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

Part of the Mechanisms and Machine Science book series (Mechan. Machine Science,volume 63)


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.


  • Electrical machines
  • Rotordynamics
  • Eccentricity
  • Unbalanced magnetic pull
  • Lagrange-Maxwell equations
  • Magnetic energy

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  1. 1.

    Note that up to this point, no further restrictions have to be considered and neither magnetic saturation, nor other effects (like slotting etc.) have been excluded. Additional assumptions have to be made, when solving the magnetic field problem using a certain method (see section results). In fact, even hysteresis effects can be covered by the proposed approach, when introducing proper generalised forces.

  2. 2.

    Here in terms of forces, torque or voltages.

  3. 3.

    Usually these expressions contain dissipative effects like viscous mechanical damping, external forces, electrical resistances or external voltages.

  4. 4.

    E.g when using the Maxwell stress tensor or the definition of the flux linkage.

  5. 5.

    Concerning this point, a special case has been considered in earlier works (e.g. [2, 6]), where the eccentrical motion has been assumed to be a circular shaped foreward whirl with the same frequency of rotation as the rotor speed. If these assumptions are met and if the equations of motion are linearised with respect to \(\varvec{x}\), it is possible to define an equivalent magnetic damping constant covering the effect of the tangential magnetic force. However, it should be noted here, that such force models can only be applied, when considering a particular solution for the eccentrical motion. If transient states, or even the stability of this particular solution (in terms of a small pertubations) shall be considered, the assumptions for the definition of an equivalent magnetic damping constant are no longer met and therefore the general force model (Eq. (5)) has to be applied.

  6. 6.

    Note that support in this context does not mean the machine support. For electro-mechanical interactions only the relative motion between stator and rotor is relevant.

  7. 7.

    The forces \(\varvec{f}_\text {imb}\) are due to mechanical imbalance. They are usually assigned to the right hand side of the equations of motion, but stem from the left hand side originally.

  8. 8.

    The orbits are extracted from the simulation data for one period. Although they seem to be closed, they certainly change their shape with time from period to period and therefore are not perfectly closed.


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

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

Fig. 7.
figure 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

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

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

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

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

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

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.

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Boy, F., Hetzler, H. (2019). A Co-energy Based Approach to Model the Rotordynamics of Electrical Machines. In: Cavalca, K., Weber, H. (eds) Proceedings of the 10th International Conference on Rotor Dynamics – IFToMM. IFToMM 2018. Mechanisms and Machine Science, vol 63. Springer, Cham.

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