MODELLING OF ELECTRICAL MACHINES WITH SKEWED SLOTS
USING THE TWO DIMENSIONAL FINITE ELEMENT METHOD
J. Gyselinck, L. Vandevelde and J. Melkebeek
Laboratory for Electrical Machines and Power Electronics, University of Gent
St.-Pietersnieuwstraat 41, 9000 Gent, Belgium
Tel: +32 9 2643424 Fax: +32 9 2643582 E-mail: gyselinck@elmape.rug.ac.be
Abstract
Some methods for considering skewed slots are brie y
discussed. It is shown that for a rigorous modelling of
skew and of the ensuing axial variation of saturation, a
2 12 D multi-slice model is a viable alternative to a fully 3D
model. The computational cost of time stepping such a
FE model connected to an external electrical circuit using
either a direct or an indirect coupling method is studied.
The latter is applied to the simulation of a squirrel-cage
induction motor, for which four dierent rotors (with
unskewed or skewed slots and with open or closed slots)
are available. A satisfactory agreement between measured and computed stator phase current waveforms has
been obtained for both no-load and load.
Keywords
skewed slots, squirrel-cage induction machine, nite element method, dynamic simulation
Introduction
The ever increasing performance of nowadays workstations allows a more and more detailed modelling of rotating electrical machines. Time stepping a machine model
built around a 2D Finite Element (FE) representation of
the cross-section has become workable. Such a model
may account for magnetic saturation, skin eect in solid
conductors, motion eects, arbitrary voltage supply (e.g.
inverter supply), an external supply or load circuit, the
kinematic equation and so on. The 3D end-eects due
to end-windings and end-rings are commonly taken into
account by inserting lumped resistances and inductances
in the electrical network.
For electrical machines with unskewed slots this 2D Single
Slice Model (SSM) may produce quite satisfactory simulation results. However, many and in particular small and
medium sized machines found in industry have skewed
slots. This is indeed an eective measure taken by the
designer to reduce the slot ripple of currents, voltages
and torque 1]. A SSM of such a machine will yield unrealistically elevated slots harmonics. A sensible prediction
of these harmonics thus requires a model that accounts
for the skew.
Time stepping a completely 3D FE model is still beyond
the practical limits of most present day workstations.
A rst approximation for obtaining a workable skewed
model then consists in neglecting in each cross-section
the peripheral current component and the ensuing ax-
ial m.m.f., i.e. in each cross-section a 2D magnetic eld
perpendicular to the machine axis is assumed.
In the classical rotating eld theory, skew is easily introduced by means of skew factors. Indeed, the air-gap uxdensity is resolved into a series of travelling waves, which
are obtained by multiplying the stator and rotor m.m.f.
waves by the permeance waves, accounting for slotting,
saturation, etc.. In case of skewed slots, the voltage induced in the stator winding by a eld wave produced by
a rotor m.m.f. component or by the rotor slotting eect
is subject to a skew factor dependent on the order of the
eld wave. Analogously, a voltage component induced in
the rotor circuit is aected by a skew factor if the inducing eld wave is produced by a stator current or by stator
slotting.
However, a rigorous application of skew factors to a time
stepped FE model is far from trivial, if not impossible. In
a FE simulation only the total induced voltage is determined, which cannot be split up unambiguously in voltage components induced by dierent ux components (of
dierent orders). Consequently, a skew factor cannot be
derived properly. Simple ltering procedures for modelling skew, as e.g. mentioned in 2] and (tacitly) used in
many other publications, naturaly result in a considerable
reduction of the slot harmonics but do not have any solid
basis and may lead to erroneous results 3].
Furthermore, in electrical machines with skewed slots,
the saturation level in a cross-section varies along the
machine axis. This eect is commonly neglected. For
instance, in the Magnetic Equivalent Circuit (MEC)
method described by Ostovic 4], skew is introduced by a
'3D' calculation of the air-gap permeances between the
stator and rotor teeth, i.e. taking into account the variation of the overlap of a stator and rotor tooth in the axial
direction. However, the reluctances of the MEC representing the stator and rotor iron are calculated on the
basis of the total tooth and yoke uxes, and not on the
basis of the local ux distribution in the cross-sections.
A feasible way for introducing skew in a time stepped FE
model in a proper way, is to consider a nite number of
cross-sections (or slices) taken at dierent positions along
the axis, i.e. modelling skew in discrete steps 3,5,6,7].
Such a model will further be referred to as a Multi-Slice
Model (MSM).
For time stepping the coupled magnetic and electrical
equations of both SSM and MSM, two distinct numerical strategies can be adopted 8,9]: the direct cou-
pling method on the one hand and the indirect coupling
method on the other hand. The direct and indirect coupling methods, and in particular the implications with
regard to accuracy and computational cost will be discussed in the following sections.
Single Slice Model
The SSM incorporates a single 2D FE model of a crosssection and an electrical network that consists of both
conductors dened in the FE model and external components such as voltage sources, resistances, inductances,
etc. .
In the FE method for 2D magnetostatic and (low frequency) magnetodynamic eld calculations, the Magnetic Vector Potential (MVP) A(x y) is commonly introduced for enforcing a divergence free magnetic induction.
In the magnetostatic case, space discretisation of the 2D
domain leads to a set of nonlinear algebraic equations
in terms of the MVP nodal values and with the exciting current density contained in the right hand side. The
dynamic case leads to a set of rst order dierential equations, in which the right hand side contains the current
excitation of stranded conductors and the voltage excitation of massive conductors.
By dening current loops and corresponding loop currents in the electrical network, Kirchho's current law automatically holds. For each external component a (simple) voltage-current relation may be assumed, whereas
the voltage-current relation for the (coupled) FE conductors is eld dependent and, in general, complicated.
Kirchho's voltage law nally leads to one equation per
current loop.
Direct coupling
In the direct coupling method 2,8] the eld and circuit
equations are solved as a single set of coupled dierential
equations. The two types of conductors, stranded conductors (coils) with a uniform current density and massive conductors (bars) displaying skin eect, are easily
included. A more comprehensive analysis, including e.g.
current sources, capacitors and nonlinear components, is
given in 10].
The dierential equations (1-4) governing the directly
coupled eld-circuit system:
S ( (t) A(t)) A(t)+G @A@t(t) = CbT R;b 1 Vb (t)+CcT Ic(t) (1)
Vb (t) = RbIb (t) + Cb @A@t(t) Ib (t) = DbIl (t) (2)
Vc (t) = Rc Ic (t) + Cc @A@t(t) Ic(t) = Dc Il (t) (3)
DbT Vb (t) + DcT Vc (t) + Rel Il (t) + Lel dIdtl(t) = El (t): (4)
are written in terms of:
A(t) (Nn 1) : MVP nodal values
Ib (t) (Nb 1) : total bar currents
Vb (t) (Nb 1) : bar voltages
,
Ic (t) (Nc 1) : coil wire currents
Vc (t) (Nc 1) : total coil voltages
Il (t) (Nl 1) : electrical network loop currents
where Nn , Nb , Nc and Nl are the number of FE nodes,
bars, coils and current loops respectively.
In the FE diusion equation (1), S (Nn Nn ) is the
stiness matrix of the FE mesh and G (Nn Nn ) is
the conductivity matrix of the FE bars. Both S and
G are sparse and Symmetric Positive Denite (SPD).
The former is dependent on magnetic saturation and the
rotor position (t), which is either specied (e.g. constant
speed) or governed by an additional kinematic equation.
Eqns. (2-3) express the voltage-current relation and the
interconnection in the electrical circuit for bars and coils
respectively. In the voltage equation (4), external voltage
sources, resistances and inductances are accounted for
through El (t) (Nl 1), Rel (Nl Nl ) and Lel (Nl Nl )
respectively. Rel and Lel are SPD.
Elimination of Ib (t), Ic (t) and Vc (t), and a nite dierence time discretisation yield the following symmetrical
Set of Algebraic Equations (SAE) to be solved for every
time step t; ! t+ : (0 < 1, t = t+ ; t; )
2
3
S + Gt CbT R;b 1 CcT Dc " A+ # " : : : #
4 R;b 1 Cb tR;b 1 tDb 5 Vb+ = : : :
Il+
:::
DcT Cc tDbT tZl
;
;
;
;
;
;
;
(5)
where Zl = DcT Rc Dc + Rel + ( t);1 Lel .
= 1 and = 0:5 produce the backward Euler and
Cranck-Nicholson schemes respectively. The right hand
side in (5) depends on the variables at t; (A; , Vb; and
Il; ) and on the loop voltages El; and El+ .
Magnetic saturation does not aect the further exposition principally. Indeed, applying the Newton-Raphson
scheme is tantamount to replacing the stiness matrix S
by its equally sparse and SPD Jacobian matrix.
Instead of tackling the complete system (5) using a general purpose algebraic solver, advantage should be taken
of the particular nature of the system matrix. Indeed,
the large (typically 1000 Nn 10000) upper diagonal
block S + G= t is sparse and SPD, the upper diagonal
2 2 block
S + Gt CbT R;b 1
(6)
R;b 1 Cb tR;b 1
is SPD, and the much smaller (typically 10 Nb + Nl 100) lower diagonal block ; tZl is symmetric negative denite. As a consequence, a number of block
elimination techniques which only produce subsystems
with SPD matrices can be deduced 10,11]. In case of
the MSM, one of these techniques elegantly leads to
a numerical decoupling of the slices and an economic
algorithm 7]. With regard to the computational cost
per Newton-Raphson iteration, this particular technique
mainly amounts to solving a number of linear SAE's with
the same system matrix S (or, more precisely, its Jacobian) and with 1 + Nb + Nlc dierent right hand sides,
where Nlc is the number of electrical current loops that
contain at least one FE coil.
;
;
Indirect coupling
In the indirect coupling method the electrical circuit
equations are time stepped while the magnetic coupling
of the machine windings (or current distributions 3]) is
calculated by means of static FE analyses. This implies
that skin eect in massive conductors (bars) cannot automatically be included. Depreciating bars to single wire
coils and using similar notations as above, the voltage
equation (4) can be rewritten as follows:
c
e
T d c
e dIl(t)
(Rl + Rl )Il (t) + Dc
dt ( (t) Dc Il (t)) + Ll dt = El (t)
(7)
where Rcl = DcT RcDc is the coil resistance matrix, and
c the coil ux linkage vector.
For a given set of currents Ic = Dc Il and a given rotor position , the ux linkage c = Cc A ( Ic ) is
obtained by a single magnetostatic eld computation:
S ( A ) A = CcT Ic :
(8)
The ux linkage calculations can be introduced in the
time stepping by means of various iterative procedures
3,12]. The latter commonly require the calculation of
'linearised' inductance matrices Mlc , typically done as
follows 3,12-14]. The local saturation level, contained
in S = S ( A), is calculated by solving (8). Subsequently, the dierent current loops which contain at least
one FE coil, are excited with a unit current while holding
the afore computed saturation, thus obtaining:
Mlc = DcT Cc (S );1CcT Dc :
(9)
The computational eort for calculating a single Mlc
amounts to solving one nonlinear SAE and Nlc linear
SAE's with the same system matrix S , where Nlc has
the same meaning as in the direct coupling method.
Numerical e ciency
From the above discussion it should be clear that it is
not sensible to simply state that either of the two methods { direct or indirect coupling { is the most ecient.
Much depends on the required accuracy and the ensuing
approximations. For instance, one of the major disadvantages of the indirect coupling method, the neglect of skin
eect in massive conductors, can be remedied by splitting up current loops into a nite number of subloops,
however at the expense of increasing Nlc.
Large computational savings, in particular when using
the direct coupling, can be obtained by truncating the
Newton-Raphson scheme at the rst solution. This is
equivalent to using the local saturation level of the previous time point and is quite justifyable if a (very) small
time step is adopted.
It should also be noted that the indirect coupling method
provides more opportunities for obvious approximations
and savings. In 3] e.g., spatial current distributions in
the squirrel-cage of an induction machine are considered,
rather than the bar currents themselves. By limiting the
order of these modal distributions, e.g. to fundamental
and third order, Nlc can be reduced considerably. Another
means for reducing computation time is not to perform
the eld analyses for every time step, but to update the
inductance matrix only when this is found to be neccesary
using some kind of criterion. In case of power electronic
supply e.g., a very small time step may be required for
the electric equations but not for the magnetic equations.
The authors believe that similar features can also be
imbedded in a direct coupling scheme, albeit in a less
straightforward way.
Multi Slice Model
The skew of rotor and/or stator slots is modelled by
means of a nite number of slices in which an axial current and a 2D magnetic eld are assumed, thus neglecting the peripheral component of the current and the axial
component of the magnetic eld.
We consider Nsl slices of equal axial length. In each slice
the rotor is shifted over the angle i = sk (2i ; Nsl ;
1)=(2Nsl ), (i = 1 :: Nsl) with respect to the average
rotor angle (t), where sk is the total skew angle.
Direct coupling
The MVP nodal values and the FE-conductors voltages
of the i-th slice are denoted Ai , Vbi and Vci respectively.
The eqns. (1-3) become, for each slice (i = 1 :: Nsl):
i
S ( (t)+ i Ai )Ai (t)+G @A@t(t) = NslCbT R;b 1 Vbi (t)+Cc Ic (t)
(10)
i (t)
@A
i
;
1
Vb (t) = Nsl (RbIb (t) + Cb @t ) Ib (t) = DbIl (t) (11)
i
Vci (t) = Nsl;1 (Rc Ic (t) + Cc @A@t(t) ) Ic (t) = Dc Il (t) (12)
where the matrices S , G, Rb, Rc, Cb and Cc have the
same dimensions and the same values as in the single slice
case. The dierential eqns. of all slices are coupled as the
FE conductors carry the same (total) current Ib (t) and
Ic (t). Furthermore, each slice contributes to the total
voltages Vb (t) and Vc (t), so (4) becomes:
D
T
b
N
X
sl
i=1
V t D
i
b( )+
T
c
N
X
sl
i=1
Vci (t) + Rel Il (t) + Lel dIdtl(t) = El (t):
(13)
The global system of dierential equations (10-13)
leads for every time step to a SAE in terms
of A^(t)T = (A1(t))T : : : (AN (t))T ], V^b (t)T =
N
1
T
T
(Vb (t)) : : : (Vb
(t)) ] and Il (t). The total number of
variables (Nn + Nb )Nsl + Nl is practically proportional
to Nsl .
The SAE takes on the same block structure as in (5), but
the block elements now have their own particular block
(diagonal) structure. For instance, the MSM global stiness matrix S^ (Nn Nsl Nn Nsl ) is equally sparse SPD
and consists of the stiness matrices of the constituent
slices:
S^ = Nsl;1 diag S (A1 + 1 ) : : : S (AN + N ) :
sl
sl
sl
sl
(14)
The global block elements have the same properties as
in the SSM, such that the above specied elimination
technique can be applied to the MSM as well. Even
better, when doing so, a numerical decoupling of the
slices naturally emerges, enabling signicant savings in
both computation time and storage requirements. As the
most time-consuming operations are performed on separate slices, computation time is practically proportional
to Nsl . Furthermore, this technique results in storage
requirements that are far less than proportional to Nsl .
Indirect coupling
In the indirect coupling method the inductance matrices
of the dierent slices are calculated and the average matrix Mlc is used for time stepping the circuit equations.
Eqns. (8) and (9) become:
S ( + i Ai) Ai = CcT Ic
i = 1 : : : Nsl
N
1 X T
c
Dc Cc (S i );1CcT Dc :
Ml = N
sl i=1
(15)
sl
(16)
Figure 1 FE mesh
It follows that computation time is proportional to Nsl ,
and that storage requirements are independent of Nsl .
Numerical e ciency
The above discussion on the numerical eciency for the
SSM also applies to the MSM, as in both cases the computation time is proportional to the number of slices considered.
Also for the MSM, the indirect coupling method provides
more opportunities for approximations. In 3] e.g., the
authors propose to equate the angle step of the discrete
rotor motion to the angle step of the discrete skew, and
to calculate for each time step only the inductance matrix of the 'leading' slice. This results, however, in the
proliferation of the leading slice saturation to the Nsl ; 1
lagging slices, and is therefore not recommended if a large
axial variation of saturation, as e.g. observed in the application example described hereafter, is to be expected
3].
Experimental Verication
We have applied the direct coupling method to the dynamic simulation of a 3 phase 3kW 4-pole squirrel-cage
induction motor. For this motor four dierent rotors to
be mounted in the same stator are available. Beside the
original (commercial) one, which has skewed (by one rotor tooth pitch: sk = 11:25) and closed slots, three
other rotors have been manufactured. Table 1 gives the
particular features of the four rotors.
rotor 1 skewed closed
rotor 2 skewed open
rotor 3 unskewed closed
rotor 4 unskewed open
Table 1 Rotor slot features
Simulation model Enforcing anti-periodicity conditions for both the magnetic eld in the FE model and
the electrical network voltages and currents, only one
pole is modelled. This naturally implies that geometric,
material and circuit asymmetries (e.g. eccentricity, magnetic anisotropy and a 'broken' rotor bar resp.) cannot
be considered.
The mesh used (3034 nodes, 5306 triangular elements 3
layers in the air-gap, the middle of which is remeshed for
every time step) and a ux plot at no-load (open slots)
are shown in Figures 1 and 2.
Figure 2 No-load ux plot
I8
ΩFE
I8
I1
I2
I3
I4
I5
I6
I7
I8
I8
Figure 3 Electrical network for the squirrel-cage
The electrical network for the stator windings (connected, single layer, 3 slots per phase belt, Nc = 9,
Nlc = 3) comprises three current loops, in each of which
a resistance and inductance are inserted in order to (approximately) account for the additional resistance and
leakage ux due to the end windings. The network for
the squirrel-cage (one pole,Nb = 8) is shown in Figure 3.
Seven current loops pass through two neighbouring bars
in opposite direction. The eighth loop, however, passes
through the rst and the eighth bar in the same direction, thus enforcing anti-periodicity for the voltages and
the currents. The end ring segments are each represented
by a resistance and inductance in series.
Measurements & simulation results Measurements
have been carried out at the rated voltage (220 V, 50Hz)
using the four rotors, at both no-load (slip < 0.0004) and
load. In the load case, a DC machine has been used to
set a load torque of about 20 Nm. This resulted in a slip
ranging from 4.6% to 4.85% (1427-1431 rpm) depending
on the rotor mounted. The corresponding no-load and
load simulations have been done assuming synchronous
A]
A]
time s]
rotor 1
A]
time s]
time s]
time s]
rotor 2
rotor 3
Figure 4 Stator phase current waveforms at no-load
full line: calculated dashed line: measured
A]
A]
time s]
rotor 1
A]
rotor 4
A]
A]
time s]
time s]
rotor 2
rotor 3
Figure 5 Stator phase current waveforms at load
full line: calculated dashed line: measured
speed and 1430 rpm respectively. Skewed slots (rotors 1
and 2) have been modelled using 5 or 10 slices.
The calculated and measured waveforms of the stator
phase current are shown in Figures 5 and 6. A satisfactory agreement between the measured and computed
waveforms is observed for all cases. In case of straight
rotor bars (rotors 3 and 4), the large slot harmonics are
quite well predicted. The reduction of these harmonics
as a result of skewing and closing of the rotor slots is also
reasonnably well calculated.
Using ten instead of ve slices has a negligible eect
on the waveforms. The computation cost per NewtonRaphson iteration and the estimated storage requirements for 1, 5 and 10 slices are presented in Table 2.
The axial variation of the saturation level (50 Hz component of the ux-density in a stator tooth) and of the
(time averaged) torque is depicted in Figures 6 and 7
respectively (load, open or closed rotor slots, Nsl =1 or
10). In case of skewed slots, a considerable increase of
the saturation and of the torque towards the leading slice
(the right side in Figures 6 and 7) is observed. This is
readily explained by the axial variation of the spatial angle
between the (fundamental) stator and rotor currents.
In all no-load cases the saturation is virtually constant
along the axis (1.82T).
The computed torque waveform for each slice (Nsl = 5)
and the average torque for rotor 1 at load are shown in
Figure 8.
time s]
rotor 4
Figure 6 Saturation level along the machine axis
: open slots 2 : closed slots
dashed line: Nsl = 1 full line: Nsl = 10
Figure 7 Torque variation along the machine axis
: open slots 2 : closed slots
dashed line: Nsl = 1 full line: Nsl = 10
Nsl = 1 Nsl = 5 Nsl = 10
computation time s]
4.1
21
43
storage requirements
1.00
1.28
1.68
Table 2 Computation time per NR-iteration
(CPU seconds on DEC AXP 3000/500)
and relative storage requirements
Figure 8 Calculated torque (load, rotor1, Nsl =5)
broken lines: individual slices
full line: average
Conclusion
For modelling electrical machines with skewed slots, the
Multi-Slice Model is shown to be a viable alternative to
a fully 3D model.
The direct and the indirect coupling methods for time
stepping a 2D FE electrical machine model coupled to
a electrical network have been studied and compared,
for both SSM and MSM. Using the indirect coupling
method, computation time clearly is proportional to the
number of slices considered. As in the direct coupling
method all dierential equations are coupled and have to
be time stepped simultaneously, computation time can
be expected to increase exponentially with Nsl . The authors have however devised a numerical technique that
equally results in a proportional computation time.
This technique has been applied to the no-load and load
simulation of a squirrel-cage induction motor. Four different rotors (with unskewed or skewed slots and with
open or closed slots) to be mounted in the same stator
have been considered. A satisfactory agreement between
measured and computed stator phase current waveforms
has been obtained for all cases. The signicant axial
variation of saturation and torque is evidenced.
Acknowledgement
The research is carried out in the frame of the InterUniversity Attraction Poles for fundamental research
funded by the Belgian State. The second author is a
research assistant with the Belgian National Fund for Scientic Research (N.F.W.O.). The authors also greatfully
acknowledge Brook Hansen (Hudderseld, U.K.) for supplying the motor and for constructing the special rotors.
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