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Equivalent Thermal Conductivities for Twisted Flat Windings
R. Glises, R. Bernard, D. Chamagne, J. Kauffmann
To cite this version:
R. Glises, R. Bernard, D. Chamagne, J. Kauffmann. Equivalent Thermal Conductivities for Twisted Flat Windings. Journal de Physique III, EDP Sciences, 1996, 6 (10), pp.1389-1401.
�10.1051/jp3:1996192�. �jpa-00249532�
Equivalent Thermal Conductivities for Twisted Flat Windings
R. Glises, R. Bernard, D. Chamagne and J-M- Kaulfmann (*)
Institut de G4nie (nerg4tique, 2
avenue Jean Moulin, Parc technologique, 90000 Belfort, France
(Received 19 April 1996, revised 4 July1996, accepted 5 July1996)
PACS.44.10.+I Heat conduction (models, phenomenological description)
PACS.44.30.+v Heat transfer in inhomogeneous media, in porous media, and through interfaces
Abstract. The authors of this paper intend to develop a method of determination of equiv- alent thermal conductivities for twisted flat windings. The conductivities determined
are radial and parallel to the principal directions of the windings. A design has been realized thanks to the
thermal modulus of the computation software Flux2D using a finite elements method. Following
that phase, numerical correlations permitting to express the radial conductivities as a function of temperature, filling rate and insulation conductivities are proposed.
R4sum&. Les auteurs de cet article se proposent de d4veIopper une 4tude de d4termina- tion de conductivit4s thermiques 4quivaIentes d'empilements de bobinages plats torsad4s. Les
conductivit4s sont d6termin4es darts Ie plan radial (perpendiculaire h I'axe des bobinages) et
parallAlement aux directions principales de la structure. La m4thode utiIis6e est exclusivement
num6rique et est r6alis6e h I'aide du Iogiciel de calculs bidimensionnels par 6I6ments finis Flux2D.
Des corr6Iations num4riques exploitables permettent d'obtenir directement Ies conductivit4s ra- diaIes en fonction du taux de remplissage, de la temp6rature du milieu et des conductivit6s des
isolants 6Iectriques.
Nomenclature
D Diameter (m)
0T/0i Temperature gradient (K m~l)
e, L, H Length (m)
#, #" Heat flux densities (W m~2)
[ Conductivities (W mK~l)
[[ Conductivities matrix (W mK~l)
q Heat flux (W)
q~/S~ Heat flux density (W m~2)
Rth Thermal resistance (K W~l)
S~ Surface (m2)
T Temperature (K)
TR, TR' Filling rates (without dimension)
X, Y Air thickness (mm)
(*) Author for correspondence
@ Les (ditions de Physique 1996
1. Introduction
The analysis and the design of electrical machines come necessarily through the study of their thermal behaviour. This study turns out to be essential because a small temperature increase beyond the normal operating level may decrease the life duration for the windings with a factor 10.
The required quality of such studies can only be obtained thanks to powerful data processing tools. Indeed, simulation coupled to experimental tests seems to be one of the most economical way, the fastest, and the most flexible of use to determine and forecast the temperature at each point with a great accuracy for different operating rates and power supplies. To valid such
a method, it is necessary to determine with a good accuracy the different thermophysical parameters of a machine like the thermal contact resistances, and first of all, the thermal
conductivities.
As all the unhomogeneous structures of the motor (iron, thermal contact resistance.... ), windings constitute a very important problem. Indeed, they are composed ofcopper,insulation
and air. Moreover, important temperature gradients are often applied to such assemblies, changing locally the value of the air conductivity. For all these reasons the computation of the local value of the conductitivity is not easy. An other reason of that difficulty is that many modes of heat transfer in the windings like conduction coupled ~vith convection and radiation
can exist. To resolve these problems, different methods have been tested.
The process currently used consists to replace the detailed model by macroscopic structures.
These lasts are often concentric cylinders, each cylinder corresponding to the real thermophysi-
cal and surface values of the presented material (air, copper and different wires insulations) [1j.
A second method developed in our laboratory has been applied to the thermal modelling of a 4 kW induction motor. Thermal conductivities of the materials and thermal contact resistances
are determined through t,vo experimental tests creating different overheatings. The first uses
a sinewave supply and predetermine the thermophysical parameters. The second uses direct current supply of the rotor and the stator to validate the parameters. In each case, validation is used to confirm the choice of the parameters by comparison with the temperature given by
thermocouples with results of modelling [2,3j.
A third method developed in this paper consists in replacing windings with a homogeneous isotropic material whom thermophysical parameters are represented with a second order tensor
[4,5j. The knowledge of that tensor is obtained by modelling. Indeed, when a heat flux is
applied to one face of that unhomogeneous material, the determination, by computation, of the temperature gradient in the axis of the heat transfer permits to estimate the macroscopic
equivalent heat conductivity with a great precision.
The determination of the equivalent conductivities tensor can be applied to every unhomo- geneous structure, knowing precisely the volume proportions and the thermophysical values of each basical component.
The work developed in that paper concerns the identification of the tensor of conductivities of twisted flat windings. These last systems are composed, in our example, of nine insulated
copper wires. Tensors oftwo extreme configurations ofthe windings are then given as a function of temperature, filling rate and electrical insulation conductivities.
2. General Heat Equation
2.1. FOURIER's LAW. The Fourier's law is used to determine the heat flux density crossing
a solid. In the case of a three dimension classical thermal study, for a homogeneous solid:
iii = -)j grad T (1)
In the location with respect to a O, z, y, z reference axis relation (1) becomes:
~ ~~~ ~~y ~xz ~~
Sx °~
) ~Y~ ~YY ~YZ ~~
(~j
Y fill
~ ~z~ ~zy ~zz °T
~~ $
Using now a O, u, ~, w reference coordinate system in which the axis are parallel to the principal directions of the solid, relation (1) becomes:
iii"
= II"] grad T (3)
whereas equation (2) gives:
u( ~ ~ ~ 0T
" au
fi o ~ o 0T
~» ~ 0~ (4)
°~
0 0 ~w 0T
~'° 0w
The thermal conductivities tensor associated to the principal directions is a diagonal mat.rix.
2-2. HEAT EQUATION TRANSFER. Knowing the thermal conductivities in the direction of the principal directions, by making a heat flux evaluation crossing a general solid, we obtain the following unlinear heat equation expressed in a general set of axis O, z, y, z:
~j S(j+dj)
~j ~j,~)
J"~<Y<~
"~<Y<~ ~~~~~~
S(j) £ ~j,~ )
+ qdzdydz
= pcpdzdydz. (5)
"~'Y>~
~
~
In the case of S(z)
= Sly)
= S(z), constant conductivities and a steady state heat transfer mode, the last equation becomes the following linear equation:
02T 02T 02T 02T 02T 02T
~~~
0x2 ~ ~"
0y2 ~ ~~~ 0z2 ~ ~~~~
0x0y ~ ~~~~ 0x0z ~ ~~~~
0y0z ~' ~~~
In the system associated to O, u, ~, w, the heat equation leads to equation (7) in thermal steady state without internal power source:
~u~(
+ ~~~(
+ ~wj
= 0. (7j
t
w~
O
Fig. I. Windings represented in the reference frame O, u, u, w.
C
ai)~~ ~
Fig. 2. Radial studied structure.
3. Determination's Principle of Thermal Conductivities Using a Finite Elements
Modelling Method
3.I. RADIAL STUDY OF A BUNDLE OF CoNDucToRs. The software used to study that
radial geometry is the Flux2D finite elements thermal calculus software. The equation treated is the linear heat transfer equation in the case of steady state thermal behaviour. Figure
shows a bundle of four wires in the set of axis O, u, ~, w.
As shown later in that paper, the determination of the equivalent conductivities in the axis O, w does not present any problem. Indeed, it requires simply the knowledge of the volume of
each component and their conductivity value.
Taking into account geometrical and heat flux symmetries, the model really studied is rep- resented in Figure 2.
To determine the conductivities along the u and ~ axis, the method consists to transfer a heat flux in a parallel direction to the two precedent axis. To force that flux in the interesting axis, it is necessary to insulate the faces as shown in Figure 3. Then, the knowledge of the
temperature gradient allows to determine the equivalent conductivities.
As shown in Figure 3, three faces of the element are insulated and the boundary conditions
are the Neumann homogeneous conditions (0T/0n
= 0). These three insulated faces "push"
the heat flux in the direction O, u. It is necessary to take a very high conductivity value for the thermal source area. Indeed, that condition allows to obtain a relatively uniform temperature profile on the interface at u
= 0.
The method imposes Dirichlet boundary conditions at ~
= L (knowledge of the temperature T). It is possible to take any temperature but the best is T
= 0 °C so that the tempera-
ture gradient becomes very easy to calculate. However it is more comfortable to take that temperature uniform on all the surface of application.
v
H Insulation
@~mm dT/dn = 0
T(K)
o L
Copper wire ° ~
EIect1icaI insulation
Fig. 3. Structure studied with FIux2D.
Knowing the average temperature at u = 0 it is then possible to calculate the equivalent conductivity along the studied axis with the Fourier's law:
~" ~ )u)
(T(u = L) T(u
= 0))' ~~~
S(u) is calculated for an unitary length along the w-axis (not represented in the 2D study).
A totally similar methodology is applied to determine the equivalent conductivity along the
~-axis. Then, if the vertical length of the structure represented in Figure 3 is H, the expression of the conductivity along the vertical axis takes the following form:
~ dq H
" At lTl~
= H) Tj~
= o))' 19)
The equivalent conductivities depend on the values of local conductivities (copper, insulation and air) used for the resolution of the heat transfer. So, many tests with different temperatures have been necessary to determine the evolution of ~u and ~» with temperature. Indeed, air
conductivity depends considerably on temperature. However,in each different studied case, the air conductivity is considered as independent of the temperature gradient. The thermophysical
parameters take range from 25 °C to 225 °C.
A second evolutionary parameter studied is the filling rate. This last represents the ratio of copper volume over the whole volume. For a 2D study, the surfaces are considered. The
filling rate depends on the assembly pressure. The lowest possible value tends to 0i~, whereas the highest tends to 100$lo, but these two values cannot be reached. Referring to Figure 2, we
can see that it is possible to obtain two equals values of filling rate with two different (a,b)
coordinates.
4. Description of the Real Studied Windings
4.I. SIMPLIFICATION OF THE MODELS. The studied winding corresponds to the stator of
an electrical wheel motor at present developed in cooperation with an industrial partner.
The considered windings are composed of flat twisted bundles like shown in Figure 4. The
winding is composed of nine insulated copper wires. These last are pressed thanks to a double
View 4
;
;
View3
3.Smm
j~i~tjtep
~
View2
view I view 2
2 3 4
8 7 s 5
view 3 view 4
View
Fig. 4. Bundles studied.
Configuration A
~
Configuration B
Fig. 5. FIux2D symmetries modelling.
thickness insulated paper with mica. The twist step along the axis is 3.4 cm long. A similar position of two different following wires is obtained each 3.8 mm length as shown in following figure.
Figure 4 shows different radial cross-sections. Each view represented in Figure 4 will not be studied. Indeed, heat flux and geometrical symmetries allow to simplify considerably the studied structures. For that reason the four views 1, 2, 3 and 4 (cross-sections) represented in
Figure 4 are replaced by the two configurations A and B in Figure 5. By taking into account these two simplified configurations, every precedent cases can be studied. Homogeneous Neu-
mann boundary conditions are then applied to each symmetry plan. The reason of the choice of these two configurations comes from the fact that they represent the extreme geometrical
area (highest and lowest compactness).
Mica Air
p D
= 0.6 ~° ~~
2 ( ( Electrical insulation
jfl I D
= 0.66
3.37 Wl~~
ConfigurationA
&~ (
~g -
lQ ~
-~--~---~
~
ConfigurationB
Fig. 6. Dimensions of the configurations.
V
C°nfig.A 3l~
.Y
.a
-
W u
b X
air V
-Y -a
ConfigB
~-~.
~ U
b X
Fig. 7. Simplification of the eight winding bundles (cases A and B).
4.2. DETAIL oF STUDIED STRUCTURE. Figure 6 gives exact dimensions of the two studied structures.
4.3. PROBLEM OF THE FILLING RATE. The filling rate is defined as the ratio of the copper volume to the whole volume of the studied structure. For a 2D study, the volumes become surfaces. If TR is the filling rate for base configurations A and B and Scopp~r, Sajr, Srrica and Sinsuiation respectively the surfaces of copper, air, mica and electrical insulation, TR is defined
as:
~
~~
Scopper + Srrica~~~~ir + Sinsulation ~~~~
It can be remarked that for the configurations A and B represented in Figure 6, the filling rate is about constant (respectively 51$lo and 52.9i~). However, each winding in the slot is made of
Table I. Conducti~ity ~aiues.
Dry air conductivity as function of temperature Wires insulation Mica
25 °C 75 °C 125 °C 175 °C 225 °C
~W~~mK~~ 0.0262 0.0303 0.03365 0.03707 0.04038 0.15 0.30 0.44 0.15
eight bundles of ~vires (eight base configurations) distributed on four lines and two columns.
In that new and complete study (slot configuration), the new filling rate is noted TR' and
becomes, with the same notations as before:
~~i Scopper
(scopper + smica + sair + sinsulation + (xja + Yj + Yb)j' ~~~~
4.4. THERMOPHYSICAL PROPERTIES OF EACH ELEMENT. The conductivities ofeach com-
ponent have been considered as constant with the temperature for every test, so the gradient of temperature during the modelling was not considered ~ersus conductivities. However, each
kind of tests has been executed considering five different average temperatures which take
range from 25 to 225 °C increasing every 50 °C. Concerning the insulation of the wires, three
conductivity values are taken while the conductivity of the mica paper stays constant. Table I gives the chosen values.
The conductivity of copper cannot be considered as a function of the temperature between 25 and 225 °C and is fixed at 386 W mK~~
4.5. ANALYTICAL DETERMINATION OF THE AXIAL CONDUCTIVITIES. For untwisted wind-
ings, the determination of the axial conductivities does not constitute any problem. Indeed, that corresponds to the case of materials arranged in parallel paths. By applying the thermal
resistance concept, we obtain the value Rth as Rth
" e/(~S) where e is the length of the con-
sidered material, ~ its local conductivity and S the surface of the radial cross-section (taken perpendicular to the w axis). If I is the total number of materials in the structure, it follows:
Rth total ~~S~
Hence
~ ~copperscopper + ~airsair + ~insulationsinsulation + ~micasm~ca
~~~~ ~~~~ j~~~
Scopper + Sair + Sinsulation + Smica
In this study, it could be interesting to calculate the equivalent axial conductivity for each axial cut. However, calculus show that the variation between extreme values does not exceed 4%. That small difference comes from the respective differences of the air and mica surfaces in different radial cross-sections.
5. Numerical Relations of Radial Conductivities
Numerical correlations are proposed in that part. It concerns the two configurations A and B.
For each case, the conductivities are given along the two principal radial axis O, u and O, ~.
So ~u and ~» are expressed as function of temperature and parameters X and Y. For both
configurations, these two last parameters are then written as a function of the filling rate.
5.I. METHOD APPLIED TO FIND NUMERICAL CORRELATIONS GIVING THE EQUIVALENT CONDUCTIVITIES. At a temperature Ti and a length X, ~u is expressed by a linear expres- sion of Y:
~u = Gi (X)Y + G2(X), (14)
Gi(X) and G2(X) are functions of X and it is supposed that a third-order polynomial is sufficient.
3
Gi(X)
= ~ ajXJ (15)
j=o 3
G2(X)
=
~ja(XJ. (16)
j=o
At a given temperature, it is necessary to know the values of ~u for 4 length X and it is
interesting to take an equidistant interval X~
= (iXmax)/4
= (I/10) mm in our case.
If we suppose that ~u varies linearly with temperature T, coefficients aj and o( depend also linearly on T. Extreme values 25 °C and 225 °C have been choosen. ~u is put in the following
form:
1 3
~u = ~ ~(akjT + bkj)X~YJ. (17)
k=o j=o And in the same way:
1 3
~» = ~ ~(akjT + bkj)Y~XJ. (18)
k=o j=o
5.2. CORRELATIONS RETAINED. Tables II and III give the akj and bkj for both configura-
tions A and B.
The last column of Tables II and III gives the maximum difference between modelling results and values obtained with correlations.
5.3. EXPRESSION OF THE FILLING RATE AS FUNCTION OF X AND Y FOR BOTH CONFIGU-
RATIONS
TR = (-7.84X + 17014)Y~ + 0.72YX~ + 16.BXY 14.758X 33.824Y + 0.51.
The maximum difference ofvalues obtained with that correlation and the real models does not exceed 2%.
6. Modelling Results and Graphic Representations
Figure 8 shows the evolution of ~u and ~» with the temperature for both configurations in the
case of X
= Y = 0 corresponding to the lowest filling rate. Parameters ucondl(T), ~condl(T), ucond2(T) and ~cond2(T) are ~u and ~» for the configurations A and B respectively. The different curves come from linear approximations of conductivities when temperature rises from 25 °C to 225 °C. Indeed, dry air conductivity rises with temperature.
Inspection of Figures 9 to 12 concerning configuration A reveals that the computed values of thermal conductivities are maximum for X
= Y
= 0 and minimum for X
= Y = 0.4 mm.
In fact, these two geometries correspond respectively to the lowest and highest air filling rates
and the air conductivity is the lowest that we can find in the structures. Temperatures of
ci ci
ci ~ ~ ~
c~ ~
c~ c~ II II II
e II E E ~ II ~
ii ~ ii 1 )
~ ~ ~ ~ ~
~li d ~li ~li d ~li
II II '( ~ ~ II f~
II ~ II
~ m ~ 4 1~ Ci
r-
x LD
g i$ ~ ~
~ Ci 4 Q
x x CQ
~
g g to
Ci Q Ci ~
~ ~
X ~t t
c~
~ ~
o~
x LD ~
~
C~ St
$$ ~i I
~ ~ L~ L~ L~
z ~ ~ ~
~
~ LD
*f x ~Q ~i #
~ § i$ Ci
,~ ~ 'qg
+J ~
~ ~
§ ~
Qb ~ c~~
~C~ ~ LD
~ x ~ ~ ~ ~
~
~ ~ ~ ~ ~
i [ I L51 $ ~
~ ~
~
§D
~i ~j
~i l~ m to
x £ x ~ i
~ m fl ~ Q
~ i
£~
°M
B ? St LD St LD
p ~t t ~t t
~ / ~ / ~ ~ ~ ~
[ ~
~ ~ ~
~ ~ ~ ~
# # ~ ~ ~ ~
ucondllT) ___----.
vcondl(T) "~~~~
___.-.
0.35 _.-.-.."'"
ucond2lT) -."'
~~i"~~~~ 0.3 -....'~"' '~'
0.25
0 100 200 300
T rc)
config A (i) config B (z)
Fig. 8. Conductivities
as function of temperature, X
= Y = 0.
Conductivity along U
~ U
X=0.4
X Y=0.4
Fig. 9. ~~(X, Y) for an overheating AT = 0 K (T
= 300 K).
300 K and 500 K (corresponding to overheatings of 0 K and 200 K for an ambient temperature
to 27 °C) are studied. For 0.44 W mK~~
as insulation conductivity, an overheating of 200 K
creates the rising of12% and 9% respectively for ~u and ~». Absolutely similar curves are
obtained for configuration B. Such results are conform to results obtained in references [2j and [3j. Indeed, concerning windings of that 4 kW motor, the estimated values of the radial
conductivities are 0.5 W mK~~ with an equivalent copper wire diameter. In that case, the
filling rate is considered as maximum (X
= Y
= 0) and the different conductivities are not taken as functions of the temperature. Otherwise, we cannot extend the comparison because
we know approximatively the value of the electrical insulation conductivity.
