BRIDGE STRESSES AND DESIGN IN IPM MACHINES Ron D Bremner, Member, IEEE

Abstract: This document describes the stresses in the bridges of three types of interior permanent magnet (IPM) machines. These are machines which had a V magnet slot, machines which have a straight slot, in machines which have a V configuration with a center post.

Formulae for calculating the bridge stresses are included for the straight and V shaped slots. The formulae are determined from a mathematical model statistically based on finite element models.

Index Terms: IPM machines, bridge stresses, laminations, rotor stresses, dimensionless analysis, FEM.

I. INTRODUCTION In interior permanent magnet (IPM) machines, there

are three basic magnet slot shapes. The magnet slots in the first type are arranged in a spoke configuration. The magnet slots in the second type are arranged in a V shape beneath the surface (see Fig. 1a), and the magnet slots in the third type are arranged in a plane perpendicular to the d-axis (see Fig. 1b).

The present paper describes machines which have the magnets in the V shaped configuration, or which have the magnets in a plane (see Fig. 1). For magnet slots with the V shaped configuration, slots with a center post are also investigated.

A previously written paper by Loveless, et al [4], discusses a similar topic. The Lovelace paper does not include rotors with either a flat or V–shaped magnet slot geometry. It also focuses on reinforcing the rotor hub rather than changing the magnet slot geometry.

Like this paper, the Loveless paper considered only the centrifugal force created by high speed rotation. Another paper, by Kab-Jae Lee [1] includes the effects of electromagnetic forces. However, as the first paper mentions, the centrifugal forces will dominate.

The calculation of bridge stresses is most accurately done using finite element analysis. This paper includes formulae for the stresses in bridges of IPM machines. These formulae are derived from the results of multiple finite element analysis (FEA) runs.

Because of manufacturing requirements, a fillet with some minimal radius is required in the manufacture of the slots. It is shown that a larger fillet at the outer corner of the bridge significantly reduces bridge stresses. The size of this fillet is included in the formulae given.

It is also shown that a center post significantly reduces stress. There are optimum ‘center-post width’ to bridge thickness ranges.

For this paper, a dimensionless analysis of 8 pole IPM machines was performed, for a large range of slot parameters. Similar methods can be used to determine the

peak stresses in motors with other pole counts. II. BACKGROUND

Many IPM machines have magnets which are

arranged as shown in Fig. 1(a) or Fig. 1(b). This paper deals with the calculation of bridge stresses in these two types of machines.

(a) (b) a

Fig. 1 - IPM Machine Topologies

The bridges of these machines hold the magnets onto the rotor. These bridges also form leakage paths from one side of the magnet to the other. These flux leakage paths will be magnetically saturated according to the properties of the material. Because thicker bridges allow more flux leakage, the bridge thickness should be minimized.

In general, the flux density of the steel in saturation is higher than the remanent flux density of the magnet. As an example, the remanence of a NdFeB magnet might be 1.4 T, and the saturation flux density might be 2.1 T. For this case, the flux from every 1 mm of leakage width requires 1.5 mm of magnet which does not contribute to torque.

The thinness of the bridge is limited by the stresses in the bridge when the machine is at the highest speed, and by manufacturing capabilities. The bridge thickness typically would not be smaller than the lamination thickness [Niazi et al, 6].

The stresses in the magnet slot region are determined by the mass of the magnets, the mass of the rotor steel outside of the magnets, the speed of the rotor, and the rotor slot geometry.

The maximum stress allowed is material dependent. Table 1 shows the yield strengths of various electrical steels.

978-1-4244-3861-7/09/$25.00 ©2009 IEEE 655

Table 1-Yield Stresses of Electrical Steels [5]

Supplier Grade Yield Strength (MPa, N/mm2)

AK Steel

M-15 FP M-38 FP

358 290

Arcelor M250-35 (�M15 420-460

M270-35 (�M19) 330-370 M300-35 (�M22)

310-350

Cogent Power M235-35A 460 M250-35A 455 M270-35A 450 M300-35A 370 M330-35A

300

JFE 35JN200 437 35JN210 432 35JN230

50JN230 401 435

35JN250 50JN250

405 407

Some motor analysis programs perform an

approximation of bridge stresses. These approximations are typically based on applications of the hoop stress formulas, assuming that the magnet and slug (pole piece) mass are concentrated in a hoop having the thickness of the bridge. They do not include stress concentrations.

To include stress concentrations present in the slots, structural finite element analysis (FEA) programs are required. This paper derives equations based on previously run finite element analyses and statistics. This allows machine design to be done without conducting structural FEA during the design process.

III. ASSUMPTIONS

The following assumptions were made:

• The density and Young’s modulus of the magnets is equal to the properties of the steel

• Modeling a single pole is sufficient (additional poles will not increase the model’s accuracy)

• Peak stresses occur in the rotor slot, near the corners • Rotor radius is 100mm

IV. METHOD

A. Modeling the Rotor

Following the method of a previous paper [Bremner, 3], the rotor of the machine was designed using dimen-sionless parameters. The magnet slot dimensions were varied within the ranges shown in Table 2.

Table 2 - Variations of Parameters Analyzed

Parameter Range Corner Radius Ratio 0.1, 0.3, 0.5, 0.7, 0.9

(CRR) Bridge Thickness Ratio (BTR)

0.01-0.101

Magnet Depth Ratio (MDR)

0.001, 0.05, 0.10, 0.15, 0.20, 0.30, 0.40

Magnet Thickness Ratio (MTR)

0.05, 0.07, 0.10, 0.14

Web Ratio (WR) 0.05-0.35 Poles 8

where Corner Radius Ratio (CRR), Bridge Ratio (BR), Magnet Depth Ratio (MDR), Magnet Thickness Ratio (MTR) , and Web Ratio (WR) are as shown in the figures below:

Fig. 2 - Corner Radius Ratio = R/R max

���� is the largest radius which will fit between slot sides L1 and L2. The Corner Radius Ratio is R/ ����.

Fig. 3 - Bridge Ratio = RBRIDGE/R rotor

The Bridge Ratio varied from 0.01 to 0.10 (1 to 10 mm), in a way that was predicted to yield linear increases in stress values. If the hoop stress formula was an accurate predictor of stress for the rotor, the stresses when plotted against the Bridge Ratio would have been fairly straight lines.

Fig. 4 - Magnet Thickness Ratio = T magnet / R rotor

1 1/100, 1/90, 1/80… 1/10

RBRIDGE

R rotor

R rotor

T magnet

L1

L2

R

Rmax

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The Magnet Depth Ratio determines how deep the magnets go into the rotor. The Magnet Thickness Ratio determines how thick the magnet is compared to the rotor radius.

Fig. 5 - Magnet Depth Ratio = DMag / DMag max

Fig. 6 - Web Ratio = � web / � pole

A MATLAB® program was written, which created an FEA model of a magnet and slot in an IPM rotor. The program used the Partial Differential Equations toolbox, using the Structural Mechanics, Plane Stress option.

The program created a quarter-rotor as shown in Fig. 8. The top of the model was constrained to have no motion in the y direction, and the left side of the model was constrained to have no motion in the x direction. This causes the quarter model to have the same strains in stresses as if it was a full rotor.

The third rotor type modeled had a center post in the magnet slots. A diagram of this magnet slot is shown below.

Fig. 7 - Magnet Slots with Center Post

The flux leakage paths consist of the bridges and the post. The total flux leakage width was:

2* 2*flux POST BRIDGE HALF POSTW T T= + (1)

where ��� - flux leakage width, �� ��������� – bridge thickness, ������� �� – half of post thickness.

For this algorithm, as the width of the post increased, the width of the bridges was decreased. This kept the total flux leakage width the same. For all magnet slots where the only variable was the post ratio, the magnet length was kept constant.

By enforcing the previous two constraints, of maintaining the magnet length as the Post Ratio varied, the flux leakage and the net flux through the magnet were maintained as constants.

All rotors were modeled with a 100mm radius. Stresses in other rotors can be calculated based on this analysis.

Fig. 8 - Simplified FEA model of quarter rotor

One half of a magnet slot was cut into the cylinder, and a magnet was modeled in the slot. A fillet was modeled in the corners of this slot.

Only one half magnet slot was modeled, as previous FEA modeling indicated that modeling additional slots did not yield improved results.

B. Modeling of Forces

The centrifugal forces in the rotor were modeled as volume forces. The x and y-direction Volume Forces were modeled as:

� � ��� (2) �! � ���" (3)

where �—the volume force in the x direction, �!—the volume force in the y direction, �—the speed of the machine in radians per second, �—the density of the material in kg/ cubic meter, and x and y—the distance from the center of the rotor (m).

The above equations model the centrifugal forces on the material in the rotor. These increase as the material is located further from the center.

C. Running and Analyzing the V-Shaped Slot

Models All combinations of the above variables were

DMagDMag max

� pole � web

R POST BRIDGE

R ROTOR

R NO BRIDGE

T HALF BRIDGE

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run for the V shaped magnet slot. This resulted in a large number of runs. The highest stresses found in the material were recorded for each run.

Plots were made of stress vs. inverse bridge thickness for all data. The inverse bridge thickness was used in an effort to achieve data which would appear linear. For all plots the maximum value on the Y axes was 4000 N/sq mm. (The initial machine ran at 11,200 RPM)

For each plot three variables were held constant. For some plots only the Magnet Depth Ratio varied, as shown in Fig. 9 and Fig. 10.

A number of variables were tried for curve fitting. Eventually, certain variables were determined to be the most critical. Curve fits were performed using these eight variables, their squares, and additional interaction terms. These interaction terms included up to four interactions. A curve fit was then done using combinations of four of these terms (see Table 3).

A second attempt at curve fitting was similar to the first. Data was analyzed separately for each setting of Radius Ratio. This resulted in a better curve fit.

D. Running the Model with the Center Post

FEA of the design with the center post was run with various Corner Ratios and Post Ratios. The radii of the outer corners and inner corners were determined using the same corner ratio. Fewer combinations of this model were run. The goal of these runs was not to determine a mathematical equation, but to gain insight into the effect of adding a center post, while maintaining a constant flux leakage.

Models with Corner Ratios of 0.1 and 0.9 were run while varying the post width from 0.0 (no center post) to nearly 1.0 (no bridges). All other variables remain the same.

V. DISCUSSION

A. Hoop Stress Calculation

Under uniform rotation, the hoop stress of a thin walled cylinder [2] is:

# � $%&'('�� (4)

where � – stress (N/)�), � – density (kg/)*), R – the distance from the center of rotation (m), and � – the speed of rotation (rad/sec). In this equation the thickness of the hoop is not considered since doubling the hoop thickness doubles the centrifugal force on the hoop. Doubling the hoop thickness also doubles the area that the force is applied to. Therefore, the stresses remain nearly the same.

One method of using the hoop stress equations to estimate the stress in the bridge is to determine the mass of the bridge, the magnets, and the rotor material outside of the magnets. The stress in the bridge can then be calculated as:

+, � -./��� �� 0.12.34

.5 �� (5)

where ), – the mass of the bridge and the mass of the material which forms a hoop having the same thickness as the bridge around the magnets. +, – the stress in the bridge, ). – the mass of the magnets, and )6 – the mass of the rotor material outside of the magnets (slug), including the (small) bridge mass.

In equation (4), the modified hoop stress is calculated by multiplying the hoop stress of equation (3) by the total mass of the bridge materials, and dividing by the mass of the hoop.

This approximation ignores the effects of stress concentration and bending. A comparison of this equation with the results of finite element analysis shows that this equation significantly underestimates the stresses in the bridge.

B. The Effect of Stress Concentration

The effects of stress concentration can be seen by looking at a graph of stresses vs. bridge thickness. In the graphs below multiple curves show the stresses achieved with different conditions of the magnet depth ratio. Note that the x axis is the inverse bridge thickness. Moving from left to right, the bridge becomes thinner, and stresses increase.

The only difference between the two graphs is the fillet radius ratio at the outer edge of the magnet slot.

The fillet radius ratio in the second graph is 0.1, while the fillet radius in Fig. 9 is 0.9. The stresses decrease from approximately 1050 N/sq mm (at 11,200 RPM) to 350 N/sq mm at the lowest stresses. The lowest stress levels when the inverse bridge thickness is 100 (1mm) are nearly 4000 N/sq mm and 2200 N/sq mm.

Fig. 9 shows a consistent increase in stresses as the Inverse Bridge Thickness increases. (For this case the Inverse Bridge Thickness of 10 represents a bridge thickness Of 10 mm. An Inverse Bridge Thickness of 100 equates to a bridge thickness of 1 mm)

Fig. 9 - Stresses with Corner Ratio = 0.9

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Fig. 10 - Stresses with Corner Ratio = 0.1

In Fig. 9, the increase in stress is fairly linear. In Fig. 10, for Magnet Depth Ratios less than 0.15, the stresses are no longer linear.

Comparing Fig. 9 and Fig. 10, it can be seen that the Corner Ratio makes a significant difference on stress levels. The axes for Fig. 9 and Fig. 10 are identical. The stresses in Fig. 10are approximately twice the stresses in Fig. 9.

This indicates that increasing the Corner Ratio causes the stresses to be more smaller, and more linear.

Roark (Young and Budynas 2001) discusses stress concentration at length. In their table 17.1, Fig. 5a and 5b, there is a shape as defined below:

(a) (b) (c)

Fig. 11 - Roark's Figures for Stress Concentration

Fig. 11a shows the dimensional symbols for the geometry. In Fig. 11b, the part is in tension, and in Fig. 11c, the part is in bending. If Fig. 11c is cut into fourths, one quarter of the figure looks somewhat like the corner formed by the magnet slot of an IPM motor.

The lower portion of the beam is similar to the bridge. This portion is exposed to both tension and bending. However, the formulas in Roark’s are complex for both bending and tension.

Fig. 12 - Quarter of Fig. 11b

For the magnet slots studied, r/h varied from 0.1 to 0.9. This corresponds to Roark’s h/r values of 10.0 to 1.1.

One of Roark’s formulas for tensile stress has four terms and is:

78 � 9: ; 9� �<�=� > ; 9* <�=� >� ; 9? <�=� >

* (5)

where (as an example):

9: �1.007+@A BC D EFEGHA BC

9� �-0.114-+EFIJI@A BC ; EFGHKA BC

9* �0.214-EFLLM@A BC D EFMNHA BC and

9? �0.134+EFINN@A BC D EFEHMA BC The above constants are correct only for

specific conditions. The equation for stresses caused by bending is identical, but the equations for the constants are different.

If the hoop stress equations were accurate for the case of bridge stress, there would be almost no change in stresses based on the Corner Ratio.

It is also obvious from the above equation that when stress concentrations exist, the hoop stress formula is overly simplistic. Cases have been observed where the hoop stress formula is off by a factor of 3 to 12.

C. The Effect of Fillet Radius Ratio

Initially, all terms (including the fillet radius

ratio) were used to create a mathematical model predicting the stresses. While good correlations could be obtained for constant fillet radius ratios, when all fillet radius ratios were included, the correlation became worse.

The plots below show the correlation between prediction and observation, using these models. The first plot below was made using the data from all models (all fillet ratios). The correlation between the model and the observed stress was poor.

Fig. 13-Stresses predicted using empirical formula

The following plot was made using the data from all models which had a fillet ratio of 0.9. The correlation between the model and the observed stress is good. The axes for both plots is the same.

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Fig. 14 - Stresses predicted using empirical formula

D. Formulas for Calculating Stress An accurate general equation for all variations of

Corner Radius Ratio was not found. However, for distinct Corner Radius Ratio values, equations could be written. Table 3 – Five term equations for stress calculations at 1,000 RPM2

Cor

ner

Rad

ius

Rat

io

Best Fit Formula

0.7 -3.1317e-001 x6 +2.5878e x2^2 -3.0604e x1 x3^2 +3.5542e-002 x1^2 x5^2 +3.7253e-001 x1 x2 x4 x6 (24.75% worst case error 7.05% average error)

0.9 -7.3655 x5 -1.9753e-003 x1^2 x4 +1.8246 x2^3 +8.7814e-004 x1^2 x4 x6 +1.1954e+001 x1 x2 x4 x5 (29.66% worst case error 5.65% average error)

The definitions for the x terms is in the following

table.

Table 4 – Terms for equations above

Term Definition x1 1/Bridge Ratio, Rad1/Bridge x2 (1-Web Ratio) x3 Magnet Depth Ratio x4 1/Corner Radius 3 x5 Magnet Thickness Ratio, LM/Rad1 x6 1/Corner Ratio

2 The number in parentheses is the maximum error for

the data used. 3 Corner Radius is the radius in mm, if the rotor has a

100 mm radius

E. Dependency of Stresses on the Magnet Thickness Ratio

Plots were also made of lines with constant Magnet Thickness Ratio. For these plots the Web Ratio, Magnet Depth Ratio, and Corner Ratio were held constant (Fig. 15 and Fig. 16).

Comparing these two plots, we see that the stresses are nearly two times higher for the Corner Ratio of 0.1 compared to a Corner Ratio of 0.9. From the previous results, this is to be expected.

As Magnet Thickness Ratio was varied, the lines remained nearly identical. This indicates that the Magnet Thickness Ratio is not a significant contributor to bridge stresses. This is partly because as the Magnet Thickness Ratio increases, the magnet length will decrease. This interaction causes the magnet mass to not vary significantly with magnet thickness.

Fig. 15-Stresses with Corner Ratio of 0.1

The figure has a larger magnet ratio, and reduced stresses.

Fig. 16 - Stresses with Corner Ratio of 0.9

F. Dependency of Stresses on the Web Thickness

Ratio In the figures below, it should be noted that

the Magnet Depth Ratio is 0.001, a basically flat magnet. The curves are curves of constant Web Ratio.

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Fig. 17- Stresses with Corner Ratio of 0.1

In Fig. 17, with an Inverse Bridge Thickness of 10 (10 mm), the Web Ratio of 0.05 had the highest stresses, and the Web Ratio of 0.35 at the lowest stresses. The Web Ratio has an obvious impact on stresses. Seeing that the spacings from each line to the others are fairly uniform, this impact appears to be linear.

Fig. 18 - Stresses with Corner Ratio of 0.9

Comparing Fig. 17 with Fig. 18, a larger Corner Radius Ratio results in a more linear stress relationship. In Fig. 17, it can also be seen that as the inverse bridge thickness increases to 100 (1 mm), the Web Ratio has no affect on stress.

The lines are not perfectly smooth. This is due to numerical calculation errors in the FEA.

G. Scaling for Speed and Rotor Size

The stresses are proportional to the square of the radius. Therefore, if the peak stress of a rotor with specific proportions is known, the stresses of a rotor with identical proportions, but different speeds and outer radius can be determined using the following formula:

�2 = �1 (R2/R1 )2 (�2 /�1 )2 (4) where �1 – the velocity of the rotor used in the FEA

analysis, (rad/ sec), �2 – the peak velocity of the rotor, R1 – the rotor radius used in the FEA analysis (100mm), R2 – the actual rotor radius �1 is the peak stress as determined from FEA at the initial speed and rotor radius, and �2 – the peak stress of the rotor at �2 and R2. The values given in Table 3 are based on 1000 RPM.

The equations above give approximations for the stress for all rotors which fall within the ranges given. For rotors which fall within the ranges above, but have corner ratios other than those given, interpolations may be made. Other equations for other fillet radii were determined, but space limitations made it impractical to print those results.

Looking again at Fig. 189, at an inverse bridge thickness of 10 (the left axis), the highest stress occurs with a magnet depth ratio of 0.4. The stresses then decrease as we proceed from 0.4 to 0.3, to 0.2, to 0.15. The stresses then begin to increase as we proceed from 0.15 to 0.1, to 0.05, to 0.001.

A magnet depth ratio of 0.4 represents the geometry with the magnets form a V which is nearly ½ of the rotor radius. This is a very deep magnet. The magnet depth ratio of 0.001 is nearly flat.4 The data in Fig. 18 indicates that to minimize the stress in the rotor, some magnet depth is preferred (about 0.15).

This is because the actual radius of the magnet slot is largest at this depth, for any given fillet ratio. In general, the geometry which allowed the largest fillet radius resulted in the lowest stresses.

The stresses in the bridge are also similar to another case in Roark [2], as shown in the figure below.

Fig. 19 - Filleted Corner in Tension

In this case, the ratio r/d was varied from 0.125 to 1.0. The stress concentrations (78) are shown inTable 5.

Table 5 - Stress Concentration for Filleted Corner r/d 0.125 0.15 0.2 0.3 0.4 0.5 0.7 1.0 78 2.50 2.30 2.03 1.7 1.53 1.4 1.26 1.20

It can be seen that the stress concentration is nearly two times higher at r/d= 0.125 than at 1.0. Comparing Fig. 9 with Fig. 10, where the r/d ratio varies from 0.1 to 0.9, we also see an approximate 2 to 1 increase in stresses as the ratio decreases. This is verification that the method is valid.

H. Slots with Center Post

If a center post is added between the two magnets, the stress can be reduced. However, if

4 A magnet depth ratio of 0.0 represents a flat

magnet. However the FEA model created did not allow for a flat magnet geometry.

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the bridge thickness remains the same, and a center post is added, the flux leakage would increase. This would change the performance of the motor. To avoid this, a new variable was added, the Post Ratio.

The Post Ratio determines how thick the center post is relative to how thick the bridge is. The post ratio was based on one half of the center post thickness. As the post became thicker, the bridge became thinner. This kept the leakage path the same width, and should maintain the same amount of leakage flux.

A Post Ratio of 0.0 is the case where there is no center post. A Post Ratio of 1.0 is the case where there is a center post and no bridge (an unrealistic design). If the Post Ratio is 0.0, the Bridge Thickness Ratio is the thickness of the bridge divided by the rotor radius. If the Post Ratio is 0.5, the bridge is half as large as if the Post Ratio was 0.0. A post ratio of 0.5 also indicates that the half post thickness is the same size as the bridge. In the figure below, the Post Ratio was varied from 0 to 1 for two different settings of Corner Ratio. The other settings are the same as in Fig. 9 and Fig. 10.

As was expected, the addition of the center post decreased stresses. The addition of a very thin center post increases stresses. The thin center post does not have enough cross section to support the slug mass. As the center post becomes thicker, the stresses decrease until they are significantly less than if the slug is supported only by bridges.

In some cases, as the bridge becomes very thin stresses again increase. These stresses are in the bridge area. For this case, the slug is being pulled away from the rotor center, and the bridge thickness is not large enough to restrain this movement.

In the graph below, for the Corner Ratio of 0.1, a wide range of Post Ratios will produce a similar stress. For the Corner Ratio of 0.9, there is a narrower range of optimum numbers. A larger Corner Ratio always reduces stresses. Therefore, in most cases there will be an optimum range for the post ratio.

Fig. 20 - Stresses vs. Post Ratio

VI. CONCLUSIONS

The following conclusions were made:

• Larger corner radii significantly reduce bridge stresses. A radius should be included at the corner of the bridge, to reduce stresses on the bridge.

• The hoop stress formulae do not include stress concentration factors, and are inaccurate when calculating stresses in the bridge.

• The use of a center post reduces stresses for the same flux leakage.

• The equations in this paper give an approximation of the stresses typically within 7%. Finite Element Analysis will give a more accurate answer.

REFERENCES

[1]. Kab-Jae Lee, Sol Kim, Seong-Yeop Lim, Ju

Lee, Bridge design of interior permanent magnet motor for hybrid electric vehicle, International Journal of Applied Electromagnetics and Mechanics 19 (2004) p 601-606

[2]. Young, W.C.; Budynas, R.G. (2002). Roark’s Formulas for Stress and Strain (7th Edition). (pp. 494). McGraw-Hill.

[3]. Bremner, R.D..; Dimensionless Optimization of a 60 kW IPM Machine, (PEMD 2008).

[4]. Lovelace, EC; Jahns, TM; Keim, TA; Lang, JH. Mechanical Design Considerations for Conventionally Laminated, High–Speed, Interior PM Synchronous Machine Rotors. IEEE Transactions on Industrial Applications, Vol. 40, No. 3, May/June 2004.

[5]. Lamination Steels Third Edition. EMERF, The Electric Motor Education and Research Foundation. March (2007).

[6]. Niazi, P; Toliyat, H.A.; Design of a Low-Cost Concentric Winding Permanent Magnet Assisted Synchronous Reluctance Motor Drive. Industry Applications Conference, 40th IAS annual meeting. (2005)

AUTHOR

Ron Bremner. Ron holds BSEE, BSME and MSME degrees from Iowa State University, USA, and is currently pursuing a PhD. He is a member of the IEEE and Society of Automotive Engineers (SAE). He has published 2 papers on electric machines, and holds 5 Patents. His primary technical interests are electric machine optimization and design.

0200400600800

100012001400

0 0.5 1

CornerRatio 0.9

Corner Ratio 0.1

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