IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014 8201207

Analysis of Overhang Effect for a Surface-Mounted PermanentMagnet Machine Using a Lumped Magnetic Circuit Model

Jung-Moo Seo1,2, In-Soung Jung1, Hyun-Kyo Jung2, and Jong-Suk Ro2

1Korea Electronics Technology Institute, Gyeonggi 420-140, Korea2Seoul National University, Seoul 151-742, Korea

This paper presents a magnetic field analysis of a surface-mounted permanent magnet machine with an overhang structure whereinthe rotor axial length exceeds that of the stator. A 2-D analytic method using a lumped magnetic circuit model (LMCM) is proposedfor more rapid computation of the overhang effect compared with a conventional 3-D or quasi-3-D finite element analysis (FEA).In the LMCM, the effective overhang length is calculated, and a leakage flux generated in the overhang region is estimated. Theaccuracy and usefulness of the proposed analytic method is confirmed through the 3-D FEA and the experimental results accordingto the diverse overhang length, the number of slot/pole, and stator core length.

Index Terms Effective overhang length, lumped magnetic circuit model (LMCM), overhang effect, surface-mounted permanentmagnet (SPM) machine.

I. INTRODUCTION

PERMANENT magnet (PM) motors are widely usedbecause of their advantages of high efficiency, highpower density, and excellent controllability. In the design ofa PM motor, the exact data of a magnetic field distributionin the air gap is essential for the correct prediction of motorperformance. A finite element analysis (FEA) with high accu-racy is useful for analyzing the magnetic field. For the analysisof a machine where the magnetic characteristics are constantin the axial direction, 2-D FEA is efficient in terms of timeconsumption. However, for the analysis of the motor with anoverhang structure where the axial length of the rotor is longerthan the stator, 3-D FEA is generally used to ensure analysisaccuracy [1], [2].

The overhang effect in 2-D FEA according to the change inthe material properties through the correction of the remanenceflux density, Br , of a PM was addressed in [3][6]. Theoverhang coefficient using a design of experiment techniquefrom 3-D FEA and employed 2-D FEA with the revised Br toconsider the overhang effect was calculated in [3] and [4].Hwang et al. [5] presented a novel coefficient modelingtechnique with 2-D FEA for the optimization of a skewedoverhang PM motor. Woo et al. [6] obtained the averagevalue of Br in all elements of a PM using 3-D FEA anddetermined the overhang parameter for the 2-D FEA througha modified permeance coefficient and demagnetization curve.These methods are not relevant 2-D approaches because theyuse a trial-and-error method to find a solution and one or more3-D magnetostatic FEAs are inherently required at the initialcomputing stage, which are still time consuming.

In [7][9], 3-D analytic methods are proposed to predict anair gap field distribution. Especially, Wang et al. [7] presentedthe flux variations according to the PM overhang for a brush-less dc motor using lumped parameter modeling. However, the

Manuscript received October 20, 2013; revised November 19, 2013;accepted November 25, 2013. Date of publication December 5, 2013; dateof current version May 1, 2014. Corresponding author: J.-S. Ro (e-mail:jongsukro@naver.com).

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

Digital Object Identifier 10.1109/TMAG.2013.2294154

specific methodology is not presented and the validity of theapplied method in comparison with the FEA is not derived.A 3-D magnetic circuit to investigate the end effect by theleakage flux was presented in [8] and [9]. However, they didnot extend the analysis to include the overhang effect.

To address those problems of the conventional analysismethods, a 2-D analytic method using a lumped magneticcircuit model (LMCM) is proposed for the accurate and rapidanalysis of the overhang effect in this paper. The evidentadvantages of the LMCM are a lower computational time thanthe FEA and a higher expandability than the analytic model bypartial differential equations [10][16]. Some considered thenonlinear BH curve of a steel core by an iterative process[8], [10], [11]. Carters coefficient has been applied [12], [13]or magnetic circuits have been reconstructed according to therelative rotor positions [14][16] to account for slotting. Thispaper presents a simplified LMCM for a surface-mountedPM (SPM) motor and approximates each flowing flux interms of the magnetic material properties and the machinedimensions. The leakage flux, fringing flux, and slottingeffect are considered simultaneously for accurate predictionof the air gap flux density distributions. The magnetic circuitis divided into overhang and nonoverhang regions, and thecorresponding circuit equations are solved to consider theoverhang effect in the LMCM. Through flux pattern analysis inthe overhang region, an effective overhang length is proposed,and magnetic reluctances considering each leakage flux arecalculated. Finally, significant values for the estimation of themotor performance, such as the air gap magnetic flux and theelectromotive force (EMF), are calculated using the proposedLMCM. The proposed analysis method is applied to diverseoverhang ratios, pole/slot numbers, and axial core lengths. Thecalculated results are also compared with 3-D FEA and theexperimental results to validate the proposed analysis method.

II. ANALYTIC MODEL FOR MAGNETICFIELD CALCULATION

A. Air Gap Magnetic Flux and Flux Density in LMCM

The SPM motor with concentrated windings of four polesand six slots is used for the verification of the proposed

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8201207 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014

Fig. 1. Cross section of the simulated PM machine.

TABLE I

SPECIFICATIONS OF SIMULATED MODEL

Fig. 2. LMCM for PM machine.

analysis method in this paper. The dimensions and specifica-tions of the motor are shown in Fig. 1 and Table I. Based on theflux distribution and the leakage flux, a simplified LMCM isconstructed and shown in Fig. 2 [13]. r /2 is the flux sourceof half of the magnet, and g/2 is the air gap flux passingthrough one half of the air gap cross-sectional area. 2Rmoand 2Rg are the reluctances corresponding to r /2 and g /2,respectively. Rs is the reluctance of the stator core, and Rris the reluctance of the rotor core. By adding Rmr (reluctancecaused by magnet-to-rotor flux leakage) and Rmm (reluctancecaused by magnet-to-magnet flux leakage) parallel to 2Rmo,the equivalent circuit can take these flux leakage effects intoaccount. Assuming that there is no magnetic saturation inthe stator and rotor core, Rs and Rr may be neglected incomparison with Rg . Therefore, Fig. 2 can be reduced toFig. 3 and the following expression for the air gap flux can

Fig. 3. Simplified circuit of Fig. 2.

be obtained by flux division

g = 2R12R1 + R2

Rmm4Rg + Rmm r (1)

where R1 is the parallel combination of 2Rmo and Rmr, andR2 is the parallel equivalent of 4Rg and Rmm. The average airgap flux density is

Bg = 2R12R1 + R2

Rmm4Rg + Rmm

AmAg

Br (2)

where Br is the PM remanence, Am = wm L andAg = (wm+w f )L are, respectively, the surface area of the PMand the air gap, where wm and w f are, respectively, the widthsof the PM and gap between adjacent magnets, and L is thelamination stack length. General expressions for the magnetand air gap reluctance are given by

Rmo = HPM0r Am

(3)

Rg = g0 Am

(4)

where HPM and g are the lengths of the PM and air gap,respectively, 0 is the permeability of free space, and r isthe relative permeability of the PM.

The magnet-to-rotor and magnet-to-magnet reluctance,Rmr and Rmm, respectively, can be determined by the circular-arc straight-line permeance model [13], [14]. Rmr and Rmmmay be expressed by

Rmr =[0 L

ln

(1 + g

HPM

)]1(5)

Rmm =[0 L

ln

(1 + g

w f

)]1. (6)

By substituting the calculated reluctance of (3)(6) into(1) and (2), the air gap magnetic flux, g , and flux density,Bg , are determined.

B. Consideration of Fringing Effect

Whenever the air gap is inserted into the magnetic path,fringing flux is induced at the gap. The magnetic flux fromthe PM is bent on the edge, and the variation changes the fluxdensity distribution in the air gap. The fringing flux using asingle smoothing function of ex/a was approximated in [17]and [18], in which x denotes the angular rotor position, andthe fringing exponent a can be expressed with the dimensionsand material properties of the PM by

a = 12

g

(g + HPM

r

). (7)

SEO et al.: ANALYSIS OF OVERHANG EFFECT FOR A SPM MACHINE 8201207

Fig. 4. Comparison of air gap flux density distribution.

C. Consideration of Slotting Effect

If there are slots in stator, the air gap length is variableaccording to the angular positions. The air gap flux densityunder the influence of stator slot, Bgs, can be expressed by [14]

Bgs() = Ksl()Bg() (8)where Ksl() is the relative permeance that modifies the airgap flux density in the neighborhood of the slots.

Taking into account the calculated air gap flux density in (2),the fringing function with (7), and the relative permeancein (8), the air gap flux distribution is calculated and comparedwith the FEA results as shown in Fig. 4. The calculated datausing the LMCM fits well with the FEA data.

D. Back EMF Calculation

From Faradays law, the EMF can be deduced by

E = ddt

= dedt

d

de= e N dteeth

de= e N d Bg

dewt L (9)

where is the linkage flux, N is the number of turns, teethis the teeth flux, and wt is the width of the teeth. e ande are the electrical position and speed, respectively. As therotor is rotating, the waveforms of the magnetic flux densitymove in rotational directions according to the relative rotorpositions over the stator. By calculating the average air gapflux density, Bg , according to the each rotor position, thecorresponding flux passing into the teeth is determined atevery position. The calculated EMF according to the teethflux variations shows a good agreement with the FEA data,as shown in Fig. 5. Through the fast Fourier transform resultsof the EMF as shown in Fig. 5(b), we also confirmed thatthe harmonics calculated by the LMCM and FEA are notsignificantly different. Consequently, it is verified that theanalysis method presented using the LMCM is accurate.

III. PROPOSED ANALYTIC MODEL FOR ANALYZINGOVERHANG EFFECT

The overhang structure is used to increase the magneticflux density in the air gap of the PM motors. Furthermore,additional sensors (i.e., Hall ICs and Hall elements) areinstalled, and the rotor is designed to be extended in one orboth axial directions in the SPM motors to detect the rotorposition and proper switching timing [2], [6].

Fig. 5. Comparison of back EMF at 1000 r/min. (a) Waveform.(b) Harmonics.

The increase of the air gap flux by the overhang structureaugments the linkage flux and the EMF. This increases thetorque and consequently the performance of the motor. How-ever, the magnetic flux does not increase proportionally to theelongation of the overhang length. In the case of magneticsaturation, the overhang effect is further reduced [6], [7].Hence, in terms of the cost, size, and weight of the motor, thedesign process is essential to determine the efficient overhanglength.

The modified LMCM is used to analyze the overhang effect.The corresponding parameters of the LMCM for the overhangstructure are estimated from the magnetic flux flowing in theradial and axial directions.

A. LMCM Considering Overhang Structure

An asymmetrical overhang structure is applied in this paperand the analysis model is separated in two parts, as shown inFig. 6. Parts I and II denote the nonoverhang and the overhangregions, respectively.

The magnetic flux density in the air gap, Bg , is complexto estimate in the overhang structure. In other words, it isdifficult to define the accurate air gap surface area, Ag . Forthis reason, the magnetic flux is a better factor than the fluxdensity for the overhang effect expression. The total magneticflux is calculated by adding the fluxes of Parts I and II as

total = part I + part II. (10)part I is instantly calculated by (1). The flux pattern of Part IIis investigated by 3-D FEA to determine part II. From the

8201207 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014

Fig. 6. Two separated regions for the overhang effect analysis.

Fig. 7. Side view: Vector plot of magnetic flux density in Part II using 3-DFEA.

Fig. 8. Simplified permeance model of axial leakage and linkage flux inPart II.

results, an effective overhang length is determined and the airgap, magnet-to-rotor, and magnet-to-magnet reluctances arerecalculated as follows.

1) Effective Overhang Length: Fig. 7 shows the vector plotof magnetic flux density from the side view of the motor. Someof the magnetic fluxes from the PM flow into the stator core.However, some of the magnetic fluxes leak out and circulatethrough the air gap and rotor core. In other words, the totalflux is not increased to the full extent predicted by the actualoverhang length of the PM, and the leakage flux is generatedin the axial direction. Hence, the effective overhang length isproposed to reflect the practical amount of the flux.

Fig. 8 shows a simplified drawing that represents the leakageand linkage flux, and the actual and effective overhang lengths.

Fig. 9. Top view: Vector plot of magnetic flux density in Part II using3-D FEA.

The total fluxes originating from the PM surface are dividedinto the up and down directions in which the rotor andstator cores exist. They are the leakage and linkage fluxes,respectively. For an arbitrary point on the PM surface, theflux from the PM flows along shorter path between the leakageand linkage flux paths. For example, because the length of theleakage flux path is significantly shorter than the linkage fluxpath at the top of the PM overhang, the flux from that point iscounted as the leakage flux. As the point moves downward, thelength of the leakage and linkage flux paths become gettinglonger and shorter, respectively. The length of the both fluxpaths would be the same at a certain point, and the effectiveoverhang length is determined at that point, which is given by

g + 2

Leff.OH = g + HPM + 32(LOH Leff.OH) (11)

where LOH and Leff.OH are the actual and the effective over-hang lengths, respectively. The proposed effective overhanglength can be deduced by

Leff.OH = 12

HPM + 34

LOH. (12)

2) Air Gap Reluctance: With the calculated Leff.OH in (12)and the permeance model from Fig. 8, the air gap permeanceof Part II can be derived by

Pg part II =Leff .OH

0

0(wm + w f )g + 2 x

dx

= 20(wm +w f )

ln

(1 + Leff.OH

2g

). (13)

By using the reciprocal relationship between reluctance andpermeance, the air gap reluctance, Rg part II, is given by

Rg part II =[

20(wm +w f )

ln

(1 + Leff.OH

2g

)]1. (14)

3) Magnet-to-Rotor and Magnet-to-Magnet Reluctance: InPart II, the leakage fluxes generated in the PM-rotor andadjacent PMs are not limited by the air gap length. This can beconfirmed by a vector plot of the magnetic flux density using3-D FEA, as shown in Fig. 9. Because of the absence of thestator core in Part II, the leakage fluxes are not restricted.Hence, the path and amount of the leakage flux in Part IIis longer and larger than in Part I, as shown in Fig. 10.

SEO et al.: ANALYSIS OF OVERHANG EFFECT FOR A SPM MACHINE 8201207

Fig. 10. Simplified permeance model of radial leakage flux in Parts I and II.(a) Magnet-to-rotor leakage flux. (b) Magnet-to-magnet leakage flux.

This means that the integration section for the calculation ofthe permeance caused by leakages of the magnet-to-rotor andmagnet-to-magnet in Part II are extended to w f /2 and wm /2,respectively, although those in Part I are g from (5) and (6).Therefore, each permeance of Part II is derived using Leff.OHas

Pmr part II =w f /20

0 Leff.OHHPM + x dx

= 0 Leff.OH

ln

(1 + w f

2HPM

)(15)

Pmm part II =wm/20

0 Leff.OHw f + x dx

= 0 Leff.OH

ln

(1 + wm

2w f

). (16)

Hence, the magnet-to-rotor and magnet-to-magnet reluctances,Rmr part II and Rmm part II, are

Rmr part II =[0 Leff.OH

ln

(1 + w f

2HPM

)]1(17)

Rmm part II =[0 Leff.OH

ln

(1 + wm

2w f

)]1. (18)

Substituting (14), (17), and (18) into (1), the air gap flux ofPart II, part II, and the total magnetic flux can be calculatedfor analyzing the overhang effect.

B. FEA Verification by Various Overhang Models

The back EMF was calculated and compared with the resultsby 3-D FEA to verify the proposed method for the overhangeffect. Fig. 11 shows four types of FEA models with overhanglengths of 0, 3, 6, and 9 mm. The overhang ratios of the modelsare 0%, 10%, 20%, and 30%, respectively.

Fig. 11. 3-D FEA models with various overhang lengths (a) 0, (b) 3, (c) 6,and (d) 9 mm.

Fig. 12. Comparison of EMF peak value with respect to the overhang length.

TABLE II

CALCULATED EMF PEAK VALUE (V)

The calculated values using the proposed analytic methodare compared with the FEA results in Fig. 12 and Table II.As the overhang length is increased, the increasing amountof the back EMF is decreased. This result implies that theoverhang structure is effective in some overhang ranges asexpected. The difference between calculated values using theanalytic method and FEA is increased as the overhang lengthis increased. However, the errors are 30% (symmetrically 60%) is unusual fora practical motor design. Hence, we verified that the proposedanalytic method is correct and reasonable.

From the result of Fig. 13, it is also confirmed that theproposed method fits well with the FEA with respect to thevarious combinations of the pole/slot numbers. Moreover, we

8201207 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014

Fig. 13. Comparison of back EMF increase rate with respect to the pole/slotnumber. The overhang ratio is constant at 20%.

Fig. 14. Comparison of back EMF increase rate with respect to the axiallength of stator core. The overhang ratio is constant at 20%.

Fig. 15. Actual and effective overhang length with respect to the axial lengthof stator core. The overhang ratio is constant at 20%.

found that even though the overhang ratio is constant, as theaxial length of the motor is increased, the overhang effect isdecreased from Fig. 14. This is caused by the variation of theratio between the leakage and linkage fluxes among the totalflux increased by the overhang structure. This result is alsoconfirmed by the fact that the ratio of the effective overhanglength over the actual overhang length is decreased as the axiallength is increased in the same structure of the motor, whichcan be found in Fig. 15.

Fig. 16. Manufactured rotors and assembled motor. (a) Rotor PM of 26 mm(nonoverhang model). (b) Rotor PM of 31.2 mm (overhang model with theoverhang ratio of 20%). (c) Assembled motor.

TABLE III

SPECIFICATIONS OF MANUFACTURED MODEL

IV. MANUFACTURING AND EXPERIMENTAL RESULTS

The prototype motor is manufactured for the validation ofthe proposed method through the experiment, as shown inFig. 16. The motor is designed for the joint module of theservice robot system. The service robot can guide people to aplace or show desired information using a voice and screen.Because the robot system is mobile, the driving motor of thejoint module requires high power density. The specific designparameters are listed in Table III. The overhang structure withan Nd-sintered PM and concentrated winding is employed toincrease the power density of the motor. Hall ICs are attachedin the rear parts of motor housing to detect the magnetic fluxby the overhang PM. Nonoverhang and overhang rotors aremanufactured to investigate the overhang effect.

The EMF waveform is calculated using the proposed analy-sis method and compared with the measured results, as shownin Fig. 17. The measured EMF values of the nonoverhang andoverhang models are 2.14 and 2.29 V0peak at a rotationalspeed of 1000 r/min, respectively. As the rotor length isincreased by 20% from 26 to 31.2 mm, the EMF is increased

SEO et al.: ANALYSIS OF OVERHANG EFFECT FOR A SPM MACHINE 8201207

Fig. 17. Comparison of back EMF at 1000 r/min. (a) Nonoverhang model.(b) Overhang model.

by 7%. It is noted that the PM is magnetized to a radialdirection in the analytic model but a parallel direction inthe manufactured model. This is because the LMCM canconsider only the radial component of the magnetic fieldinherently, whereas a general-purpose magnetizing coil forparallel magnetization is used in the manufactured model toreduce the manufacturing cost and time. Therefore, the topshape of the waveform from the analytic model is flat andslightly different from the measured shape. In the overhangmodel, the difference between the analytic and measuredresults is greater than the nonoverhang model because of themagnetic saturation caused by the increased flux. However,the calculated results show a good agreement with the exper-imental data, which verifies the validity and usefulness of theproposed method.

V. CONCLUSION

The proposal of the rapid and accurate 2-D no-load analysismethod using an LMCM for the motor with an overhangstructure is the primary contribution of this paper. Remarkably,a useful parameter, which is the effective overhang length, isproposed in this paper, enabling the quantitative estimation ofthe overhang effect and motor performance.

The proposed analysis method can be widely used for thedesign of the motor because the accuracy of the proposedmethod is verified through the applications of the diverseoverhang lengths, combinations of pole/slot numbers, and axiallengths of the motor.

This paper dealt with the magnetic field distribution undera no-load condition. Under a load condition, the reluctance ofthe stator core cannot be neglected and the armature reactionfield should be considered. In future paper, an extendedLMCM considering the load condition should be employedfor torque and iron loss calculations.

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