On-Load Field Component Separation in Surface Mounted Permanent Magnet Motors using an Improved Conformal Mapping Method

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1 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < On-Load Field Component Separation in Surface Mounted Permanent Magnet Motors using an Improved Conformal Mapping Method Karim Abbaszadeh Farhad Rezaee Alam Department of Electrical Machines Drives, K.N. Toosi University of Technology, Tehran, Iran This paper introduces an Improved Conformal Mapping method for separation of the on-load air gap magnetic field components in surface mounted permanent magnet SPM motors under loading conditions. The method can model the magnetic induction inside a permanent magnet PM due to the armature reaction the relative recoil permeability. The method can also consider the simultaneous influences of the armature reaction, slotting effect, magnetic saturation, relative recoil permeability in the determination of operating points of the PM in its different parts. Therefore, this proposed method can be useful for on-load performance analysis of SPM motors. Three conformal mappings are used to reach the main canonical domain: two logarithmic complex functions the Schwartz-Christoffel SC mapping. The field solution in the slotless domain is then mapped back into the slotted domain using the complex air gap permeance. The field solution in the slotted domain is used to calculate the on-load air gap field components due to the PMs the armature reaction. These on-load field components allow the calculation of the on-load PM flux-linkage, the on-load PM Back-EMF, the on-load torque components. The influences of electric loading on the on-load torque components the on-load PM Back-EMF are studied using the method. The on-load cogging torque periodicity is confirmed to change due to the magnetic saturation the harmonic content of torque components the PM Back-EMF is also altered due to the electric loading. The accuracy of this developed model is verified by comparing the results obtained through with the respective results obtained through Finite Element Analysis FEA the Frozen Permeability Method. Index Terms Back-EMF, Improved Conformal Mapping, Magnetic saturation, Permanent Magnet PM, Torque. I. INTRODUCTION T different methods currently available for modeling electrical machines, such as the Finite Element Method FEM [], Winding Function Theory WFT [], Magnetic Equivalent Circuit MEC [], Field Reconstruction Method FRM [], Subdomain SD Model [], Conformal Mapping method [], all share a common deficiency; namely, the inability to separate the on-load air gap field components. This separation is necessary for calculation of the on-load components of the torque the Back-EMF, such as the cogging torque the PM Back-EMF. Conventionally, the cogging torque the PM Back-EMF are considered to be electric loading-independent i.e., they are usually considered as the open circuit characteristics. However, saturation of the machine iron under loading conditions is likely to influence these components []. The PM air gap field calculated by finite element analysis FEA software does not include the influence of electric loading. Therefore, FEA software cannot segregate the onload air gap field components. The Frozen Permeability method is the only method that has been introduced to separate the on-load air gap field components [-]. acts based on FEM, but it is a time-consuming cumbersome method. This paper presents an Improved Conformal Mapping method that segregates the on-load air gap field components. Unlike, is a fast userfriendly method. acts based on complex analysis. One of the first applications of the method was the calculation of the well-known Carter coefficients that consider the slotting effect in electrically excited machines []. Zhu introduced a -D HE Manuscript received December, ; revised March July, ; accepted October,. Corresponding author: Farhad Rezaee Alam. farhad.rezaee.alam@gmail.com. Digital Object Identifier inserted by IEEE relative permeance function for the slotting effect using [-]; this is a predetermined function only one point of its waveform in the center of the slot opening is calculated by. Moreover, this function is unable to predict the tangential component of air gap flux density []. Zarko resolved these issues by proposing a complex relative permeance at every point in one slot pitch single slot model [,-]. The main assumptions in [,-] were an infinitely deep slot opening, ignoring the interaction effect between adjacent slots, neglecting the deformation of magnet shape circular contour to predict the air gap flux density. The numerical Schwarz-Christoffel mapping the Matlab SC Toolbox were to relax these assumptions [-]. The Schwarz-Christoffel SC method was used to analyze the thrust force in a permanent-magnet linear motor PMLM [], in a tubular permanent magnet actuator [] for magnetic field analysis of inset surface mounted permanent magnet synchronous motors [] using the image method for field calculation in the canonical domain. O Connell Krein in [-] Boughrara et al. in [] used three s, including SC mapping two logarithmic complex functions, for field calculation in an annular domain main canonical domain using Hague s solution. The method has also been used to improve the other modeling methods of electrical machines, such as MEC [-]. Two common deficiencies appear in most previous publications regarding application of the method for modeling electrical machines. First, the operating point of the PMs was assumed to be constant over their entire volumes did not vary with rotation of the rotor. Second, the saturation effect was neglected. The present paper has two main novelties: the first is to introduce the method, which can consider the core saturation effect the working point variation of PMs throughout their volumes. The other novelty - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See Downloaded from

2 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < is the separation of the on-load air gap field components calculation of the on-load PM flux-linkage, the on-load PM Back-EMF, the on-load torque components using this developed method. This paper is organized as follows: Section II introduces the analytical numerical s used in this paper. The modeling of the permanent magnet, stator winding, magnetic saturation are introduced in sections III, IV, V, respectively. The simulation results of the on-load air gap field component separation, the on-load PM flux-linkage BackEMF calculation are presented in section VI the on-load torque component separation is given in VII. Section VIII presents the conclusions. A C are the unknown integration constants, n is the number of polygon corners, w,,wn are the points in the canonical domain in the w-plane corresponding to the polygon corners, αk are the interior angles of polygon. TABLE I MAIN PARAMETERS FOR THE PERMANENT MAGNET SYNCHRONOUS MOTOR Parameter Value Unit Number of pole pairs, p Number of slots, Q s Magnet remanence, Br. T Recoil Relative permeability, µr. Rated frequency, f Hz Motor topology Internal rotor Magnetization Radial Stator outer diameter mm Stator inner diameter mm Active length, L mm Air gap length, g mm Magnet thickness. mm Pole arc to pole pitch ratio, αp. Winding turns per coil, Nc CONFORMAL MAPPING The method is an analytical numerical tool for solving all kinds of D fields magnetostatic, electrostatic, etc.. This combined analytical numerical method offers far more facilities than other analytical semi-analytical methods. In this paper, three s are used to reach the main canonical domain annular domain [-]. - - This transforms the curved geometry in the s-plane into multilateral geometry in the z-plane. Figs. - show the motor geometry in the s-plane the z-plane for a typical PMSM with slots poles Table I. B. Schwarz-Christoffel Mapping The second is the numerical Schwarz- Christoffel SC maps one canonical transformation. SC mapping domain e.g., a rectangle, disk, bi-infinite strip, or the upper lower half-plane into the interior exterior of the respective polygon. SC mapping is defined as follows [-, -]: Real z Fig.. The motor geometry in the z-plane The analytical solution of the SC integral in is very difficult for geometries with more than three vertices []. Therefore, the SC Toolbox is used for numerical solution of. The SC Toolbox provides a library of comm-line functions. The key functions used in this Toolbox are described briefly in [, -] in more detail in []. In this paper, the SC Toolbox is used to map the rectangle canonical domain in the w-plane to the air gap, which is elongated due to the stator slots. For the studied PMSM, the canonical rectangle in the w-plane is shown in Fig.. Imw - A. Logarithmic Conformal Mapping I The first is a logarithmic complex function, as follows: Imag z II. Fig.. The motor geometry in the s-plane - Rew Fig.. The motor geometry in w-plane - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

3 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < C. Logarithmic Conformal Mapping II The final is also a logarithmic complex function of the following form [-]: are the length the width of the canonical rectangle in the w-plane. The interior of the annulus in the ψ-plane is mapped into the canonical rectangle in the w-plane using the aforementioned. This annular domain is used to solve the problem the field solution in the slotless domain is then mapped back to the desired domain. The annular domain is used as the main canonical domain for two reasons. First, the right left edges of the canonical rectangle are mapped on together in the annular domain Fig... Therefore, the boundary condition is applied automatically by using. Second, Hague s field solution is known in the annular domain. The air gap flux density in the ψ-plane can then be calculated as: [ ] Similarly, the air gap flux density is calculated for all line currents due to permanent magnets armature windings. Conformal maps preserve the scalar magnetic potential, but they cannot conserve the vector field potential, because is coupled with through the gradient operator, which is coordinate system-dependent. The air gap flux density in the desired domain s-plane is calculated as []: a The complex permeance for the slotted air gap is defined as: c I b Fig.. The main canonical domain in the ψ-plane A single line current of magnitude I, located at in an annulus, is shown in Fig.. The stator rotor radii in the ψ-plane are a b, respectively. The relative permeabilities of the stator, air gap, rotor in the ψ-plane are,,, respectively. The scalar magnetic potential in the air gap annulus, as a function of r, θ, is given by Hague [] as: Circumferential position Mech deg. a - { t { are the radial the tangential components of the slotted air gap complex permeance. Fig. shows the components of complex air gap permeance for the analyzed PMSM in this paper, without considering the saturation effect. As shown in Fig., the complex permeance is quite similar in all slot pitches its period is one slot pitch. r,, - Circumferential position Mech deg. b Fig.. a The radial b the tangential components of complex air gap permeance without saturation effect - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

4 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < III. PERMANENT MAGNET MODELING A. PM Operating Point So far, the working point of permanent magnets has been assumed constant throughout their volume in the method. Some previous publications approximated the relative recoil permeability of permanent magnets as [, ]. The general form of the BH characteristic of PM materials is shown in Fig.. is the PM radial thickness, is the number of wires in each PM side, is the PM equivalent surface current at each segment of the PM sides. Fig.. The equivalent surface currents for one PM with radial magnetization B R BR P BP D BD HD HP H Fig.. The demagnetization characteristic of a PM with the operating line Boules minimized this error by assuming an average magnetization for a PM with []. Furthermore, Markovic at first chose the middle point between R D in Fig. as the operating point for all PM parts []. Gysen Markovic then introduced similar formulas for the flux density in the PM operating point in terms of relative recoil permeability of the PM [, ]. However, they did not consider the slotting the armature reaction effects when determining the PM operating point. For magnets with high relative recoil permeability, the PM operating point is not constant for different parts inside PM it also varies with the rotation of the rotor, while considering the slotting, the armature reaction, the magnetic saturation effects. B. PM Equivalent Currents A PM body can be replaced with two equivalent currents: The surface currents with the current density is the unit normal vector to body surface. The volume currents with the current density is the PM magnetization vector can be defined as: is due to the external fields armature reaction inside PMs. For radial PM magnetization, the surface currents are shown in Fig.. The current magnitude of wires in PM sides with radial magnetization is defined as: For magnets with high, the second term in is not negligible. Therefore, the magnetization level differs throughout the PM volume the PM equivalent volume currents have to be considered in calculation of air gap flux density distribution. The PM equivalent volume currents are calculated by dividing all PMs into a large number of segments segments, as shown in Fig.. The equivalent volume current is then calculated for each PM segment having the magnetization level in its all vertices, using in the finite difference form. + is the area of each PM segment. Fig.. The equivalent volume currents for a typical PM IV. STATOR WINDING MODELING The studied PMSM has a single-layer stator winding. The total mmf per slot is divided into the number of wires wires. The matrix connection between the phase the slot currents is defined as: [ ] The matrix has columns, because the air gap polygon is started from centerline of slot Fig.. The relationship between the punctual slot currents the phase currents is introduced as: is the winding turns per coil, is the wire number per slot, i is the phase current vector, which is defined as:, A representation of PM armature equivalent currents in the z-plane is shown in Fig.. The positions of these line currents are also mapped to an annular domain in the ψ-plane. V. SATURATION EFFECT MODELING The basic assumption of the method is the infinite relative permeability of the core. The saturation effect in the - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

5 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < method has been assumed to be negligible in all previous publications other than []. Hafner in [] proposed a technique to include the saturation effect on the armature flux density in the method by finite element reparameterization in frequency domain, but this method has two main defects: first, it does not consider the effect of the no-load flux density second, this proposed technique is FE-dependent. Moreira in [] Faiz in [-] modeled the air gap length of induction machines in sinusoidal form to include the magnetic saturation effect. Magnetic saturation usually happens on the tooth surface its tips. Ojaghi, in [], only considered the saturation effect on the tooth surface while assuming the stator rotor slots are open. -. is the tooth flux, is the tooth length in radial direction, are the average values of magnetic flux density magnetic field intensity in the particular tooth, respectively. Calculation of the mmf drops in the stator yoke should consider the following magnetic circuit: y y y t t y y t t Fig.. The magnetic circuit of the stator yoke In the above circuit, the tooth fluxes are known. This circuit includes nodes one loop. Therefore, an equation system can be created, as follows: Armature Currents -. Volume Currents Surface Currents { Real z -. Fig.. A zoomed view of the air gap polygon with PM armature equivalent currents in the z-plane A. Saturation Effect on Tooth Surface The magnetic saturation of the tooth surface is due to the main flux-linkage can be included by introducing an additional air gap permeance. This is equivalent to increasing the air gap length under each tooth by the coefficient. For PM synchronous machines, the coefficient is defined as []: is the air gap flux density, g is the air gap length, are the mmf drops in two adjacent teeth, are the mmf drops in the stator rotor yokes between two adjacent teeth, respectively. The additional term on the right-h side of is the ratio between the mmf drops in the iron parts air gap for two neighboring teeth. The mmf drops the saturation factor are calculated anew for every rotor position. The mmf drops in stator teeth are calculated when having the radial component of the, as follows: total air gap flux density This equation system is solved for every rotor position. Therefore, calculation of the magnetic flux is possible in. The mmf drop is different parts of the yoke calculated for every part of the yoke, as follows: is the mmf drops in the respective segment of the stator yoke, is the yoke length between two adjacent teeth, are the average values of the magnetic flux density magnetic field intensity in the respective segment of the stator yoke. Fig. shows one complete time cycle which is applied to a tooth ms of the saturation factor surface. The saturation factor for the other teeth surface is similar is only phase shifted compared to the plot shown in Fig.. Saturation Factor Ksat Imagz Time ms Fig.. The saturation factor applied to a tooth surface - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

6 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < B. Saturation Effect on Tooth Tip The magnetic saturation in the tooth tips is largely due to the stator leakage flux. Saturation of the stator tooth tips results in a decrease in the relative permeability of the tooth heads. This effect is equivalent to increasing the stator slot opening width. The magnetic saturation due to the stator leakage flux can be approximated by considering Norman s method. According to the Norman method, the mmf of single stator slot can be expressed as []: is the slot current, is the winding turns per coil, is the number of stator winding layers, is the number of stator winding parallel paths, is the stator coil pitch, is the pole pitch, is the stator slot number, are the stator winding pitch factor the stator winding factor for the fundamental component, respectively. The fictitious leakage flux density due to that saturates the tooth tips is calculated as []: is the stator slot pitch. The saturation factor due to the leakage flux as a function of can be approximated as: is the slot opening width of unsaturated tooth tips. Therefore, the slot opening widths are calculated for saturated tooth tips using the aforementioned algorithms. Fig. shows one complete time cycle of the increments in slot opening width in one pole pitch. slot opening wso deg Start Sinusoidal ɸ current Input Yes Calculate the unsaturated complex permeance Calculate the on-load field components Calculate the on-load PM total flux-linkage slot opening Apart from some papers, such as [-], the influence of electric loading magnetic saturation on the on-load PM field has been neglected in most previous investigations. The on-load PM armature fields are coupled together while considering the magnetic saturation. Therefore, finding suitable techniques to separate the on-load PM the on-load armature fields becomes important for calculating the on-load torque Back-EMF components. To this end, the method is introduced in this paper, as it can include the influence of magnetic saturation electric loading. In this paper, the balanced sinusoidal three phase currents with RMS value A are injected into the stator windings of the analyzed PMSM. Under this condition, the influence of magnetic saturation electric loading is investigated on the on-load PM armature field components, the on-load PM Back-EMF, the on-load torque components. Fig. shows a simulation flowchart. No Calculate the saturated complex permeance The slot opening width, with the magnetic saturation taken into account, is defined as: slot opening ON-LOAD AIR GAP FIELD ANALYSIS If t= The increment in slot opening width is defined as: VI. Time ms Fig.. The increment in slot opening width due to saturation effect Calculate the on-load torque components If t=tend No Yes Calculate the on-load PM total Back-EMF end Fig.. A simulation flowchart A. Frozen Permeability Method The Frozen Permeability Method acts based on FEM. The algorithm used in this paper is as follows: The magnetic saturation is ignored in the rotor yoke for SPM motor. The stator core is divided into elements, as shown in Fig.. The PM armature excitations are applied simultaneously. The average value of the magnetic flux density is calculated for all elements of the stator core in one complete time cycle ms. - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

7 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE <. The amplitude of the fundamental component of the onload PM field is decreased due to the magnetic saturation. r The average value of the magnetic field intensity is calculated for all elements of the stator core in one complete time cycle ms by using the BH curve. The relative permeability is calculated for all elements of the stator core in one complete time cycle ms. The Curve Fitting Toolbox of Matlab software is used to obtain the suitable Fourier series for the function of relative permeability of all stator core elements in terms of simulation time. The suitable materials with the relative permeabilities obtained through the Curve Fitting Toolbox are created in the Maxwell software. These materials are then assigned to the respective stator elements. Under the aforementioned condition, the on-load radial tangential components of air gap field due to PMs armature reaction are calculated while considering the PM excitation only the armature excitation only, respectively. These on-load field components allow calculation of the on-load PM flux-linkage, the on-load PM BackEMF, the on-load torque components. Circumferential position Mech deg. a Radial component t Circumferential position Mech deg. b Tangential component Fig.. The components of the saturated complex air gap permeance. Bmr T Circumferential position Mech deg. a on-load PM field C. On-load air gap field components The on-load air gap PM, armature, total field components are compared with the respective results obtained through or FEM, as shown in Figs. -. A good agreement is achieved between the FEM/ results. The influence of electric loading on the on-load air gap field components was investigated by comparing the Fourier spectrum of the radial component of the on-load PM field in the middle of the air gap calculated by to the corresponding result obtained through, as shown in Fig. Bar T B. Saturated Complex Permeance The radial the tangential components of slotted air gap complex permeance, considering the magnetic saturation in a moment of simulation time, are shown in Fig.. These components are required to calculate the on-load air gap field components. As shown, the saturated complex air gap permeance components are not similar in different slot pitches their periods are one pole pitch due to the magnetic saturation. Circumferential position Mech deg. b on-load Armature field FEM Br-total T Fig.. The model Circumfrential position Mech deg. c Total field Fig.. The radial component of on-load PM, armature total field in the middle of air gap. - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

8 To allow the reader to assess the level of saturation in this particular SPM motor, Fig. shows one complete time cycle of the average levels of flux density in the stator teeth different parts of the stator yokes in one pole pitch. The material used in the stator core is Steel_. Fig. shows the BH curve of Steel_. Bmr T./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < D. On-load PM Back-EMF The on-load PM flux-linkage the on-load PM BackEMF having the on-load air gap PM field the turn function of phases can be calculated as follows: is the radius of circular integration contour in the air gap, is the core length in axial direction, is the turn is the radial component of onfunction of phase A, load air gap PM flux density, is the mechanical angle, is the on-load PM flux-linkage, is the on-load PM Back-EMF, both of phase A.. Frequency order Bt. Bt. Bt Time ms a Tooth flux density Circumferential position Mech deg. a on-load PM field.. Yoke Flux Density T Bmt T.. Bat T Fig.. The amplitude spectrum of the radial air gap PM field Tooth Flux Density T By. By By Time ms b Yoke flux density Fig.. The average flux density levels in different parts of the stator in one pole pitch Circumferential position Mech deg. b on-load armature field. Bt-total T B T FEM..... H KA/m Circumfrential position Mech deg. c Total field Fig.. The tangential component of the on-load PM, armature, total field in the middle of air gap Fig.. The BH curve for Steel_ The turn function of phase A is shown in Fig.. Similar turn functions exist for other phases with respective phase shifting. Fig. shows the on-load PM total fluxlinkage the on-load PM total Back-EMF of phase A obtained through /FEM with input RMS current A. - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

9 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < The influence of electric loading magnetic saturation on the on-load PM Back-EMF was studied by comparing the Fourier spectrums of no-load on-load PM Back-EMF of phase A calculated by, as shown in Fig.. The amplitude of the fundamental component of the on-load PM Back-EMF decreases due to the magnetic saturation the magnetic induction inside the PMs. Turn function VII. Circumferential position Mech deg. Fig.. The turn function of phase A. m Wb In this paper, the Maxwell Stress Tensor MST method having the on-load air gap field components is used to calculate the on-load torque components. According to the MST method, the total electromagnetic torque can be defined as follows:. SEPARATION OF ON-LOAD TORQUE COMPONENTS. -., -. Time ms total Wb a FEM Time ms Frequency order Fig.. The Fourier spectrums of PM Back-EMF of phase A - The total torque can be divided into three components, as follows: - Time ms c Total Back-EMF volts FEM Time ms d Fig.. The on-load PM total flux-linkage Back-EMF of phase A Good agreement is evident between results the respective results obtained through FEM/. A. On-Load Cogging Torque The cogging torque is usually calculated under an open circuit condition. Fig. compares the cogging torque waveforms obtained through FEM, without considering the armature reaction the magnetics saturation. Cogging torque N.m PM Back-EMF volts b L is the core length of the motor in the axial direction, Rg is the radius of the circular integration contour in the air gap, θ is the mechanical angle. A reliable result is perhaps best obtained by setting the MST integration contour as far away from any active material boundary as possible []. In this paper, the contour is set in the middle of the air gap. PM Back-EMF volts -.. FEM Rotor position Electrical deg. Fig.. The open circuit cogging torque obtained through FEM - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

10 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < The on-load cogging torque is due to the interaction effect between the on-load air gap PM Field the saturated stator teeth, while considering the electric loading the magnetic saturation, as follows: Cogging Torque N.m are the radial the tangential components of the on-load air gap PM field. Fig. shows the on-load cogging torque waveforms under two different loading conditions. The on-load cogging torques obtained through are also compared under the input RMS current A, as shown in Fig.. As shown, the cogging torque periodicity changes its amplitude increases due to the electric loading the magnetic saturation. - I= I= - I= in tooth surface tips. The nd order harmonic is the reluctance torque ripple due to the magnetic saturation the electric loading. B. On-Load PM Torque The on-load PM torque is the main component of the produced torque is due to the interaction effect occurring between the on-load air gap PM armature fields, as follows: The on-load PM torques obtained through under input RMS current A are compared in Fig.. Fig. also compares the Fourier spectrums of the PM torques obtained through methods. As seen from Fig., the main component of PM torque ripple is th order harmonic due to slotting effect. The amplitudes of DC component th order harmonic are also decreased slightly due to the magnetic saturation. C. Armature Torque The on-load armature torque is calculated as follows: Rotor position Electrical deg. Fig.. On-load cogging torque PM Torque N.m The on-load armature torques calculated by are compared as shown in Fig.. Fig. also compares the Fourier spectrums of armature torques obtained through methods, while considering the pure sinusoidal input currents with RMS value A. Rotor position Mech deg. Fig.. On-load PM torques for input RMS current A Frequency order Fig.. The Fourier spectrum of on-load no-load cogging torques As expected, the no-load cogging torque calculated by has only a th order harmonic because its period is equal to electrical degrees. The period of the no-load cogging torque waveform is calculated as follows: No-load cogging torque period=π/lns, p LNs, p is the least common multiple of the stator slot number the rotor pole number. However, the on-load cogging torque calculated by has components other than the th order harmonic. The DC component is the average reluctance torque due to the time variation of air gap reluctance, owing to the saturation effect PM Torque N.m Cogging Torque N.m The reluctance variation due to the saturation effect is independent of flux direction; therefore, the period of on-load cogging torque waveform is expected to equal electrical degrees. The Fourier spectrums of on-load no-load cogging torques obtained through methods are compared, as shown in Fig.. Frequency order Fig.. The Fourier spectrums of PM torques considering without considering the magnetic saturation. The armature torque obtained through method has only a nd order harmonic as the armature torque calculated by method has also other harmonics due to the magnetic - c IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

11 ./TMAG.., IEEE Transactions on Magnetics > FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G., AB- DOUBLE-CLICK HERE < Armature Torque mn.m saturation. The DC component is the average value of reluctance armature torque which is only applied to the stator core. The amplitude of the nd order harmonic also increases owing to the magnetic saturation. In general, the amplitude of the armature torque can be ignored over other torque components. Lastly, the developed method was validated by comparing the total torques obtained through the method FEM, as shown in Fig.. This figure shows good agreement between the results. the electric loading. In contrast to the no-load cogging torque, the on-load cogging torque periodicity changes other harmonic components, such as the average reluctance torque DC component the reluctance torque ripple nd order harmonic, are added to its Fourier spectrum. The PM torques obtained through the methods have almost the same Fourier spectrums, but the amplitude of the DC component the th order harmonic is decreased slightly due to the magnetic saturation with the method. The armature torque component is purely sinusoidal when the magnetic saturation is not considered, but its Fourier spectrum alters due to the magnetic saturation. On the other h, the armature torque can be ignored because its amplitude is much smaller than the other on-load torque components. REFERENCES [] - Rotor position Mech deg. [] Armature Torque mn.m Fig.. On-load armature torque [] [] [] Frequency order [] Total Torque N.m Fig.. The Fourier spectrums of armature torques obtained through methods. [] FEM [] [] Time ms [] Fig.. The total torques obtained through FEM [] VIII. CONCLUSION This paper introduces the method for separation of the on-load air gap field components. The method can consider the simultaneous influences of the slotting effect, the armature reaction, the magnetic saturation, the relative recoil permeability in determining the PM operating point in its different parts in different rotor positions. The use of on-load PM armature field components allows calculation of the on-load PM flux-linkage, the on-load PM Back-EMF, the on-load torque components using. All results were verified through comparison with the respective /FEM results. As shown, the harmonic content of these on-load components is altered due to the magnetic saturation [] [] [] [] D. Wang, X. Wang S. Y. Jung, Cogging torque minimization torque ripple suppression in surface-mounted permanent magnet synchronous machines using different magnet widths IEEE Trans. Magn., vol., no., pp. -, May.. J. Faiz I. Tabatabaei, Extension of winding function theory for nonuniform air gap in electric machinery, IEEE Trans. Magn., vol., no., pp. -, Nov.. S. A. Saied, K. Abbaszadeh M. 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