Journal of Shanghai University (English Edition )
I S S N 1007-6417, Vol.4, N o . 2 ( J u n . 2 0 0 0 ) , pp 133--139
Direct Modeling of Induction Motors with Skewed Rotor Slots
Using 2-D Multi-Slice Model and Time Stepping FEM
F U Wei-nong,
JIANG Jian-zhong
School of Electromechanical Engineering and Automation, Shanghai University, Shanghai 200072, China
The geometrical feature of the skewed rotor slots in induction motors makes the 2-dimensional (2-D) finite element method
(FEM) not directly applicable. Based on the multi-slice model in this paper, a time stepping 2-D eddy-current FEM is described to study
the steady-state operation and the starting process of induction machines with skewed rotor slots. The fields of the multi-slices are solved in
parallel, and thus the effects of skewed slots and eddy-current can be taken into account directly. The basic formulas for the multi-slice
model are derived. Special technique to reduce computation time in solving the coupled system equations is also described. The results obtained by using the program developed have very good correlation with the test data.
Abstract
Key words i.~ductionmotors, finite element method
1
Introduction
its axial length produces variations in the magnetic
field. Note, however, that,
compared with the 3-D
The traditional method to study the performances of
F E M , the 2-D FEM has the advantages of simple mesh
induction motors is based on the concept of equivalent
generation, short computing time and small computer
circuit.
The parameters in the equivalent circuit are
storage. So in practical applications it is highly desirable
usually obtained by magnetic circuit computation. In re-
that a 2-D model can be applied to motors with skewed
cent years it has become practical to use finite element
rotor slots.
methods ( F E M )
with 2-dimensional ( 2 - D ) magneto-
Williamson pointed out that one possible technique is
static model and 2-D eddy current FEM in frequency do-
to represent a motor with skewed rotor slots by a 2-D
mains to estimate the parameters [1'2]. However, the
multi-slice model [51. A set of 2-D models for non-
precision of this method is limited by the concept of the
skewed rotor slots,
equivalent circuit. The following effects are difficult to
taken at different position along the axis of the ma-
be included:
chine, is used to model the skewed rotor slots. In order
(1) The rotation of the rotor;
each corresponding to a section
to ensure that the currents flowing in the bars of one
(2) The non-sinusoidal quantities;
slice are the same as those flowing in the bars of every
(3) The end windings and field;
other slice of the same rotor bars, it will be necessary to
(4) The skewed rotor slots.
find the field solutions in parallel. This will pose signifi-
The first two problems have been successfully solved
cant difficulties in software programming.
with 2-D time stepping FEM which is developed quickly
Another method is to use circuit models instead of
in recent years. Even the external circuit equations are
eddy-current models, in which the slice models may be
mutually coupled, the third problem can also be consi-
solved separately, rather than simultaneously. The dis-
dered with certain accuracy to small motors, although 3D FEM is still needed to large motors [3'4l .
advantage is that the eddy-current effect is not included
The fourth problem will present an enormously diffi-
which is especially important in the starting process,
cult problem if 2-D FEM is used, because the change in
orientation of the stator with respect to the rotor along
in the field solution [6] . That is to say, the skin effect,
cannot be taken into account directly.
This paper presents the authors' experience in solving
eddy-current muhi-slice model in parallel. It is an excel-
Received May. 11, 1999; RevisedDec.8, 1999
lent 2-D model for induction machines with skewed rotor slots. The skew effect, the skin effect, the satura-
Journal of Shanghai University
134
tion, the rotor rotation and the non-sinusoidal quantities
E (~)
=
l (re) 9A (k)
at '
_
can all be included directly in the system equations.
When the applied terminal voltage is known, the currents, torque,
etc. , can be computed directly. In this
where
(3)
l (k) is the axial length of the k-th slice. The total
current density j(k) in a conductor of the k-th slice is
paper the basic formulas for multi-slice model are derived. Special techniques needed in the mesh generation
= l(k)
a (-
j(~)
and the software programming for salient structure are
u (k) + E(k))
a ((k)
also described. The CPU time needed and the compu-
-
-
IG )
l(k) aA(k))
u
+
at
(4)
'
ting accuracy are compared when different number of
slices are used. The test results of an 11 kW skewed
cage induction motors are used to verify the computed
where u (k) is the potential difference between the two
ends of a conductor in the k-th slice and a is the conductivity of the material.
results.
Integrating the current density j(k) over the cross-
2
B a s i c E q u a t i o n s of M u l t i - S l i c e M o d e l
section of the conductor gives
It is first assumed that there are no leakage fluxes in
the outer surface of the stator core and in the inner sur-
_
i (k)
__(su(k) + J l(~) aA(~)
)
a t dg2 ,
G
(5)
l?~-)
a2(k)
face of the rotor core. The stator outside circuit and the
effect of the rotor end rings are considered by coupling
where i(k) is the total current in the conductor, s =
da2 is the cross-sectional area of the rotor bar.
the electrical circuits into the FEM equations. The leakage inductances at the end regions of the stator winding
and at the end rings of the rotor cage are obtained by
analytical methods.
By putting l(k) in Eq. (5) to the left hand side, and
letting k = 1,2 . . . . . M ( M is the number of slices) and
to group all these M equations together, one has
The motor in the axial direction is considered as comM
posing of multi-slices [6] . In each slice the magnetic vec-
N~ l(,,,) i(m)
tor potential has an axial component only. The magnetic
m = I
M
fields are present in planes normal to the machine axis.
Hence the characteristic of the electromagnetic field of
each slice is 2-dimensional. The electrical relationships
M
--G(SEU(m)
+ , ~ ff l(rn) 3n(m,daQ)ot
m=l
=
as those flowing in the bars of every other slice.
i = i(l)=
i(2)
i(M)
.....
(7)
Replacing i (m) in Eq. (6) with i, one obtains
The Maxwell's equations applied to the domain under
i =
investigation will give rise to the following equation:
V x (v V x A ) : - ] ,
(6)
According to the assumption the rotor bar current is
between the various slices are based on the principle that
the currents flowing in the bars of one slice are the same
.
~(m)
~
Su +
M
=
(1)
l ('') 3A(m)dg]
3t
)
(8)
g~(,n)
M
where A is the magnetic vector potential having only
the axial component; v is the retuctivity of the material.
2.1
In air-gap and iron core domains
m-I
M
u = ~ u (m) is the potential differences between the two
m-1
Because the iron cores are laminated, the eddy currents in the iron cores are neglected in the mathematical
model. Therefore, in the air-gap and the iron core domain,
ends of a bar.
Because the items on the right hand side of Eqs. (5)
and (8) are equal, one has
u (k'
j = 0.
2.2
where l = x~ l (m) is the axial length of the bar and
-
l(k)
l
-u
+
l(k)~Z~_l~f
lS
(2)
In rotor conductor domain
l~fl(k) aA(k)
at
l(m)
=
OA(
at m)dX2
-
da2.
The induced electromotive force between the two ends
of a conductor in the k-th slice is
Substituting Eq. (9) into Eq. ( 4 ) , one has
(9)
Vol. 4 No. 2 Jun. 2000
FU W. N. : Direct Modeling of Induction Motors with Skewed Rotor Slots Using'"
electromotive force in one phase is therefore
a~ +
j(k) = -- ( 1 u + a aA(k)
o" '@ ff .(,,,)aA ('')
"+Z-'~-IJm>l- m
at
M
a
612-
8 A (k>
~da).
ff
S1m~--~jl(m)[
:l
E-
f aA d12 +
Substituting Eq. (10) into Eq. (1) and noticing that
j(k), A(k) and 12(~) can be written simply as j , A and
Y], one obtains the basic electromagnetic field equation
w
""ff
s(m)
w+2
1
)
(m)
S2w
M
- - 1~=1l('~)(~
aWA d f 2 _
a(+~
~f a8 tA d 1 2 // , ( 1 5
a (~
)
Tu + Vi-t +
S f fa ~O- Ad£2 "
where 12(+m) = S~,,) + s(m) + ... + s(m) and 12(Y) =
(11)
Another relationship between i and u can be obtained
by the circuit equations for the rotor end windings [2] .
Because all rotor bars are connected with rotor rings,
the relationship must be expressed in the matrix equation:
= [Z][/],
(12)
where [ u ] and [ i ] are column matrices, each element
means the voltage difference and the current of the rotor
bar, respectively; [ Z ] is the impedance matrix of the
end network of the rotor cage.
Eqs. ( 8 ) , (11) and (12) will give rise to the basic
formulas in the rotor bar domain. The unknown potential difference u can then be eliminated first.
2.3
dAd12 ]
jj
aa
'::r ~,~_~;
/S
:,a(,, , , ('+)OA('',
at d a -
[u]
ff
"+
(fsf+ aAd12~ + .II__ aAdg2
in the rotor conductor domain as:
a
dAd12
&
+
S(I 'n)
(10)
Vx(vqxA)=
135
In stator conductor domain
S(m)
+ c(m)
+ ... + e(m)
w+l
'Jw+2
~'2w
coils are S~ + l S w + 2 . . . . .
S 2 w , and the total crosssectional area of one turn at one side is equal to S. The
circuit equations are
Substituting Eq. ( 1 5 ) into
Eq. ( 1 4 ) , and noticing that R = 2wl/aS and the total
conductor area of one phase is 12( m) = ~ (+m) + 1-2(m) =
2wS,
one obtains
G
2wl
[Su + ~ l ( m ) ( ~ a Ad12~ - ~f a Ad12]l
.... ,
f2(m)
+
D(m)
,.
(16)
Notice that j = i/S. The electromagnetic field equation in Eq. (1) becomes
2wl
[u + ~=ll(m)(ff
-
Because the stator winding consists of fine wires, the
current density j in the stator conductor is assumed uniform. Assuming that the stator winding of one phase
consists of w turns in series, the cross-sectional areas of
each conductor at one side of the coils are $1, $ 2 , . . . ,
S ~ , the areas of each conductor at the other side of the
•
D(÷m)
aO~tAd12_ ff Vf-t
a a d121/]"
t2(m)
(17)
Substituting Eq. (13) into Eq. ( 1 6 ) , one has
i---2w/
a
[
(
S us + Rfi + L ~
ff
di)
+
d12- f f a A d a
(is)
,
di
u, = u - R~i - L~ -~,
u = E - Ri,
(13)
(14)
where Us is the applied voltage; i is the phase current;
R, and L, are resistance and inductance at the end of
winding, respectively; u is the total potential difference
of each element in the stator winding; R is the resistance of each element in the stator. The total induced
Eqs. ( 1 7 ) and ( 1 8 ) constitute the basic formulas
governing the stator conductor domain.
By using finite element formulation and coupling the
electromagnetic field equations together with the electrical circuit equations, one obtains the following large
nonlinear system of equations:
i
+ [Q
R]
0i
= [P],
(19)
136
Journal of Shanghai University
where the unknowns [ A ] and [ i ] are the axial components of magnetic vector potentials and the currents, respectively, that are required to be evaluated. [ K ] ,
[ C 1, [ Q 1, [ R ] are the coefficient matrices and [ P l is
the column matrix associated with the input voltages.
The mechanical equation of the motor is also required
to be coupled,
dZ0
Jm dt 2 -
T e - Tf = T,
(20)
where Jm is the moment of inertia; 0 is the angular position of the rotor; Te is the electromagnetic torque,
and Tf is the load torque. The Maxwell stress tensor is
used for electromagnetic torque computation [z] . Because
the magnetic flux density B of the air gap in the elements contacted with iron is liable to low accuracy, it
will give more reliable results if only the B in the middle
elements is used to compute the torque. Therefore, the
air gap is divided into three layers. The middle layer is
chosen as an integration cross sectional area (integration
ring). The electromagnetic torque can be obtained by
moved with the rotation of the rotor. The position of
the rotor mesh is determined by Ok, [ Kk ], [ Ck 1,
[ Qk 1, [Rk ] will change with the rotation of the rotor
mesh. In the iteration process for the solution of the
equation coupling Eqs. ( 2 2 ) , (23) and ( 2 4 ) , the rotor
mesh will be changed repeatedly. This will certainly
give rise to difficulties in the programming.
The proposed method is, instead of using Eq. ( 2 2 ) ,
using the Euler's method to obtain an initial guess of wk
as follows:
o~(ko)
Tk-a
= 6Ok_1 -[- T A t .
(25)
During the process in solving Eq. ( 2 4 ) , the rotor
mesh is fixed. After the solution of Eq. ( 2 4 ) , Tk can be
obtained, wk can be computed again by using the Backward Euler's method according to Eq. ( 2 2 ) . The difference between cok(°) and cok indicates the discretization
nents of the flux density.
error which is dependent on the At step size [7]. If this
error is larger than allowed, the step size At will be reduced automatically.
At each time step, the iterative solver is most suitable
in solving the system of large equations. This is because
the good initial value, which can be obtained easily from
the result of the previous step, can be used to reduce the
number of iterations. In this paper the Newton-Raphson
method coupled with the incomplete Cholesky-conjugate
gradient algorithm is used to solve the system of large
nonlinear equations.
3
4
T~
=
/10(rs
1
rr)%
frB~B~ds,
(21)
where/~0 is the permeability in vacuum; rs and rr are
the outer and the inner radii of the integration ring respectively, and Sag is the cross-sectional area of the integration ring; Br and B~ are the r- and the p-compo-
Solution of System of Equations
For time stepping process in F E M , the Backward Euler's method is used. In the steady-state problems, the
complex FEM model is solved first to give an initial
guess of A (0) and i ( 0 ) .
The Backward E u l e r ' s method is used to discretize
the time variable. If the solution at the ( k - 1 ) - t h step is
known, then at the k-th step one has
cok = ~'k-x + f ~ At,
(22)
Ok = Ok-1 + wkz~t,
(23)
Q~ Ck + ~R-~] [ Aikk l =
[Kk + ~
[~k AtRkJ[A'-lJ+
ik_l
[P,]
(24)
where o) is the rotor speed. The rotor FEM mesh is
Results
The presented method has been used to simulate the
operation of an 11 kW, skewed rotor cage induction motor. The details of the motor is shown in Table 1. The
stator of the motor is wounded with full-pitch singlelayer windings.
4.1
Load operation
The programs are run on a personal computer Pentium/150 MHz. The size of each time step is 0. 039 ms.
The computed and measured stator phase currents are
shown in Fig. 1. The comparison of the CPU time of
the computation of one step, as well as the error between computed and measured stator currents when using different slice numbers is shown in Table 2.
Vol.4
No.2
Table 1
Jun.2000
FU W.N. :
Direct Modeling of Induction Motors with Skewed Rotor Slots U s i n g ' "
Main parameters of tested motor
Table 2
137
Comparison of C P U time and error using different slice
numbers
Rated power (kW)
11
Rated voltage (V)
380
Number of slices
Unknowns
CPU time(s)
Error( % )
1
2712
38
19.67
2
5399
76
7.77
3
8086
128
5.96
4
10773
209
5.26
Rated current (A)
22.61
Rated frequency (Hz)
50
Connection
A
Number of phases
3
Number of pole pairs
2
Number of stator slots
48
5
13460
313
5.17
6
16147
446
5.12
Number of rotor slots
44
Stator diameter (ram)
240
Rotor diameter (ram)
157
Air-gap length (ram)
0.5
Core length (mm)
165
Skew
1.3 stator slot pitch
Table 3
Comparison of currents and torques at locked-rotor test
Computed result
30
20
Phase current
Torque (Average)
(RMS) (A)
(N'm)
148.2
145.3
Test result of motor I
150.4
158.4
Test result of motor II
149.5
155.8
10
150
100
50
-10
-20
<-<- 0
-30
;
10
t/ms
1'5
20
-50
-100
-151
(a) Computed
fo
2b
3'o
4'O
5'o
4b
5b
t/ms
(a) current
20
600
10
0
400
-10
-20
-301
0
Z
i
5
,
10
t/ms
1'5
200
i
20
b~ -200
400
(b) Measured
Fig. 1
2'0
3'0
t/ms
Stator phase current waveform at load test
4.2
R o t o r locked operation
The comparison of the results between the computed
and the measured data for locked-rotor operation at
steady-state is shown in Table 3. If the results of the
complex FEM model are used as the initial data for the
time stepping F E M model, the computed stator currents
and electromagnetic torques for rotors with non-skewed
and skewed rotor-bars are shown in Figs. 2, 3, and 4,
respectively. Because the solutions are computed before
the currents reach their steady-states, there are very obvious torque ripples in the torque waveforms.
fo
(b) Torque
Fig. 2
Current and torque at locked-rotor test
( n o n skewed rotor-bars)
4.3
Starting process
The method has also been used to simulate the starting operation of the motors. The computed stator phase
current (phase A ) and the torque at the no-load condition are shown in Fig. 5, while the measured current
(phase A ) is shown in Fig. 6. The applied voltages are
va = V m c o s (wt - ~ r / 3 ) ,
vn = V,~cos (cot - ~r/3 - 2~r/3),
vC
= Vmcos (:o~ot - re/3 - 47r/3),
Journal of Shanghai University
138
150
100
50
150
100
-50
-100
-150
AAAAAAA
VVVVVvv'
50
40
g0
-50
10
20
30
40
-100
5'0
t/ms
-150
~o
16o
(a) Current
'~0
3'0
4'0
b~
io
0
"1000
50
100
Fig. 5
( s k e w e d rotor-bars with 1 . 3 stator slot pitch)
20
30
40
(b) Torque
Computed current and torque at starting process
(skewed rotor-bars with 1 . 3 stator slot)
<%:
10
150
t/ms
Current and torque at locked-rotor test
-100
-150
250
~_~
oa 200100
(b) Torque
-50
200
3oo
t/ms
0
2'50
400 -
1'0
150
100
50
25o
(a) Stator phase current
300
~ . 250
200
150
100
0
50
Fig. 3
1~o
t/ms
150
100
-100
IAAAAAAA
VVVVVVVV
-150
50
50
5'0
t/ms
160
150
200
230
t/ms
(a) Current
Fig. 6
250 [
5
Measured stator phase current at starting process
Conclusions
150
100
50
00
10
20
30
40
#0
t/ms
(b) Torque
Fig. 4
Current and torque at locked-rotor test
(skewed rotor-bars with 1 . 0 stator slot pitch)
where Vm = 380 r ~ V. The result of computed current
using the proposed method shows very good correlation
with the test data.
The computed currents and torques when the rotor is
non-skewed and skewed 1 stator slot are shown in Fig.
7 and Fig. 8, respectively. The computed magnetic
flux distributions during starting process are shown in
Fig. 9.
The skewed geometry of the rotor slots in induction
motors has great influences upon the performances of
the machines. The multi-slice model for skewed rotor
slot motors, being solved in parallel using time stepping
2-D eddy current F E M , can take into account the skew
effect, the eddy-current effect, the rotor rotation and
the non-sinusoidal quantities directly in the system equations. It is an excellent 2-D model to solve an essentially
3-D problem. The proposed method thus has a significant contribution in applying time stepping FEM models
to study practical electrical machines with skewed slots.
Vol. 4
No. 2
Jun. 2000
F U W . N. :
Direct Modeling of Induction Motors with Skewed Rotor Slots U s i n g ' "
139
IOO
°hAAAAAAAAA
'
A
50
---,ojVV!vvvvvv
0
A^,
:11~00
5t
160
150
200
250
t/ms
(a) Current
(a) t = 0.800 ms
800
600
b".200
-400
-6000
5'0
100
150
t/ms
200
250
(b) Torque
Fig. 7
Computed current and torque at starting process
Fig. 9
(b) t =4.609 ms
Computed flux distributions during starting process
(non-skewed rotor-bars)
References
150
100
50
--< o
[I]
IAAAAAAAAAA
-50
-100
VVVVVVVVVVV
V
-150
I
I
0
100
I
150
t/ms
[2]
I
250
[3]
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[5 ]
If
"
I
I
I
I
100
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t/ms
[6]
(b) Torque
Fig. 8
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