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

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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

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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



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