influence of arrangement and sizes of … of arrangement and sizes of magnets upon ... their...

Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)

43

Želmíra Ferková Technical University of Košice, Slovakia Faculty of Electrical Engineering and Informatics

INFLUENCE OF ARRANGEMENT AND SIZES OF MAGNETS UPON

COGGING TORQUE AND EMF OF TWO-PHASE PMSM

Abstract: Discussed in the paper are two-phase permanent magnet synchronous motors, and explored is

influence of the magnet sizes as on induced voltage (EMF) so on cogging torque (CT). Induced voltage and

the cogging torque have been calculated in ANSYS/Maxwell program. In more detail contemplated is

influence of the magnet size, their positioning and orientation to the CT in two basic rotor types. The cogging

torque waveform for critical dimensions has been considered using the FFT analysis. The first rotor type is the

one with semi-arc shaped magnets located on the rotor surface. The other motor type features magnets

embedded in its rotor. Shape of the stator magnetic circuit has been predetermined.

Kewords: cogging torque, EMF, permanent magnets

1. Introduction

In the matter of several recent years, utilisation

of permanent magnet synchronous motors has

dramatically increased. The advantage of

excitation with the permanent magnets (PM) is

the absence of Joule losses in excitation, and

especially then elimination of the sliding

contact. The disadvantage is that excitation is

uncontrollable, whilst the recent growing price

of magnets is an issue too.

Analysed in the present article is a two-phase

two-pole motor with permanent magnets

(NdFeB, Hc= -849000 A/m, Br=1.125 T). Used

was a ready-made stator stack. Shape and

number of stator slots could not be changed.

2. Permanent Magnet Synchronous Motor

In the steady state, behaviour of a two-phase

synchronous machine can described by the use

of the below equations:

(1),(2)

where:

R1 is phase resistance,

Xd is d-axis synchronous reactance,

Xq is q-axis synchronous reactance,

U1- input voltage, I1- one phase current,

I1d a I1q – d and q axis stator currents,

Uf – voltage (EMF) induced by magnetic

excitation flux of the rotor.

Phasor diagrams of over-excited and under-

excited motors are presented in Fig. 1.

The equations clearly imply induced voltage Uf of the synchronous machine influences its

behaviour. This depends on the number of turns

and the arrangement of the stator winding as

well as on the arrangement and size of magnets.

Moreover, placement of magnets affects the

reactance Xd and Xq.

a) b)

Fig. 1. Phasor diagrams of PMSM,

a) overexcited motor, b) under-excited motor

For the PMSM electromagnetic torque it holds

that:

−+

⋅= )sin(

XX

Usin

X

UUT

dqd

f

synch

eθθ

ω2

11

2

2 2

(3)

where:

U1, and Uf are effective values of phase voltage

and the voltage induced by excitation, θ is the

loading angle [2]. Te is the electromagnetic

torque that neglects CT.

qd

fqqdd jXjXR

111

11111 ,

III

UIIIU

+=

+++=


44

3. Dynamic Model of PMSM The d-q dynamic model is expressed in

a rotating frame that moves at synchronous

speed ωr. The general equation of PMSM

working as motor are following:

[ ]qdqdqe

q

r

dr

q

d

q

q

q

qq

qr

d

q

d

d

d

dd

ii)LL(ΦipT

L

pΦip

L

L

L

Ri

L

u

dt

di

ipL

L

L

Ri

L

u

dt

di

−+=

−+−=

+−=

2

3

ωω

ω

(4)

where id and iq are the d and q axis stator

currents, Ld and Lq are the d and q axis

inductances, ud and uq are d and q axis stator

voltages, R is phase resistance, p is the number

of pole pairs, ωr is the rotor angular speed, Φ is

excitation flux created by the PM.

Electromagnet torque with CT considered is:

Te´=Te+Tc (5)

Tc depends on stator/rotor angular position.

Explored by ANSYS/Maxwell was the impact

of the permanent magnet volumes and

arrangement on the voltage induced and on the

cogging torque. Chosen were 2 types of

designs, when both motors were two-pole ones.

Type A has two poles composed of on the

motor surface positioned magnets. The magnets

have semi-arc shapes (rotor-surface-mounted

magnets). With this type it can be anticipated

that the reactance Xd and Xq are identical.

Type B has two poles made of six magnets

embedded in rotor (Fig. 2). Altering was the

width, thickness and angles among magnets of

one pole.

a) b)

Fig. 2. Design types explored: a) A – motor with rotor surface-mounted magnets, b) B – motor with

tangentially embedded magnets. Dimension of rotor: Drot=55 mm, rotor_length=26 mm,

air_gap=0.5 mm

4. Impact of the Magnet Geometry upon Cogging Torque

When designing 2-pole and 2-phase

synchronous motors with permanent magnets

emerging is the cogging torque (CT) issue in a

greater extent than in the three-phase machines

with PM. In this case, the situation becomes

worse due to the fact that magnetic circuit of

the stator cannot be changed. The stator has a

concentric winding.

Oscillation frequency of the cogging torque

depends on the number of slots, and is given by

the following equation [2]:

fp

zfc ⋅= 1 (6)

z1 is the number of stator slots, z1=24

p is the number of pole pairs

f is the input frequency, f=50Hz

Analytical methods for calculating the cogging

torque usually neglect the magnetic flux

through the stator slot and saturation of the

magnetic circuit. The cogging torque can by

computed by the derivative of magnetic energy

W produced by permanent magnets with respect

to the rotor position angle α:


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αd

dWT

c= (7)

Upon adjustment and introduction of coordinate

x that represents the air gap length we have

arrived to the below equation:

dx

dWDT out

c2

2= (8)

α=2x/D2uot,

It is anticipated that the rotor outer diameter is

roughly equal with the stator inner diameter

D2out≈D1in.

In addition to fc frequency in the timeline of the

CT found can be frequencies that are related to

the arrangement of magnets.

Waveform of CT can be, if neglected is

saturation of the magnetic circuitry, described

by equation [5]:

)zksin(M)(Mk

kkc

ϕαα +⋅⋅= ∑∞

=1

1

(9)

Where z1 is the number of stator slots, Mk is the

related harmonic’s amplitude, ϕk is phase shift

of the respective harmonic, and α is the rotor

position.

The cogging torque can by minimised by the

proper design [2]. Measures taken to minimise

the torque ripple by motor design include

• elimination of stator slots

• skewed slots

• special shape slots and stator

laminations

• selection of the number of stator slots

with respect to the number of poles

• decentred magnets

• skewed magnets

• shifted magnet segments

• selection of magnet with

• direction-dependent magnetization of

permanent magnets

Calculated in ANSYS/Maxwell 2D was

cogging torque for the given magnetic circuit of

stator, for two principal rotor types (types

A and B; Fig. 2), and for a varying geometry of

magnets.

Changing in type A, which has two arched

magnets on surface of the rotor, was the

thickness (HM) and also the magnet central

angle represented by the pole embrace. Pole

embrace=1 corresponds to a central angle of

180° and a pole embrace=0.5 corresponds to

90°.

Changing in case of type B, which is made of 6

magnets forming 2 poles, were the magnet

thickness (HM), magnet width (SM), and

arrangement of magnets that was characterised

by an angle between one pole magnets. (Fig.

2b)

In both cases, the magnet length equals length

of the magnetic circuit.

Type A

Dependence of CT from pole embrace is

shown in Fig. 3. Dashed line shows multiples

of the groove spacing (slot pitch). The smallest

values and the minimum CT are for:

Pole embrace ≈ (π-k.2π/z1+αod)/π.

Where:

2π/z1 is slot pitch in degrees (2π/z1 =15°), k-

integer (k=<1,5>), αod corresponds to 1/4 of the

angle corresponding with the slot opening.

Fig. 3. Average value of the absolute values of

CT as depending on the pole embrace. (HM-

thicknesses of magnets)

The smallest CT value can be seen with

pole embrace = 0.59. Corresponding with

this is the magnet central angle of 106.2°.

Differing content of higher harmonics in

the cogging torque waveform can be for

selected pole embrace seen in Fig. 4.

Prevailing in local max values is the

frequency of 1200Hz. (Equation (6))


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Fig. 4. Cogging torque as function of time for

different „pole embrace”. Rotor speed =3000rpm.

(pole embrace: 0.54, 0.59,0.635, 0.675, 0.72, 0.82)

Fig. 5. Harmonic components of the waveform

(mag(CT)) given in Fig. 4.

(pole embrace: 0.54, 0.59,0.635, 0.675, 0.72, 0.82)

Type B

In this motor type is each pole composed of

three magnets. In symmetric distribution the

angle between magnets is 60°. Analysis has

shown that arrangement of magnets is of

significant impact on the cogging torque, and

that there exists an angle at which is the

cogging torque minimal. CT average value

(avgabs(CT)), as depending on the angle

between magnets is shown in Fig. 6. The

parameters are: the magnet widths (SM) and

thicknesses (HM), respectively. The curves

show the minimum for the angle of 55°. It is

the angle between two magnets of one and the

same pole.

The CT waveform dramatically changes for

angles to the left and to the right of the

minimum. The CT time waveforms for angles

of 53°, 55°, and 57° for magnet thicknesses of 3

mm-s and for magnet width of 18mm and 19

mm, respectively, are shown in Fig. 7.

Fig. 6. The average value of the CT (absolute

values) depending on the angle between PM.

(HM-heights of magnets, SM –widths of

magnets)

Fig. 7. Fig. 8 Cogging torque as function of

time for different width of PM and the angle

between PM. Rotor speed =3000 rpm


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In the harmonic analysis shown in Fig. 9 seen

can be distribution of individual harmonics.

Prevailing are multiples of 1200Hz. The

1200Hz frequency is given by the number of

slots (equation (6)).

Fig. 9. Harmonic components of the wavwform

(mag(CT)) given in Fig. 7.

5. Impact of Magnet Geometry on the Permanent Magnet Induced Voltage

Type A

The no-load rms voltage induced in one phase

of the stator winding (EMF) by PM excitation

flux for varying pole embraces, and for magnet

thicknesses of 1 and 2 mm are shown in Fig.

10.

Fig. 10. Dependence of the induced voltage on

the pole embrace and the magnet thicknesses

Type B

The calculated values of by permanent magnet

flux induced voltage (EMF) are shown in Fig.

11. A slight drop in induced voltage is brought

about by leakage flux within magnets.

The induced voltage is dependent mainly on the

size of magnets.

Fig. 11. Dependence of EMF on the magnet

thicknesses (HM) and widths (SM), respectively

6. Conclusion

The paper presents dependencies of the cogging

torque and induced voltage on the dimensions

and geometry of placement of magnets. Fourier

analysis of CT waveforms shows changing

spectrum that changes with arrangement of

magnets. Prevailing in both cases are

frequencies of fc=1200Hz, which are

determined by the number of slots, and also

multiples of fc. In areas with a minimum value

of CT, these frequencies are significantly

suppressed.

Results better than have been presented can be

reached by using skewing of magnets. The

issue has been resolved but it is not discussed in

this article.

Effect of size and storage of magnets on the

induced voltage has a logical course. In type B,

if the magnets are not arranged symmetrically,

anticipated is adjustment of the magnetic circuit

so that Lq≈Ld.

0.50 0.60 0.70 0.80 0.90pole embrace [-]

0.00

50.00

100.00

150.00

200.00

rms(I

nd

uce

dV

olta

ge

) [V

]

Uf

Curve Info

rms(InducedVoltage)Setup1 : TransientHM='1mm'

rms(InducedVoltage)Setup1 : TransientHM='2mm'


48

7. Bibliography

[1]. Pyrhönen L., Jokinenm T., Hrabovcová V.:

Design of Rotating Electrical Machines. John Wiley

&Sons,Ltd, 2008

[2]. Gieras J.,Wing M.: Permanent Magnet Motor

Technology. Marcel Dekker 2002, USA

[3]. Gieras J: Analytical Approach to cogging

Torque Calculation of PM Brushless Motors, IEEE

Transactions on Industry Applications, Vol.40,

No.5, 2004

[4]. Glinka T.: Maszyny elektryczne wzbudzane

magnesami trwałymi, Wydawnictwo Politechniki

Śląskiej, Gliwice 2002

[5]. Glinka T. Jakubiec M.: Silniky elektryczne z

magnesami trwałymi umieszconymi na wirniku,

Zeszyty problemowe – Maszyny Elektryczne no

71/2005, pp. 103-, Katowice 2005

[6]. Luke Dosiek, Pragasen Pillay: Cogging Torque

Reduction in Permanent Magnet Machines, IEEE

Transactions on Industry Applications, Vol.43,

No65, 2007

[7]. Tudorache T.,Trifu I., Ghita C., Bostan V.:

Improved mathematical model of PMSM taking into

account cogging torque oscilations, Advances in

Elevtrocal and Computer Engineering, Vol. 12, No.

3, 2012, pp.59-66

[8]. Rosa R., Pistelok P.: The algotithm for the

electromegnetic calculations of the concentrated

winding permanent magnet synchronous motors,

Zeszyty problemowe – Maszyny Elektryczne Nr

4/2013, BORME Komel, pp. 161-169, Katowice

2013

[9]. Mynarek P., Kowol M.: Field-circuit analysis

of permanent magner synchronous motors, Zeszyty

Problemowe – Maszyny Elektryczne Nr 4/2014,

BORME Komel, pp. 73-76, Katowice 2013

[10]. Sekerák, P., Hrabovcová, V., Pyrhönen, J.,

Kalamen, L., Rafajdus, P., Onufer, M. : Comparison

of synchronous motors with different permanent

magnet and winding types, IEEE, Transaction on

Magnetics, Vol. 49, no. 3 (2013), 2013, ADC, pp.

1256-1263, 0018-9464

[11]. Dúbravka, P., Rafajdus, P., Hrabovcová, V.,

Mušák, M., Peniak, A.: Improvement of Switched

Reluctance Motor Fault Tolerances Using Novel

Design by Means of 3D FEM Parametrical Model,

SMM21-Soft Magnetic Materials, Conference

Program and Book of Abstracts, Budapešť, 2013,

09, 1.-4., AFG, str. 319

Acknowledgement

This work was supported by the Slovak

Research and Development Agency under the

contract No APVV-0138-10. (90%) and SK-

CZ-2013-0065.

Author

Želmíra FERKOVÁ (Assoc. Prof. Ing. CSc.)

graduated in electrical engineering from the

Technical University Košice in 1982 and

received her PhD degree in 1994. At present

she is active as associated professor at the

Department of Electrical Engineering and

Mechatronics, Faculty of Electrical Engineering

and Informatics TU Košice. Her professional

area covers design of electrical machines and

their performance.

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