the armature current I AC is not fed. The load condition means when both the I DC and the I AC are excited. Considering the allowable current densities of the prototype, the former varies from A dc to 16A dc and the latter from A rms to 213A rms. The rated load condition is when the field current is 16A dc and the armature current is 213A rms. The wide current range was employed to validate whether the method could reflect the degree of saturation of the core materials or not. The variable armature current angle β was also investigated. All the analysis conditions achieved for this verification were as follows. The total element number was maintained at the similar level (approximately 5,. Case 1 I DC = 1A dc and I AC = A rms (no-load condition Case 2 I DC = 1A dc and I AC = 1A rms (β = 3 o (load condition Case 3 I DC = 16A dc and I AC = A rms (no-load condition Case 4 I DC = 16A dc and I AC = 213A rms (β = 3 o (rated load condition Figs. 13 and 14 compare the flux linage waveforms of the Electromagnetic Torque [Nm] Field Current : 16 A dc, Armature Current : 213 A rms Field Current : 1 A dc, Armature Current : 1 A rms Electrical Angle [ o ] Fig.. Comparison of the torque waveforms (Case 2 and Case 4 TABLE II COMPARISON OF AND (CASE 4 Division Error Field Current [Adc] 16 - Armature Current [Arms] Armature Current Phase [ o ] 3 - Max. Flux Linage [mwb] % Avg. Torque [Nm] % Total Element Number 495,7 495,264.1 % Solving Time [min] %

8 Case Case Case Axial Position [] Axial Position [] Axial Position [] Axial Position [] Case Fig. 16. Magnetic flux density distribution at the air-gap produced by (Case 1, Case 2, Case 3, and Case Case Case Axial Position [] Axial Position [] Axial Position [] Axial Position [] Case Case Fig D vector plot of the magnetic flux density of the rotor produced by (Case 1, Case 2, Case 3, and Case Fig. 17. Magnetic flux density distribution at the air-gap produced by (Case 1, Case 2, Case 3, and Case 4 phase winding (Case 1, Case 2, Case 3, and Case 4. Fig. compares the electromagnetic torque waveforms (Case 2 and Case 4. It is shown that their amplitudes and phases from 3-D FEA and are in good agreement. The reason why they are in good agreement is based on the precise calculation of the permeances and the magneto-motive forces. Figs. 16 and 17 that compare the magnetic flux density distributions at the air-gap helps understand the reliability of the. The method reduced 8.9% of the solving time, compared with. It is because the former is based on the MSP calculation whereas the latter the MVP computation. Consequently, the additional time saved is expected through the method. Figs. 18 and 19 compare the vector plots of the rotor magnetic flux density (Case 1, Case 2, Case 3, and Case 4. Figs. 2 and 21 compare the rotor magnetic flux density distributions (Case 1, Case 2, Case 3, and Case 4. Figs. 22 and 23 compare the vector plots of the stator magnetic flux density (Case 1, Case 2, Case 3, and Case 4. Fig D vector plot of the magnetic flux density of the rotor produced by (Case 1, Case 2, Case 3, and Case 4

9 Fig D magnetic flux density distributions of the rotor produced by (Case 1, Case 2, Case 3, and Case 4 Fig D vector plot of the magnetic flux density of the stator produced by (Case 1, Case 2, Case 3, and Case 4 Fig D vector plot of the magnetic flux density of the stator produced by (Case 1, Case 2, Case 3, and Case 4 Fig D magnetic flux density distributions of the rotor produced by (Case 1, Case 2, Case 3, and Case 4 Figs. 24 and 25 compare the stator magnetic flux density distributions (Case 1, Case 2, Case 3, and Case 4. Such Figs. prove that the results from and are in good agreement in terms of the magnitude and the directions of the magnetic flux density. Fig. 26 compare the core loss values obtained from two method where the maximum error was 5%. In addition, Figs. 27 and 28 compare the internal magnetic flux density distribution of the PM (Case 4. It can be used to judge its demagnetiation based on its demagneti-.

10 Fig. 27. Magnetic flux density distribution of the PM produced by 3-D FEA. It can be used for judging the PM demagnetiation (Case 4 Fig D magnetic flux density distributions of the stator produced by (Case 1, Case 2, Case 3, and Case 4 Fig. 28. Magnetic flux density distribution of the PM produced by 3-D EMCN. It can be used for judging the PM demagnetiation (Case 4 Fig D magnetic flux density distributions of the stator produced by (Case 1, Case 2, Case 3, and Case 4 B. Experimental Verification The experiment was conducted to verify the reliability of the phase BEMF and the electromagnetic torque results. Fig. 29 shows the experimental setup for the no-load and load tests. Fig. 3 describes a graph comparing the phase BEMF values from the 3 methods. The maximum errors with and experimental data were 1.23% and 4.66%. It can be considered that the 3% error is caused during the manufacturing process. Figs. 31 and 32 compares the electromagnetic torque values according to the armature current and its current phase. The maximum errors with the experimental data were 2.34% and 3.91%, respectively. The method was validated in the various experimental conditions as above and the error was reasonable for its further applications. ation curve. There is only a slight difference of the magnetic flux density distributions for both computational methods. 5 Core Loss [W] % 2.38% % 2.71% 5 Case 1 Fig. 26. Case 2 Case 3 Case 4 Comparison of the calculated core loss values (Case 1, Case 2, Case 3, and Case 4 Fig. 29. Experimental setup for the no-load and load tests

11 Fig. 3. Phase BEMF [V rms ] Electromagnetic Torque [Nm] Armature Current : A rms, Armature Current Phase : o Field Current [A dc ] Experiment Comparison of the phase BEMF values as the field current changes Field Current : 6 A dc, Armature Current Phase : o Armature Current [A rms ] Experiment Fig. 31. Comparison of the average electromagnetic torque values as the armature current changes Electromagnetic Torque [Nm] Field Current : 16 A dc, Armature Current : 213 A rms Armature Current Phase [ o ] Experiment Fig. 32. Comparison of the average electromagnetic torque values as the armature current angle changes V. CONCLUSION This paper developed an infinitesimal hexahedron-elementbased method to evaluate the performance of the PM-assisted CPSM. The numerical method, which is based on the MSP calculation, palliated the computational burden much more than. It also improved its accuracy by applying the infinitesimal hexahedron-element to the PM-assisted CPSM modeling. The method is able to yield the specific magnetic flux density distribution, the flux linage, the BEMF, and the electromagnetic torque. The results were in good agreement with data and experimental data. It is expected that the proposed method is eventually being efficient in the iterative design procedure. In contrast, the hexahedron elements used in the proposed method did not show the ability to fan in or out the meshing as opposed to the tetrahedron elements normally used in. This limited meshing adjustability induced the excessive number of elements in less critical regions than high stored energy regions such as air-gap. Thus, the flexible mesh generation techniques are needed to enhance the suggested method in the future wor. ACKNOWLEDGEMENT This research was supported by the MSIT (Ministry of Science and ICT, Korea, under the ITRC (Information Technology Research Center support program (IITP supervised by the IITP (Institute for Information & counications Technology Promotion. REFERENCES [1] Y. Shen, Z. Q. Zhu, J. T. Chen, R. P. Deodhar, and A. Pride, Analytical modeling of claw-pole stator SPM brushless machine having SMC stator core, IEEE Trans. Magn., vol. 49, no. 7, pp , Jul [2] J. Cros and P. Viarouge, New structures of polyphase claw-pole machines, IEEE Trans. Ind. Appl., vol. 4, no. 1, pp , Jan./Feb. 24. [3] S. H. Lee, S. O. Kwon, J. J. Lee, and J. P. Hong, Characteristic analysis of claw-pole machine using improved equivalent magnetic circuit, IEEE Trans. Magn., vol., no. 1, pp. 7-73, Oct. 29. [4] A. K. Jain and V. T. Ranganathan, Hybrid LCI/VSI power circuit-a universal high-power converter solution for wound field synchronous motor drives, IEEE Trans. Ind. Electron., vol. 58, no. 9, pp , Sep [5] A. Ibala and A. Masmoudi, Accounting for the armature magnetic reaction and saturation effects in the reluctance model of a new concept of claw-pole alternator, IEEE Trans. Magn., vol. 46, no. 11, pp , Nov. 21. [6] F. Zhang, H. Zhang, and G. Liu, 3D finite element analysis and experiment for axial sectional claw pole machine with permanent magnet outer rotor, in Proc. IEEE ICEMS, Oct. 21, pp [7] C. Tong, P. Zheng, Q. Wu, J. Bai, and Q. Zhao, A brushless claw-pole double-rotor machine for power-split hybrid electric vehicles, IEEE Trans. Ind. Electron., vol. 61, no. 8, pp , Aug [8] Y. Ni, Q. Wang, X. Bao, and W. Zhu, Optimal design of a hybrid excitation claw-pole alternator based on a 3-D MEC method, in Proc. IEEE ICEMS, Sep. 25, pp [9] Y. Guo, J. Zhu, and D. G. Dorrell, Design and analysis of a claw pole permanent magnet motor with molded soft magnetic composite core, IEEE Trans. Magn., vol., no. 1, pp , Oct. 29. [1] Y. G. Guo, J. G. Zhu, and H. Y. Lu, Effects of armature reaction on the performance of a claw pole motor with soft magnetic composite stator by finite-element analysis, IEEE Trans. Magn., vol. 43, no. 3, pp , Mar. 27. [11] D. S. Jung, S. B. Kim, K. C. Kim, J. Ahn, S. C. Go, Y. G. Son, and J. Lee, Optimiation for improving static torque characteristic in permanent magnet stepping motor with claw poles, IEEE Trans. Magn., vol. 43, no. 4, pp. 77-8, Apr. 27. [12] C. Kaehler and G. Henneberger, Transient 3-D FEM computation of eddy-current losses in the rotor of a claw-pole alternator, IEEE Trans. Magn., vol. 4, no. 2, pp , Mar. 24. [13] Y. Chen, Z. Q. Zhu, and D. Howe, Three-dimensional lumpedparameter magnetic circuit analysis of single-phase flux-switching permanent-magnet motor, IEEE Trans. Ind. Appl., vol. 44, no. 6, pp , Nov./Dec. 28.

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