End Winding Characterization .pdf
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CAD Package for Electromagnetic and Thermal
Analysis using Finite Elements
FLUX 10
®
3D Application
End winding characterization
Copyright - February 2008
FLUX is a registered trademark.
FLUX software
FLUX technical papers
:
:
COPYRIGHT CEDRAT/INPG/CNRS/EDF
COPYRIGHT CEDRAT
FLUX Quality Assessment
(Electricité de France, registered number AQMIL013)
This technical paper was edited on 7 February 2008
Ref.: K305-L-10-EN-02/08
CEDRAT
15 Chemin de Malacher - Inovallée
38246 Meylan Cedex
FRANCE
Phone: +33 (0)4 76 90 50 45
Fax: +33 (0)4 56 38 08 30
Email: cedrat@cedrat.com
Web: http://www.cedrat.com
Foreword
About the
technical paper
The aim of this technical paper is the computation of end windings
inductance. It contains the general steps and all data needed to describe the
model of asynchronous machine.
Required
knowledge
Before preceding this technical paper, the user must have under control all
functionalities of the Flux software. The user can gain the required knowledge
in the Generic tutorial, which explains in detail all actions to build the
geometry and mesh of the study domain.
Path
The files corresponding to the different cases studied in this technical paper
are available in the folder: …\DocExamples\Examples3D\EndWinding\
Command files
and Flux files
The files provided for this technical paper are:
• command files,
come in handy to build the Flux projects
• Flux files
already built project files
The use of files is explained in the table below.
the user can
To describe …
the 2D geometry
and mesh
the 3D geometry
and mesh
the physics
case 1
case 2
case 3
case 4
follow
execute the
command file
recover the Flux file*
§2
2DGeoMesh.py
2D_GEO_MESH.FLU
§3
3DGeoMeshPhys.py
3D_GEO_MESH_PHYS.FLU
§4
§5
§6
§7
3DCase1A_1B.py
3DCase2A_2B.py
3DCase3A_3B.py
3DCase4A_4B.py
3D_CASE1A.FLU, 3D_CASE1B.FLU
3D_CASE2A.FLU, 3D_CASE2B.FLU
3D_CASE3A.FLU, 3D_CASE3B.FLU
3D_CASE4A.FLU, 3D_CASE4B.FLU
the user can
To describe …
coil 1
coil 2
…
coil 12
follow
§ 3.3.7
execute the
command file
coil_1.py
coil_2.py
…
coil_12.py
recover the Flux file*
3D_GEO_MESH_PHYS.FLU
* Flux files are ready to be meshed and then solved.
FLUX®10
TABLE OF CONTENTS
TABLE OF CONTENTS
1. General information .................................................................................................................1
1.1.
1.2.
1.3.
Overview of the sample problem ..................................................................................................3
1.1.1. Foreword.........................................................................................................................4
1.1.2. Description of the device ................................................................................................6
1.1.3. Studied cases ...............................................................................................................10
Strategies of the geometry description, mesh generation and physical description...................11
1.2.1. Main phases for geometry description and mesh generation of the motor ..................12
1.2.2. Main phases for physical description of the motor .......................................................14
Computation of end windings inductance: principle & method ...................................................15
1.3.1. Inductance computation principle .................................................................................16
1.3.2. Operating mode with Flux 3D application.....................................................................19
1.3.3. Self inductance or cyclic inductance computation?......................................................21
1.3.4. Symmetry and periodicity .............................................................................................22
2. Motor 2D geometry and mesh ...............................................................................................23
2.1.
2.2.
2.3.
2.4.
Geometry and mesh of the 2D stator slot ...................................................................................25
2.1.1. Create geometric parameters and coordinate systems................................................26
2.1.2. Create points and basic lines .......................................................................................28
2.1.3. Create transformations .................................................................................................29
2.1.4. Create lines by propagation..........................................................................................30
2.1.5. Build faces ....................................................................................................................30
2.1.6. Create and assign mesh points ....................................................................................31
Geometry and mesh of the 2D rotor slot.....................................................................................33
2.2.1. Create geometric parameters and coordinate systems................................................34
2.2.2. Create points and basic lines .......................................................................................36
2.2.3. Create transformations .................................................................................................37
2.2.4. Create lines by propagation..........................................................................................38
2.2.5. Build faces ....................................................................................................................38
2.2.6. Create and assign mesh points ....................................................................................39
Geometry of the motor 2D model................................................................................................41
2.3.1. Import into a new project ..............................................................................................42
2.3.2. Create coordinate systems ...........................................................................................42
2.3.3. Create symmetries........................................................................................................43
2.3.4. Create transformations .................................................................................................43
2.3.5. Create lines and faces by propagation .........................................................................44
2.3.6. Add an infinite box ........................................................................................................46
2.3.7. Create points and lines .................................................................................................47
2.3.8. Build faces ....................................................................................................................49
Mesh of the motor 2D model.......................................................................................................51
2.4.1. Create and assign mesh points ....................................................................................52
2.4.2. Generate the mesh .......................................................................................................54
2.4.3. Create the basic 2D motor object .................................................................................55
3. Motor 3D geometry, mesh and physical description..............................................................57
3.1.
3.2.
Geometry of the motor 3D model................................................................................................59
3.1.1. Import into a new project ..............................................................................................60
3.1.2. Create coordinate systems ...........................................................................................60
3.1.3. Create symmetries / periodicities .................................................................................61
3.1.4. Create transformations .................................................................................................62
3.1.5. Create volumes by extrusion ........................................................................................62
3.1.6. Add an infinite box ........................................................................................................63
3.1.7. Create lines...................................................................................................................64
3.1.8. Build faces and volumes...............................................................................................64
Mesh of the motor 3D model.......................................................................................................65
3.2.1. Assign mesh points.......................................................................................................66
3.2.2. Create and assign mesh lines ......................................................................................67
3.2.3. Create and assign the mesh generator ........................................................................68
3.2.4. Generate the mesh .......................................................................................................69
END WINDING CHARACTERIZATION
PAGE A
TABLE OF CONTENTS
3.3.
FLUX®10
Physical description of the motor 3D model................................................................................71
3.3.1. Define the physical application .....................................................................................72
3.3.2. Define physical aspects of symmetries / periodicities ..................................................72
3.3.3. Create materials............................................................................................................73
3.3.4. Create and assign volume regions ...............................................................................73
3.3.5. Create geometric parameters and coordinate systems for non-meshed coils .............77
3.3.6. Create electric components ..........................................................................................78
3.3.7. Create non-meshed coils..............................................................................................79
3.3.8. Define sensors ..............................................................................................................92
4. Case 1: reference computation ............................................................................................. 93
4.1.
4.2.
Case 1: physical properties (continued) and solving process.....................................................94
4.1.1. Complete 3D simulation: Modify the physical properties..............................................95
4.1.2. Define the solving scenario...........................................................................................95
4.1.3. Simulation in 2D conditions: Modify the physical properties.........................................96
Case 1: results post-processing..................................................................................................97
4.2.1. Inductance computation: main results ..........................................................................98
4.2.2. Inductance computation: additional results ..................................................................99
5. Case 2: single-phased computation .................................................................................... 101
5.1.
5.2.
Case 2: physical properties (continued) and solving process.................................................. 102
5.1.1. Complete 3D simulation: Modify the physical properties........................................... 103
5.1.2. Define the solving scenario........................................................................................ 104
5.1.3. Simulation in 2D conditions: Modify the physical properties...................................... 105
Case 2: results post-processing............................................................................................... 107
5.2.1. Inductance computation: main results ....................................................................... 108
5.2.2. Inductance computation: additional results ............................................................... 109
6. Case 3: effect of the magnetic saturation ............................................................................ 111
6.1.
6.2.
Case 3: physical properties (continued) and solving process.................................................. 112
6.1.1. Complete 3D simulation: Modify the physical properties........................................... 113
6.1.2. Define the solving scenario........................................................................................ 114
6.1.3. Simulation in 2D conditions: Modify the physical properties...................................... 115
Case 3: results post-processing............................................................................................... 117
6.2.1. Inductance computation: main results ....................................................................... 118
6.2.2. Inductance computation: additional results ............................................................... 119
6.2.3. Compute and display isovalues of the magnetic flux density on volume regions ..... 120
7. Case 4: influence of the rotor .............................................................................................. 123
7.1.
7.2.
Case 4: physical properties (continued) and solving process.................................................. 124
7.1.1. Complete 3D simulation: Modify the physical properties........................................... 125
7.1.2. Solving scenario......................................................................................................... 125
7.1.3. Simulation in 2D conditions: Modify the physical properties...................................... 126
Case 4: results post-processing............................................................................................... 127
7.2.1. Inductance computation............................................................................................. 128
8. Inductance computed values for a Flux 2D study ............................................................... 129
8.1.1.
8.1.2.
PAGE B
Physical description ................................................................................................... 130
Electric circuit description .......................................................................................... 131
END WINDING CHARACTERIZATION
FLUX® 10
1.
General information
General information
Introduction
This chapter describes the device and introduces the theoretical aspects of the
end winding reactance in asynchronous machines and inductance computation
principle.
Contents
This chapter contains the following topics:
Topic
Overview of the sample problem
Strategies of the geometry description, mesh generation and
physical description
Computation of end windings inductance: principle & method
Bibliography
See Page
3
11
15
Complementary information is available in the following documents:
• “Contribution à la modélisation de la machine asynchrone à cage par
logiciels d’éléments finis 2D et 3D” - thesis of Abdelhalim TAIEB
BRAHIMI – 1992 - INPG (Contribution to the modeling of the
asynchronous induction machine by software of finite elements 2D and 3D )
• “Contribution à la modélisation des moteurs asynchrones par la méthode
des éléments finis” - thesis of Eric VASSENT – 1990 - INPG (Contribution
to the modelling of asynchronous motors by the finite elements method)
• “Etude électromagnétique des parties frontales des alternateurs en régime
permanent et transitoire” - thesis of Stephanie RICHARD – 1997 – INPG
(Electromagnetic study of the frontal parts of alternators in steady state and
transitory mode)
• “Etude tridimensionnelle des effets d’extrémité dans les parties frontales
des machines asynchrones” - thesis of Christine SILVA – 1994 – INPG
(Three-dimensional study of the end winding effects in the frontal parts of
asynchronous machines)
• “Calcul des impédances de tête de bobine de machines asynchrones à
partir de FLUX3D” – by Jean Pierre DUCREUX – EDF technical note
HM-18/0235 – 1993 (Computation of the impedances of end winding of
asynchronous machines in FLUX3D)
• “Induction Machines” – by Philippe ALGER – 1969
• “Theory of end winding leakage inductances” – by VB HONSINGER –
IEEE Transactions on magnetics, pp 417-426, – 1959
• “Proceedings ICEM” by WILLIAMSON – pp 480-484 – 1990 – Boston
“Calcul des machines électriques” by LIWSCHITZ – Tome 1, ed DUNOD –
1967 (Computation of electric machines)
END WINDING CHARACTERIZATION
PAGE 1
General information
PAGE 2
FLUX®10
END WINDING CHARACTERIZATION
FLUX® 10
1.1.
General information
Overview of the sample problem
Introduction
This section is an overview of the sample problem. It contains a brief
description of the device and of studied cases.
Contents
This section contains the following topics:
Topic
Foreword
Description of the device
Studied cases
END WINDING CHARACTERIZATION
See Page
4
6
10
PAGE 3
FLUX®10
General information
1.1.1. Foreword
Equivalent
scheme of
STEINMETZ
Since the invention of asynchronous machines, the manufacturers have tried
to improve analysis techniques in order to predict the performance of their
machines: voltage-current characteristic, output power, torque, power factor,
losses and efficiency.
One of them is the well-known equivalent scheme of STEINMETZ that
represents the asynchronous machine by an equivalent electrical circuit per
phase.
R1
Vi
with:
Vi:
R1:
R2/g:
Xm:
X1:
X2:
Leakage
reactance
X1
X2
Xm
R2/g
Voltage phase i
Stator phase resistance
Squirrel cage resistance reduced to one phase
Magnetizing reactance represents the useful flux between stator and
rotor
Magnetic leakage stator reactance
Magnetic leakage rotor reactance
The two leakage reactances can be divided in several reactances in series
according to the physical origin of the leakages:
• The stator and rotor slot leakages:
concern the magnetic flux lines that shut around the slots without crossing
the air gap.
• The stator and rotor differential leakages:
concern the zigzag flux that goes through one tooth to another without
crossing the air gap.
• The leakages due to skew:
flux which results of the winding imperfections and which does not
contribute to the useful sinusoidal field in the air gap.
• The coil end leakage:
leakages in the air created by the looping of conductors in the extremities of
machines.
These leakages are divided into:
- the end winding stator leakages
- the end ring of squirrel cage leakages
Continued on next page
PAGE 4
END WINDING CHARACTERIZATION
FLUX® 10
Flux
computation of
leakage
reactance
General information
All the parameters of the STEINMETZ scheme can be determined with tests,
analytic computations or finite element studies with Flux in 2D, except the
coil end leakage.
For the end winding and the end ring inductances, because of their geometry,
only a 3D model can give good results.
This technical paper details methods for the computation of end winding
stator inductance with Flux software in 3D application.
The results will be compared with the main analytic formulas. This technical
paper also explains how to implement them in Flux 2D application for other
studies on asynchronous machines.
END WINDING CHARACTERIZATION
PAGE 5
FLUX®10
General information
1.1.2. Description of the device
Studied device
The device to be analyzed is a three-phase induction (asynchronous) motor.
The motor under study consists of two parts:
• an outside stationary stator
having 12 coils, each passes
Rotor
through four stator slot
• an inside squirrel-cage rotor
Rotor
attached to the output shaft
slots
Stator
Stator
slots
Shaft
Coils
Operating
principle
The operation of an induction motor is explained below:
• The coils supplied with AC current produce a rotating magnetic field.
• The secondary current is induced onto the rotor.
• This current in the rotor conductors will therefore induce a magnetic field
which will be attracted to the rotating magnetic field in the stator and the
rotor will turn.
Continued on next page
PAGE 6
END WINDING CHARACTERIZATION
FLUX® 10
Geometry
General information
The asynchronous motor is composed of a stator and a rotor.
The stator includes 48 stator slots. The dimensions of the whole stator and
the stator slot are presented in the figures below.
3.51
240
150
22.2
30
2.5
3.21
1.28
3.2
4.6
The rotor includes a shaft and 40 rotor slots. The dimensions of the rotor and
the rotor slot are presented in the figures below.
149
58
0.4
5.25
1
2.425
8.4
2.04
26.3
0.89
Continued on next page
END WINDING CHARACTERIZATION
PAGE 7
FLUX®10
General information
Materials
The different parts of the asynchronous motor are made of the following
materials:
• the material of the rotor and stator is iron (Fe V1000)
• the material of the bars of the squirrel cage is aluminum
• the material of the shaft is steel
Electrical
characteristics
The electrical characteristics of the asynchronous motor are presented in the
table below.
Characteristics
Rated power
Power supply voltage
Rated current
Speed
Resistance of ring extremities
Inductance of ring extremities
Resistance of end windings
Winding
characteristics
Value
18.5 kW
220 V
37 A
1450 rpm
7.25 10-7 Ω
3.7 10-9 H
0.225 Ω
The winding characteristics are presented in the table below.
Characteristics
Type of winding
Average length of end windings
Average length of the straight part of
end windings
Number of pairs of poles
Number of slots per pole and per phase
Number of turns in series per phase
Diameter of a spire
Number of winding in parallel per phase
Distribution coefficient
Value
Concentric with consequent poles
231 mm
31 mm
2
4
136
1.3 mm
2
0.957
Continued on next page
PAGE 8
END WINDING CHARACTERIZATION
FLUX® 10
The winding is concentric with consequent poles. It is represented in the
figure below.
Winding 1
END WINDING CHARACTERIZATION
Winding 2
V- V+
48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8
7
6
5
4 3 2 1
Winding
diagram
General information
PAGE 9
FLUX®10
General information
1.1.3. Studied cases
Studied cases
To calculate the end windings inductance, four cases are carried out in a
Magneto Static application:
• case 1: reference computation
• case 2: single-phased computation
• case 3: effect of the magnetic saturation
• case 4: influence of the rotor
Case 1
The first case is a reference computation.
This computation can be considered as a reference computation since it is
based on the experimental test record of the end windings inductance
recognized by the IEC norms. The norms do not take the rotor into account;
we need to make computations only in the regions of the stator.
To simplify the model, we use small values of current with linear
approximation for the B(H) characteristic of magnetic materials.
Case 2
The second case is a single-phased computation.
This computation is focused on the self-inductivity, that is why only the first
phase is modeled.
Case 3
The third case is a parametric computation taking into account the effect of
magnetic saturation.
In order to evaluate the influence of the magnetic saturation, the real B(H)
characteristic of the materials is used. Two simulations with two different
current values will be made to compute the inductance of nonlinear materials
with the definition L=∆Φ/∆I.
Case 4
The fourth case is a computation taking into account the rotor influence.
To take the rotor influence into account, the computations are made in all the
regions of the asynchronous motor.
PAGE 10
END WINDING CHARACTERIZATION
FLUX® 10
1.2.
General information
Strategies of the geometry description, mesh
generation and physical description
Introduction
This section explains the strategies of the geometry, mesh and physical
description of the motor finite element model.
Contents
This section contains the following topics:
Topic
Main phases for geometry description and mesh generation of
the motor
Main phases for physical description of the motor
END WINDING CHARACTERIZATION
See Page
12
14
PAGE 11
FLUX®10
General information
1.2.1. Main phases for geometry description and mesh generation of
the motor
2D motor
outline
An outline of the 2D geometry building and mesh generation is presented in
the table below.
Stage
Description
1
The geometry description and
mesh preparation of a stator
slot object
2
The geometry description and
mesh preparation of a rotor slot
object
3
Importation of the stator slot
object and the rotor slot object
4
Creation of symmetries to
model a quarter of the motor
5
Creation of faces by
propagation
6
Creation of infinite box
7
Faces building
8
Creation of mesh tools,
assignment of mesh
information and meshing of the
motor 2D geometry
• X-axis symmetry
• Y-axis symmetry
Continued on next page
PAGE 12
END WINDING CHARACTERIZATION
FLUX® 10
3D motor
outline
General information
An outline of the 3D geometry building and mesh generation is presented in
the table below.
1
Importation of the 2D motor
object
2
Creation of one symmetry
and one periodicity to model
one eighth of the motor
original size
3
Creation of faces and
volumes by extrusion
4
Creation of infinite box
5
Faces and volumes building
6
Creation of mesh tools,
assignment of mesh
information and meshing of
the motor 3D geometry
END WINDING CHARACTERIZATION
• XY-plane symmetry
• angular periodicity with rotation
about Z-axis
PAGE 13
FLUX®10
General information
1.2.2. Main phases for physical description of the motor
3D motor
outline
An outline of the physical description process of the 3D motor is presented in
the table below.
Stage
Description
1
Definition of the
application
2
Definition of physical
aspects
3
Creation of materials
• 3D Magneto Static
• symmetry:
Tangent magnetic field
• periodicity:
Odd (anticyclic boundary conditions)
• ALUMINIUM – material for squirrel
cage with a linear B(H) characteristic
• STEEL – material for rotor shaft with a
linear B(H) characteristic
• LINEAR_IRON – material for stator
and rotor with a linear B(H)
characteristic
• IRON_FEV1000 – material for stator
with a nonlinear B(H) characteristic
INFINITE
4
Creation and
assignment of volume
regions
EXT_AIR
(invisible)
STATOR
SQUIRREL_CAGE
SLOTST
AIR_GAP
SHAFT
ROTOR
5
Creation of sources
12 non-meshed coils
PAGE 14
END WINDING CHARACTERIZATION
FLUX® 10
1.3.
General information
Computation of end windings inductance: principle &
method
Introduction
Flux gives the possibility to model the extremity of the asynchronous
machine and to represent correctly the geometry of end windings. After the
resolution, the value of magnetic field is available in each node of our finite
element model.
Then, there are several methods (based on energy and flux computation) to
extract from Flux 3D simulations the expected inductance. These methods
and the operating mode in Flux are detailed in this section.
Contents
This section contains the following topics:
Topic
Inductance computation principle
Operating mode with Flux 3D application
Self inductance or cyclic inductance computation?
Symmetry and periodicity
Hypothesis
See Page
16
19
21
22
The general working hypotheses are as follows:
• there are no eddy currents and no skin effects in conducting parts:
⇒ magneto static application
• the influence of the rotor is negligible:
⇒ the rotor is not modeled
• there is no saturation of magnetic materials:
⇒ linear approximation of B(H) characteristic
• the power supply is homopolar: the same current I passes through all the
phases (I=Itot/N)
• the values of self and mutual inductances are identical for windings of the
same phase and for all the phases.
END WINDING CHARACTERIZATION
PAGE 15
FLUX®10
General information
1.3.1. Inductance computation principle
Introduction
This paragraph deals with inductance computation principles (based on
energy or flux computation).
It presents the various possible approaches to compute end windings
inductance with post processing quantities available in Flux.
Definitions
With linear materials, if one phase is modeled, relations between flux (Φ),
energy (W), current (I) and inductance (L) could be written as:
Φ = L.I
⇒
W = 1/2.L.I² ⇒
Computation
with Flux
L = Φ/I
(1)
L = 2.W/I2
(2)
It is possible to calculate inductance with FLUX using equations 1 or 2 since
Flux proposes the calculation of:
• the magnetic flux viewed by a coil (Φ)
• the magnetic energy (W) in volume regions
by integrating B*H in the corresponding volumes
• the current I injected in the windings (input data)
The problem is to determine which energy (or which flux) will
give you the expected inductance.
Various approaches can be considered. They are listed hereafter (in the
following blocks) and detailed (practical applications) in this document.
First approach
With the first approach we assume that W in formula (1) is the energy in the
air around the end windings. The computation of the energy in the volume
regions corresponding to the air around the machine is representative of the
magnetic energy lost by the system, that is to say leakage energy.
W = WAIR
(3)
This method assumes that the extremity leakages are only due to the windings
at the external of the machine but in the reality, the conductors located at the
extremity of the straight part of the machines also contribute to the extremity
leakages.
Continued on next page
PAGE 16
END WINDING CHARACTERIZATION
FLUX® 10
Second
approach
General information
With the second approach, we will try to take into account extremity
leakages due to the conductors located at the extremity of the straight part.
In this case, we assume that W in formula (1) could be written as:
W = WTOTAL – WSTRAIGHT_PART
(4)
where:
• WTOTAL is the energy computed in all the regions modeled in 3D, that is to
say, the air around the end windings and also the extremities of the straight
part of the stator (magnetic circuit and air gap)
• WSTRAIGHT_PART is the energy computed in the straight part of the motor
without taking into account any extremity leakage (computation in 2D
conditions: see details hereafter)
We can also use the flux relation and write:
Φ = ΦTOTAL – ΦSTRAIGHT_PART
(5)
where:
• ΦTOTAL is the total flux viewed by a phase
• ΦSTRAIGHT_PART is the 2D flux in the straight part of the machine viewed by
a phase (computation in 2D conditions: see details hereafter)
Third approach
In the third approach, we use the relation between energy (or flux) and iron
length of the machine.
We assume that:
• the energy (or field) corresponding to bi-dimensional field evolves linearly
with the length of iron (WSTRAIGHT_PART = a * liron)
• the energy W in formula (1) is constant (W = b)
We can write in this case:
WTOTAL = a * liron + b
(6)
It’s possible to calculate the (origin coordinate) of the WTOTAL (liron) straight
line with two computations with two iron lengths (see diagram in following
block).
Continued on next page
END WINDING CHARACTERIZATION
PAGE 17
FLUX®10
General information
Graphical
aspects
The different approaches to calculate W are represented on the following
diagram.
WTOTAL
WSTRAIGHT_PAR T
Energy (W)
WTOTAL - WSTR AIGHT_P ART
(W3D - W2D method)
Extrapolation to zero
0
10
20
30
40
50
60
70
Iron length (l)
PAGE 18
END WINDING CHARACTERIZATION
FLUX® 10
General information
1.3.2. Operating mode with Flux 3D application
Introduction
To calculate the end windings inductance with the three previous approaches,
you need three simulations in Flux 3D application. The simulation conditions
and the available results are presented in this paragraph.
Complete 3D
simulation
A first simulation is made with the straight part of the motor, the external air
around the end windings and windings. We choose a first arbitrary length of
iron (l1).
We compute the following energies:
• WAIR(l1) on the external volume
regions – air around the end
windings
• WTOTAL(l1) on all volume regions
– air around the end windings,
magnetic circuit and air gap
Simulation in
2D conditions
A second simulation is made taken into account only the straight part of the
machine. We are working with the same length of iron (l1). In order to have
the magnetic energy corresponding only to a bi-dimensional field, we have to
calculate the energy in the motor as if we were in the middle of the straight
part. We have to deactivate all the volume regions corresponding to the
external air and to extend the conductors in order to eliminate every edge
fringing.
We compute the following energy:
• WSTRAIGHT_PART(l1) on all straight
part volume regions – magnetic
circuit and air gap
This computation can also be made with Flux 2D application, it will give us the
possibility to validate our results by comparing Flux 3D and Flux 2D results.
Continued on next page
END WINDING CHARACTERIZATION
PAGE 19
FLUX®10
General information
Complete 3D
simulation
modifying the
iron length
A third simulation, similar to the complete 3D simulation, is made modifying
the iron length (l2).
We compute the following energy WTOTAL(l2) on all volume regions – air
around the end windings, magnetic circuit and air gap.
Summary
The table below summarizes the three simulations (simulation conditions and
results) presented in this document.
Complete
3D simulation
Conditions
Simulation
in 2D conditions
inactive volume regions of
external air and extended size
of conductors
l1
Complete
3D simulation
l2
Iron length
l1
Energy computation
WAIR(l1)
WTOTAL(l1)
WSTRAIGHT_PART(l1)
WTOTAL(l2)
Flux computation
ΦTOTAL(l1)
ΦSTRAIGHT_PART(l1)
ΦTOTAL(l2)
Final results
The table below summarizes the different methods for end windings
inductance computation starting from the previous results.
Method
Energy in the air around end
windings
Energy: contribution of 2D
energy and 3D energy
Flux: contribution of 2D field
and 3D field
Use of W(liron) straight line
function: extrapolation to zero
Caution
Formula
Computation
(3)
W = WAIR(l1)
(4)
W = WTOTAL(l1) – WSTRAIGHT_PART(l1)
(5)
Φ = ΦTOTAL(l1) – ΦSTRAIGHT_PART(l1)
(6)
W = (l2.WTOTAL(l1) – l1.WTOTAL(l2)) / (l2 – l1)
To reduce the size of the Flux project, only part of the device is represented
(to take into account of symmetry and periodicity). It is thus necessary to
introduce some corrector coefficients into the formulas. These aspects are
described in § 1.3.4.
PAGE 20
END WINDING CHARACTERIZATION
FLUX® 10
General information
1.3.3. Self inductance or cyclic inductance computation?
One phase or
three phases
In the equivalent scheme of STEINMETZ, the value of inductances
corresponds to the cyclic inductances (L-M) per phase (L being the self
inductance of the considered phase and M being the mutual inductance
coefficient) so that it is the value we will try to obtain directly. But, it could
be interesting to calculate the two values.
With the
energetic
method
If only one phase is modeled:
W = 1/2*L*I² ⇒ L = 2*W/I²
If the three phases are modeled:
W=1/2*L1*I1²+1/2*L2*I2²+1/2*L3*I3²+M12* I1* I2+M23* I2* I3+M13* I1* I3
We make the hypothesis that L1 = L2= L3= L and M12 = M13 = M23= M.
With the following values for the current: I1=Imax, I2 = I3 = -Imax/2, we obtain
directly (L-M).
W= 3/4*(L-M)* Imax ² ⇒ (L-M) = 4/3*W/Imax ²
The calculated value must be multiplied by 4/3 in order to take the 3 phases into
account.
With the flux
method
If only one phase is modeled:
Φ = L*I ⇒ L = Φ/I
If the three phases are modeled, and with the triplet (Imax;-Imax/2;-Imax/2), we
have:
Φ = (L-M)*Imax ⇒ (L-M) = Φ/Imax
END WINDING CHARACTERIZATION
PAGE 21
FLUX®10
General information
1.3.4. Symmetry and periodicity
Periodicity
The motor has 2 pairs of poles, the periodicity planes allow to represent only
one coil pitch, that is to say one quarter of the machine, 12 slots for the stator
and 10 slots for the rotor.
Then, anticyclic conditions of periodicities on the lateral faces of our study
domain have to be imposed in order to respect the physical reality.
The winding is constituted of 2 ways in parallel per phase with 2 coils in
series in each way. Though one coil per phase is entirely represented in the
quarter of the motor.
Energetic
method
In this case the energy of one coil per phase is only computed.
So that, the calculated inductance for one coil with the total current Iphase is
equivalent to the inductance of the entire phase that is the expected value.
L
I phase
Flux method
L
L
L
≡
L
I phase
If we compute magnetic flux through a coil conductor, the calculated flux is
the flux for the coils belonging to the chosen coil conductor in the study
domain.
As for the energy, if the coil in the study domain is supplied with the current
Iphase, the computation in the quarter is equivalent to the computation on the
entire machine.
Symmetry
The motor has a symmetry plane, it is possible to represent one half of the
device, and to set appropriate symmetry conditions on this symmetry planes.
The calculated value must be multiplied by 2 in order to take the 2 extremities
of the machine into account.
PAGE 22
END WINDING CHARACTERIZATION
FLUX® 10
2.
Motor 2D geometry and mesh
Motor 2D geometry and mesh
Introduction
This chapter describes the main steps of the geometry building and the mesh
generation of the 2D finite element model of asynchronous motor.
Contents
This chapter contains the following topics:
Topic
Geometry and mesh of the 2D stator slot
Geometry and mesh of the 2D rotor slot
Geometry of the motor 2D model
Mesh of the motor 2D model
END WINDING CHARACTERIZATION
See Page
25
33
41
51
PAGE 23
Motor 2D geometry and mesh
PAGE 24
FLUX®10
END WINDING CHARACTERIZATION
FLUX® 10
2.1.
Motor 2D geometry and mesh
Geometry and mesh of the 2D stator slot
Introduction
This section explains the geometry description and mesh preparation of the
2D stator slot object.
3.51
22.2
2.5
3.21
1.28
3.2
4.6
Project name
The project is saved under the name 2D_STATOR_SLOT.FLU.
Contents
This section contains the following topics:
Topic
Create geometric parameters and coordinate systems
Create points and basic lines
Create transformations
Create lines by propagation
Build faces
Create and assign mesh points
END WINDING CHARACTERIZATION
See Page
26
28
29
30
30
31
PAGE 25
FLUX®10
Motor 2D geometry and mesh
2.1.1. Create geometric parameters and coordinate systems
Goal
16 geometric parameters and 3 coordinate systems are created to describe the
geometry of the stator.
Data (1)
The geometric parameters and their characteristics are presented below.
RAD2_SLOTST
DPT_SLOTST
RAD1_SLOTST
H2_SLOTST
INTER_SLOTST/2
H1_SLOTST
WID_SLOTST/2
L2_SLOTST
Geometric parameter
Name
ANG
TOT_LEN
DIAST_OUT
DIAST_IN
LEN_IRON
NBR_SLOTST
DPT_SLOTST
H1_SLOTST
H2_SLOTST
L2_SLOTST
WID_SLOTST
RAD1_SLOTST
RAD2_SLOTST
PERI_ST
SLOTST_GAP
INTER_SLOTST
Comment
Angle of the stator part to design
Total motor length
Outer stator diameter
Inner stator diameter
Motor height
Number of stator slots
Stator slot depth
Height of stator slot neck
Height of stator slot bottom
Length of stator slot bottom
Width of stator slot opening
Fillet radius of stator slot bottom
Radius of stator slot top
Inner stator perimeter
Angle between two stator slots
Curvilinear interval between two
stator slots
Expression
90
225
240
150
15
48
22.2
1.28
3.21
2.3
3.2
2.5
3.51
Pi()* DIAST_IN
360/NBR_SLOTST
(PERI_ST-NBR_SLOTST*WID_SLOTST)/
NBR_SLOTST
Continued on next page
PAGE 26
END WINDING CHARACTERIZATION
FLUX® 10
Motor 2D geometry and mesh
Data (2)
The coordinate systems and their characteristics are presented below.
CENT2 SLOTST
STATOR
SLOTST
Cylindrical coordinate system defined with respect to the Global coordinate system
Name
Comment
Units
STATOR
Coordinate system
for the stator
millimeter
/degree
Origin coordinates
first (R)
second (θ)
0
Rotation angle
about Z
0
0
Cartesian coordinate system defined with respect to the Local coordinate system
Name
Comment
SLOTST
Coordinate system
for the stator slot
Parent
coord.
system
first (X)
STATOR
DIAST_IN/2
Origin coordinates
second (Y)
180*(INTER_SLOTST
+WID_SLOTST)/
(Pi()* DIAST_IN)
Rotation
angle
about Z
0
Cylindrical coordinate system defined with respect to the Local coordinate system
Name
Comment
Parent
coord.
system
CENT2_
SLOTST
Coordinate system
for the stator slot
extremity
SLOTST
END WINDING CHARACTERIZATION
Origin coordinates
Rotation angle
first (R)
second (θ)
about Z
DPT_SLOTSTRAD2_SLOTST
0
0
PAGE 27
FLUX®10
Motor 2D geometry and mesh
2.1.2. Create points and basic lines
Goal
First, the points of the 2D stator slot are created. Then, the line segments and
arcs of the 2D stator slot are defined as shown in the figure below.
Data (1)
The characteristics of the points are presented in the tables below.
Point defined by its parametric coordinates
Number
Coordinate
system
1
2
STATOR
Local coordinates
first (R)
second (θ)
DIAST_IN/2
0
180/(Pi()*DIAST_IN)*
DIAST_IN/2
(INTER_SLOTST)
Point defined by its parametric coordinates
Number
Coordinate
system
3
4
5
SLOTST
Local coordinates
first (X)
second (Y)
0
0
H1_SLOTST
-WID_SLOTST/2
H2_SLOTST
-L2_SLOTST
Point defined by its parametric coordinates
Number
6
7
Coordinate
system
CENT2_
SLOTST
Local coordinates
first (R)
second (θ)
RAD2_SLOTST
0
RAD2_SLOTST
-90
Continued on next page
PAGE 28
END WINDING CHARACTERIZATION
FLUX® 10
Data (2)
Motor 2D geometry and mesh
The characteristics of the lines are presented in the tables below.
Segment defined by starting and ending points
Number
1
2
Starting point
2
5
Ending point
4
7
Arc defined by its center coordinates, starting and ending points
Number
3
4
Coordinate
system
STATOR
STATOR
Starting
point
1
2
Ending
point
2
3
Center point coordinates
first
second
0
0
0
0
Arc defined by its radius, starting and ending points
Number
Coordinate system
Arc radius
5
6
SLOTST
CENT2_SLOTST
RAD1_SLOTST
RAD2_SLOTST
Starting
point
4
7
Ending
point
5
6
2.1.3. Create transformations
Goal
One geometric transformation is created to describe the geometry of the 2D
stator slot.
Data
The characteristics of the transformation are presented in the table below.
Affine transformation with respect to a line defined by 2 points
Name
Comment
First
point
Second
point
Scaling
factor
SLOTST_MIRROR
Affine transformation to
build the stator slot
3
6
-1
END WINDING CHARACTERIZATION
PAGE 29
FLUX®10
Motor 2D geometry and mesh
2.1.4. Create lines by propagation
Goal
The other lines of the 2D stator slot are created by propagation from lines.
Data/Action
The SLOTST_MIRROR transformation is applied once to the 6 previously
created lines.
⇒
Line created with command Propagate lines
Number
Transformation
7 – 12
SLOTST_MIRROR
Reference line
1, 2, 3, 4, 5, 6
(select all)
Number of times
1
2.1.5. Build faces
Goal
The face of the 2D stator slot is built using the algorithm of automatic
construction.
Action
The face is automatically built.
⇒
PAGE 30
END WINDING CHARACTERIZATION
FLUX® 10
Motor 2D geometry and mesh
2.1.6. Create and assign mesh points
Goal
One mesh point is created and assigned to define the mesh of the 2D stator
slot.
Data
The characteristics of the mesh point are presented in the table below.
Mesh point
Name
SLOTST
Action
Comment
Mesh point of the stator slot
Unit
millimeter
Value
1.8
Color
Green
The SLOTST mesh point is assigned to the points as shown in the figure
below.
END WINDING CHARACTERIZATION
PAGE 31
Motor 2D geometry and mesh
PAGE 32
FLUX®10
END WINDING CHARACTERIZATION
FLUX® 10
2.2.
Motor 2D geometry and mesh
Geometry and mesh of the 2D rotor slot
Introduction
This section explains the geometry description and mesh preparation of the
2D rotor slot object.
0.4
5.25
1
2.425
8.4
2.04
26.3
0.89
Project name
The project is saved under the name 2D_ROTOR_SLOT.FLU.
Contents
This section contains the following topics:
Topic
Create geometric parameters and coordinate systems
Create points and basic lines
Create transformations
Create lines by propagation
Build faces
Create and assign mesh points
END WINDING CHARACTERIZATION
See Page
34
36
37
38
38
39
PAGE 33
FLUX®10
Motor 2D geometry and mesh
2.2.1. Create geometric parameters and coordinate systems
Goal
14 geometric parameters and 4 coordinate systems are created to describe the
geometry of the rotor.
Data (1)
The geometric parameters and their characteristics are presented below.
WID_SLOTROT
INTER_SLOTROT/2
H1_SLOTROT
H2_SLOTROT
RAD1_SLOTROT
H3_SLOTROT
RAD2_SLOTROT
DPT_SLOTROT
RAD3_SLOTROT
Geometric parameter
Name
DIAROT
NBR_SLOTROT
DPT_SLOTROT
DIASHAFT
H1_SLOTROT
H2_SLOTROT
H3_SLOTROT
RAD1_SLOTROT
RAD2_SLOTROT
RAD3_SLOTROT
WID_SLOTROT
SLOTROT_GAP
PERI_ROT
INTER_SLOTROT
Comment
Rotor diameter
Number of rotor slots
Rotor slot depth
Shaft diameter
Height of rotor slot opening
Height of rotor slot top bar
Height of rotor slot neck
Radius of rotor slot top bar
Radius below the neck
Radius of rotor slot bottom bar
Width of rotor slot opening
Angle between two rotor slots
Rotor perimeter
Curvilinear interval between two rotor
slots
Expression
149
40
26.3
58
0.4
5.25
8.4
2.425
2.04
0.89
1
360/NBR_SLOTROT
Pi()*DIAROT
(PERI_ROT-NBR_SLOTROT*
WID_SLOTROT)/NBR_SLOTROT
Continued on next page
PAGE 34
END WINDING CHARACTERIZATION
FLUX® 10
Motor 2D geometry and mesh
Data (2)
The coordinate systems and their characteristics are presented below.
SLOTROT
CENT2 SLOTROT
CENT3 SLOTROT
ROTOR
Cylindrical coordinate system defined with respect to the Global coordinate system
Name
ROTOR
Comment
Coordinate system
for the rotor
first (R)
second (θ)
Rotation
angle
about Z
0
0
0
Origin coordinates
Units
millimeter/
degree
Cartesian coordinate system defined with respect to the Local coordinate system
Name
SLOTROT
Comment
Coordinate system
for the rotor slot
Parent
coord.
system
first (X)
ROTOR
DIAROT/2
Origin coordinates
second (Y)
180*
(INTER_SLOTROT
+ WID_SLOTROT)/
(Pi()*DIAROT)
Rotation
angle
about Z
180
Cylindrical coordinate system defined with respect to the Local coordinate system
Name
CENT2_
SLOTROT
CENT3_
SLOTROT
Comment
Coordinate system
for the rotor slot
bottom beginning
Coordinate system
for the rotor slot
bottom extremity
END WINDING CHARACTERIZATION
first (R)
second (θ)
Rotation
angle
about Z
SLOTROT
H3_SLOTROT+
RAD2_SLOTROT
0
0
SLOTROT
DPT_SLOTROTRAD3_SLOTROT
0
0
Parent
coord.
system
Origin coordinates
PAGE 35
FLUX®10
Motor 2D geometry and mesh
2.2.2. Create points and basic lines
Goal
First, the points of the 2D rotor slot are created. Then the line segments and
arcs of the 2D rotor slot are created as shown in the figure below.
Data (1)
The characteristics of the points are presented in the tables below.
Point defined by its parametric coordinates
Number
Coordinate
system
1
2
ROTOR
Local coordinates
first (R)
second (θ)
DIAROT/2
0
180/(Pi()*DIAROT)*
DIAROT/2
(INTER_SLOTROT)
Point defined by its parametric coordinates
Number
Coordinate
system
3
4
5
6
SLOTROT
Local coordinates
first (X)
second (Y)
0
0
H1_SLOTROT
WID_SLOTROT/2
H2_SLOTROT
WID_SLOTROT/2
H3_SLOTROT
WID_SLOTROT/2
Point defined by its parametric coordinates
Number
7
Coordinate
system
CENT2_
SLOTROT
first (R)
Local coordinates
second (θ)
RAD2_SLOTROT
90
Point defined by its parametric coordinates
Number
8
9
Coordinate
system
CENT3_
SLOTROT
Local coordinates
first (R)
second (θ)
RAD3_SLOTROT
0
RAD3_SLOTROT
90
Continued on next page
PAGE 36
END WINDING CHARACTERIZATION
FLUX® 10
Data (2)
Motor 2D geometry and mesh
The characteristics of the lines are presented in the tables below.
Segment defined by starting and ending points
Number
1
2
3
Starting point
9
6
4
Ending point
7
5
2
Arc defined by its center coordinates, starting and ending points
Coordinate
system
ROTOR
ROTOR
Number
4
5
Starting
point
1
2
Ending
point
2
3
Center point coordinates
first
second
0
0
0
0
Arc defined by its radius, starting and ending points
Number
Coordinate system
Arc radius
6
7
8
SLOTROT
CENT2_SLOTROT
CENT3_SLOTROT
RAD1_SLOTROT
RAD2_SLOTROT
RAD3_SLOTROT
Starting
point
5
7
8
Ending
point
4
6
9
2.2.3. Create transformations
Goal
One geometric transformation is created to describe the geometry of the 2D
rotor slot.
Data
The characteristics of the transformation are presented in the table below.
Affine transformation with respect to a line defined by 2 points
Name
Comment
First point
Second
point
Scaling
factor
SLOTROT_
MIRROR
Affine transformation
to build the rotor slot
8
3
-1
END WINDING CHARACTERIZATION
PAGE 37
FLUX®10
Motor 2D geometry and mesh
2.2.4. Create lines by propagation
Goal
The other lines of the 2D rotor slot are created by propagation from lines.
Data/Action
The SLOTROT_MIRROR transformation is applied once to the 8 previously
created lines.
⇒
Line created with command Propagate Lines
Number
Transformation
9 – 16
SLOTROT_MIRROR
Reference line
1, 2, 3, 4, 5, 6, 7, 8
(select all)
Number of times
1
2.2.5. Build faces
Goal
The face of the 2D rotor slot is built using the algorithm of automatic
construction.
Action
The face is automatically built.
⇒
PAGE 38
END WINDING CHARACTERIZATION
FLUX® 10
Motor 2D geometry and mesh
2.2.6. Create and assign mesh points
Goal
Two mesh points are created and assigned to define the mesh of the rotor slot.
Data
The characteristics of the mesh points are presented in the table below.
Mesh point
Name
SLOTROT_INT
SLOTROT_EXT
Comment
Mesh point of the rotor
slot top
Mesh point of the rotor
slot bottom
Unit
Value
Color
millimeter
1.3
Cyan
millimeter
2.8
Turquoise
Action (1)
The SLOTROT_INT mesh point is assigned to the points as shown in the
figure below.
Action (2)
The SLOTROT_EXT mesh point is assigned to the points as shown in the
figure below.
END WINDING CHARACTERIZATION
PAGE 39
Motor 2D geometry and mesh
PAGE 40
FLUX®10
END WINDING CHARACTERIZATION
FLUX® 10
2.3.
Motor 2D geometry and mesh
Geometry of the motor 2D model
Introduction
This section shows how to build the complete geometry of the motor 2D
finite element model by importing the two previous projects into a new
project.
Project name
The project is saved under the name 2D_GEO_MESH.FLU.
Contents
This section contains the following topics:
Topic
Import into a new project
Create coordinate systems
Create symmetries
Create transformations
Create lines and faces by propagation
Add an infinite box
Create points and lines
Build faces
END WINDING CHARACTERIZATION
See Page
42
42
43
43
44
46
47
49
PAGE 41
FLUX®10
Motor 2D geometry and mesh
2.3.1. Import into a new project
Goal
The two preliminary projects – 2D_STATOR_SLOT.FLU and
2D_ROTOR_SLOT.FLU – are imported into the new project
2D_GEO_MESH.FLU.
Result
After importation, the geometry of the project looks like the figure below.
2.3.2. Create coordinate systems
Goal
One coordinate system is created to describe the geometry of the 2D motor.
Data
The coordinate systems and their characteristics are presented below.
2D
Cylindrical coordinate system defined with respect to the Global coordinate system
PAGE 42
Name
Comment
Units
2D
Coordinate system
for the 2D motor
millimeter
/degree
Origin coordinates
first (R) second (θ)
0
0
Rotation angle
about Z
0
END WINDING CHARACTERIZATION
FLUX® 10
Motor 2D geometry and mesh
2.3.3. Create symmetries
Goal
Two symmetries are created to describe one quarter of the motor 2D
geometry.
Data
The characteristics of the symmetries are presented in the tables below.
Symmetry versus X-axis
Name
(automatic)
SymmetryXaxis_1
Geometrical aspects
Type
Y offset position
Versus X-axis
0
Physical
aspects*
-
Symmetry versus Y-axis
Name
(automatic)
SymmetryYaxis_1
Geometrical aspects
Type
X offset position
Versus Y-axis
0
Physical
aspects*
-
*
Physical aspects of the symmetries are defined in the section concerning physical
description.
2.3.4. Create transformations
Goal
Two geometric transformations are created to describe the geometry of the 2D
motor.
Data
The characteristics of the transformations are presented in the table below.
Rotation defined by an angle and an existing pivot point
Name
Comment
SLOTST_
ROTATION
SLOTROT_
ROTATION
Rotation transformation
to build the 2D stator
Rotation transformation
to build the 2D rotor
END WINDING CHARACTERIZATION
Coord.
system
Pivot point
coordinates
first
second
Rotation angle
about Z
2D
0
0
SLOTST_GAP
2D
0
0
SLOTROT_GAP
PAGE 43
FLUX®10
Motor 2D geometry and mesh
2.3.5. Create lines and faces by propagation
Goal
The other faces / lines of the 2D motor are created by propagation from faces
/ lines.
Data/Action (1)
The SLOTST_ROTATION transformation is applied 11 times to the stator
slot face.
⇒
Face created by command Propagate Faces
Number
Transformation
Reference face
Number of times
3 – 13
SLOTST_
ROTATION
1
11
Data/Action (2)
Building options
Add faces and
associated linked
mesh generator
The SLOTROT_ROTATION transformation is applied 9 times to the rotor
slot face.
⇒
Face created by command Propagate Faces
Number
Transformation
Reference face
Number of times
14 – 22
SLOTROT_
ROTATION
2
9
Building options
Add faces and
associated linked
mesh generator
Continued on next page
PAGE 44
END WINDING CHARACTERIZATION