PM Axial Flux Machine Design for Hybrid Traction

16 Pages·1.967 MB·English

by  Barrière O. et al.

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Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 1, pp. 203-218
Copyright ©2009, Institut français du pétrole
DOI: 10.2516/ogst/2009058
IFP International Conference
Rencontres Scientifiques de l’IFP
Advances in Hybrid Powertrains
Évolution des motorisations hybrides
PM Axial Flux Machine Design for Hybrid Traction
O. de la Barrière, S. Hlioui, H. Ben Ahmed, M. Gabsi and M. LoBue
SATIE, ENS Cachan, CNRS, UniverSud, 61 avenue du Président Wilson, 94230 Cachan - France
e-mail: [email protected] - [email protected] - [email protected]
[email protected] - [email protected]
Résumé— Conception d’une machine à flux axial à aimants permanents pour la traction hybride
— La traction hybride semble actuellement un des moyens les plus prometteurs pour réduire la
consommation de carburant des véhicules. Ce procédé consiste à associer un moteur électrique au moteur
thermique traditionnel. Pour une telle application embarquée, le rendement, ainsi que le couple massique,
sont des critères de conception de première importance. Dans ce contexte, le recours à une machine
synchrone à aimants permanents, reconnue pour satisfaire ces deux critères, semble être approprié. Vu
que le volume alloué à la machine électrique est de forme discoïde, les topologies à flux axial semblent
les plus intéressantes. L’objectif de cet article est de proposer une méthodologie de pré-dimensionnement
de tels actionneurs, en ayant fixé au préalable le volume maximal permis ainsi que le cahier des charges
de la machine.
Abstract— PM Axial Flux Machine Design for Hybrid Traction— Hybrid traction seems to be one of
the best ways in order to reduce fuel consumption for vehicles. It consists of associating an electric motor
next to the classical thermal machine. In this embedded system, the efficiency and the torque per unit
mass are very important. So it seems to be a good idea to use a permanent magnet synchronous machine,
which is recognized for satisfying these two objectives. Since the allocated volume is rather flat, axial flux
topologies are interesting. This paper’s goal is to propose an optimal first design method for such
structures, given the allocated volume and the machine requirements.
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INTRODUCTION 2 THE OPTIMIZATION STRATEGY
For both economic and environmental reasons, it is becoming We are going to describe our optimization method. More
necessary to reduce fuel consumption of cars. The associa- precisely, given the application requirements, and data about
tion of an electric motor to the thermal one, which is called the materials which will be used, the purpose consists on
hybrid traction, is one of the most interesting solutions for reversing the mathematical model in order to provide the best
that. We are going to use an axial flux permanent magnet geometrical parameters for chosen optimization criteria.
machine, for its flat shape. After the presentation of the
machine, we would like to propose an optimization strategy
2.1 Optimization Problem Presentation
for the design of the electric machine, based on the use of
evolutionary algorithms. Then, we will describe briefly the Every optimization problem is clearly defined once the following
model used for the optimization. To finish, a few optimization points are given:
results will be presented, and analyzed. – a single or several objectives, which will have to be
minimized;
– parameters, evolving on a permitted domain;
1 PRESENTATION OF THE MACHINE
– restrictions to be observed so as to get allowable solutions.
These three points have to be detailed for our problem.
There are a lot of axial flux permanent magnet machines
structures [1]. The basic one is composed of a rotor with
2.1.1 Optimization Objectives
alternated polarity permanent magnets, placed in front of a
laminated stator carrying a multiphase winding. There are Since the motor is designed for an embedded system, it seems
also multi-layers structures, but most of them can be seen as to be important to minimize both its mass and its working
a composition of single-layers ones. losses, which are two contradictory criteria. So it is a multi-
We are just going to study the single layer machine in this objectives optimization problem. Multi-objective optimiza-
article. The detailed topology is drawn in Figure 1a. Water tion results are Pareto’s curve, composed of all the closest
cooling is employed, so as to remove the loss in a more effi- point to the ideal point, which is the zero mass and zero
cient way than simple air natural convection. As we will see, losses point, and is of course impossible to reach. Each point
since flux weakening capabilities are required, we choose a on this curve represents a different, but optimal compromise
rotor with salient poles (Fig. 1b). The stator is also shown in between the two objectives.
Figure 1c, and the two parts together are represented in The optimization of the hybrid system must be carried out
Figure 1d. for a cycle of points [3] in the torque vs speed plan that the
electric machine must be able to reach (Fig. 2).
The best thing to do is certainly to carry out the mass and
losses minimization strategy for a working cycle of the
Rotor
Magnet ferromagnetic
Cooling tooth
Massive system
magnetic Frame T (Nm)
circuit Representative
Lmamaginnaetteicd Windings Rotor yoke points of the cycle
circuit
Magnet
a) b)
Slot
Stator
ferromagnetic
0
tooth
Ω (rd/mn)
Frame
Stator yoke
Winding
c) overhang d) Airgap
Figure 1
Figure 2
Presentation of the studied machine. a) Sectional representation
of the machine. b) The salient rotor. c) The stator. d) The The cycle in the torque vsspeed plan, and two representative
whole machine. points.
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T (Nm) 2.1.3 Optimization Constraints
Tb There are three kinds of optimization constraints. The first
one comes from the motor’s design requirements, which con-
sist of being able to reach the points (T , Ω ), and (T , Ω )
b b w w
required in the torque vsspeed plan.
T
w
Ω (rd/mn) Others constraints come from physical criteria, such as no
0 Ω Ω saturation for ferromagnetic parts, or limited temperature
b w
rises, or also avoiding demagnetization effects in the magnets.
Figure 3
(See Appendix for more details concerning these parameters).
Simple, but significant cycle in the torque vs speed plan.
The last constraint comes from the maximum allocated
(See Appendix for the numerical values of these
volume for the machine (see Appendix for data about the
requirements).
maximal allocated volume).
2.2 Solving the Problem with Evolutionary
Algorithms
hybrid vehicle. Nevertheless, such a work needs to take into
account the thermal transient states, which is not permitted
Once the optimization problem has been presented, we have
for the moment by our thermal model. That is why we have
to choose a solving method. In such complex and multi-
decided to perform the optimization for a very simple cycle, objectives problems, stochastic methods such as evolutionary
composed of two significant points in the torque vs speed algorithms are known to be efficient [1]. This is justified by
plan: one point characterized with a high torque and low the fact that the objective functions are not convex, so the use
speed (T , Ω ), and another point with a low torque and high of a stochastic algorithm is necessary to avoid local minima.
b b
speed named (T , Ω ) (Fig. 3), and these two points must be This method consists on choosing at the beginning a random
w w
reached during the same amount of time during the driving population composed of N individuals, each individual
indiv
cycle. This is a huge simplification of the problem. being a parameters array representing a machine topology.
Then the objectives’ values are calculated for each machine,
Moreover, the two points in the plan have to be reached in
and the “best” individuals with regard to these objectives will
continuous rate work.
get more chances to reproduce for giving the next generation.
Some good individuals can also be cloned, and random muta-
2.1.2 Optimization Parameters
tions can also occur with a low probability. After an impor-
The parameters will be the independent geometrical parameters tant number of generations, the optimal individuals are going
of the machine (See Appendix for more details concerning to be reached. The illustrated principle, for machine design,
these parameters). is given in Figure 4.
DATA:
– Materials Obj. 1
– Cycle in the torque vs speed plan
– Maximal volume Pareto’s
front
MACHINE’S MODEL Obj. 2
0
GEOMETRICAL
PARAMETERS EVOLUTIONNARY
Obj. 1: MASS
GEOM FUNCTION ALGORITHM END
Init.
TEST
LOSSES’
Min (Obj. 1, Obj. 2)
MINIMIZATION ON
Obj. 2: LOSSES (UNDER CONSTRAINTS)
THE CYCLE BY
THE COMMAND PHYSICAL
CONSTRAINTS
Figure 4
The machine’s design strategy.
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In fact, it is a bit more complicated (Fig. 5). It is true that
Geometry
the first objective, the mass, can be easily evaluated from the
geometrical parameters. Nevertheless, the losses are not only
a function of the geometry, but also a function of the way the
machine is commanded. It is why, in the model, there is also
a local optimization loop which can, given the geometry, Thermal Electromagnetic
determine the best command parameters. This command resistances model
parameter is in fact the phase displacement Ψ between the
0
electromotive force of the magnets, and the active current Magnetic flux
(Fig. 7). The other command parameter, which is the modu- density (B)
lus nI of the active current in a slot, is related to the angle Ψ Magnetic field
0 0 (H)
given the wished torque (2). The electromagnetic model will
ϕ P P
be presented in Section 3.1, for the calculation of the mag- a d q
netic field H, and induction B. The losses model will be
Resultant Torque
introduced in Section 3.4, for the evaluation of the copper
field
losses P, and the iron losses P . The calculus of thermal Command
c iron &
resistance is given in Section 3.5, as well as the thermal law Ω
Command
model, for the determination of the temperatures T in the calculus
different parts of the machine. Section 4.1 will present a
sensibility study on the nature of the command law.
Losses model
– iron losses (P )
iron
– copper losses (P)
3 THE MODEL OF THE MACHINE c
P and P
c iron
This paragraph’s purpose is to present an analytic model of
the axial flux permanent magnet synchronous machine. So it Thermal model
temperature
will be possible to carry out an optimal design, by submitting
this model to an optimization tool.
3.1 Simplifying Hypothesis
Are constraints
No respected? Yes
As it was said before, the stator ferromagnetic circuit is
laminated, so as to reduce eddy current losses. The windings
are put in slots. We only consider the electromagnetic active
parts, which are the ferromagnetic parts for the stator and the Command acceptation.
rotor, the windings and of course, the magnets. These active Return of the losses
and the constraints values.
parts are represented on a mean radius drawing in Figure 6.
– –
Let call R this mean radius, and τ = π/p.R the pole pitch at
this mean radius, where p is the number of pole pairs. The
– Losses (P + P )
relative length of permanent magnets with regard to the pole c iron
– Physical constraints
pitch is named β . This winding is composed of three – (temperature, B, H)
a1 Figure 5
phases, with, for simplification, only one slot per pole. The
number of turns per pole is called n, so each slot is filled with The detailed procedure.
2.n wires. The mechanical angular speed of the machine is
called Ω.
As it is often the case in electrical engineering, we assume – we consider that the motor is equivalent to a linear actuator
the following points, in order to simplify the problem: which would be obtained by developing the axial flux
– no saturation effect is taken into account, by limiting the machine at its mean radius.
induction in ferromagnetic parts under its saturation value. Under such hypothesis, the well-known Blondel’s theory
The magnetic permeability of the ferromagnetic parts is applied to salient poles synchronous machines permits us to
assumed to be infinite; draw the two electric equivalent circuits of Figure 7. There
– end effects are neglected; are two different equivalent circuits, since, in the case of
– we suppose that continuous rate has been reached, and salient poles machines, it is necessary to use the Park trans-
that the machine is driven with sinus current waves; formation, consisting on projecting the complex equations of
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O de la Barrièreet al. / PM Axial Flux Machine Design for Hybrid Traction 207
w The resistance rof one phase of the machine is very easy to
e
compute, as the leakage permeance P, which is classically
f
STATOR Tooth slot ehCeS eNveavluerattheedl ebsys, tthhee motahgenr eptaicr aemneetregrys , sstuocrehd a isn ϕ tfhMeAX ,s Ptadtoarn sdl oPtsq,.
and even r, will be more difficult to compute, and we need
AIRGAP a c ’ b a ’ c b ’ a e f
MAGNETS for that an electromagnetic model.
AND POLES ha
ROTOR N S eCR 3.2 Torque Calculation of the Machine
a1 By a simple power balance equation performed in Figure 7, it
The rotor moves with the speed is possible to compute the torque of the machine:
a2 V = RΩ at the mean radius R.
ϕ
Figure 6 T =3p fMAX nIo +3p(P −P )nIo .nIo
q d q d q
Sectional representation of the machine seen at the mean 2 (2)
radius. T =3pϕfMAX nnIocos(Ψ )+ 3 p(P −P )nIo2sin(Ψ )
0 d q 0
2 2
P p Ωnl P p Ωnlo
f• • d d• • d Since, thanks to the Park theory of electrical machines, we
r nloq have: ⎧⎪nIo =nIo⋅sin(Ψ )
⎨ d 0 (3)
nlq nlfq ⎩⎪nIo =nIo⋅cos(Ψ )
q 0
V r e The first term of Equation (2) is called the hybrid torque,
q f f
while the second term is often called the reluctant torque,
only present in salient pole machines. In fact, our electromag-
netic model, through the parameters ϕ , P and P , will
fMAX d q
a) permit to calculate this torque.
P p Ωnl
f• • q
r nl nlo 3.3 The Electromagnetic Model
d d
The goal of the electromagnetic model of the machine is to
nlf
d compute the electromagnetic field in the machine, which is
Vd rf Pq•p•Ωnloq useful both for knowing the iron losses and the non-saturation
conditions. Then, an integral of the field may give parameters
of the equivalent electric circuits of Figure 7, as the magnets
b)
flux, and the inductances. As we said, we are going to use an
Figure 7 exact analytical model [3], which is suitable for optimization,
Park equivalent electric circuits of the salient pole because it is faster to compute than numerical methods.
synchronous machine. a) The electric equivalent circuit for As it is usually the case in electromagnetism, we are going
the qaxis. b) The electric equivalent circuit for the daxis.
to compute the vector potential, rather than computing
directly the induction levels. Thanks to the simplifying
hypothesis, the problem can be reduced to the determination
of the following vector potential:
the machine along two different axis, which are usually (cid:2)(cid:3) (cid:3)
called the daxis, and the qaxis. Instead of directly using the A(x,y)=A(x,y).uz (4)
real supply voltage and current, we have chosen to use the
(x, y, and zare the space coordinates defined in Fig. 8).
voltage per turn, and the currents in a whole coil, in order to
We present here the problem to solve. Figure 8 represents
be independent of the real supply voltage value. Iron losses
the machine over one pole pitch. For simplicity, the number
are being taken into account, by a resistance r.
f q of stator slots per pole is chosen to be equal to 3, but its
s
In Figure 7, the electromotive force caused by the magnets value can be changed in the computation procedure.
in the stator winding, effective on the qaxis of the Park base,
We compute both the magnetic field produced by the
is called e. Saying that ϕ is the maximal flux under a
f fMAX magnets and the one due to the stator currents. In this way, a
machine’s phase, e is equal to:
f general expression is derived where, if we only need the no
ϕ
e = fMAX pΩ (1) load magnets’ flux, the currents’ values must be set to zero.
f 2 For the stator reaction flux, a zero value is given to the
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τ (pole pitch), with q , stator stlots The last Neumann condition is due to the infinite perme-
s
Y ability of iron. We solve the problem using developments in
the eigen functions of the potential [4, 5]. The expression, in
STATOR YOKE
zone I (for a slot number l, l being between 1 and q), has
s
been found as: (Eq. 7).
Slot 1 Slot 1 Slot qs The numbers a (l)and a (l)will have to be determined by
0 m
the application of boundary conditions.
Zone I
The airgap potential has been determined in [5] as:
Current Current Current
denj1sity denjlsity dejnqssity –dd–An– = 0 A(II)(x,y)=∑bk(y)cos⎛⎝⎜(2k−1)πτ x⎞⎠⎟
k≥1 (8)
Stotaottohr Zone II –dd–An– = 0 +∑bk'(y)sin⎛⎝⎜(2k−1)πτ x⎞⎠⎟
k≥1
Rotor tooth Zone III MAGNET Thanks to the Laplace equation, we get:
⎛ ⎞ ⎛ ⎞
0 X b (y)=b(1)cosh⎜(2k−1)πy⎟+b(2)sinh⎜(2k−11)πy⎟ (9)
Displacement ROTOR YOKE k k ⎝ τ ⎠ k ⎝ τ ⎠
X
d
And:
τ (pole pitch) b ′(y)=b(3)cosh⎛⎜(2k−1)πy⎞⎟+b(4)sinh⎛⎜(2k−−1)πy⎞⎟(10)
Figure 8 k k ⎝ τ ⎠ k ⎝ τ ⎠
Machine’s geometry used in the electromagnetic field
The magnets zone potential (in Zone III) was found in [5]
computation.
under the following form (Eq. 11)where the M numbers are
n
dependant on the magnets remanent induction B as:
r
⎛ β ⎞
magnets’ remanent induction B. Taking into account the
statorslotting effect is important,r since it permits to perform M = 4 Br 1sin⎜⎜n a1 π⎟⎟
n πμ n ⎜ β 2⎟
a local and accurate computation of the machine electromag- 0 ⎝ a ⎠
2
netic behaviour, even the slots leakage flux lines.
In [5], all the unknown numbers in Equations (7, 9, 10)
Calling M(x) the magnetization of the magnet which a
and (11) were found by applying the boundary conditions of
remanent induction value B, the vector potential A(x, y) is
r electromagnetism, and forming a Kramer system. Once the
given by the following equations:
vector potential is known, the induction components can be
⎧⎪ΔA(III) =μ ∂M in zone III evaluated thanks to the formulae:
⎪ 0 ∂x ⎧ ∂A
⎨⎪ΔA(II) =0 in zone II (5) ⎪⎪Bx = ∂y
⎪ ⎨ (12)
⎩⎪⎪ΔA(Il) =μ0jl in zone I (slot l) ⎩⎪⎪By =−∂∂Ax
We now propose a verification using finite elements
Moreover, the assumption of ideal ferromagnetic materials
methods. We are going to plot the airgap induction component
imposes a Neumann condition along the iron’s edges (both
B(x, y = ha+ e), xvarying between 0 and one pole pitch τ.
for the stator and rotor): y
We performed a calculation using the analytical method, and
∂A =0 (6) another with a finite element software. Finally, the method
∂n can be tested by comparing the results with a finite element
iron
⎛ ⎞ ⎛ ⎞
A(I,l)(Xl,Yl)=a0(l)+∑am(l)cosh⎝⎜mwπ (y−(ha+e+he)))⎠⎟cos⎝⎜mwπ (x−(l−1)(we+wd))⎠⎟−−12μ0jl(y−(ha+e+he))2 (7)
m≥1 e e
A(III)(x,y)=c +∑⎡⎢c cosh⎛⎜n π y⎞⎟−μ βa2τ∑Mnn cos⎛⎜n π (x−X )⎞⎟⎤⎥cos⎛⎜nn π (x−X )⎞⎟ (11)
0 n≥1⎣⎢ n ⎝ βa2τ ⎠ 0 π n≥1 n ⎝ βa2τ d ⎠⎦⎥ ⎝ βa2τ d ⎠
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0.8
1.5
0.6
1.0
0.4
B (T)y0.5 B(T)y 0.2
ANA
0 ANA 0
FEM
FEM
-0.2
-0.5 0 0.02 0.04
0 0.02 0.04 0.06 Pole pitch τ (m)
Pole pitch τ (m)
Figure 9 Figure 10
Induction component By(x, y= ha+ e) caused by the magnets Induction component By(x, y = ha + e) caused by a stator
over one pole pitch. efficient current nI0d= 1000 Atr over one pole pitch, the rotor
displacement being equal to X = 0.
d
software. For this test, the chosen machine has the following why, as the saturation phenomena is not easy to take into
geometry: account with an analytical model, and also for optimal com-
– a magnet’s thickness of h = 3 mm, with a remanent mand parameters determination, this phenomena will not be
a
induction B = 1.2 T; taken into consideration. More precisely, the airgap induction
r
– an airgap of e= 0.002 m; under the non saturated hypothesis is deduced, and then the
– a pole pitch of τ= 0.05 m; flux in the ferromagnetic parts is calculated using flux con-
servations law. At the end, the maximal induction in the fer-
– the stator and rotor yoke have the same thickness e = e
cr cs
romagnetic parts will be constrained to non-saturation values.
= 0.01 m;
This constraint is not a physical constraint like the tempera-
– the slots’ dept is h = 0.03 m, and the relative slot opening
e ture limit, but that is rather a “model constraint”, so as to
is 2/3. This signifies that, if we call w the slot width and
e keep valid the developed model.
w the tooth width, we have the ratio w/(w + w ) = 2/3.
d e e d
Moreover, the stator presents three teeth and slots over the Once the induction in the airgap is known, the inductions
pole pitch; in the ferromagnetic parts of the machine can be evaluated by
– the parameter β is equal to 0.8; flux concentration considerations [6]. We take the example
a2
of the stator yoke flux calculation. In the stator teeth, the
A (0z) length H is supposed to be long enough for
induction component is assumed to have just a single compo-
applying the 2D approximation.
nent, located along the (0y) axis. The calculation method is
The results are shown in Figure 9 for the magnets excita-
illustrated in Figure 12. Once, for each coordinate on the
tion, and in Figure 10 for an effective current per slot of nI
0 tooth Y, the induction component B(Y) is computed using
=1000Atr, the rotor displacement being equal to X = 0. The t t t
d flux concentration consideration: the vector potential compo-
analytical results are in good accordance, and the computa-
nents on the left A(Y) of the tooth, and A(Y) on the right of
tion time is nearly divided by two with the analytical model. l t r t
the tooth are taken from the previous airgap analytical model.
Nevertheless, this analytical model does not permit to take
Then, the following formula is applied:
the saturation into account. The same induction component
induction component By(x, y = ha+ e) obtained with the ana- B (Y )= Al(Yt)−Ar(Yt) (13)
lytical model and saturated finite elements are plotted in t t w
e
Figure 11. So as to reach saturation in the iron parts, the
effective current per slot of nI was put to = 3000 Atr. It can In the stator yoke, another method has been used, based
0
be seen that the induction levels are in the saturated case on the continuity condition on the potential between the air-
overestimated by the linear analytical model (by integration gap and the ferromagnetic yoke [6]. The calculation principle
we find that there is an error of 20% on the total flux). That is is explained in Figure 13.
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210 Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 1
Y
NO SAT. cs
SAT.
1.5
Tooth pitch (width τ )
s
A = 0
e
cs
N Stator yoke
1.0
T) X
B (y 0cs We Q Wd R cs
0.5 Tooth 1
A
A (X) A (X)
OQ RS
0
0 0.01 0.02 0.03 0.04 0.05
X
Half pole pitch (m) 0 cs
Figure 11
Figure 13
Induction component B(x, y = ha + e) caused by a stator
y Calculation of the (B, B) induction components in the stator
efficient current nI = 3000 Atr over one pole pitch, the rotor x y
0d yoke using flux conservation laws.
displacement being equal to X = 0, obtained with the linear
d
analytical model on one hand, and saturated finite elements
on the other hand.
0.15
Y
t
0.10
Tooth pitch (width τ)
s
0.05
Stator yoke
Tooth
T) ANA
B(Y) B (y 0 FEM
t t
A(Y) A(Y)
l t r t -0.05
Y
t
X
0 t -0.10
t
Tooth width we -0.15
Figure 12
-0.1 0 0.1
Calculation of the Byinduction in the stator teeth using flux Bx (T)
conservation laws.
Figure 14
Calculation of the no-load (B(t), B(t)) induction locus in the
x y
middle of the stator yoke using the analytical model and
finite elements.
On the stator yoke segment [OQ] between the stator yoke
and the slot, the evolution of the potential is known, from the
calculations in the airgap. On the segment [QR] between the
stator yoke and the tooth, the evolution of the potential is be found. As an example, the no-load flux locus (B(t), B(t))
x y
assumed to be linear, since the B induction component in the in the middle of the rotor yoke has been computed, both with
y
tooth was assumed to be uniform. Outside the yoke, the the analytical model and finite elements. The results are
potential is assumed to be zero. Then, the potential is known shown in Figure 14, and the agreement between the two
at every point of the boundary of the stator yoke. Given the methods is satisfying. This method has the advantage of
Laplace equation of the vector potential, the eigen function giving the two components (B (t), B (t)) of induction in the
x y
method permits to find the vector potential in the yoke by stator yoke.
solving the Dirichlet problem, and then the two components A similar method is used in the rotor yoke and saliency.
(B, B) of the induction at each point of the yoke. This calcu- This is useful for writing a correct no-saturation condition of
x y
lation has been repeated for each elementary position of the the rotor ferromagnetic parts. Nevertheless, the time varia-
rotor over one pole pitch, and then the time dependency can tion of the rotor induction is neglected, and iron losses are
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O de la Barrièreet al. / PM Axial Flux Machine Design for Hybrid Traction 211
computed only in the stator. It can be seen that the induction The simplifying assumptions of the thermal model are the
waveforms are not at all sinusoidal (the induction locus following:
presented in Figure 14 is not elliptic). So an iron losses – continuous rating is assumed to be reached;
model giving the iron losses even under non-sinusoidal – since the machine is water-cooled, we suppose that the
waveform is required. water circulation is the only way to cool the machine. All
other convective heat transfer, such as direct air cooling of
the winding overhangs, are assumed to be negligible. The
3.4 Copper and Iron Losses
convective thermal resistance between the stator yoke and
We are going to compute the copper losses and the iron water is assumed to be zero (perfect thermal contact);
losses. – heat radiation transfer is also neglected;
Copper losses are given by the formula: – the thermal model, just as the electromagnetic one, is
( ) assumed to be bi-dimensional, and developed at the
PC =3.r nId2+nIq2 (14) machine’s mean radius;
The resistance rincludes the one of the winding overhangs. – the heat transfer in the ferromagnetic stator yoke known
We recall that the currents have been evaluated thanks to the for having a good thermal conductivity λ = 72 W/(K.m),
iron
optimization of the command, once the geometry is given. is assumed to be uni-dimensional along the (0y) axis. This
The deduction of iron losses per unit volume at the point assumption will be verified using thermal finite elements;
(x, y) will be made by using Bertotti’s classical formulation – the slot zone is assumed to be homogeneous. The thermal
[8, 9], taking into account the non-sinusoidal properties of the conductivity of this zone composed of the wires, the insulator
waveforms, as well as the minor loops: around them, and air between the wires is evaluated
p (x,y)=k .ΔB .f +k .ΔB 2.f experimentally. A numerical value of λslot = 0.04 W/(K.m)
iron H pp H pp has been found;
1 2
∑
+2. k .ΔB .f +k .ΔB 2.f – it is also assumed that copper losses are more difficult to
H pp H pp
1 ML 2 ML (15) evacuate than the iron losses, so that the hottest point of the
minor loopps
stator yoke is located in the middle of the slot (Fig. 15);
⎛dB⎞2 ⎛dBB⎞32
+α .⎜ ⎟ +k .⎜ ⎟ – the losses are assumed to be uniform in the stator ferro-
p ⎝ ⎠ exc ⎝ ⎠
dt dt
eff eff magnetic tooth and yoke, taken equal to their mean value
where k and k , α and k are constant representative of given by the iron losses model.
h1 h2 p exc
respectively hysteresis losses, classical losses and hysteresis Assuming an unidirectional heat transfer, the thermal
losses in the ferromagnetic material. resistances R and R are computed using the classical for-
THy THi2
To take into account the two dimensional property of the mulae of conductive heat transfer, calling λ = 0.143 W/(K.m):
iso
induction, as it is often done in electrical engineering, we
1 e
R = cs (16)
independently compute the losses in the two directions with THy λ 1 (w +w )⋅ΔR
Equation (15), and then sum up the losses along the two axis. iron 2 e d
This problem does not arise in the teeth zone, since it was And:
assumed in the model that the induction was only along the R = 1 ( eiso ) (17)
(Oy) axis. Nevertheless, it is useful in the stator yoke losses THi2 λiso we −e ⋅ΔR
computation. 2 iso
The total iron losses P in the stator are then evaluated The other thermal resistances are more difficult to compute,
iron
by integration on all the stator volume. This value is useful since they involve bi-directional heat transfer. The calcula-
for the computation of the resistance r in Figure 7. tion is performed using the eigen function development of
f
the Poisson equation describing the conduction heat transfer.
The constant used in this model about the materials are
Calling p and p the losses volume densities in the slot and
given in the appendix. C T
in the tooth, these equations are:
⎧
p
3.5 Thermal Model ⎪ΔT =− C in the slot
λ
⎪⎪ slot
We are going to build a thermal model with the assumption ⎨ΔT =0 in the insulatorr (18)
of continuous rate. So it will be possible to calculate tempera- ⎪
ture rises in the different parts of the machine. Then, the cop- ⎪ΔT =− pT in the tooth
⎩⎪ λ
per wires temperature is going to be constrained to an upper iron
value during the optimization process, so as to avoid the After solving this differential system, the thermal resistances
destruction of the wires insulator material. can be evaluated.
17_ogst09045 18/02/10 12:47 Page 212
212 Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 1
Water circulation for cooling Then, using the lumped thermal model of Figure 15 (p ’ is
C
the copper losses for a half slot, p is the losses for a half
T
Tw tooth and pYrepresents the yoke losses in a half tooth pitch),
the maximal temperature of the copper can be calculated.
P YOKE
y R To verify this thermal model, a 2D thermal simulation has
e THy
cs been carried out. We take the following geometry:
– the airgap dimensions are the same as the ones used for
the electromagnetic simulations;
INSULATOR R – the stator yoke thickness is e = 0.01 m;
THi2 cs
– the losses densities in the yoke, the slot, and the tooth are
taken equal to, respectively: p = 103 W/m3, p = 105
Y C
W/m3, and p = 104W/m3;
T
R
THd1 – an insulation thickness e = 2⋅10-4m;
R iso
THc2
– the slot dept h varies from w to 10w. This will permit to
e e e
understand what happens if the temperature drops along
the ferromagnetic parts is not taken into account, that is to
say if λ is taken as infinite.
h -e iron
e iso R R R
THc1 THi1 THd2 For each geometry, the maximal temperature of the copper
T
c
is calculated. The result is given in Figure 16.
Although the 1D heat transfer assumption in the stator
P’ P
c T yoke, the results obtained with thermal finite elements and
the analytical model are very similar. Moreover, it becomes
clear that, even if the iron thermal conductivity is high, taking
HOTTEST
it as infinite may lead to significant error, particularly if the
POINT COPPER WIRES TOOTH
slot dept h becomes high. This is very important in an opti-
e
mization purpose, since such an error may lead the optimiza-
W /2 e W /2
e iso d tion algorithm to propose very narrow slot geometries,
although this leads in reality to excessive heating of the
wires.
Figure 15
Thermal model of a half slot and a half tooth. 4 RESULTS PRESENTATION AND ANALYSIS
In this paragraph, we are going to present, and discuss some
240 optimization results.
In a first part, we are going to study the influence of the
chosen command law on the machine design. Then, the
220
nature of the magnets is going to be studied. No inverter
restriction is being taken into account.
C)200
(°X 4.1 Influence of the Machine’s Command Law
A
M
Tc180 As we said in the previous paragraph, the local optimization
procedure of the machine can compute the optimal com-
ANA.λ Inf
iron
mand law for the two working points given the geometrical
160 ANA.λiron = 72 W•m–1•K–1 parameters. This optimal command law finds the command
FEM
parameters satisfying all the physical restrictions and the
140 torque requirements, while giving the minimum sum of iron
0 2 4 6 8 10 and copper losses. We wonder what would happen if, as it is
K = h /τ
e s often done for simplicity, we choose to minimize only the
Figure 16 copper losses, and not the sum of iron losses and copper
losses. We call Ψ the phase displacement between the
Evaluation of the maximal temperature of the copper, using 0
analytical methods, and thermal 2D finite elements. active current nI0 and the magnets electromotive force ef.