9. NUMERICAL TECHNIQUES
1
Reduced Thermal Model for Stator Slot L. Idoughi1 , X. Mininger1 , F. Bouillault1 and E. Hoang2 1
LGEP (CNRS(UMR 8507) ; SUPELEC ; Univ Paris-Sud ; UPMC Paris 6), Plateau de Moulon, 11 rue Joliot-Curie ; F-91192 Gif sur Yvette Cedex ; France
[email protected] 2 SATIE, ENS Cachan, CNRS UMR 8029, UniverSud, 61 av President Wilson, F-94230 Cachan Cedex, France
Abstract— This paper presents a method to get a reduced thermal model simplifying the calculation of different temperatures in an electrical machine winding. The equivalent thermal conductivity is deduced from a homogenization of the winding, and a discretization is achieved using the Finite Integration Technique. The model is then reduced, and the corresponding results are compared with finite element simulations.
II. E QUIVALENT THERMAL CONDUCTIVITY OF THE SLOT In this study, we consider a slot, made up of only two materials. The first one corresponds to the copper conductors (thermal conductivity λ1 ), randomly distributed, and placed in a second material, that can be air or resin (thermal conductivity λ2 ). Considering an isotropic distribution of 2D cylindrical conductors, [1] have shown that an estimation of the effective conductivity can be obtained with the estimation of Hashin and Shtrikman. The equivalent thermal conductivity λeq is expressed: (1 + τ )λ1 + (1 − τ )λ2 λeq = λ2 . (1) (1 − τ )λ1 + (1 + τ )λ2 with τ the occupancy rate of the conductors in the slot. Fig. 1 presents the corresponding results, considering either resin or air in the slot. The increase of the equivalent thermal conductivity with the occupancy rate is important with the resin, because of its high thermal conductivity comparing to the air one. In the next parts, an occupancy rate of 55% is considered, with resin around the copper (λeq = 0.87).
Equivalent thermal conductivity (W/m/K)
I. I NTRODUCTION A current tendency in electrical engineering is the use of electrical actuators in rough conditions, particularly in high temperature. For such applications, precise thermal models integrating thermal material (electric insulates and magnetic materials) properties are necessary to describe the system behavior. Thus, one of the main points in the thermal study of electrical machines concerns their winding, where the temperature rises to the maximum. However, the use of numerical tools like finite element methods to estimate the hot spot in the slot can lead to excessive simulation time, due to the heterogeneous structure and the presence of electrical conductors with small geometric dimensions compared to the machine ones. In this paper, a reduced thermal model is proposed to determine different temperatures (maximal, average...) in the slot. The first step corresponds to an thermal homogenisation of a simple geometry slot. Next, the Finite Integration Technique (FIT) is used to establish the reduced thermal model.
2.5
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1.5
resin
1
air
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Occupancy rate
Fig. 1.
Equivalent thermal conductivity as a function of the occupancy rate
III. R EDUCED
THERMAL MODEL OF THE SLOT
A usual solution to get simple thermal models of electrical machines is the nodal approach, considering the elementary volume as isotherm at the temperature of the associated node. The thermal models often consider only one node in the slot, giving one temperature of the winding [2]. The main difficulty of this problem is the determination of the conductance between the two nodes, which depends on geometrical dimensions and thermal material properties, with the choice of the length and surface that have to be considered to get a correct representation of the thermal flux distribution. Fig. 2 shows a static finite element simulation corresponding to the thermal problem (2, with ϕ the thermal flux, P the heat losses), where Dirichlet boundary condition (T = 0) is imposed on the upper side of the structure, and Newmann boundary condition on the other sides. div(~ ϕ) = P (2) −−→ ϕ ~ = −λ.gradT Due to the high iron conductivity, the corresponding part of the machine is quite isotherm, and can therefore be right represented with only one node. On the other hand, the temperature gradient is important in the slot, where the maximum temperature is reached. Thus, the model has to be more detailed in this area. To treat this problem, we use the FIT method, which is well adapted to highlight the concept of thermal resistances of the different parts of the slot. The FIT method transforms the thermal equations in their integral form into a set of matrix equations on dual grids pair [3]:
9. NUMERICAL TECHNIQUES
Fig. 2.
2
Fig. 3.
Reduced thermal model
Fig. 4.
Nodal discretization of a half slot
Temperature distribution, FE method
CΦ = P and H = C T T
(3)
where C is the discrete divergence matrix and −C T the gradient one. Φ is the thermal flux vector through the facets of the dual mesh, H the thermal grid voltage and T the temperature at the nodes of the primary mesh. To complete the system, we add a behavior law (H = M Φ). In the case of orthogonal grids, the M matrix is diagonal and mii is equal to: Z 1 1 ~τ .d~γ (4) mii = SL L i λ where λ is the local thermal conductivity, Si the surface of facet i, Li the lenght of edge i, ~τ the unit tangent vector along the edge i. From (3) and the behavior law, the system becomes: [C][M ]−1 [C T ][T ] = [G][T ] = [P ]
(5)
If the domain relative to the thermal fluxes is the slot, the unknown temperatures near the boundary are linked to those in the iron, supposed to be at the same temperature Tf . If we suppose that Joule losses P0 are uniform in the slot, P can be expressed as [P ] = P0 [D] with Di = Si /S, S being the total surface of the slot. Equation 5 can be written [Ts ] = −Tf [Gs ]−1 [Gf ] + P0 [Gs ]−1 [D]
(6)
Due to the property of matrix [G], [Gs ]−1 [Gf ] is a vector [I] with all elements equal to 1, and the system becomes: [Ts ] = −[I]Tf + P0 [R]
(7)
R is the column vector giving the equivalent thermal resistance between the considered point in the slot and the one in the iron. The different equivalent thermal models of the devices using two nodes are represented Fig. 3. The resistance Rf includes the effect of conduction in the core and the convection exchange with the exterior, supposed to be at temperature Te . Considering a 9-nodes model for a half slot, the resulting nodal locations are shown Fig. 4. Depending on the choice for the node i, the temperature Ti can have different meanings:
maximum temperature in the slot (node 7) mean temperature of the slot (obtained by the mean of all the equivalent thermal conductances) The corresponding results are presented in table I, with a comparison with FE ones. The relative error is lower than 5 %. • •
TABLE I COMPARISON OF ANALYTICAL AND
Reduced model FE Method Relative error (%)
FEM RESULTS
∆Tmax ( ˚ C) 18.02 18.24 1.2
∆Tmean ( ˚ C) 10.96 11.43 4.3
IV. C ONCLUSION The results obtained in the static case with the reduced model presents a good accuracy, with low computation needs. In the full paper, an extension of the model for transient analyses will be established in order to get the temperature variations versus time. Ongoing works will concern the adaptation of this reduced model to structures more similar to usual electrical machines. R EFERENCES [1] L. Daniel, and R. Corcolle, A note on the effective magnetic permeability of polycrystals, IEEE Trans. on Magnetics, vol. 43, no. 7, pp. 3153-3158, July 2007. [2] P.H. Mellor, D. Roberts, and D.R. Turnerpoulos, Lumped parameter thermal model for electrical machines of TEFC design, Proc. Inst. Elect. Eng., pt. B, vol. 138, no. 5, pp. 205-218, Sep. 1991. [3] T. Weiland, A discretization method for the solution of Maxwell’s ¨ vol. 31, equations for six-component fields, Electron. Commun. AEU, no. 3, pp. 116-120, Sep. 1977.
