Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT
Study regarding end winding inductance of three phase A.C. windings
in a single layer
OLIVIAN CHIVER, LIVIU PETREAN, LIVIU NEAMT and ZOLTAN ERDEI
Electrical Engineering Department
North University of Baia Mare
Victor Babes 62/A, Baia Mare, Maramures
ROMANIA
olivian.chiver@ubm.ro http://www.ubm.ro
Abstract: - This paper presents a study on the end winding inductance of stator windings used in the threephase A.C. machines. It refers to the windings in a single layer with coils ends in two and three plans
respectively. End winding inductance is determined by numerical analysis based on finite elements method
(FEM). Based on the analysis of more than 80 models designed by the authors, representing the stator of some
asynchronous machines, a FEM-Analytic comparative analysis will be realized. Also, for one of the analyzed
model, model that represents the stator of an asynchronous machine from our laboratory, measurements are
made in order to compare the results.
Key-Words: - End Winding Inductance, FEM, Single Layer, A.C. Machines
3D form of coils ends, some of them taking in
consideration also the eddy currents, have been
presented in several papers [1], [2], [3], [4].
A more accurate determination of the end
winding magnetic field is possible using numerical
methods. One of these, most utilized lately, is finite
elements method (FEM) because it can solve
complicated structures with reasonable assumptions
and reliable results.
Some papers that deal with the end winding
magnetic field and the method of determining the
corresponding inductance based on FEM 3D can be
specified [2], [5], [6], [7], [8], [9], [10].
In order to determine experimentally the leakage
inductance of stator winding in case of A.C.
machines, the rotor is removed from the stator. Also
in case of simulations, the numerical model will be
realized without the rotor.
In order to separate the end winding inductance
from the total inductance, both 2D and 3D models
are realized for the same machine, and the difference
between 3D and 2D total inductance gives the end
winding inductance [6], [7], [8].
1 Introduction
The stator of the A.C. rotating electrical
machines presents a winding, generally three-phase,
which is the support for the currents that produce the
rotating magnetic field. Functionally, this winding
includes two different regions. In the active region
the useful energy is transferred between the stator
and the rotor, and it corresponds axially to the length
of the ferromagnetic material. In this region, the
coils sides are placed in the slots. Another region
represents the end winding and the coils sides are
placed in air and through them flow the same
currents as in the active region. In the active region,
rotating magnetic field can be considered plane,
while in the end winding region rotating magnetic
field is three-dimensional.
The magnetic field produced by the coils ends
represents a leakage field, the corresponding
inductance being named end winding inductance.
The determination of this inductance as accurate as
possible is very important because it influences the
starting current and starting torque.
In the design phase, the leakage inductances in
the active region are determined analytically with a
satisfactory accuracy, only end winding inductance
is determined with less accuracy. This error is due to
the complicated form of the flux lines in the end
winding region, which determines higher
discrepancies between the simplified model used to
determine the analytical relations and the real form
of the magnetic field.
Improved analytical methods for end winding
inductance determination, taking into account the
ISSN: 1790-5095
2 FEM-Analytic comparative analysis
In order to realize a comparative analysis between
FEM and analytical values of the end winding
inductance, a software developed by the authors for
computer aided design of asynchronous machines
has been used. The software has been realized in
Visual Basic, because this program is recognized by
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ISBN: 978-960-474-181-6
Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT
the MagNet 6.25, the tool used for magnetic field
computation.
The software has been created to allow the design
computation of the asynchronous machines, and
realizes the required 2D and 3D numerical models.
The numerical models are realized on the basis of
data that results from the design computation or, in
case of an existing machine, on the basis of user’s
data.
Fig.1. Model with end winding in two plans
2.1 Numerical simulations
The initially realized models corresponds to a
half of the machine, the stator winding having got
the coils ends in two and three plans (Fig. 1 and Fig.
2). The numerical model also includes the carcass
and an air volume that surrounds the machine and
has got the length higher than the coils ends.
In order to decrease the necessary time for
analysis, from the initial model only the part
corresponding to a single pole has been selected. On
the two radial faces “Odd periodic” boundary
condition has been imposed (Fig. 3). On all other
boundaries “Flux tangential” condition has been
imposed.
2.2
Fig.2. Model with end winding in three plans
Analytical computation of end winding
inductance
In the design phase, the end winding inductance
has been determined using relation (1) for coils ends
in two plans and (2) for coils ends in three plans
respectively [11].
N2
L f = 1.34 µ 0
(l f − 0.64τ )
(1)
p
N2
L f = 0.94 µ 0
(l f − 0.64τ )
(2)
p
where L f is the end winding inductance, µ 0 is the
Fig.3. 3D numerical model with air volume and
“Odd Periodic” boundary conditions
air magnetic permeability, N is the phase number of
turns, p is the pole pairs number, l f is the end coil
length and τ represents the polar pitch.
For each designed model, the end coil length has
been determined in the same time with the 3D
numerical model, in terms of the spatial coordinates.
Fig.4. “A” parameter
determined both analytically and FEM. Finally the
ratio of these values was computed and graphically
represented in terms of distance “A” from the
stator’s yoke to the plan where the formation of
frontal ends begins (Fig. 4). In case of 5.5 kW
machine, with coil ends in two plans the variation of
this ratio is shown in Fig. 5.
2.3 The results of the analysis
2D and 3D numerical models have been carried
out for more than 80 asynchronous machines. For
these machines end winding inductance has been
ISSN: 1790-5095
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ISBN: 978-960-474-181-6
Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT
In the follows some aspects will be mentioned. In
case of a machine with a certain power, if the stator
winding is carried out for different values of
distance “A” and all other geometrical parameter of
coils ends are unchanged, the analytical value of end
winding inductance differs from FEM value as much
as the distance is smaller.
Also, for the same value of “A”, in case of higher
power machines the difference between FEM and
Analytical values is more important, while for the
same power the difference is higher in case of
several poles machine.
The explanation of these results is the following:
analytical value of end winding inductance is a
linear function of parameter “A”, while the FEM
value is not linear. For the smaller values of distance
“A” the influence of ferromagnetic stator material
on the end winding inductance is more important.
This fact is not reflected in analytical relations.
However can be noticed that analytical results
differs from FEM results up to 30-35% only for
values of parameter “A” smaller than 15 mm in case
of higher power machines. For the values of “A”
used in practice (20-50 mm) in correlation with
power of the machine, the average difference
between FEM and analytical values is generally
smaller than 10%, only in a few cases being up to
15%.
Fig.5. Variation of Lf[FEM]/Lf[Analytic] ratio for
5.5 kW machine, coils ends in two plans
Fig.6. Variation of Lf[FEM]/Lf[Analytic] ratio for
15 kW machine, coils ends in three plans
3 Practical measurements
Measurements have been carried out for a stator
of an asynchronous three-phase machine, the main
data are presented in table 1 and the machine is
shown in Fig. 9.
In the rotor space a control coil has been placed
(Inner coil in Fig. 9.), the coil span being equal with
the polar pitch. The active sides of the coil are
placed above the corresponding slots. The coil ends
are connected to a multimeter to measure the rms
value of the induced voltage.
Fig.7. Variation of Lf[FEM]/Lf[Analytic] ratio for
the models with coils ends in two plans
Table 1 – Main data of test machine
Parameter
Value
Rated power [kW]
0.37
Phase voltage [V]
230
Stator core length [mm]
75
Outer diameter of stator [mm] 106.5
Inner diameter of stator [mm] 70
Pole pairs number
2
Stator slots number
36
Tooth width [mm]
2.75
Number of turns in a coil
133
Number of wires in parallel
1
Fig.8. Variation of Lf[FEM]/Lf[Analytic] ratio for
the models with coils ends in three plans
In case of 15 kW machine with coils ends in
three plans the variation of the same ratio is shown
in Fig. 6.
For all analyzed models with coils ends in two
plans and in three plans respectively, the variation of
Lf[FEM]/Lf[Analytic] average ratio in terms of “A”
is shown in Fig.7 and Fig.8 respectively.
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ISBN: 978-960-474-181-6
Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT
To carry out the simulations, 2D and 3D
numerical models have been made. 2D
magnetostatic analysis allows finding out the
magnetic vector potential distribution on the inner
diameter of the stator. This distribution is shown in
Fig.11 in case of phase current amplitude of 0.862A.
Fig.9. – The tested machine and two search coils
Fig.11. Magnetic vector potential distribution of the
stator inner diameter, Imax=0,862 A
Fig.10. – Induced voltage as a function of the stator
phase current
Using a personal developed MATLAB program,
the fundamental component of the magnetic vector
potential A1 has been obtained and then Lb
inductance:
1
N
(5)
Lb = 2
A1lk w
Nb
2I
The measured values are shown in table 2, where
Iphase represents the average value of the currents,
Uphase is the average value of the phases’ voltages,
Ub and Uend are the induced voltage in the “Inner
coil” and the “End coil” respectively and P is the
absorbed power.
The analytical, FEM and measured values are
presented in table 3.
Another coil (End coil in Fig. 9.) is a closed loop
along a coil end and near to the end of the stator.
The voltages induced in the two coils have been
measured for different values of stator currents
(Fig.10). It can be noticed that removing the rotor
from inside the stator determines almost a linear
behavior of the stator yoke in case of currents that
do not exceed to much the nominal value.
In terms of the induced voltage in the “Inner
coil” the inductance corresponding to the useful
magnetic flux from the rotor space has been
determined [12]:
1 U b Nk w
Lb =
(3)
ω I Nb
Table 2 - Measured values
Iphase[A] 0.61 0.507 0.393
Uphase [V] 38
31.3 24.47
Ub [mV] 22.7 18.7 14.7
Uend[mV] 3.6
3.1
2.5
P [W]
45.5 31.5 19.5
where ω is the angular frequency, U b is the induced
voltage in the “Inner coil”, k w is the winding factor,
I is the phase current and N b is the turns number of
the “Inner coil”.
Analytically, Lb inductance can be determined
with relation [12]:
τ
Table 3 – Inductances’ values
Inductance
Measured
FEM Analytic
Lt [mH]
145.89
142
Lb [mH]
45.92
46.3 58.8
Lσ [mH]
99.97
95.7 Lf [mH]
33.11 26
15 N 2 k w2 f (l + )
1
6 10 −8 [ H ] (4)
Lb =
ω
p
where l is the stator length [cm], τ [cm] and f is the
current frequency.
The total leakage inductance represents the
difference between total inductance per phase and Lb
inductance.
ISSN: 1790-5095
0.237
15.33
9.2
1.8
8
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ISBN: 978-960-474-181-6
Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT
[2] Hsieh M. F., Hsu Y. C., Dorrell D. G., and Hu
K. H., Investigation on end winding inductance
in motor stator windings, IEEE Transactions on
Magnetics, vol. 43, no. 6, June 2007;
[3] Schramm A. and Gerling D., Analytical
calculation of the end winding leakage
inductance based on the solution of Neumann
integral”, IEEE International Symposium on
Industrial Electronics (ISIE) 2005 Conference,
20-23 June 2005, Dubrovnik, Kroatien;
[4] Williamson S., Mueller M. A., Induction motor
end winding leakage reactance calculation using
the Biot-Savart method, taking rotor currents
into account, Proceedings of ICEM’90, Boston,
August 1990;
[5] Brahimi A. T., Foggia A., Meunier G., End
winding reactance computation using a 3D finite
element program, IEEE Transactions on
Magnetics, vol. 29, no. 2, March 1993;
[6] Chiver O., Micu E., Barz C., „Stator winding
leakage inductances determination using Finite
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on Optimization of Electrical and Electronic
Equipment OPTIM'08, Braşov, România, May
22-24, 2008;
[7] Cox T., Eastham F., Proverbs J., End turn
leakage reactance of concentrated modular
winding stators, IEEE Transactions on
Magnetics, vol. 44, no. 11, November 2008;
[8] Lin R., Arkkio A., Calculation and analysis of
stator end-winding leakage inductance of an
induction machine, IEEE Transactions on
Magnetics, vol. 45, no. 4, April 2009;
[9] Y.B. Li, S.L. Ho, W.N. Fu and W.Y. Liu, An
interpolative finite-element modeling and the
process simulation of a large solid pole
synchronous machine, IEEE Transactions on
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[10] Lin R., Haavisto A. and Arkkio A., Validation
of a time-harmonic numerical model for solving
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[11] Cioc I., Nica C., „Proiectarea maşinilor
electrice”, Ed. D. P., Bucureşti, 1994;
[12] Drăgănescu O. Gh., Încercările maşinilor
electrice rotative, Ed. Tehnică, Bucureşti, 1987;
In table 3 Lt is the total inductance per phase when
the rotor is removed and Lσ is the total leakage
inductance per phase.
If relation (3) is used to determine end winding
inductance in terms of the induced voltage in “End
coil”, obtained value is 29.9 mH. Using FEM this
value is 33.11 mH, and analytically – 26 mH. In this
case the FEM/Analytic ratio is 1.27. This value is
expected since, in case of this machine, the distance
“A” is 5mm.
The smaller value obtained on the basis of the
induced voltage in “End coil” is due to the fact that
not the entire magnetic flux produced by coils ends
is closed trough this coil. A part of this field is
closed trough the stator, and does not induce voltage
in the “End coil”. Thus the value of end winding
inductance is higher than that determined using
“End coil” induced voltage. Finally, the FEM value
can be considered more accurate than other values
referred to in this paper.
4 Conclusion
This paper presents a comparative analysis
regarding end winding inductance of a single layer
winding. Over 80 models were analyzed and the
FEM results have been compared with analytical
ones.
Also, for a 0.37 kW asynchronous machine from
our laboratory, measurements have been carried out
and again the values have been compared with FEM
and analytical values.
For Lt and Lσ inductances, the FEM values are
close to the measured values.
In conclusion, the FEM simulations allow the
determination of the end winding inductance with
enough accuracy. Also, in some cases, analytical
results can be less accurate regarding this
inductance.
References:
[1] Ban D. Zarko D., Mandic I., Turbogenerator
End Winding Leakage Inductance Calculation
Using a 3-D Analytical Approach Based on the
Solution of Neumann Integrals, Research Report,
Wisconsin Electric Machines & Power
Electronics Consortium, Iulie 2003;
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