System Model

System Model#

This document describes the continuous-time models of the motulator_bearingless.model module. The conventions are those of motulator: peak-valued complex space vectors, the superscript s marks the stationary coordinates, and space vectors without a superscript are in rotor coordinates. Uppercase M denotes mechanical quantities.

Vector-Space Decomposition#

The symmetrical six-phase winding consists of the phases \(k = 1, \ldots, 6\), whose magnetic axes are at \(60^\circ\) intervals. The winding is described by means of the vector-space decomposition [Ahmed et al., 2023]

(1)#\[ \mathbf{x}_{\mathrm{s}h}^\mathrm{s} = \frac{1}{3}\sum_{k=1}^{6} x_k \e^{\jj h (k - 1)\pi/3}\]

where the plane \(h = 1\) is the two-pole plane and \(h = 2\) the four-pole plane. The zero-sequence components are omitted. The first three-phase set consists of the phases 1, 3, 5 and the second set of the phases 2, 4, 6, see motulator_bearingless.utils.vsd(). The six-phase converter of motulator_multiphase.model.MultiThreePhaseConverter feeds the two sets.

Machine Model#

The machine model motulator_bearingless.model.BearinglessSynchronousMachine is represented in rotor coordinates, where the space vectors of the planes are \(\ish{1} = \e^{-\jj\thetaM}\ish{1}^\mathrm{s}\) and \(\ish{2} = \e^{-2\jj\thetaM}\ish{2}^\mathrm{s}\), \(\thetaM\) being the mechanical rotor angle. The rotor displacement is \(\qM = \e^{-\jj\thetaM}\qM^\mathrm{s}\) and the radial force \(\fM = \e^{-\jj\thetaM}\fM^\mathrm{s}\), whose real and imaginary parts in stator coordinates correspond to the i and j axes. The stator voltage equations are

(2)#\[ \frac{\D\psish{1}}{\D t} = \ush{1} - \Rs\ish{1} - \jj\omegaM\psish{1} \qquad \frac{\D\psish{2}}{\D t} = \ush{2} - \Rs\ish{2} - 2\jj\omegaM\psish{2}\]

where \(\omegaM\) is the mechanical rotor speed. A magnetic model gives the currents, the radial force, and the torque as functions of the flux linkages, the displacement, and the rotor angle. It is lossless, i.e., it follows from the magnetic energy \(W(\psish{1}, \psish{2}, \qM, \thetaM)\) [Das et al., 2026]. The electromagnetic torque is

(3)#\[ \tauM = 3\,\mathrm{Im}\{\ish{1}\psish{1}^*\} + 6\,\mathrm{Im}\{\ish{2}\psish{2}^*\} + \mathrm{Im}\{\qM\fM^*\} - \frac{\partial W}{\partial\thetaM}\]

where the last term is due to the spatial harmonics.

Magnetic Models#

The magnetic model is selected via the machine parameters, which correspond to those of motulator:

The maps can be gradient networks, see Gradient-Network Magnetic Maps, or any other function with the same signature.

Radial Dynamics#

The rotor is either held at a given radial position by external bearings, see motulator_bearingless.model.ExternalRotorPosition, or it levitates, see motulator_bearingless.model.RadialMechanics. In the latter case, the rotor is modeled as a point mass \(m\) in stator coordinates

(4)#\[ m\frac{\D\vM^\mathrm{s}}{\D t} = \fM^\mathrm{s} + \mathbf{f}_\mathrm{ext}^\mathrm{s} - c\vM^\mathrm{s} \qquad \frac{\D\qM^\mathrm{s}}{\D t} = \vM^\mathrm{s}\]

where \(\mathbf{f}_\mathrm{ext}^\mathrm{s}\) is the external force (e.g., gravity) and \(c\) is the viscous damping coefficient. The tilting and axial motion are not modeled. The rotational dynamics are modeled by the mechanical system models of motulator.