Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using...

$: Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using Schwarz\u0026#x2013;Christoffel Transformation$
3166 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 8, AUGUST 2009

Magnetic Field Analysis of Inset and Surface-Mounted Permanent-MagnetSynchronous Motors Using Schwarz–Christoffel Transformation

K. Boughrara�, D. Zarko�, R. Ibtiouen�, O. Touhami�, and A. Rezzoug�

Ecole Nationale Polytechnique (LRE-ENP), Algiers, BP182, 16200, AlgeriaFaculty of Electrical Engineering and Computing, Department of Electrical Machines, Drives and Automation,

University of Zagreb, HR-10000 Zagreb, CroatiaFaculté des Sciences, Université Henri Poincaré, BP 239 54506 Vandoeuvre lès Nancy, France

In this paper, we apply an original numerical Schwarz–Christoffel (SC) transformation to analyze magnetic field originating from per-manent magnets and the armature winding currents in a slotted air gap of an inset permanent-magnet synchronous motor. We obtainedthe solution of the SC integral numerically using Matlab SC Toolbox. We used this field solution to calculate both cogging torque andelectromagnetic torque by integrating the Maxwell stress tensor inside the air gap. The case without inter-polar piece, which is equiva-lent to a surface-mounted permanent-magnet motor, is also treated. The accuracy of the developed method is verified by comparing itsresults with those obtained from the developed numerical finite-element models.

Index Terms—Finite-element method, inset permanent-magnet synchronous motor, magnetic field, Schwarz–Christoffeltransformation.

I. INTRODUCTION

T HE technological developments in the manufacturing ofpermanent magnets with high energy product and power

semiconductors have led to an increasing interest in permanent-magnet excited machines. Permanent-magnet synchronous mo-tors (PMSMs) have been increasingly used in machine tools androbotic applications which require motors with low inertia, hightorque to volume ratio, and high torque to stator current ratio [1],[2]. These features have made brushless dc motors one of thebest choices to replace conventional brushed type dc motors inrobotic applications [3], [4]. The rare-earth permanent magnetsallow for different structures of synchronous motors [5], [6].Nevertheless, their relatively high cost implies the minimiza-tion of their size while preserving acceptable performances. Thepossible minimization degree depends on the rotor design typeand the inverter type.

An accurate magnetic field analysis of the motor, which in-cludes iron saturation, can be performed using the finite-elementmethod (FEM). However, it is time consuming [7] and can beused only at the final stage to verify the proposed model. In thispaper, an advanced method is proposed, which solves the mag-netic field in a rotary motor with slots. The method defines mag-netic field strength as a function of a complex variable. In orderto determine the magnetic field and torque characteristics, firstthe permanent magnets and the armature winding currents instator slots are substituted with their equivalent currents. Next,a sequence of two conformal maps (CMs) is applied: the firstone converts a rotating configuration of the motor into a linearone, whereas the second one opens the interior of a polygoncontaining rotor and stator slots into a rectangle. Using the fieldsolutions of equivalent currents and evaluating numerically the

Manuscript received September 19, 2008; revised December 10, 2008.Current version published July 22, 2009. Corresponding author: K. Boughrara(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2009.2016559

inverse mapping and the derivative of the SC map leads to thefield solution in the air gap of the motor. Thus, calculated ra-dial and tangential components of the flux density allow for cal-culation of cogging and electromagnetic torque characteristicsby integrating the Maxwell stress tensor inside the air gap. Thestructure (one pole pitch) of the inset permanent-magnet syn-chronous motor fed by trapezoidal currents in which arc-shapedradially magnetized permanent magnets are inset into the rotoriron surface so that the outer surface of the rotor is cylindricalis shown in Fig. 1(a). From the stator viewpoint the magnet ap-pears as a large air gap in the direct axis while the inter-polariron presents a small air gap in the quadrature axis. This struc-ture can be interesting when we research a good price to perfor-mance ratio [6] and a maximum torque per Ampere.

The numerical Schwarz–Christoffel analysis of magneticfields in electrical machines has already been used by O’Con-nell and Krein [8] for the analysis of magnetic field and forcein a variable reluctance machine. The method has also beenused by Krop [9] for a linear permanent-magnet motor andGysen [10] for a tubular permanent-magnet actuator. In [11],O’Connell and Krein confirm the efficiency of the method inoptimization of electric machine design.

The numerical Schwarz–Christoffel method is an alternativemethod to the analytical one, which is limited to simple ge-ometries like a single slot model [12]–[15]. The transformationof a single slot can provide accurate field solutions in surfacemounted PM motors. However, it fails in the case of an insetPM motor, because the presence of a slot in the rotor requiresan SC transformation of the air-gap region throughout the entirepole pitch, which cannot be carried out analytically. The use ofnumerical SC transformation presented in this paper can yieldan accurate field solution in such a structure.

II. NUMERICAL SCHWARZ–CHRISTOFFEL TRANSFORMATION

The SC transformation maps the real axis ofcoordinate system onto the boundary of the polygon in

the coordinate system, so that the upper half planein is mapped into the interior of the polygon in .

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BOUGHRARA et al.: MAGNETIC FIELD ANALYSIS OF INSET AND SURFACE-MOUNTED PERMANENT-MAGNET SYNCHRONOUS MOTORS 3167

Fig. 1. Study model in the � plane �� : (a) with inter-polar piece topology(IPM), (b) without inter-polar piece topology (SPM). (c) Direction ofmagnetization.

The SC mapping is given by the SC integral [16]

(1)

where and are the unknown integration constants (to befound from the geometry of the polygon), is the number ofthe polygon corners, are the points on the real axisof the coordinate system corresponding to the polygon cor-ners , and are the interior polygon angles. The for-mula in (1) can be modified to allow maps from disks, bi-infinitestrips, and rectangles (all referred to as canonical domains) to apolygon. The theorem’s utility is in allowing the user to solvea difficult boundary value problem on an arbitrary polygon-

Fig. 2. Study model in direct position of rotor of inset permanent-magnet motorin the � plane �� .

shaped domain by first solving the problem in a canonical do-main and then mapping the result to the desired domain. Unfor-tunately, for problems with more than three vertices, in general,the SC integral in (1) has no closed-form solution. In addition,in problems of this size the locations of points , called prever-tices, are unknown a priori. In other words, the target polygonis known, but the points in the canonical domain correspondingto its vertices are not. Thus, the SC method requires three nu-merical steps: (a) finding the point , which is known as theparameter problem, (b) calculating the SC integral in (1), and(c) inverting the map. These steps have been difficult to imple-ment numerically for meaningful problems. Recently, each of(a)–(c) has been coded into Matlab SC Toolbox, in which theuser specifies the canonical and source domains, and the toolboxcalculates the SC map. The current version of SC Toolbox wasreleased in 2005. A key improvement in this version is the ad-dition of the CRDT algorithm [17]. This algorithm facilitatesmapping to multiply elongated regions, which are normally verydifficult to map due to crowding. Most typical motor air-gappolygons have multiple elongations due to the stator and rotorslots, so this new functionality is the key in making the toolboxuseful for motor design. The toolbox provides a library of com-mand line functions. The key functions used in the paper aredescribed in [18].

A. Motor Model in Complex Planes

The original motor configuration is placed in the complexcoordinate system , with the origin in the motor center (Fig. 1).The first CM to be used is given by the complex function

(2)

where . The link between the coordinates in the andplanes is

(3)

This complex logarithmic conformal transformation creates alinear configuration (given in the coordinate system) instead



Fig. 3. Canonical rectangle in the � plane.

Fig. 4. Study model of inset permanent-magnet motor in the � plane �� with rotation of rotor by one half slot pitch.

of the rotating one (given in the coordinate system), with re-gard to the motor radius (curvature). After the transformation,the stator and rotor surfaces become as shown in Fig. 2. At thisstep of the study, the slots are considered as rectangular (in theactual motor they are semi-closed). This structure is much sim-pler than the original one, but the magnetic field is still impos-sible to solve analytically.

The magnetic field is calculated in the middle of the air gapin the plane, where is the radius of

the stator inner surface and is the radius of the rotor outersurface. In the plane, the middle of the air gap is given by

where (4)

The coordinates of the stator coils and permanent-magnet cur-rent carrying conductors in the plane are given by

(5)

(6)

B. Canonical Rectangle in the Domain

The air gap in Fig. 2 is a polygon with 32 vertices, numberedin counterclockwise order. This numbering convention definesan array of vertex coordinates. The coordinates are definedin the complex domain where the real part of the coordinatespecifies the abscissa and the imaginary part the ordinate. TheSC Toolbox is then used to create the polygon object by callingin Matlab Command Window

(7)

Next, an SC map from the interior of the closed rectangulardomain, shown in Fig. 3, to the interior of this polygon is calcu-lated by calling

(8)

The array alpha specifies which of the original 32 verticesshould map the four corners of the rectangle. In the case ofFig. 2, vertices 1, 6, 7, and 32 are indicated in alpha. Note that,due to the previously mentioned parameter problem, the dimen-sions of the rectangle are unknown a priori and are calculatedwhen is calculated. These dimensions depend on the specificgeometry of .

In the plane, the middle of the air gap is given by

(9)

and the coordinates of the coil and magnet currents are

(10)

(11)

In Fig. 2, an example of representation of the permanent magnetcurrent sheet and slot currents using 18 conductors is given (forthe calculation of magnetic field higher number of conductorsis used to represent magnet current sheets and slot currents).These conductors are shown in Fig. 3 after the transformationinto canonical rectangle. The number of conductors in theplane is apparently 14. This difference occurs because of thecrowding phenomenon. The actual number of conductors in the

plane is also 18, but 4 out of 12 conductors of the coils aremapped into almost identical coordinates. The number of per-manent-magnet currents in the plane is 6, which means thatthe coordinates of the mapped currents, unlike coil currents, areall distinguishable. Since crowding affects the accuracy of thesolution [10], [17], it is best to avoid it. In [10], the authors preferto model the currents in one slot with only one point (one cur-rent), but a more detailed study is required to show if such ap-proach has advantage in this particular case.

III. DETERMINATION OF MAGNETIC FIELD

After the mapping of the motor, the task of mapping the cur-rents, which are the field excitation, remains. If a current isplaced at a point , the same current should beplaced at the point in the transformed configuration.



Fig. 5. Canonical rectangle in the � plane with rotation of rotor by one halfslot pitch.

Fig. 6. Study model of inset permanent-magnet motor in the � plane �� with rotor in the quadrature position.

Fig. 7. Canonical rectangle in the � plane when rotor is in the quadratureposition.

A. Magnetic Field Due to the Current in Stator Slots

The total current per slot has to be divided into a dis-crete number of punctual line currents. Let this number be ,and hence each of these punctual currents is . Then, each

of these currents is mapped from (Fig. 2) to coordinatesystem (Fig. 3).

The matrix connection between slots and phase currents is

(12)

with

being the vector of phase currents.In the plane, the slot line currents are distributed in the

length , where is the radius of the slotbottom and is the radius of the stator inner surface. In theplane that length is .

The punctual currents in the slots are

(13)

where is the number of conductors in one slot. In the steadystate at , two phases are fed, i.e.,

(14)

where is the phase current.

B. Magnetic Field Due to Permanent Magnets

The PM equivalent currents can be determined using thetheory of electromagnetism. The magnetic field produced bya body with the magnetization is the same as the magneticfield produced by a system of two equivalent currents [19]:

• volume current with the density ;• surface current with the density , where is a

unit vector normal to the surface of the body.These equivalent currents “flow” in the air, which replaces the

magnetic material.The magnetization is in direction, with a constant inten-

sity . Therefore, there are no equivalent volumecurrents. The equivalent surface currents flow only on the PMlateral sides, where and are mutually normal. It means thatthe equivalent surface current density is . The total cur-rent flowing on a lateral side is

(15)

In the plane, and in the plane, where is the radius of the outer rotor



Fig. 8. Flux and potential lines for open slot model (direct position of rotor).

surface and is the radius of the internal rotor surface at themagnet bottom.

The surface currents on both lateral sides have to be dividedinto a discrete number of punctual currents (Fig. 2). Let thisnumber be , and each of these punctual currents . Inthe plane that current is

(16)

Finally, each of the equivalent punctual currents is mappedfrom to coordinate system.

For modern permanent-magnet materials, the - character-istic is a straight line in the second quadrant, and is given by

(17)

Since by definition , it follows from (17) that

(18)

If the PM permeability is equal to the permeability ofair , the second right-hand-side term in (18) is zero,and the magnetization is constant everywhere, regardless of thefield. In that case

(19)

In our study, we assume that .

C. Magnetic Field in the Plane

Every conformal transformation should lead to a simple con-figuration in which an exact analytical formula for the magnetic

TABLE IPARAMETERS OF THE INSET PERMANENT-MAGNET MOTOR

Fig. 9. Radial �� and tangential �� flux density due to magnets alone at� � � calculated using SC transformation and FEM.

field is known. This simple configuration is usually a conductorwith current placed above an infinite iron plane, but sometimesit is a conductor placed between two infinite iron planes, whichis the case in our study.



Fig. 10. Radial �� and tangential �� flux density due to slot currentsalone at � � � calculated using SC transformation and FEM.

In the configuration where conductors are placed in the airgap of length between two infinite iron planes, the field orig-inating from the magnet is given by [20]

(20)

The field of slot currents is given by

(21)

The complex potential due to magnets is

(22)

Fig. 11. Total radial �� and tangential �� flux density on loadat � � � calculated using SC transformation and FEM.

The complex potential due to slot currents is given by

(23)

The flux between two points and in the coordi-nate system is given by

(24)

(25)

where is the stack length and is the number of pole pairs ofthe machine.

With it means that permeances and inductances areinvariant after a CM [20]. In that case, inductances (in directposition of the rotor) and (in quadrature position of the rotor)can be calculated easily for the inset PM motor as

and (26)



Fig. 12. Radial �� flux density due to slot currents alone at � � � calcu-lated using SC transformation and FEM. The SC transformation is carried outwith three different distributions of slot currents.

D. Magnetic Field in the Plane

A complex analysis theorem gives a connection between thefields in the and coordinate systems [12], [13], [20]. Thefield due to magnet currents is given as

(27)

The field due to slot currents is given as

(28)

where is determined using the SC Toolbox function as

(29)

Then, the radial and tangential flux density due to magnets aloneare

(30)

The radial and tangential flux density due to slot currents are

(31)

The total radial and tangential flux density due to magnet andslots currents (on load) are

(32)

IV. DETERMINATION OF COGGING AND ELECTROMAGNETIC

TORQUE BASED ON MAXWELL STRESS THEORY

According to Maxwell’s theory, it is possible to calculate thetotal force on a rigid body placed in the electromagnetic field by

Fig. 13. Flux and potential lines for the rotor shifted by one half slot pitch inregards to direct position.

integrating the magnetic stress on the closed surface around thebody. The surface which encloses the rotor of inset PM motor isin the shape of a cylinder placed entirely inside the air gap. Thecogging torque equation in the integral form can then be writtenas

(33)

and the total electromagnetic torque as

(34)

The radius inside the air gap at which the integration surfaceis positioned is arbitrary, but for calculation purposes it can bedone in the middle of the air gap.

A. Modeling the Movement in Inset Permanent-Magnet MotorWith Polar-Piece

The cogging and electromagnetic torque are calculated ateach position of the rotor in regard to the fixed position ofthe stator. For simulation of movement, the positions of theequivalent currents of the magnets and the coordinates of thepolar piece in the and domains are modified for eachposition of the rotor. The shape of the polygon in the planeis modified, but with the same number of vertices, as shownin Fig. 4 for rotation of rotor by one half slot pitch. Thesecoordinates are then mapped to the domain (Fig. 5) whichis different from the domain given in Fig. 3 where the rotoris aligned in the direct position. To implement the movementcorrectly, care is to be taken when the position of the currentsheets of the magnet and the polar piece changes so that theytranslate beyond the domain (one pole, e.g., north pole).Each current sheet and polar piece coordinate leaving throughone edge of the domain has to enter through the opposite edge(south pole where magnet remanence changes the sign).



Fig. 14. Radial flux density due to magnets �� and slot currents �� fordifferent positions of the rotor at � � � calculated using SC transformation.

Fig. 15. Flux and potential lines in the quadrature position of the rotor.

In the quadrature position of the rotor, as shown in Fig. 6,the polygon in the plane is modified with the same numberof vertices as in the direct position of the rotor (Fig. 2). Thecanonical domain in the quadrature position (Fig. 7) is modifiedas well. This position corresponds to the case when two halvesof poles pitches (north and south) are present in the model withcurrent sheets of two different magnets having the same negativesign.

In the case of the inset permanent-magnet machine, the shapeof the polygon and the dimensions of the canonical domain in

Fig. 16. Radial flux density due to magnets �� and slot currents �� in thequadrature position of the rotor at � � � calculated using SC transformation.

the plane change with the rotation due to the presence of thepolar piece in the rotor.

B. Modeling the Movement in Inset Permanent-Magnet MotorWithout Polar-Piece

In this case, the movement is easy to implement. Only po-sitions of equivalent magnet current sheets are modified in the

and planes. The shape of the polygon in the plane andthe dimensions of the canonical domain as well are not modi-fied. When the position of the magnet currents changes so thatthey translate beyond the domain (one pole, e.g., north pole),each current sheet leaving through one edge of the domain hasto enter through the opposite edge (south pole where magnet re-manence changes the sign).

V. RESULTS AND DISCUSSION

A. Inset Permanent-Magnet Motor With Polar-Piece

The objective of this section is to verify the accuracy of themodel in predicting both the flux density waveform as well asthe torque characteristics. This is accomplished by comparingthe results of the model with results obtained from the finite-el-ement model. The dimensions and parameters of the six-polemachine are given in Table I. The flux and potential lines areshown in Fig. 8. The flux lines run horizontally while the poten-tial lines are vertical.



Fig. 17. Cogging torque at � � � calculated using SC transformation andFEM.

Fig. 18. Total electromagnetic static torque at � � � calculated using SCtransformation and FEM.

Fig. 19. �� curve of stator and rotor core laminations.

The field solution due to magnet and slot currents in themiddle of the air gap is shown in Figs. 9–11, where theyare compared to the finite-element solution obtained usingfirst-order triangular elements. There is a very good agreementbetween the SC results and the results of FE simulations. Sinceconformal mapping inherently assumes that iron is infinitelypermeable, for better comparison of SC and numerical resultthe same assumption has been made in the FE simulationas well. The mesh size has been adjusted to reduce errors inthe critical areas, namely the tooth tips. The most noticeable

Fig. 20. Flux and potential lines for semi-closed slot model.

Fig. 21. Cogging torque at � � � for open slot and semi-closed slot modelscalculated using SC transformation.

Fig. 22. Flux and potential lines for semi-closed slot model with notched teeth.

difference between SC and FE results exists in the case of radialflux density due to slot currents alone, which is caused by thecrowding effect in SC transformation. This difference can bereduced (Fig. 12) if distribution of slot currents is considerednot between and , but between and , where

. If , which is the proposed currentdistribution in a slot (only one current) by [10], the peak mag-nitude of flux density due to slot currents is equal to the value



Fig. 23. Cogging torque at � � � for semi-closed slot model with andwithout notches calculated using SC transformation.

Fig. 24. Study model of inset permanent-magnet motor without inter-polarpiece in the � plane �� .

Fig. 25. Canonical rectangle in the � plane.

given by FEM, but the waveform has a small deformation. InFig. 12 one can see that mm is a good tradeoff valuewhere the magnitude and the waveform are in good agreementwith those obtained by FEM.

Fig. 13 shows the flux lines and potential lines when the rotoris shifted by one half slot pitch in regards to direct position.

The radial flux density due to magnets and slot currents fordifferent position of rotor is given in Fig. 14. For the rotor inquadrature position, which corresponds to a rotation by in

Fig. 26. Flux and potential lines for inset magnet motor without inter-polarpiece.

Fig. 27. Radial �� and tangential �� flux density due to magnets alone at� � � calculated using SC transformation and FEM.

regards to the direct position, the flux lines and potential linesare shown in Fig. 15. For this position of the rotor the radial fluxdensity due to magnets and slot currents is given in Fig. 16.

The cogging torque and electromagnetic torque are evalu-ated at consecutive rotor positions using Maxwell stress tensormethod. The SC and FEM solutions are compared in Figs. 17and 18. The FEM solutions are given for both linear and non-linear cases. The characteristic of the core laminations isgiven in Fig. 19. Small differences exist between the SC map-ping and the FEM results, but they are due to simplified perma-nent-magnet and slot current models and crowding effect.



Fig. 28. Radial �� and tangential �� flux density due to slot currents at� � � calculated using SC transformation and FEM.

The results indicate that saturation in the motor does not sig-nificantly affect the torque solutions, which is in favor of the SCsolution, since the assumption that the iron is infinitely perme-able is essential for its implementation. However, in the case ofa buried PM motor where magnets are magnetized tangentiallyand generate flux concentration in the rotor, the effect of satu-ration cannot be neglected. That is the case where the SC trans-formation cannot provide an accurate field solution. Therefore,our solution is limited only to inset and surface PM motors withradial magnetization.

It is apparent from comparison of the SC results with thosefrom the FE simulation that the SC solution is able to predictcorrectly both radial and tangential components of flux density,cogging torque, and electromagnetic torque in terms of bothmagnitude and waveform. Therefore, one can use the SC modelto analyze the effect of slot shape on cogging torque, as shownin Figs. 20 and 21. The effect of teeth notches on the coggingtorque is also given in Figs. 22 and 23.

B. Inset Permanent-Magnet Motor Without Polar-Piece

In this section, the SC and FEM study is given for the insetpermanent-magnet motor without polar-piece . Afterapplying the logarithmic conformal mapping, the stator androtor surfaces become as shown in Fig. 24. The polygon in the

plane has 28 vertices and the canonical rectangle in thedomain is given in Fig. 25.

Fig. 29. Total radial �� and tangential �� flux density on loadat � � � calculated using SC transformation and FEM.

Fig. 30. Cogging torque at � � � calculated using SC transformation andFEM.

Fig. 26 shows flux and potential lines. The field solutionsdue to magnet and slot currents in the middle of the air gap areshown in Figs. 27–29, where they are compared to the finite-el-ement solution obtained using first-order triangular elements.There is a very good agreement between the SC results and theresults of FE simulations. The cogging torque and electromag-netic torque are evaluated at consecutive rotor positions usingMaxwell stress tensor method. The SC and FEM solutions arecompared in Figs. 30 and 31.



Fig. 31. Total electromagnetic static torque at � � � calculated using SCtransformation and FEM.

Fig. 32. Flux and potential lines for semi-closed slot model.

Fig. 33. Cogging torque at � � � for open slot and semi-closed slot modelscalculated using SC transformation and FEM.

The effect of slot shape on cogging torque is shown in Figs. 32and 33. The effect of teeth notches on cogging torque is alsogiven in Figs. 34 and 35.

VI. CONCLUSION

The paper shows an implementation of SC mapping to deter-mine the magnetic field and torque in inset permanent-magnetmotors. The method is completely based on the theory of thecomplex functions. Due to the complex geometry of the air gap

Fig. 34. Flux and potential lines for semi-closed slot model with notched teeth.

Fig. 35. Cogging torque at � � � for semi-closed slot model with andwithout notches calculated using SC transformation and FEM.

with elongated regions, the numerical SC mapping is applied.To solve the SC integral, the Matlab SC Toolbox functions havebeen used. The CM method has drawbacks (e.g., the iron shouldbe ideal), but it offers several advantages compared to the FEM:The Matlab programs are very compact, the simulations are veryfast, and the parameters can be easily modified (in order to per-form a parametric analysis). The results are in accordance withthe FEM. In addition, the developed method provides accurateresults for cogging torque and electromagnetic torque both inmagnitude and waveform. This method is particularly helpfulfor properly selecting the magnet, rotor teeth and stator teethdimensions in order to achieve optimum performance of insetand surface mounted permanent-magnet synchronous motors.

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Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using...

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