# Numerical simulation of electric motor*

## 1. Objectives and goals

The main objective of our task is to develop a thermal model for finding the temperature distribution of different components of an electric motor during its various operating conditions and to determine the heat removal by natural convection from the machine surface.

The application allows the predictions of simultaneous heat transfer in solid and fluid media with energy exchange between them. The prediction of the temperature distribution inside an operating electric motor is required at the machine design stage. Excessive temperature in the motor can cause electrical insulation failure, demagnetization of the magnets and increase in Joule losses.

## 2. Geometry of the analysed Motor

Fig.1. Temperature test set up for the prototype machine. As shown in Fig. 1, the electric motor model has a very complicated geometry. Obviously, it is not possible to perform simulation on the entire PM motor, due to the large number of volumes and calculations required. Therefore, a part of the PM motor consisting of each part (front cover, stator core, stator windings, rotor, permanent magnets, shaft, external housing and the air-gap between these components are modelled for this study as shown on the Fig. 2. Fig. 3. shows a detailed view of the main components of the modeled electric motor.

Fig.2. View of the entire motor under analysis. Fig.3. View of the components that form the electric motor under analysis. Model detail includes; (1) the front cover, (2) and (5) the external housing geometry, (3) the stator core geometry including slot detail and the stator windings, (4) the rotor geometry including permanent magnets and shaft. In order to simplify the model, the winding region was modeled as a solid section and the end winding region was modeled as a torus. This assumption made this motor element easier to mesh.

As shown in Fig. 4, the mesh for the motor and the air-gap was created based on tetrahedral elements, as these are the only elements capable of meshing that domain.

Fig.4. A general view of the final Fig. 5. shows a cut-view of the internal air-gap section Fig.6. Internal air-gap section ## 3. Thermal modeling

We are interested in solving a conjugate heat transfer problem in a domain Ω which consists of a set of N sub-domains that represent fluid and solid regions.

The strategy to solve conjugate heat transfer problems is based on:

• modeling the heat transfer in solid

• modelling flow and heat transfer in the fluid

• the solutions are coupled at the fluid-solid interfaces using the continuity of temperature and heat-flux.

The fluids (internal and external) are modeled using the incompressible Navier-Stokes equations with the Boussinesq approximation. The solids are modeled with the heat equation.

The coupling of the fluid domain and the solid domain is accomplished through the transfer of temperature and heat flux at interfaces.

The Navier-Stokes equations with the Boussinesq approximation are solved to compute the fluid flow and it can be expressed as:

• The continuity:

$\nabla\cdot V = \frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} +\frac{\partial w}{\partial z}$
• momentum equations:

$\rho \left( \frac{\partial u}{\partial t} +u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} +w\frac{\partial u}{\partial z} \right) = - \frac{\partial p}{\partial x} +\frac{\partial}{\partial x} \left(\mu \frac{\partial u}{\partial x} \right) +\frac{\partial}{\partial y} \left(\mu \frac{\partial u}{\partial y} \right) +\frac{\partial}{\partial z} \left(\mu \frac{\partial u}{\partial z} \right)$
$\rho \left( \frac{\partial v}{\partial t} +u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y} +w\frac{\partial v}{\partial z} \right) = - \frac{\partial p}{\partial y} +\frac{\partial}{\partial x} \left(\mu \frac{\partial v}{\partial x} \right) +\frac{\partial}{\partial y} \left(\mu \frac{\partial v}{\partial y} \right) +\frac{\partial}{\partial z} \left(\mu \frac{\partial v}{\partial z} \right)$
$\rho \left( \frac{\partial w}{\partial t} +u\frac{\partial w}{\partial x} +v\frac{\partial w}{\partial y} +w\frac{\partial w}{\partial z} \right) = - \frac{\partial p}{\partial z} +\frac{\partial}{\partial x} \left(\mu \frac{\partial w}{\partial x} \right) +\frac{\partial}{\partial y} \left(\mu \frac{\partial w}{\partial y} \right) +\frac{\partial}{\partial z} \left(\mu \frac{\partial w}{\partial z} \right) +\rho_\text{ref} g(1-\beta(T-T_\text{ref}))$

The variables p, T, andare the pressure, temperature and density. is the coefficient of thermal expansion.

The energy equation is solved to compute the fluid and solid temperatures:

$\rho C_p \left[ \frac{\partial T}{\partial t} +u\frac{\partial T}{\partial x} +v\frac{\partial T}{\partial y} +w\frac{\partial T}{\partial z} \right] -k \left[ +\frac{\partial^2 T}{\partial x^2} +\frac{\partial^2 T}{\partial y^2} +\frac{\partial^2 T}{\partial z^2} \right] = Q$

Whereis the thermal conductivity, Cp is the specific heat capacity and is is the internal heat generation.

### 3.1. Conjugate heat transfer algorithm:

The algorithm starts with the initialization of the of temperature at the solid/fluid interfaces.

#### 3.1.1. Solving a Dirichlet type problem in fluid domain:

The fluid solver utilizes the wall temperature (solid/fluid interfaces) for the evaluation of flow and heat transfer in the fluid. This step allows us to establish the continuity of the temperature at solid/fluid interfaces.

• Based on the results of the previous step the heat flux and the heat transfer coefficient between solid and fluid parts are determined.

#### 3.1.2. Solving a Neumann type problem in the solid domains:*

The computed temperature distribution of fluid and the heat transfer coefficient are transfered to the heat solid solver and then we solve the conduction problem in solid domain with a boundary condition on fluxes. This step allows us to establish the continuity of the heat flux at solid/fluid interfaces.

This process is repeated until the convergence is achieved.

## 4. HEAT SOURCE MODEL

The heat losses produced in a PM synchronous motors are the results of different losses (electrical, mechanical, eddy current..). The main heat sources of the electrical losses which comprise the conduction losses of the stator windings and the iron losses, including eddy current loss and hysteresis loss of the stator core.

Thus, one goal of this part is to develop method for calculation of the Joule heat only using a Finite Element Method and the other types of losses are neglected. The output results of the loss calculation is used in the thermal analysis.

### 4.1. Electrostatics

The electrical potential field in a conductor is governed by Maxwell’s equation of conservation of electrical charge.

Steady-state case: the equations can be written as follows:

$E=-\nabla\phi \quad \text{in} \Omega$

The constitutive equation:

$J= \sigma E \quad \text{in}\Omega$

where $E$ is electrical field, $\Phi$ is the electrical potential, $\sigma$ is the electrical conductivity, and $\sigma$ is the electrical current density vector.

The balance equation expresses conservation of charge (Gauss’ law):

$\nabla J= 0 \quad \text{in}\Omega$

Elimination of the intermediate variables and in yields the scalar Poisson’s equation

$\nabla(\sigma\nabla\phi)= 0 \quad \text{in}\Omega$

The boundary conditions are as follows:

• a prescribed electrical potential: $\phi=\phi_0$, on $\Gamma_\phi$

• imposition of an electrical current density: $J_n = -J.n = J_0$

with denoting the local outward unit normal vector.

Using Green’s (divergence) theorem, the weak or global form of the Poisson’s equation is:

$\int_\Omega\nabla\phi\nabla Vd\Omega - \int_{\partial\Omega}\sigma\nabla\phi nVd\Gamma = 0$

,

$\int_\Omega\nabla\phi\nabla Vd\Omega + \int_{\partial\Omega}(J.n) Vd\Gamma = 0.$

The volume heat source associated with resistance heating is given by Joule’s law

$P=\sigma J.J = \sigma\nabla\phi\nabla\phi$