Embed Size (px)
Citation preview
Thermal homogenisation of electrical machinewindings applying the multiple-scales method
Pietro RomanazziDepartment of Engineering Science
University of OxfordOxford, U.K., OX1 3PJ
Email: [email protected]
Maria BrunaMathematical InstituteUniversity of Oxford
Oxford, U.K., OX2 6GGEmail: [email protected]
David A. HoweyDepartment of Engineering Science
University of OxfordOxford, U.K., OX1 3PJ
Email: [email protected]
ABSTRACT
Low order thermal models of electrical machines are fundamental for the design and management of electricpowertrains since they allow evaluation of multiple drive cycles in a very short simulation time, and implementationof model-based control schemes. A common technique to obtain these models involves homogenisation of theelectrical winding geometry and thermal properties. However, incorrect estimation of homogenised parameters hasa significant impact on the accuracy of the model. Since the experimental estimation of these parameters is bothcostly and time consuming, authors usually prefer to rely either on simple analytical formulae or complex numericalcalculations. In this paper we derive a low order homogenised model using the method of multiple-scales and showthat this gives an accurate steady state and transient prediction of hot-spot temperature within the windings. Theaccuracy of the proposed method is shown by comparing the results with both high order numerical simulationsand experimental measurements from the literature.
1 IntroductionWith the increasing hybridisation and electrification of vehicle powertrains needed to meet efficiency and emissions
standards, models that can accurately estimate the thermal behaviour of electrical machines in a short computational timeare crucial, since torque density and machine lifetime are strongly temperature dependent [2]. Accurate and fast models willallow industry to speed up their design processes and push machines closer to their limits, reducing time to market and costs.The temperatures within an electrical machine strongly impact the machine efficiency and lifetime [3]. Studying the thermalbehaviour of a machine over multiple different drive cycles is as important as refining the electro-magnetic design and thecontrol strategy [4]. The hot-spot temperature estimation within the electrical windings ultimately limits the available outputtorque according to the insulation class and can be used for real-time control [5]. However, it is not practical to measurehot-spot temperature experimentally in most commercial machines, since a large number of temperature sensors would berequired. For this reason, validated mathematical models that can predict the entire temperature distribution are key. Thisis not a trivial task since the windings are a complex composite domain, composed of insulated conductors usually bondedtogether with varnish or epoxy, see fig. 1(a).
A common approach to finding the hot-spot temperature is to solve the thermal problem in the full domain whereevery single wire is modelled individually, using for example the Finite Element (FE) method [6]. One must solve for thetemperature T satisfying the heat equation in each of the phases
ρpcp∂T∂t
= kp∇2XT + q, (1)
HT-16-1066 - Howey 1
(a)
0 0.2 0.4 0.6 0.8 1X1
L
0
0.2
0.4
0.6
0.8
1
Full FE model - square lattice
Homogenised model through MS method
l
L
(b)
Fig. 1. (a) Cross section of an electrical winding (courtesy of the Clarendon Laboratory, Oxford). (b) Normalised steady-state temperatureΘ = T/Tmax in a simulation of the mid-cross-section of a winding with internal heat generation and fixed temperature at the boundaries,obtained from the full model (1) and the homogenised model (2). In (b), l/L indicates the relative distance between adjoining wires
where kp, cp and ρp are the thermal conductivity, heat capacity and density of the pth phase respectively, q is the sourceterm and ∇2
X stands for the Laplacian operator. However, the mesh sizes required to capture the thermal gradients betweeneach wire are unsuitable for fast simulations (see fig. 1(b)). The standard approach to overcome this is to homogenise thecompound and treat the mixture of conductors, insulation and bonding material as a continuum with equivalent thermalparameters [7–9], satisfying the following “homogenised heat equation”
ρeqceq∂T∂t
= keq∇2XT + qeq, (2)
where ρeq, ceq and keq are the equivalent thermal parameters of the homogenised domain and qeq is the effective heat source.The advantage of (2) over (1) is that, with the correctly homogenised parameters, it can be solved quickly with a low orderapproach across the entire domain of a slot while still predicting the hot-spot temperature accurately, as shown in fig. 1(b).This particular figure will be thoroughly analysed in the forthcoming sections. The complexity of the homogenisation proce-dure lies in correctly calculating the equivalent thermal parameters in (2) based on the bulk geometrical design, the materialproperties and filling ratio, in particular the equivalent anisotropic thermal conductivity keq [10, 11]. In the literature varioustechniques have been applied to evaluate keq of this compound, each with strengths and weaknesses. Experimentally, it ispossible to directly measure the conductivity in various directions [8,12–14], however this requires appropriate bespoke testequipment and suitably manufactured composite winding specimens, making the approach expensive and time-consuming.Alternatively, to estimate keq many authors have relied on numerical methods or analytical formulae (which must of course bevalidated using the kinds of experimental methods just described) [1, 15, 16]. The simplest and oldest formula, proposed byWiener [17], gives upper and lower bounds limiting the effective thermal conductivity, calculated respectively with paralleland series arrangements of conductor and filling phases. This simple calculation has been subsequently improved to accountfor the shape of the conductors, for example the Hashin and Shtrikman (HS) [18] formula accounts for circular conductors,whereas the Milton [19] formula, introducing the parameter ξ, accounts for different geometries.
However all of these approaches neglect the thin layer of insulation around the copper, which results in a more complexthermal problem. This electrical insulation, due its very low thermal conductivity (see table 1), acts as a good thermal insula-tor. This impacts the domain’s temperature field and accordingly influences the estimation of the homogenised conductivitykeq. Attempts by Kanzaki [20], Simpson [1] and the “multi layer” method by Nategh [9] to include a third phase for theinsulation only slightly improved the accuracy. It has been demonstrated [21] that the HS formula is reasonably good atestimating the homogenised keq for impregnated windings (when compared with experimental data), but it is not accurate fornon-infiltrated windings. This is because the material properties of the infiltration material (epoxy) and insulation materialare quite similar, whereas the properties of air are significantly different to insulation material.
The key limitation of these analytical formulae is their strong dependence on a specific wire geometry, wire distribution,and a set of material properties. Any modifications may only be valid within very limited ranges of geometries, materialproperties or filling ratios. To our knowledge, the only accurate and reliable way to obtain the effective parameters in (2) isto solve the full model (1) and this view is supported by others [1, 20]. In this paper therefore we develop a completely newtechnique, applying the Multiple-Scales (MS) method [22,23] to systematically derive the homogenised thermal problem (2)in a principled way. This in turn enables us to define a general, accurate and efficient technique to estimate the equivalentthermal features of composite electrical windings, regardless of their specific shape and distribution. In the next section we
HT-16-1066 - Howey 2
will present the MS method, showing the solution procedure for a domain with circular conductors. We will then validate themodel by comparing it to the solution of the full problem (1) using FE and experimental measurements from the literature.
2 Derivation of the homogenised modelThe method of Multiple-Scales (MS) relies on the separation of length scales between the microscopic and the macro-
scopic domains, and typically assumes periodicity in the microstructure. The microscopic length scale measures variationswithin a periodic cell (i.e. from copper to insulation to local impregnation in the case of an electrical machine winding) andthe macroscopic length scale measures variations within the macroscopic region of interest (i.e. across the whole slot wherea winding is located). Starting from a full model such as (1), the MS method yields a homogenised equation of the type(2) with the effective coefficients containing the microscopic information such as filling volume and wire arrangement. Themethod has been widely applied in various engineering branches for the study of equivalent properties (thermal, structural,etc.) of periodic structures including composite materials, polymers and nuclear reactor cores [24–27].
2.1 Problem set-up and governing equations
L c
i
f
ωX ∈
(a)
1
f
i
c
νc
νi
1
Y∈
(b)
Fig. 2. (a) Macroscopic domain ω, with insulated conductors in a square periodic lattice separated by a distance l L. (b) Microscopicdomain or unit cell Ω with one conductor at its centre of radius νc = εc/l surrounded by insulation to radius νi = εi/l
We model the windings in the first instance as insulated circular conductors distributed within a periodic square lattice,fig. 2(a). For simplicity, we assume that all the conductors are equally spaced with inter-wire distance l and have the sameexternal radii, εc and εi for the conductor and the insulation respectively. This means that the microstructure is perfectlyperiodic, which is the standard assumption of the MS method as discussed above; however, this assumption can be relaxedas discussed in [28] if one wanted to account for gradual variations in the material distribution. We use the subscripts c, iand f to refer to the conductor, insulation and filling phases, respectively, according to fig. 2. Finally, we assume that thereis perfect contact between the materials. In general, to achieve an accurate homogenisation we require a good separationof lengthscales i.e. L l, where L represents the characteristic dimension of the total winding compound or slot size.The quality of the homogenisation is then a function of the ratio δ = l/L and the matching improves asymptotically withδ→ 0 [29].
The starting point is the heat equation (1) defined in the original windings domain. The heat source term q in (1) isassumed to be from ohmic heating i.e. q = J2/σ where J is current density in the conductors (which we take to be constant)and σ is electrical conductivity. Therefore the source term appears in the conductor phase only, q = 0 if ‖X− c j‖ > εc forall j, where c j represents the coordinates of the jth conductor.
On the boundaries between the phases we assume continuity of temperature and heat flux, namely
kc∇XT ·n = ki∇XT ·n on ‖X− c j‖= εc, (3a)ki∇XT ·n = k f ∇XT ·n on ‖X− c j‖= εi, (3b)
where n stands for the outward unit normal (outward of Ωc in (3a) and outward of Ωi in (3b)). We scale the variables viaT = T T , X = LX, t = t t, where T is the characteristic temperature and t is the characteristic time. Using these scaling
HT-16-1066 - Howey 3
Table 1. Reference values
Variable Value
L 2.21×10−2 m
l 2×10−3 m
εc 0.8×10−3 m
εi 0.835×10−3 m
σ 5.96×107 Ωm
kc 385W/mK
cc 386J/kgK
ρc 8890kg/m3
ki 0.26W/mK
ci 1000J/kgK
ρi 1440kg/m3
k f 0.85W/mK
c f 1700J/kgK
ρ f 1766kg/m3
factors in (1) and (3), we obtain the dimensionless equations
αp∂T∂t
= βp∇2XT +Ψp, (4a)
βc∇XT ·n = βi∇XT ·n on ‖X− c j‖= εc, (4b)
βi∇XT ·n = β f ∇XT ·n on ‖X− c j‖= εi, (4c)
where we introduce the dimensionless groups
αp =L2
tcpρp
k f, βp =
kp
k f, Ψp =
L2
TJ2
σk fδpc
for p= c, i, f , and δpc stands for the Kronecker delta (so that Ψp = 0 for p= i or f ). The values of the dimensional parametersare given in table 1. From now on we will refer to dimensionless quantities dropping hats for ease of notation.
2.2 The Multiple-Scales methodWe now use the MS method to derive an averaged model for the temperature T , valid over many wires in the limit δ 1.
We retain X as the macroscale variable, spanning the entire compound, and introduce Y = X/δ as the microscale variable,measuring distance over the scale of the wire separation. As usual in the method of MS, we treat these two variables asindependent and impose that the solution is exactly periodic in Y; small variations from one wire to the next are therebycaptured through the macroscale variable X. We define νc = εc/l and νi = εi/l as the conductor and insulator external radiirelative to the unit cell Y ∈Ω = [−1/2,1/2]2 (see fig. 2(b)).
After introducing the two scales and using the chain rule, spatial derivatives transform according to
∇X→ ∇X +1δ
∇Y. (5)
HT-16-1066 - Howey 4
Under the transformation (5), equations (4) become
δ2αp
∂T∂t
= βp(δ
2∇
2XT +2δ∇X ·∇YT +∇
2YT)+δ
2Ψp, (6a)
βc (δ∇XT +∇YT ) ·n = βi (δ∇XT +∇YT ) ·non ‖Y‖= νc, (6b)
βi (δ∇XT +∇YT ) ·n = β f (δ∇XT +∇YT ) ·non ‖Y‖= νi. (6c)
Following the standard MS method, we now seek an asymptotic solution in the limit of small δ to (6) of the formT = T (X,Y, t) that is periodic in Y, while treating X and Y as independent. Expanding in powers of δ as T (X,Y, t) =T (0)(X,Y, t)+ δT (1)(X,Y, t)+ δ2T (2)(X,Y, t)+ · · · , we find that the leading-order temperature T (0) is independent of themicroscale variable Y. Due to the linearity of the problem, the general solution at order δ can be written as:
T (1) =−∇XT (0)(X, t) ·ΓΓΓ(Y), (7)
where ΓΓΓ(Y) is a vector function whose components Γr satisfy the following so-called cell problem [23]
βp∇2YΓr = 0, (8a)
βc (∇YΓr− er) ·n = βi (∇YΓr− er) ·n on ‖Y‖= νc, (8b)βi (∇YΓr− er) ·n = β f (∇YΓr− er) ·n on ‖Y‖= νi, (8c)
and Γr periodic on Y ∈ ∂Ω. Here er stands for the rth vector in the standard basis. As we will see later, the solution of thecell problem contains all the required microscopic information to obtain an accurate homogenised model.
Proceeding to the order δ2 terms in the expansion of (6a) in each of the phases leads to:
αp∂T (0)
∂t=βp∇X ·
(∇XT (0)+∇YT (1)
)+βp∇Y ·
(∇XT (1)+∇YT (2)
)+Ψp.
(9)
Integrating (9) over the corresponding domain Ωp, using (7) in the second term and the divergence theorem in the third termyields
αp‖Ωp‖∂T (0)
∂t= βp∇X ·
(∫Ωp
[I− Q(Y)
]dY)
∇XT (0)
+βp
∫∂Ωp
(∇XT (1)+∇YT (2)
)·n dS+Ψp‖Ωp‖, (10)
where ‖Ωp‖ denotes the area of Ωp, ∂Ωp indicates the boundaries limiting Ωp, and~n is the outward unit normal to Ωp. I isthe 2×2 identity matrix, and (Q)i j = ∂Γ j/∂Yi is the transpose of the Jacobian matrix of ΓΓΓ, the solution of the cell problem(8).
2.3 Homogenised heat equationWe now add up the three integrated equations (10) (one equation per phase) to obtain a homogenised equation over
the whole unit cell Ω = Ωc ∪Ωi ∪Ω f . Using the order δ2 of the boundary conditions (6b) and (6c) and the periodicityassumption, we find that all the boundary integrals in (10) cancel out [30]. The resulting nondimensional heat equation forthe electrical windings reads
A∂T (0)
∂t= ∇X ·
(B∇XT (0)
)+Ψc
‖Ωc‖‖Ω‖
, (11a)
HT-16-1066 - Howey 5
Fig. 3. Solution of the two components of the cell problem (8) Γ1 and Γ2 according to the properties in table 1
where A is the equivalent heat capacity and B is the 2×2 thermal conductivity matrix which are defined as
A =1‖Ω‖∑
pαp‖Ωp‖, (11b)
B =1‖Ω‖∑
p
[βp
∫Ωp
I− Q(Y)
]dΩp, (11c)
where the sum is over p = c, i, f .We find that the equivalent thermal capacity A can be calculated by just area weighting the contributions of the three
materials; in this case the disposition of the wires has no effect. As we mentioned before, the equivalent thermal conductivityB is obtained from the solution ΓΓΓ of the cell problem (8). For simple geometries (like the one depicted in fig. 2(b)) this canbe computed analytically [31]; however, the method is general and the resulting homogenised model (11a) is valid for anyunit cell geometry. So in general B will be computed numerically by solving (8) using FE software (see fig. 3). For isotropicmaterial B becomes diagonal and, for the case of circular inclusions, the diagonal entries are identical, that is B11 = B22.
3 Model validationTo validate the homogenised model (11a), in this section results are compared against a full FE model including every
single insulated wire (using commercial software COMSOL Multiphysics v5.1), and against experimental results availablefrom the literature.
We first consider the matrix of 11×11 wires arranged in a regular square lattice as depicted in fig. 2(a) with the propertiesand dimensions listed in table 1; the reference material properties are taken from [1]. This configuration leads to a fillingratio of µ = 0.5, defined as the ratio of the conductor total area to the whole available area
µ =nwπε2
c
L2 ≡ ‖Ωc‖, (12)
with nw denoting the total number of wires.In fig. 3 we present the solution of the cell problem (8). We note that the solution is unique up to a constant, but this has
no effect to the homogenised model since it is the Jacobian of ΓΓΓ that enters in B. To ensure the uniqueness of the solutionwe applied the constraint ∑p
∫Ωp
Γr = 0. To obtain the dimensional equivalent thermal conductivity we need to restore thedimensions with keq = k f B11, leading to
keq,S = 2.05W/mK,
where we use the subscript S to denote the square lattice configuration.In fig. 1(b) we compare the mid-cross-section temperature distribution obtained with the solution of the full model
(1) with the FE method (≈ 140k elements) and the homogenised model (11a), for δ = 1/11, internal heat generation (q =1×107 W/m3) and fixed temperature (T = 0 C) at the boundaries. The homogenised model is solved with a Chebyshevspectral collocation method [32] using 25 grid points. The solution on the Chebyshev points (shown by squares in fig. 1(b))is then interpolated with the Matlab embedded interp2 function. We plot the normalised temperature Θ = T
Tmaxversus the
HT-16-1066 - Howey 6
0 0.2 0.4 0.6 0.8 1X1
L
0
0.2
0.4
0.6
0.8
1
Θ
Full FE model - square lattice
Homogenised model through MS method
(a)
δ
10−4
10−3
10−2
10−1
Θfull-Θ
MS
1
31
1
21
1
11
1
9
1
7
1
5
(b)
Fig. 4. (a) Temperature distribution obtained from the full model Θ f ull and the homogenised model ΘMS in the mid-cross-section of wiresdistributed in a square lattice with δ = 1/31, with internal heat generation and Dirichlet boundary conditions. (b) Relative error Θ f ull−ΘMSover a range of δ
Ωf
1
Ωi
Ωcνc
νi
ΩY∈
(a) (b)
Fig. 5. (a) Microscopic domain Y ∈ Ω related to the hexagonal lattice wires and (b) Γ1 distribution for the case hexagonal lattice withµ = 0.5
normalised coordinate X1L , where the reference maximum temperature Tmax is taken from the solution of the full model. We
observe a very good agreement between the two solutions, in particular in the central area of the domain in correspondencewith the hot spot. As expected, the limitation of the homogenised solution is that is unable to capture the localised temperatureplateaus due to each individual wire.
The homogenised model (11a) is asymptotic in the limit of small δ and, as such, the solution improves as δ→ 0 asshown in fig. 4(a), where we take δ = 1/31. In fig. 4(b) we compare the relative error Θ f ull−ΘMS in the central temperatureestimation between the full and homogenised model. As expected, the error decreases asymptotically as δ decreases.
3.1 Effects of wire distribution in the macro scale domainWe now examine the effects that the microscopic wire configuration has on the effective windings properties. First the
MS model is applied with a different periodic configuration - on a hexagonal lattice whose unit cell is depicted in fig. 5(a).This configuration achieves the highest filling ratio µ with circular wires, as is likely to be found in a real winding environment(fig. 1(a)). The first component Γ1 of the cell problem (8) in an hexagonal lattice is presented in fig. 5(b) for the case ofµ = 0.5. This leads to an equivalent thermal conductivity of
keq,H = 2.03W/mK,
where the subscript H refers to the hexagonal lattice configuration.In fig. 6(a) we compare the effect of the square and hexagonal lattice over a range of filling ratios in terms of equivalent
thermal conductivities obtained from the MS method. The differences between the effective coefficients keq,S and keq,H in asquare or hexagonal configuration respectively only become important for filling ratios µ > 0.5. This is consistent with theresults in [28] in the context of an effective diffusion coefficient in porous media.
One could argue that the assumption of perfectly periodic wire distributions (arranged on a square or hexagonal lattice)may not hold in practice. To assess the discrepancy that may rise due to a random distributions of wires, we solved the fullFE model with uniformly distributed and non-overlapping wires and we evaluated 100 randomised configurations for each
HT-16-1066 - Howey 7
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Filling ratio - µ
1
2
3
4
5keq-[W
/mK]
MS method, keq,SMS method, keq,Hk−
eq,HS
k−
eq,Wiener
keq,R, 100 random distributions
(a)
0 0.2 0.4 0.6 0.8 1X1
L
0
0.2
0.4
0.6
0.8
1
Θ
Solution with MS method, keq,SSolution with keq,HS
Average solution of 100 random cases
Solution with highest hot-spot
Solution with lowest hot-spot
(b)
Fig. 6. (a) Effective thermal conductivity keq over a range of µ using different approaches, (b) Steady-state temperature distribution forrandom distributions at µ = 0.5
filling ratio µ. To generate the random uniform distributions, we initialised the system at a given µ with a periodic squarelattice and let the wires move randomly (according to a Brownian motion) within the domain, using 300 · nw steps of theMetropolis-Hastings (MH) algorithm [33]. Repeating the process starting from a hexagonal rather than square lattice givesthe same results, indicating that enough MH steps have been taken to lose any initial effects.
For each configuration we computed the equivalent thermal conductivity keq,R, where the subscript R stands for randomdistribution. In the FE simulations a thermal gradient is applied to the domain, setting fixed temperature conditions on twoopposite boundaries and no-flux condition on the remaining two boundaries. The equivalent thermal conductivity is thencalculated with
keq,R =Q
∆TLA, (13)
where Q is the total heat flux evaluated on a cross section of area A perpendicular to the domain and ∆T is the imposedthermal gradient. We plot the average together with the 95% error bars in fig. 6(a). As we can see, the differences betweenthe effective conductivity from the random distribution, keq,R, and the periodic ones are barely noticeable. Interestingly, theset of data follows the behaviour of the periodic square lattice distribution quite closely. Therefore, this suggests that theMS approach to estimate keq may be a suitable method to obtain accurate estimates for real windings, not just theoreticalperiodically arranged windings.
For comparison purposes, in fig. 6(a) we also include two commonly used analytical formulae, namely the lower HSbound [1] and the lower Wiener bound [17]:
k−eq,HS = k f(1+µ)kc +(1−µ)k f
(1−µ)kc +(1+µ)k f, (14)
k−eq,Wiener =
(µkc
+1−µ
k f
)−1
. (15)
The 2-phase HS formula lies always above the correct estimation, which leads to an unacceptable underestimation of thehot-spot temperature. This is clearly shown in fig. 6(b), where we compare the homogenised solutions using k−eq,HS and keq,Swith the full solutions using random wire configurations at µ = 0.5. We show the average distribution obtained averagingover the 100 configurations, as well as the distributions of the two configurations that gave the highest and lowest hot-spottemperatures. The MS estimation follows the average random solution very well. This result indicates that the MS methodcan provide an accurate estimation of the macroscopic thermal behaviour of the electrical windings, therefore leading to areliable approximation of the hot-spot temperature.
3.2 Experimental data from the literatureFinally, we validate the approach with measured data available on the literature. Simpson [1] presents the equivalent
measured thermal conductivity of electrical windings for both copper and aluminium conductors, arranged in cubes impreg-
HT-16-1066 - Howey 8
0.45 0.5 0.55 0.6
Filling ratio - µ
1.7
2
2.3
2.6
keq-[W
/mK]
(a)
0.51 0.55 0.59
Filling ratio - µ
2.1
2.2
2.3
2.4
2.5
keq-[W
/mK]
(b)
Fig. 7. Comparison of keq estimated via the MS method, using the square lattice cell, with experimental measurements from [1] with (a)copper and (b) aluminium conductors at various filling ratios
nated with epoxy. The thermal conductivity of aluminium is assumed to be kAl = 234 W/m2, and all the remaining relevantproperties are as presented in table 1.
As simulated in the previous section, the equivalent conductivity is measured by injecting a known heat flux into a cubeboundary, with the opposite boundary in contact with a cooling plate. In this manner the heat flows in a single direction,assuming all the remaining boundaries are well insulated. The equivalent conductivity is then evaluated with (13), knowingthe temperature gradient between the hot and cold boundaries.
For the MS method, the square lattice cell was used, in accordance with fig. 6(a). Fig. 7 shows that the mismatch betweenmeasurements and estimations is less than 4%. This error could be due to the assumption of perfect contact between thephases (no airgaps in potting, for example), or there could also be an error related to the systematic and random measurementuncertainties.
3.3 Transient resultsAfter proving the reliability of the MS method for steady state problems, we now investigate its performance over
transients. We refer to the case of µ = 0.5 with wires distributed on a square lattice as in fig. 2(a) where δ = 1/11. Theequivalent heat capacity can be estimated restoring the dimensions of A, given by (11b), with:
Ceq = Ak f tL2 = 3.12×106 J/m3K.
As mentioned before, this quantity is independent of the wire distribution and only depends on the relative filling of eachphase.
To solve the homogenised model (11a), we used an explicit forward Euler method in time and the same spectral methodin space as before. Accordingly, for a fixed time step λ, at each instant we solve the following linear system
[T ]τ =λ
Ceq[qeq]+
λ
Ceqkeq,S[D][T ]τ−1 +[T ]τ−1, (16)
where [T ]τ is the vector of nodal temperatures at the τth instant, [D] the discretisation matrix and [qeq] the equivalent heatgeneration vector.
In fig. 8 we compare the normalised temperature Θ in the middle of the domain obtained from the homogenised and fullmodel as function of the Fourier number τ = t keq,S
CeqL2 and Θ is again normalised using the steady-state maximum temperatureof the full model. We set the initial condition to 0 C and let the domain heat up applying a uniformly distributed internalheat generation. Dirichlet boundary conditions (T = 0 C) are imposed on the four lateral edges as in the previous tests.
The RMS relative error between the two estimations is√∑
mτ=1(Θτ
full−ΘτMS
)2
m= 1.3%, (17)
which once more confirms the high accuracy of the MS method.
HT-16-1066 - Howey 9
0 0.05 0.1 0.15 0.2 0.25
τ
0
0.2
0.4
0.6
0.8
1
Θ
Solution with full FE model - square lattice
Solution with MS method, keq,S and Ceq
Fig. 8. Transient normalised temperature Θ = T/Tmax as a function of the Fourier number τ = t keqCeqL2 in centre of the domain obtained
with the MS homogenised model the full model using a square lattice wire distribution with δ = 1/11
4 ConclusionsIn this paper we successfully applied the MS method to obtain the homogenised heat equation in a electrical winding
compound. The MS method provides a systematic and efficient way to estimate the equivalent thermal properties of thehomogenised material. The resulting homogenised model is much less computationally expensive than the original modelincorporating the full geometry of the windings, while still capturing the essential microscopic features. The method hasbeen shown to provide an accurate estimation of the equivalent thermal conductivities when compared with numerical valuesfrom the full model and experimental values in the literature.
Although the method assumes a periodic distribution of the wires, the hot spot estimated with the model homogenisedthrough the MS method is still accurate when compared to random configurations of the conductors. On the other hand, theemployment of analytical formulae, such as the HS (14) or Wiener (15) equations leads to a significant error in the assessmentof the hot spot due to an error in the calculation of the homogenised conductivity. This may lead to a wrong estimation ofthe remaining parameters related to a full machine thermal model when trying to match parameters to experimental results.This in turn will reduce the reliability of the model against varying loads and cooling conditions even after the calibration.
AcknowledgmentThe authors would like to thank the European Union for their funding to this research (FP7 ITN Project 607361 ADEPT).
HT-16-1066 - Howey 10
NomenclatureA areaC ρ · cJ current densityL macro-lenght(Q)i j ∂Γ j/∂YiT temperatureX space variable in the macro-domainY space variable in the micro-domainc heat capacityk thermal conductivityl micro-lenghtm number of instantsn unit vectorq heat fluxt time
Greek SymbolsΓΓΓ vector function in the micro-domainΘ normalised temperatureΨ dimensionless heat generationΩ micro-domainα dimensionless heat capacityβ dimensionless thermal conductivityδ ratio l/Lε radius in the macro-domainλ fixed time stepµ filling ratioν radius in the micro-domain ε/lρ densityτ dimensionless timeω macro-domain
SubscriptsAl aluminiumMS multiple-scalesH hexagonal latticeHS Hasin - ShtrikmanS square latticec conductoreq equivalent quantityf filling materiali insulationp phase
HT-16-1066 - Howey 11
References[1] Simpson, N., Wrobel, R., and Mellor, P. H., 2013. “Estimation of Equivalent Thermal Parameters of Impregnated
Electrical Windings”. IEEE Transactions on Industry Applications, 49(6), pp. 2505–2515.[2] Bennion, K., Cousineau, E., Feng, X., King, C., and Moreno, G., 2015. “Electric Motor Thermal Management R &
D”. In IEEE Power & Energy Society General Meeting.[3] Huang, Z., Marquez-Fernandez, F. J., Loayza, Y., Reinap, A., and Alakula, M., 2014. “Dynamic Thermal Modeling
and Application of Electrical Machine in Hybrid Drives”. In International Conference on Electrical Machines (ICEM),pp. 2158–2164.
[4] Popescu, M., Staton, D., Boglietti, A., Cavagnino, A., Hawkins, D., and Goss, J., 2016. “Modern heat extractionsystems for electrical machines - A review”. IEEE Transactions on Industry Applications, 52(3), pp. 2167–2175.
[5] Zhang, H., 2015. “Online Thermal Monitoring Models for Induction Machines”. IEEE Transactions on Energy Con-version, 30(4), pp. 1279–1287.
[6] Mellor, P., Wrobel, R., and Simpson, N., 2014. “AC Losses in High Frequency Electrical Machine Windings formedfrom Large Section Conductors”. In Energy Conversion Congress and Exposition (ECCE), pp. 5563–5570.
[7] Boglietti, A., Staton, D., and Dipartimento, T., 2015. “Stator Winding Thermal Conductivity Evaluation : an IndustrialProduction Assessment”. In Energy Conversion Congress and Exposition (ECCE), pp. 4865–4871.
[8] Ayat, S., Wrobel, R., Goss, J., and Drury, D., 2016. “Estimation of Equivalent Thermal Conductivity for ImpregnatedElectrical Windings Formed from Profiled Rectangular Conductors”. In 8th IET Intlernational Conference on PowerElectronics, Machines and Drives (PEMD), pp. 1–6.
[9] Nategh, S., Wallmark, O., Leksell, M., and Zhao, S., 2012. “Thermal analysis of a PMaSRM using partial FEA andlumped parameter modeling”. IEEE Transactions on Energy Conversion, 27(2), pp. 477–488.
[10] Takatsu, Y., Masuoka, T., Nomura, T., and Yamada, Y., 2016. “Modeling of Effective Stagnant Thermal Conductivityof Porous Media”. Journal of Heat Transfer, 138(1), p. 012601.
[11] Kundalwal, S. I., Suresh Kumar, R., and Ray, M. C., 2015. “Effective Thermal Conductivities of a Novel FuzzyFiber-Reinforced Composite Containing Wavy Carbon Nanotubes”. Journal of Heat Transfer, 137(1), p. 012401.
[12] Wrobel, R., and Mellor, P. H., 2010. “A General Cuboidal Element for Three-Dimensional Thermal Modelling”. IEEETransactions on Magnetics, 46(8), pp. 3197–3200.
[13] Wrobel, R., Mlot, A., and Mellor, P. H., 2012. “Contribution of End-Winding Proximity Losses to TemperatureVariation in Electromagnetic Devices”. IEEE Transactions on Industrial Electronics, 59(2), pp. 848–857.
[14] Baker, J. L., Wrobel, R., Drury, D., and Mellor, P. H., 2014. “A methodology for predicting the thermal behaviour ofmodular-wound electrical machines”. In Energy Conversion Congress and Exposition (ECCE), pp. 5176–5183.
[15] Idoughi, L., Mininger, X., Bouillault, F., Bernard, L., and Hoang, E., 2011. “Thermal Model With Winding Homoge-nization and FIT Discretization for Stator Slot”. IEEE Transactions on Magnetics, 47(12), pp. 4822–4826.
[16] Galea, M., Gerada, C., Raminosoa, T., and Wheeler, P., 2012. “A Thermal Improvement Technique for the PhaseWindings of Electrical Machines”. IEEE Transactions on Industry Applications, 48(1), jan, pp. 79–87.
[17] Wiener, O. H., 1912. Die theorie des mischkorpers fur das feld der stationaren stromung. 1. abhandlung: Die mittelw-ertsatze fur kraft, polarisation und energie. BG Teubner.
[18] Hashin, Z., and Shtrikman, S., 1963. “A variational approach to the theory of the elastic behaviour of multiphasematerials”. Journal of the Mechanics and Physics of Solids, 11(2), pp. 127–140.
[19] Milton, G. W., 1981. “Bounds on the Transport and Optical Properties of a Two-Component Composite Material.”.Journal of Applied Physics, 52(8), pp. 5294–5304.
[20] Kanzaki, H., Sato, K., and Kumagai, M., 1992. “A Study of an Estimation Method for Predicting the EquivalentThermal Conductivity of an Electric Coil”. Heat transfer Japanese research.
[21] Siesing, L., Reinap, A., and Andersson, M., 2014. “Thermal properties on high fill factor electrical windings : Infiltratedvs non infiltrated”. In International Conference on Electrical Machine (ICEM), pp. 2218–2223.
[22] Bensoussan, A., Lions, J.-L., and Papanicolaou, G., 1978. Asymptotic analysis for periodic structures. North-Holland.[23] Sanchez-Palencia, E., 1980. Non-homogeneous media and vibration theory. Springer-Verlag.[24] Yu, Q., and Fish, J., 2002. “Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and
temporal scales: A coupled thermo-viscoelastic example problem”. International Journal of Solids and Structures, 39,pp. 6429–6452.
[25] Asakuma, Y., and Yamamoto, T., 2013. “Effective thermal conductivity of porous materials and composites as afunction of fundamental structural parameters”. Computer Assisted Methods in Engineering and Science, 20, pp. 89–98.
[26] Hales, J., Tonks, M., Chockalingam, K., Perez, D., Novascone, S., Spencer, B., and Williamson, R., 2015. “Asymptoticexpansion homogenization for multiscale nuclear fuel analysis”. Computational Materials Science, 99, pp. 290–297.
[27] White, J., 2015. “Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method”.Journal of Heat Transfer, 137(7), p. 071301.
[28] Bruna, M., and Chapman, S. J., 2015. “Diffusion in spatially varying porous media”. SIAM Journal on Applied
HT-16-1066 - Howey 12
Mathematics, 37(2), pp. 215–238.[29] Cioranescu, D., and Paulin, J. S. J., 2012. Homogenization of reticulated structures. Springer Science & Business
Media.[30] Mei, C. C., and Vernescu, B., 2010. Homogenization methods for multiscale mechanics. World scientific.[31] Joesson, H., and Halle, B., 1996. “Solvent diffusion in ordered macrofluids: A stochastic simulation study of the
obstruction effect”. Journal of Chemical Physics, 104(17), pp. 6807–6817.[32] Trefethen, L. N., 2000. Spectral Methods in MATLAB. Siam.[33] Hastings, W. K., 1970. “Monte Carlo sampling methods using Markov chains and their applications”. Biometrika,
57(1), pp. 97–109.
HT-16-1066 - Howey 13
List of Figures1 (a) Cross section of an electrical winding (courtesy of the Clarendon Laboratory, Oxford). (b) Normalised
steady-state temperature Θ = T/Tmax in a simulation of the mid-cross-section of a winding with internal heatgeneration and fixed temperature at the boundaries, obtained from the full model (1) and the homogenisedmodel (2). In (b), l/L indicates the relative distance between adjoining wires . . . . . . . . . . . . . . . . . 2
2 (a) Macroscopic domain ω, with insulated conductors in a square periodic lattice separated by a distancel L. (b) Microscopic domain or unit cell Ω with one conductor at its centre of radius νc = εc/l surroundedby insulation to radius νi = εi/l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Solution of the two components of the cell problem (8) Γ1 and Γ2 according to the properties in table 1 . . . 64 (a) Temperature distribution obtained from the full model Θ f ull and the homogenised model ΘMS in the
mid-cross-section of wires distributed in a square lattice with δ = 1/31, with internal heat generation andDirichlet boundary conditions. (b) Relative error Θ f ull−ΘMS over a range of δ . . . . . . . . . . . . . . . 7
5 (a) Microscopic domain Y ∈ Ω related to the hexagonal lattice wires and (b) Γ1 distribution for the casehexagonal lattice with µ = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6 (a) Effective thermal conductivity keq over a range of µ using different approaches, (b) Steady-state temper-ature distribution for random distributions at µ = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Comparison of keq estimated via the MS method, using the square lattice cell, with experimental measure-ments from [1] with (a) copper and (b) aluminium conductors at various filling ratios . . . . . . . . . . . . 9
8 Transient normalised temperature Θ = T/Tmax as a function of the Fourier number τ = t keqCeqL2 in centre of
the domain obtained with the MS homogenised model the full model using a square lattice wire distributionwith δ = 1/11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
HT-16-1066 - Howey 14
List of Tables1 Reference values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
HT-16-1066 - Howey 15
