A Co-energy Based Approach to Model the Rotordynamics of Electrical Machines

Abstract

New technological fields of application, as for example electric vehicles and closely related lightweight design increase the sensitivity of electrical machines towards torsional and lateral rotor oscillations. The modelling of such electro-mechanical processes is a challenging multiphysical task. In this context, a vast majority of scientific publications use direct approaches to model the problem. These methods derive the equations of motion from Newton’s and Kirchhoff’s laws. In contrast to that, this work proposes a fully coupled indirect approach to the problem using Lagrange-Maxwell equations and the involved magnetic co-energy functional. Such an indirect approach provides for distinct advantages concerning energetical consistency, electro-mechanical coupling and computational effectiveness. Modelling implications like the dependency of the magnetic force on the mechanical motion are outlined and the applicability is shown for a transient simulation of a cage induction machine.

Keywords

A Derivation of the Magnetic Co-energy Using the Permeance Harmonic Method

Fig. 7.

Definitions to derive the magnetic co-energy $W_\text {m}^*$. (a) Machine air gap and involved kinematics, (b) Winding distributions $N_1$ and $N_4$ for the 1st stator phase and for the 4th rotor mesh, considering 200 harmonics.

Choosing the classical PHM approach here [18], the co-energy reads

$$\begin{aligned} W_\text {m}^*(\varvec{x}, \varvec{i}) = \frac{\ell _\text {m}r_1}{2}\int _0^{2\pi } B(\varvec{x}, \varvec{i},\theta ) F(\varvec{x}, \varvec{i}, \theta ) d\theta . \end{aligned}$$

(10)

Here B is the radial component of the magnetic flux density in the air gap, F is the magneto motive force (MMF) and $\theta$ is the air gap circumferencial coordinate (see Fig. 7(a)). The definition above presumes several assumptions, which are discussed in detail in [20]. The most strict ones among them are that no eddy currents are considered, that the stator and rotor iron parts are assumed to be perfectly permeable and that the magnetic field is orientated straight radially. Furthermore end-field effects are included as stray load losses only.

The magnetic flux density can be expressed as

$$\begin{aligned} B(\varvec{x}, \varvec{i},\theta ) = \varLambda (\varvec{x},\theta ) F(\varvec{x}, \varvec{i}, \theta ), \quad \text {where} \quad \varLambda (\varvec{x},\theta ) = \frac{\mu _0}{\delta (\varvec{x},\theta )} \end{aligned}$$

is the air gap permeance and $\delta$ is the actual air gap width depending on the momentarily lateral rotor position, and the circumferencial coordinate.

Expressing both the air gap permeance and the MMF in terms of Fourier series yields

$$\begin{aligned} \varLambda (\varvec{x}, \theta ) = \sum _{\kappa = -\infty }^{\infty } \underline{\lambda }_\kappa (\varvec{x}) e^{-j\kappa \theta }, \, F(\varvec{x}, \varvec{i}, \theta ) = F_0(\varvec{x}, \varvec{i}) + \sum _{\nu = -\infty }^\infty \underline{f}_\nu (\varvec{x},\varvec{i}) e^{-j\nu \theta }. \end{aligned}$$

(11)

The permeance harmonics are given by [18]

$$\begin{aligned} \underline{\lambda }_\kappa (\varvec{x}) = \frac{(1-\sqrt{1-\rho ^2})^{\vert \kappa \vert }}{\sqrt{1-\rho ^2}\rho ^{\vert \kappa \vert }}e^{-j\kappa \gamma }, \quad \text {with} \quad \rho = \frac{\sqrt{x_\text {M}^2+y_\text {M}^2}}{\delta _0} \quad \text {and} \quad \tan \gamma = \frac{y_\text {M}}{x_\text {M}}. \end{aligned}$$

Here $\delta _0$ is the nominal air gap width, when the rotor is centered.

For perfectly permeable iron, the MMF is a linear function of the currents, reading

$$\begin{aligned} F(\varvec{x},\theta ) = F_0(\varvec{x}, \varvec{i}, \theta ) + \varvec{N}^\top (\varvec{x},\theta ) \varvec{i}. \end{aligned}$$

In this context, $\varvec{N}$ is the winding distribution. Two of its components (a stator phase and a rotor mesh component) are exemplarily shown in Fig. 7 (b). The constant of integration $F_0$ is used to ensure the condition $\nabla \cdot \mathbf{B} = 0$.

Gathering all the information from above, Eq. (10) can be written in terms of harmonics as

$$\begin{aligned} W_\text {m}^* = \sum _{\mu =-\infty }^{\infty } \sum _{\nu =-\infty }^{\infty } \underline{\lambda }_{(\mu -\nu )}(\varvec{x}) \underline{f}_\nu (\varvec{x},\varvec{i}) \overline{\underline{f}}_\mu (\varvec{x},\varvec{i}). \end{aligned}$$

(12)

Using this functional and its generalised gradients

$$\begin{aligned} \varvec{F}_\text {m} = \frac{\partial W_\text {m}^*}{\partial \varvec{x}}, \quad \varvec{\psi }= \frac{\partial W_\text {m}^*}{\partial \varvec{i}} \end{aligned}$$

with respect to $\varvec{x}$ and $\varvec{i}$, the $\text {el-m. }$coupling can be calculated very efficiently.