XXIV Symposium
Electromagnetic Phenomena in Nonlinear Circuits
June 28 - July 1, 2016 Helsinki, FINLAND
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3D FINITE ELEMENT COMPUTATION OF ELECTROMAGNETIC FORCES
AND STRESS ANALYSIS ON MOTOR END WINDING
Sabin Sathyan, Anouar Belahcen, Juhani Kataja*
Aalto University, Dept. of Electrical Engineering and Automation
PO Box 13000, FIN-00076, Aalto, Finland, e-mail: [email protected], [email protected]
*CSC-IT Center for Science, PO Box 405, FIN-02101, Espoo, Finland, email: [email protected]
Abstract – A high voltage induction motor generates high
magnitude currents during starting transient mode and in case
of faults. These large currents produce excessive electromagnetic
forces on end windings and induce stresses and vibrations. This
paper presents an efficient way to compute the electromagnetic
forces on end windings by 3D Finite Element Method (FEM)
using Virtual Work (VW) principle. Besides, the computed
forces are used for stress and vibration analysis by means of a
Magneto-elastic coupling in an open source software Elmer. The
force and stress distribution on end windings are imperious
factors in the design process and normal and faulty operation
analysis of motors.
I. INTRODUCTION
The electromagnetic forces acting on the end windings of
electrical motors and generators are significant factors in
machine design and analysis. High currents during starting
transients and faults can induce excessive amount of forces
on the end windings, and hence, for the design of insulation
and supporting structures, knowledge of stresses and
vibrations due to magnetic forces is needed. In this study,
the magnetic forces are calculated by local application of the
Virtual Work principle to the nodes of a 3D finite element
mesh [1]-[4]. The stress and displacements are analysed by
dint of a direct coupling of the magnetic forces to the
mechanical solver.
Biot-Savart law describes the magnetic field generated by
an electric current by relating the magnetic field to the
magnitude, direction, length and proximity of electric
current. This law is utilized for the force calculation in [5]
and [6]. The magnetic forces produced on end windings due
to current flow can be categorized as Lorentz forces [7]. This
same principle is followed in [8]. In this paper, the magnetic
forces at the nodal level are precisely calculated by
differentiating the magnetic energy with respect to virtual
displacement. This method enables the use of the calculated
nodal forces for further mechanical analysis for vibration
and stress computations.
II. COMPUTATIONAL METHODOLOGY
The virtual work method for force computation is
implemented in the open source software Elmer [10] and the
elasticity solver of the same package is coupled to the
magnetic solution.
The nodal magnetic forces Fe can be calculated by
differentiating the magnetic energy Ws with respect to
virtual displacement 𝝓𝒔 depending on a real parameter s so
that lim 𝝓𝑠 (𝒙) → 𝒙.
𝑠→0
𝜕𝑾
𝒔
The derivative
is obtained utilizing change of
𝜕𝑠
variables and the Piola transformation for the magnetic flux
density as follows:
𝜕𝑾𝒔
𝜕
= ∫ 𝑯(𝑩) ∙ 𝑱𝒔 ∙ 𝑩 −
𝜕𝑠
𝜕𝑠
𝑩
𝜕 det 𝑱𝒔
(𝑯(𝑩) ∙ 𝑩 − ∫ 𝑯 ∙ 𝑑𝑩) 𝑑𝑥
𝜕𝑠
(1)
where 𝑱𝒔 is the Jacobian of 𝝓𝒔 . Choosing 𝝓 (𝒙) = 𝒙 +
𝑠𝒖𝑁(𝒙), where 𝒖 is a unit vector and 𝑵(𝒙) is a nodal shape
function defined on the computational mesh, it holds that
𝜕𝑾𝒔
= ∫ 𝑯(𝑩) ∙ 𝒖 ∇𝑵 ∙ 𝑩 − 𝒖
𝜕𝑠
𝐵
∙ ∇𝑵 (𝑯(𝐵) ∙ 𝑩 − ∫ 𝑯 ∙ 𝑑𝑩) 𝑑𝑥
(2)
The displacement due to forces can be analyzed as linear
elasticity problem using Navier equations. The dynamic
equations for elastic deformation of solids can be expressed
as
𝜌
𝜕2𝒅
− ∇. 𝜏 = ⃗𝒇
𝜕𝑡 2
(3)
where ρ is the mass density, d is displacement field, f given
volume force and 𝜏 the stress tensor. The stress tensor is:
𝜏 𝒊𝒋 = 𝐶 𝑖𝑗𝑘𝑙 𝜀𝑘𝑙 − 𝛽 𝑖𝑗 (𝑇 − 𝑇0 )
(4)
where ε is the strain, C is the elastic modulus, 𝛽 is the heat
expansion tensor and the reference temperature of the stress
free state is T0 .
III. FEM ANALYSIS RESULTS
The magnetic solutions from the finite element simulation
gives the distribution of flux density, volume current and
nodal magnetic forces. The specifications of the induction
motor is given in Table I. Fig. 1 shows the magnetic flux
density and force distribution on the end winding. The
simulation corresponds to a state where there is high current
flow in the windings during the starting mode. The results
shown here are from a static FE simulation, where the
currents are imposed as a body force in the winding.
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The magneto-elastic model is simulated hierarchically,
assuming that the coupling is weak, first solving the
magnetic system and then providing the elasticity solver
with the magnetic nodal forces as the volume force on the
right side of (3).
The FE software Elmer has the advantage of mesh
partitioning and the simulations can be carried out using
efficient parallel computation [9].
TABLE I
SPECIFICATIONS OF ANALYSED MODEL
Parameter
Rated Power
Rated Voltage
Number of phases
Connection type
Number of poles
Number of stator slots
Number of rotor slots
Rated current
Value
770 kW
6.6 kV
3
star
2
42
34
138 A
From the stress distribution pattern in the winding, one
can compare the magnitudes of the maximum stress with the
yield stress of copper, insulations and supporting materials.
This data is decisive in the mechanical design including
insulation and supporting rings. On the other hand, the
amplitude of vibrations are very important in the design of
retaining structure.
The machine end surfaces comprising of laminated core
and rotor are not included in the presented simulation. The
error due to the neglect of end surfaces are generally ±10 %
[11]. Furthermore, the rotor currents can markedly influence
the end winding forces [5]. Especially, the radial component
of the forces due to rotor currents near the core can be as
large as 50 % of that due to stator currents [11]. Besides, for
different transient simulations compared to static case, the
force calculation results will be different and the force
distribution can be very complicated [12].
III. CONCLUSIONS
The proposed method and study facilitates an effective
and clear-cut way to compute the electromagnetic forces in
end windings using 3D FEM and for analysing the stresses
in case of faults and starting transients. The knowledge of
distribution of electromagnetic forces from the magnetic
solution itself can give light on the possible vibrations and
stresses, which occur in the motor end windings.
REFERENCES
Fig.1. Distribution of flux density and nodal magnetic force vectors in the
end winding.
The magnetic forces in the inner layer are higher than that
in the outer layer. The forces on different part of the same
coil are different in magnitude. These forces produce
stresses, which in turn result in displacements of the
winding. The stress distribution and displacements are given
in Fig. 2.
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[10] https://www.csc.fi/web/elmer
[11] P. J. Lawrenson, “Forces on turbogenerator end windings” Proc. IEE,
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Fig.2. Stress distribution and displacement. The displacement is zoomed
by 200.
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Proceedings of EPNC 2016, June 28 - July 1, 2016 Helsinki, FINLAND