Download influence of arrangement and sizes of magnets upon

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.
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.
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:
U1 = R1I1 + jX d I1d + jX q I1q + U f ,
I1 = I1d + I1q
a)
(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 underexcited motors are presented in Fig. 1.
b)
Fig. 1. Phasor diagrams of PMSM,
a) overexcited motor, b) under-excited motor
For the PMSM electromagnetic torque it holds
that:
Te =
2
ω synch
U ⋅U f

U 2  1
1 
sin θ +
−
sin( 2θ ) (3)

2  X q X d 

 X d
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.
Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)
44
3. Dynamic Model of PMSM
Electromagnet torque with CT considered is:
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:
did ud Rid Lq
=
−
+
pωr iq
dt Ld Ld
Ld
diq
dt
Te =
=
uq
Lq
−
Riq
Lq
+
Ld
pωr
pωr id − Φ
Lq
Lq
3
p Φiq + ( Ld − Lq )id iq
2
[
(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.
a)
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.
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]:
fc =
z1
⋅f
p
(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:
Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)
Tc =
dW
dα
(7)
Upon adjustment and introduction of coordinate
x that represents the air gap length we have
arrived to the below equation:
Tc =
D2 out dW
2 dx
(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]:
∞
M c ( α ) = ∑ M k ⋅ sin( k ⋅ z1α + ϕ k ) (9)
k =1
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.
45
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°), kinteger (k=<1,5>), αod corresponds to 1/4 of the
angle corresponding with the slot opening.
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
Fig. 3. Average value of the absolute values of
CT as depending on the pole embrace. (HMthicknesses of 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
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))
•
•
•
•
•
•
•
•
•
46
Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)
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. 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. 6. The average value of the CT (absolute
values) depending on the angle between PM.
(HM-heights of magnets, SM –widths of
magnets)
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
Fig. 7. Fig. 8 Cogging torque as function of
time for different width of PM and the angle
between PM. Rotor speed =3000 rpm
Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)
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)).
47
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
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.
Uf
200.00
rms(InducedVoltage) [V]
150.00
Curve Info
rms(InducedVoltage)
Setup1 : Transient
HM='1mm'
rms(InducedVoltage)
Setup1 : Transient
HM='2mm'
0.00
0.50
0.60
0.70
pole embrace [-]
0.80
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.
100.00
50.00
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.
0.90
Fig. 10. Dependence of the induced voltage on
the pole embrace and the magnet thicknesses
48
Zeszyty Problemowe – Maszyny Elektryczne Nr 4/2014 (104)
7. Bibliography
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Acknowledgement
This work was supported by the Slovak
Research and Development Agency under the
contract No APVV-0138-10. (90%) and SKCZ-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.