Research Papers: Heat Transfer in Manufacturing

Thermal Homogenization of Electrical Machine Windings Applying the Multiple-Scales Method

[+] Author and Article Information
Pietro Romanazzi

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: pietro.romanazzi@eng.ox.ac.uk

Maria Bruna

Mathematical Institute,
University of Oxford,
Oxford OX2 6GG, UK
e-mail: bruna@maths.ox.ac.uk

David A. Howey

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: david.howey@eng.ox.ac.uk

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received February 8, 2016; final manuscript received July 22, 2016; published online August 30, 2016. Assoc. Editor: Gongnan Xie.

J. Heat Transfer 139(1), 012101 (Aug 30, 2016) (8 pages) Paper No: HT-16-1066; doi: 10.1115/1.4034337 History: Received February 08, 2016; Revised July 22, 2016

Low-order thermal models of electrical machines are fundamental for the design and management of electric powertrains since they allow evaluation of multiple drive cycles in a very short simulation time and implementation of model-based control schemes. A common technique to obtain these models involves homogenization of the electrical winding geometry and thermal properties. However, incorrect estimation of homogenized parameters has a significant impact on the accuracy of the model. Since the experimental estimation of these parameters is both costly and time-consuming, authors usually prefer to rely either on simple analytical formulae or complex numerical calculations. In this paper, we derive a low-order homogenized model using the method of multiple-scales (MS) and show that this gives an accurate steady-state and transient prediction of hot-spot temperature within the windings. The accuracy of the proposed method is shown by comparing the results with both high-order numerical simulations and experimental measurements from the literature.

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Grahic Jump Location
Fig. 1

(a) Cross section of an electrical winding (courtesy of the Clarendon Laboratory, Oxford, UK). (b) Normalized steady-state temperature Φ = T/Tmax in a simulation of the middle cross section of a winding with internal heat generation and fixed temperature at the boundaries, obtained from the full model (1) and the homogenized model (2). In (b), l/L indicates the relative distance between adjoining wires.

Grahic Jump Location
Fig. 2

(a) Macroscopic domain ω, with insulated conductors in a square periodic lattice separated by a distance l≪L. (b) Microscopic domain or unit cell Ω with one conductor at its center of radius νc=εc/l surrounded by insulation to radius νi=εi/l.

Grahic Jump Location
Fig. 3

Solution of the two components of the cell problem (8)Γ1 and Γ2 according to the properties in Table 1

Grahic Jump Location
Fig. 7

Comparison of keq estimated via the MS method, using the square lattice cell, with experimental measurements fromRef. [14] with (a) copper and (b) aluminum conductors at various filling ratios

Grahic Jump Location
Fig. 4

(a) Temperature distribution obtained from the full model Θfull and the homogenized model ΘMS in the middle cross section of wires distributed in a square lattice with δ = 1/31, with internal heat generation and Dirichlet boundary conditions. (b) Relative error Θfull−ΘMS over a range of δ.

Grahic Jump Location
Fig. 5

(a) Microscopic domain Y∈Ω related to the hexagonal lattice wires and (b) Γ1 distribution for the case hexagonal lattice with μ = 0.5

Grahic Jump Location
Fig. 6

(a) Effective thermal conductivity keq over a range of μ using different approaches and (b) steady-state temperature distribution for random distributions at μ = 0.5

Grahic Jump Location
Fig. 8

Transient normalized temperature Θ=T/Tmax as a function of the Fourier number τ=t(keq/CeqL2) in the center of the domain obtained with the MS homogenized model and the full model using a square lattice wire distribution with δ = 1/11



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