The Heat Transmission Equation in FF-CM (consult [1, 3, 5]) is expressed in Eq. (1) as:

$$\begin{array}{}{\displaystyle {\left[G\right]}^{T}\left\{\left[{M}_{\lambda}\right]\left[G\right]\left[\tau \right]\right\}+\left[{M}_{Cp}\right]\frac{d\tau}{dt}=\left[W\right]}\end{array}$$(1)

Two constitutive matrices are observed: [*M*_{λ}] and [*M*_{Cp}]. The constitutive matrix [*M*_{λ}] belongs to Fourier’s law of the heat conduction equation. The constitutive matrix [*M*_{Cp}] belongs to the Heat transmission equation with change of temperature.

Fourier’s law of the heat conduction equation applied to stator-airgap-rotor is:

$$\begin{array}{}{\displaystyle {-\left[G\right]}^{T}\left\{-\left[{M}_{\lambda}\right]\left[G\right]\left[\tau \right]\right\}=\left[\stackrel{~}{D}\right]{q}_{\lambda}}\end{array}$$(2)

The heat sources, corresponding to Joule effects and magnetic hysteresis and eddy currents, are represented as the vector [*W*].

The heat produced by mechanical friction is not studied in this work.

The changes in the internal energy, dependent on the component materials of the asynchronous machines, are represented in the following expression:

$$\begin{array}{}{\displaystyle \left[{M}_{Cp}\right]\frac{d\tau}{dt}}\end{array}$$(3)

We establish an analogy between thermal conduction and electrical conduction as is indicated in Eq. (4).

$$\begin{array}{}{\displaystyle}& {-\left[G\right]}^{T}\left\{-\left[{M}_{\lambda}\right]\left[G\right]\left[\tau \right]\right\}=\left[\stackrel{~}{D}\right]\left[{q}_{\lambda}\right]\\ & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\iff {-\left[G\right]}^{T}\left\{-\left[{M}_{\sigma}\right]\left[G\right]\left[v\right]\right\}=\left[\stackrel{~}{D}\right]\left[I\right]\end{array}$$(4)

The Fourier heat transmission equation is treated in 2-D and 3-D. The cartesian absolute coordinates of the cell are converted to local standardized coordinates. The origin is one of the four vertices of the tetrahedron used as cell [1,2,3]. The dual variables are projected to the primal using geometrical methods. Planar symmetries and axisymmetric symmetries are used. Quadratic interpolations are applied to obtain distributions of temperature [1,2,3,4, 5].

The main parameters used in this work are the following:

Assuming the analogy indicated in Eq. (4) gives:

$$\begin{array}{}{\displaystyle \left[{M}_{\lambda}\right]\iff \left[{M}_{\sigma}\right]}\end{array}$$(5)

Using the expression proposed by [6], the matrices exposed in Eq. (5) are topological equivalent:

$$\begin{array}{}{\displaystyle \lambda \left(\frac{1}{{V}_{t}}{\overrightarrow{\stackrel{~}{S}}}_{i}{\overrightarrow{\stackrel{~}{S}}}_{j}\right)\iff \sigma \left(\frac{1}{{V}_{t}}{\overrightarrow{\stackrel{~}{S}}}_{i}{\overrightarrow{\stackrel{~}{S}}}_{j}\right)}\end{array}$$(6)

Where matrix [*S*] is defined according to (8), following Specogna and F. Trevisan [6],

$$\begin{array}{}{\displaystyle [S{]}_{6\times 6}=\left[\begin{array}{cccccc}{\overrightarrow{\stackrel{~}{S}}}_{0}{\overrightarrow{\stackrel{~}{S}}}_{0}& \cdots & \cdots & \cdots & \cdots & {\overrightarrow{\stackrel{~}{S}}}_{0}{\overrightarrow{\stackrel{~}{S}}}_{5}\\ \vdots & \ddots & & & & \vdots \\ \vdots & & {\overrightarrow{\stackrel{~}{S}}}_{2}{\overrightarrow{\stackrel{~}{S}}}_{2}& & & \vdots \\ \vdots & & & \ddots & & \vdots \\ \vdots & & & & \ddots & \vdots \\ {\overrightarrow{\stackrel{~}{S}}}_{5}{\overrightarrow{\stackrel{~}{S}}}_{0}& \cdots & \cdots & \cdots & \cdots & {\overrightarrow{\stackrel{~}{S}}}_{5}{\overrightarrow{\stackrel{~}{S}}}_{5}\end{array}\right]}\end{array}$$(7)

is the dot product of the vectors that define the dual faces of the cell and (*V*_{T}) is the volume of the tetrahedron, see Figure 1.

Figure 1 Dual faces vectors of a tetrahedrical cell with baricentric

The matrix [*M*_{σ}] must be calculated for the electro-thermal phenomena occurring in the asynchronous machine. So why do the calculations twice? Several numerical experiments were designed to confirm this assumption.

The obtained results were contrasted with the results obtained from the same model developed with Finite Element Methods (FEM). The particular caseof 2-D was also studied.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.