Identifying Magnetic Saliency with the Finite Element Method

M.Sc. Lars Ottar Kv̊ale

Rogerts gt. 1

7016 Trondheim

kvale@stud.ntnu.no

Ph.D. candidate Sigurd Øvrebø

Energy Conversion Group

Dept. Electrical Power Engineering

NTNU

N-7491 Trondheim, Norway

Prof. Robert Nilssen

Energy Conversion Group

Dept. Electrical Power Engineering

NTNU

N-7491 Trondheim, Norway

October 2, 2003

Abstract– In this paper redesign of a surface

mounted permanent magnet synchronous ma-

chine is evaluated in order to enable sensorless

control at low and zero speed. The non lin-

ear permeability of iron is exploited in order to

produce position dependent inductance in the

machine. The sources of magnetic flux in the

model are both stationary and time-changing.

To estimate the inductance for various designs

and rotor positions with the non linear iron,

the non linear solver in FEMLAB is used. Su-

perimposing the flux from both stationary and

time-changing sources is done with the multi-

physics environment. The simulation also ac-

counts for induced currents in iron parts. Three

possible re-designs are evaluated and promising

simulation results are presented for sensorless

operation.

1 Introduction

Sensorless control has been a research topic for morethan two decades. The objective of this work is toeliminate the position sensor as the sensor itself, ca-ble and connections has been a source of failure insuch systems. For small drives the sensor contributesconsiderable to the overall cost. Several methods canbe used in order to obtain sensorless control at allspeeds [1] [2]. Additional excitation gives a current re-sponse that depends on the rotor position. The rotorposition dependency in the current response may haveseveral reasons, but saturation in the iron parts is themost common for surface mounted permanent magnetsynchronous machines (PMSM’s). The ratio of the po-sition dependent inductance compared to the averageinductance describes the easiness of implementing sen-sorless control. In the axial flux PMSM evaluated in

this paper the position dependent part of the induc-tance is too small for sensorless control. Hence, severalre-designs are evaluated in order to enable such con-trol.

2 Basic magnetic properties

The axial flux wheel motor can be described as a wheelsplit into three discs in which the centre disc is thestator surrounded by two rotor discs on both sides.The permanent magnets seen in blue in figure 1 areattached to the rotor yoke. Making an axial incision of

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stator rotor assembleda b c

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axial_skisse1.emf (2062x843x24b emf)

Figure 1: Principal design of the axial flux permanentmagnet machine

the motor produces the geometry shown in figure 2. Inthe same figure the current is directed perpendicularlyinto and out of the yellow regions , giving a magneticfield parallel to the paper. The arrows indicate themagnetization vector of the permanent magnets.

In magnetic circuit theory the reluctance of a ma-terial is analogous to the resistance of a conductor inelectric circuit theory. The reluctance describes the re-sistive force in a material subjected to magnetic flux.The permeability of a material is inversely propor-

1

tional to the reluctance and can be said to describe howwell magnetic flux is conducted. More often used inthe literature is the term relative permeability, whichis the ratio of the permeability in a material to thatof free space. Iron is generally considered to have arelative permeability of 5000, while permanent mag-nets have a relative permeability of 1, hence the samepermeability as air.Looking again at figure 2, stator magnetic flux, tak-ing the path of least resistance, will cross over the airgap, through the permanent magnets and into the firstrotor yoke (made of iron), before returning throughneighbouring stator core and making the same waythrough the second rotor yoke. Some of the statorflux, called leakage flux, will simply cross straight overto the neighbouring core. The stator flux path is illus-trated for one stator winding in figure 2.

However, since the permeability of the permanentmagnets is the same as that of air, the magnets canmagnetically be regarded as air gap for the stator flux.Thus at first sight, no rotor position will present anychange in the stator flux path permeability. On thecontrary, if change in rotor position affects the statorflux path permeability, the motor is said to have rotormagnetic saliency.

Figure 2: Incision of the principal axial flux machine

Very simplified, the self inductance of a winding isthe ratio of the flux linking the winding to the currentin the winding as given in 1.

L =ψ

I(1)

The inductance of a magnetic circuit is proportionalto the ampere-turns N · I in the winding squared, andinversely proportional to the reluctance R of the circuitas in 2.

L =(N · I)2

R(2)

The reluctance of a uniform segment of a magneticcircuit is defined as in 3.

R =l

µrµ0A(3)

where:

l = mean length of magnetic circuit segment (m)A = cross sectional area (m2)

µ0 = permeability of free space (Wb/A− t ·m)µr = relative permeability of a material to µ0

In real life the permeability of a ferromagnetic ma-terial is a non linear quantity which is explained in fol-lowing sections. Calculating the reluctance by hand,the permeability is approximated as a constant quan-tity. Looking at figure 2 a magnetic circuit as in fig-ure 3 can be deduced. In figure 3 Rcore,Rgap,Rpm and

N•I

Rleak

Rgap

Ryoke

Rcore

Rcore

Rleak

Rcore

Ryoke

2Rgap

2Rcore

2Rgap

Ryoke

Rgap

2Rgap

2Rcore

2Rgap

Ryoke

Rpm 2Rpm2Rpm

Rpm 2Rpm2Rpm

RleakRleak

FgapFleak

Figure 3: The magnetic circuit of a stator winding

Ryoke is the reluctance across the stator winding core,air gap, permanent magnet and yoke iron respectively.Evident in figure 3, the stator flux can be split up intotwo components, namely air gap flux Φqap, reachingthe rotor yoke and leakage flux Φleak , crossing directlyto neighbouring stator cores. This way the total re-luctance of the respective flux paths can be more eas-ily calculated. Detailed geometric measures will notbe disclosed and therefore the remaining calculation isomitted. The resulting total self inductance per wind-ing per unit depth into plane (1m) is given in 4.

Lwinding = Lgap + Lleak = 3.606 [mH ] (4)

2

3 Saliency phenomena in axial

flux PMSM

In the axial flux PMSM there is no basis for inherentsaliency. Observed from the stator flux, the permanentmagnets can be regarded as air in terms of their fluxconducting abilities. Saliency in the machine is presentas a result of saturation. There are two regions wheresaturation is most likely to occur. One possible regionof saturation might be in the stator powder core, andthe second possible region may be in the rotor yoke be-tween adjacent magnets [3]. This saturation is due tothe main flux from the permanent magnets. The firstpossibility of saturation in the stator cores, is not verylikely as the machine was designed to give maximumtorque at nine times rated current. The second satu-ration region is of greater possibility and is illustratedwith dark dots in figure 2 on the page before.

Most sensorless methods makes use of a high fre-quency excitation signal in stator windings. The mainflux produced from this excitation will take the pathof leakage flux as described above. As a result of thisthe re-design process aims to make modifications inthe leakage flux path to produce saturation inducedsaliency.

4 Saturation modelling

In ferromagnetic materials there is a nonhomogeneousrelationship between magnetic flux density B [T] andmagnetic field intensity H [A-t/m]. This relationshipis usually depicted in B−H-curves where the three re-gions called linear, knee and saturation become clear.The knee region is of interest in this work. When aferromagnetic material reaches this region a certainincrease in magnetic field intensity leads to limitedincrease in magnetic flux density. Eventually all themagnetic dipole moments in the ferromagnetic mate-rial will be aligned in the direction of the magneticfield. The material is then said to be in complete sat-uration. Increase the magnetic field intensity beyondthis level and the material will act as it’s permeabilitywas the same as that of air. [5]

This effect is of interest since the magnetic field frompermanent magnets is very strong. Given the rightdesign it is likely that the rotor permanent magnetfield will saturate parts of the stator core. Havingpartly saturated the stator core in a given position, thepermeability of the core is changed. Hence the stator

flux path reluctance will be sensitive to the position ofthe rotor permanent magnet.

The stator core is made out of somaloy550. Thisis electrically isolated powder iron which limits lossesdue to induced currents. In field element analysis offerromagnetic materials taking into account the non-linear aspect, the normal magnetization curve of thematerial is used. This curve is obtained by reversingthe direction of the magnetic field through the mate-rial at different levels of field intensity H , and loggingB and H [6]. This will produce a hysteresis loop andresulting normal magnetization curve as illustrated infigure 4.

B

H

Figure 4: Illustrated hysteresis loop and the normalmagnetization curve as the dotted curve.

The permeability of a material can be expressed asthe ratio of magnetic flux density to magnetic fieldintensity as in 5.

µ = µ0 · µr =∆B

∆H(5)

In the steep/linear region of the B − H-curve a rel-atively small increase in field intensity leads to greatcontribution to the flux density which is the same ashigh permeability. However in the flat regions of theB−H-curve the permeability is considerably lower. Atlow levels of magnetic field intensity the normal mag-netization curve is nearly flat, hence low permeabilityin this region. This is true for time varying fields oflow quantity, so called minor loops, or first time mag-netization of a material. It would however be wrongto assume that this is the case in the stator core sub-jected to the field from permanent magnets. For this

3

reason the ferromagnetic materials in the motor is as-sumed to have constant high permeability at low levelsof magnetic field intensity as illustrated in figure 5.

0 0.5 1 1.5 2 2.5 30

100

200

300

400

500

600

700

800

900

Flux density [B]

Rel

ativ

e pe

rmea

bilit

y (m

ur)

Figure 5: µr vs. B, relative permeability curve of so-maloy550 with constant high permeability at low fluxlevels (green) compared to non modified original curve(dotted blue).

5 Induced currents

Faraday’s law describes the mechanism which leads toinduced currents or eddy currents in a conductive ma-terial acting as core for time-changing magnetic flux.Circulating currents are induced in the perpendicularplane to the direction of the magnetic flux. These cur-rents in turn, generate flux opposing the initial fluxand thereby reducing the total time-changing flux inthe core. This effect is effectively limited in the statorcore because of the structure of the powder iron soma-loy550. But in the rotor yoke, being a solid iron discthe effect of eddy currents is present.

Most sensorless control methods known to take ad-vantage of rotor magnetic saliency make use of highfrequency signals in the stator windings. This resultsin high frequency flux trying to penetrate into the rotoryoke. The phenomena of induced currents is supportedin FEMLAB’s ”Perpendicular currents quasi-statics”application mode.

6 Modelling in FEMLAB

Modelling the wheel motor in FEMLAB the geometryis frozen for two distinct rotor positions. These po-

sitions are extremes when observing from the statorcore. At one position (case 1) the core is directly fac-ing the center of a permanent magnet, at which mostof the flux from the PM goes through the entire core tothe other rotor yoke. At the second position (case 2)the core is facing the air gap between adjacent perma-nent magnets, at which most of the flux from the PMwill simply dip into the core ending and go right backup through the adjacent PM. Case 1 and 2 is soughtillustrated in figure 6.

Iron yoke

Permanent Magnet

Iron powder core

Windings

Air

Symmetry of rotor

Overlapping symmetry lines

Case 2Case 1

Figure 6: All models are simulated in both case 1 andcase 2.

In the wheel motor the magnetic field set up by thepermanent magnets is referred to as the fundamen-tal excitation. This magnetic field is a static mag-netic field. But in the application of a possible sensor-less control algorithm high frequency currents in statorwindings will excite time-changing flux referred to asstator excitation. The frequency of this flux fulfills thecriteria validating a quasi-static simulation.

However, a challenge arises when modelling a staticand time-changing field in the same simulation. In the”transient” submode harmonic time-changing sourcesare not supported. In the ”time harmonic” submodeall sources change with the selected frequency. Dealingwith this fact FEMLAB’s multiphysics environmentcan be used to specify two identical application modeswith different submodes. This way the fundamentaland stator excitation are separated and specified onlyin each one of the application modes.

The main goal in modelling the motor is to see theimpact of rotor position on stator winding magneticfield. Linking the static rotor magnetic field to thetime-changing stator field is accomplished if both ap-plication modes share the same nonlinear permeabilityfor all subdomains. In other words, during simula-tion the sum magnetic flux density of both applicationmodes is used as input for interpolating the permeabil-ity value of a node. This node and whole subdomain

4

for that matter, then have the same permeability inboth application modes.

In plotting the B−H-curve from data sheet it is clearthat the curve is unnecessarily rugged. The ruggedness(see blue curve in figure 5 on the preceding page) is at-tributed a high number of data points and also possibleinaccuracy of some points. A smoother curve will easethe nonlinear solving process considerably. By first in-terpolating the data creating a smooth curve, selectionof as few as possible points on this new curve maintain-ing shape is performed. Interpolating between thesenew points creates a very smooth relative permeabilitycurve of required accuracy. The result of this techniqueis shown in figure 5 on the page before and in figure 7where the first derivative of the un-interpolated andtwo times interpolated permeability curves are shown.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3000

-2000

-1000

0

1000

2000

3000

4000

5000

Figure 7: First derivative of unmodified and two timesinterpolated relative permeability curves.

With a complete FEM solution available there areseveral methods in which to solve for the inductance.The key is evaluating the magnetic energy. Given themagnetic energy produced by the stator excitation, theself inductance can be calculated by 6 [7].

L = 2Wmag

I2(6)

Finding the magnetic energy is best done by evaluatingthe integral of J ·A over the winding subdomain of tworeasons.

Wmag =1

2

∫Si

( J · A )dSi (7)

In 7 J is the specified current density and A the mag-netic vector potential. Firstly, the integration domainis limited to only the winding subdomain of constantsize, and in this way being insensitive to changes intotal geometry size. Secondly, the magnetic vector po-tential A is the dependent variable which the simula-tion is solved for, hence no derivation is needed.

Apart from the non linear permeability curve simu-lations were performed from within FEMLAB’s graph-ical user interface. The simulation running times weregreatly reduced by providing an initial guess with thelinear solver and enabling differentiation of all vari-ables, hence differentiating the non linear permeabil-ity curve. In post simulation the solution was exportedto the Matlab environment where a m-file performinginductance calculations was run.

7 Results

In the work summarized in this paper four models ofthe axial flux motor were simulated, in which three ofthose included various modifications to the machine.Modification 1, the surrounding disc and modification2, the slot wedge make use of an additional soft fer-romagnetic material in the stator. The permeabilitycurve of this material is of the same shape as seenbefore, but has a lower maximum permeability. Mod-ification 3 labeled embedded PM is characterized bythe permanent magnets embedded into the rotor yoke.All three modifications are illustrated for case 1 in fig-ure 8.

Iron yoke

Permanent Magnet

Iron powder core

Windings

Air

Soft ferromagnetic material

Slot wedgeSurrounding disc Embedded PM

Figure 8: Machine modifications modelled and simu-lated in FEMLAB.

The resulting inductances in both case 1 and 2, andinductance variations for all models are presented intable 1 on the following page.

5

Model Case 1 Case 2 % δLNon modified 3.4 mH 3.2 mH 5.8 %Mod.1: Disc 5.6 mH 5.0 mH 10.7 %Mod.2: Slot wedge 4.4 mH 3.8 mH 13.6 %Mod.3: Embedded 3.2 mH 4.6 mH 30 %

Table 1: Simulation results for initial and modifiedmodels

In lack of space only one screen shot of the magneticfield of modification 3 is provided in figure 9.

Figure 9: Mod. 3, embedded PM in case 1.

8 Future work and Conclusion

In case of the results presented in this paper is to befurther verified by field element analysis several im-provements are suggested. Modelling the winding sub-domain more accurately by drawing cross section ofeach wire is of interest. This includes the skin effectwhich becomes more prominent at high frequencies.While a 3D simulation of the motor still might be tooambitious for some time, simulating the model at otherlevels of axial incisions might give a more complete pic-ture of the magnetic field in the machine.

Seeking to find a modification of the axial flux ma-chine that introduces magnetic saliency modification3, the embedded PM, is identified as the most suited.The variation of inductance is found to be 1.4[mH ] or30 % of the largest inductance. It is believed that thisvariation will enable sensorless control of the axial fluxPMSM. As a way of verifying the simulated model ahand calculation of the stator winding inductance wasperformed. The hand calculated inductance was foundto be 3.606[mH ] and the corresponding inductancefound by simulation was 3.4[mH ]. Although resultsbased on simulations might not be of exact quantity,it is believed that the simulations suffice in displayinginductance variations.

References

[1] M. Schroedl: ”Sensorless Control of AC Machines

at Low Speed and Standstill based on the Inform

Method”, IEEE IAS Annual Meeting San Diego(1996), Vol.1, pp.270-277.

[2] Jansen P.L., Lorenz R.D.: ”Transducerless Field

Orientation Concepts Employing Saturation-Induced

Saliencies in Induction Machines”, IEEE Transac-tions on Industry Applications No.6, Vol.32, (1996),pp.1380-1393.

[3] Sigurd Øvrebø: ”Comparison of excitation signals for

low and zero speed estimation of the rotor position in

an axial flux PMSM”, Institute of Electrical PowerEngineering, NTNU 2002.

[4] Christian M. Hartmann: ”Optimal Design av Aksial-

magnetiserte PM-Motorer”, M.S. thesis at Institutefor Electrical Power Engineering, NTNU, December1999, Page(s): 10 -23

[5] Stanley V. Marshall, Richard E. DuBroff, Gabriel G.Skitek: ”Electromagnetic concepts and applications”,Prentice-Hall International (UK) Limited, London,1996, Fourth Edition, Page(s): 403-406

[6] Sheppard J. Salon: ”Finite Element Analysis of

Electrical Machines”, Kluwer Academic PublishersGroup,330 AH Dordrecht, The Netherlands, 1995

[7] Robert Nilssen: ”Modellering av elkraft kompo-

nenter”, compenduim in subject with same name,http://www.elkraft.ntnu.no/ sie1040/, chapter 5

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