Relation to motulator-multiphase

The package motulator-multiphase models and controls multi-three-phase machines by means of the decoupled multi-stator (DMS) transformation [Rubino et al., 2021]: the three-phase sets are mapped to a common mode (CM), which is a three-phase machine, and to differential modes (DMs), which are resistive-inductive circuits. The package motulator-bearingless describes a symmetrical six-phase bearingless synchronous machine by means of the vector-space decomposition (VSD) [Ahmed et al., 2023]: the winding is mapped to a two-pole plane and a four-pole plane, which are coupled by the rotor displacement. This document shows how the transformations and the models of the two packages are related. The control principle of the multi-three-phase drives, i.e., vector control of the main mode and current control of the other modes, is applied to the bearingless machine in the current-based force control, see Control.

The notation is that of System Model. In addition, real vectors and matrices are denoted by bold symbols.

Transformations

The winding has the phases \(1,\ldots,6\) at \(60^\circ\) intervals. The first three-phase set consists of the phases \(1, 3, 5\) and the second set of the phases \(2, 4, 6\). Each set is described by its standard space vector in its own coordinates,

\[ \mathbf{x}_\mathrm{A} = \frac{2}{3}\left(x_1 + \alpha^2 x_3 + \alpha^4 x_5\right) \qquad \mathbf{x}_\mathrm{B} = \frac{2}{3}\left(x_2 + \alpha^2 x_4 + \alpha^4 x_6\right)\]

where \(\alpha = \e^{\jj\pi/3}\). The a-phase axis of the second set is at the angle \(\theta\) with respect to that of the first set; here \(\theta = 60^\circ\) (symmetrical six-phase winding), while the default of motulator-multiphase is the asymmetrical arrangement \(\theta = 30^\circ\) (motulator_multiphase.utils.default_set_angles()).

For two sets, the DMS transformation (motulator_multiphase.utils.dms_matrix()) of the set vectors expressed in common coordinates is

\[ \mathbf{x}_\mathrm{cm}^\mathrm{s} = \frac{1}{2}\left(\mathbf{x}_\mathrm{A} + \e^{\jj\theta}\mathbf{x}_\mathrm{B}\right) \qquad \mathbf{x}_\mathrm{dm}^\mathrm{s} = \frac{1}{2}\left(\mathbf{x}_\mathrm{A} - \e^{\jj\theta}\mathbf{x}_\mathrm{B}\right)\]

The VSD with peak-value scaling maps the six phase quantities to the planes \(h = 1, 2\) according to

\[ \mathbf{x}_{\mathrm{s}h}^\mathrm{s} = \frac{1}{3}\sum_{k=1}^{6} x_k\alpha^{h(k-1)}\]

which in terms of the set vectors reads (motulator_bearingless.utils.vsd())

(1)\[ \mathbf{x}_\mathrm{s1}^\mathrm{s} = \frac{1}{2}\left(\mathbf{x}_\mathrm{A} + \alpha\mathbf{x}_\mathrm{B}\right) \qquad \mathbf{x}_\mathrm{s2}^\mathrm{s} = \frac{1}{2}\left(\mathbf{x}_\mathrm{A} - \alpha\mathbf{x}_\mathrm{B}\right)^*\]

Hence, for \(\theta = 60^\circ\), the DMS and the VSD are the same transformation up to a complex conjugate,

\[ \mathbf{x}_\mathrm{cm}^\mathrm{s} = \mathbf{x}_\mathrm{s1}^\mathrm{s} \qquad \mathbf{x}_\mathrm{dm}^\mathrm{s} = \left(\mathbf{x}_\mathrm{s2}^\mathrm{s}\right)^*\]

The conjugate results from the fact that the four-pole plane sees the phase sequence of each set reversed, since \(2\cdot 120^\circ = 240^\circ = -120^\circ\). Since the conjugation does not commute with the rotation of the coordinates, the identity holds in stator coordinates. Equivalently, if the set vectors are conjugated and \(\theta = 120^\circ\) (the set angle seen as an electrical angle in the four-pole plane) is used, the CM equals the four-pole plane and the DM the conjugated two-pole plane. Which plane is called the CM is thus a matter of naming; the physical difference between the packages is in the models attached to the modes. For \(\theta = 30^\circ\), the DMS does not correspond to the VSD of (1), since the asymmetrical winding is a different winding.

Machine Models

Multi-Three-Phase Model

In motulator-multiphase, the CM is a three-phase machine whose sets are connected in parallel: the voltage and the flux linkage of the equivalent machine are those of the CM, its current is the sum of the set currents, and the resistances and inductances of a set are divided by the number of sets (motulator_multiphase.utils.to_parallel()). Each DM is a resistive-inductive circuit, \(\D\boldsymbol{\uppsi}_\mathrm{dm}^\mathrm{s}/\D t = \mathbf{u}_\mathrm{dm}^\mathrm{s} - R_\mathrm{s}\boldsymbol{\uppsi}_\mathrm{dm}^\mathrm{s}/L_{\ell\mathrm{s}}\), which produces no torque and is not coupled to the CM. The modes are decoupled by assumption.

Bearingless Machine Model

The four-pole rotor produces the torque with the four-pole plane (\(h = 2\)) and the radial force with the two-pole plane (\(h = 1\)). With linear magnetics, the flux linkages in rotor coordinates are \(\boldsymbol{\uppsi}_\mathrm{s} = \mathbf{L}_\mathrm{s}(\mathbf{q}_\mathrm{M})\mathbf{i}_\mathrm{s}\), where \(\mathbf{i}_\mathrm{s} = [i_\mathrm{s1d}, i_\mathrm{s1q}, i_\mathrm{s2d}, i_\mathrm{s2q}]^\T\), \(\mathbf{q}_\mathrm{M} = [q_i, q_j]^\T\), and the inductance matrix is [Sokolov et al., 2019], [Das et al., 2026]

(2)\[\begin{split} \mathbf{L}_\mathrm{s} = \begin{bmatrix} L_1\mathbf{I} & \mathbf{L}_{21}^\T \\ \mathbf{L}_{21} & \operatorname{diag}(L_\mathrm{d}, L_\mathrm{q}) \end{bmatrix} \qquad \mathbf{L}_{21} = \begin{bmatrix} m_\mathrm{d} q_i & -m_\mathrm{d} q_j \\ m_\mathrm{q} q_j & m_\mathrm{q} q_i \end{bmatrix}\end{split}\]

Here \(L_1\) is the inductance of the two-pole plane, isotropic for the four-pole rotor, \(L_\mathrm{d}\) and \(L_\mathrm{q}\) are the inductances of the four-pole plane, and \(m_\mathrm{d}\) and \(m_\mathrm{q}\) are the radial force constants. The radial force is \(\mathbf{f}_\mathrm{M} = \frac{3}{2}\mathbf{i}_\mathrm{s}^\T(\partial\mathbf{L}_\mathrm{s}/\partial\mathbf{q}_\mathrm{M})\mathbf{i}_\mathrm{s}\), which can be written in the complex form

(3)\[ \mathbf{f}_\mathrm{M} = 3\mathbf{k}\,\mathbf{i}^*_\mathrm{s1} \qquad \mathbf{k} = m_\mathrm{d}i_\mathrm{s2d} + \jj m_\mathrm{q}i_\mathrm{s2q}\]

The force is bilinear in the currents of the two planes, and it does not depend explicitly on the displacement, since \(\mathbf{L}_\mathrm{s}\) is linear in \(\mathbf{q}_\mathrm{M}\). The torque is \(\tau_\mathrm{M} = 3\operatorname{Im}\{\mathbf{i}_\mathrm{s1}\boldsymbol{\uppsi}^*_\mathrm{s1}\} + 6\operatorname{Im}\{\mathbf{i}_\mathrm{s2}\boldsymbol{\uppsi}^*_\mathrm{s2}\} + \operatorname{Im}\{\mathbf{q}_\mathrm{M}\mathbf{f}^*_\mathrm{M}\}\), and the voltage equations of the planes are

\[ \frac{\D\boldsymbol{\uppsi}_\mathrm{s1}}{\D t} = \mathbf{u}_\mathrm{s1} - R_\mathrm{s}\mathbf{i}_\mathrm{s1} - \jj\omega_\mathrm{M}\boldsymbol{\uppsi}_\mathrm{s1} \qquad \frac{\D\boldsymbol{\uppsi}_\mathrm{s2}}{\D t} = \mathbf{u}_\mathrm{s2} - R_\mathrm{s}\mathbf{i}_\mathrm{s2} - \jj 2\omega_\mathrm{M}\boldsymbol{\uppsi}_\mathrm{s2}\]

A force that is constant in stator coordinates results from a two-pole field rotating at the electrical angular speed \(2\omega_\mathrm{M}\) in stator coordinates, i.e., at the same speed as the four-pole field. Hence, the force-producing two-pole quantities rotate at \(\omega_\mathrm{M}\) in their own rotor coordinates.

Relation of the Models

For a centric rotor, \(\mathbf{L}_{21} = \mathbf{0}\), and the model is block diagonal: the four-pole plane is a three-phase synchronous reluctance machine with two pole pairs, and the two-pole plane is a resistive-inductive circuit with the inductance \(L_1\). In terms of the DMS model with \(\theta = 60^\circ\), the four-pole plane is the (conjugated) DM and the two-pole plane the CM. The roles are thus opposite to the multi-three-phase model: the torque is produced in the DM, and the CM is a resistive-inductive circuit. The four-pole plane is exactly the parallel-connected equivalent machine of motulator_multiphase.utils.to_parallel() with \(n_\mathrm{p} = 2\), the current of which is \(2\mathbf{i}_\mathrm{s2}\), i.e., the sum of the set currents, and whose torque \(6\operatorname{Im}\{\mathbf{i}_\mathrm{s2}\boldsymbol{\uppsi}^*_\mathrm{s2}\}\) is the total torque (motulator_bearingless.utils.fourpole_pars()).

The force arises solely from the displacement-dependent coupling \(\mathbf{L}_{21}\), which the DMS model assumes to be zero. Therefore, the multi-three-phase model cannot represent a bearingless machine, and the bearingless model is a generalization of the DMS model in which the modes are coupled via the displacement. In motulator-bearingless, the magnetic model is the gradient of a magnetic energy function, so that the linear model (2) can be replaced with a saturated one without changing the control laws. The table below summarizes the correspondences.

	motulator-multiphase	motulator-bearingless
Transformation	DMS, \(\mathbf{x}_\mathrm{cm}^\mathrm{s}\) and \(\mathbf{x}_\mathrm{dm}^\mathrm{s}\)	VSD, \(\mathbf{x}_\mathrm{s1}^\mathrm{s} = \mathbf{x}_\mathrm{cm}^\mathrm{s}\) and \(\mathbf{x}_\mathrm{s2}^\mathrm{s} = \left(\mathbf{x}_\mathrm{dm}^\mathrm{s}\right)^*\)
Torque-producing mode	CM, three-phase machine	Four-pole plane, synchronous reluctance machine
Other mode	DM, \(R_\mathrm{s}\) and \(L_{\ell\mathrm{s}}\)	Two-pole plane, \(R_\mathrm{s}\) and \(L_1\)
Coupling of the modes	None	\(\mathbf{L}_{21}(\mathbf{q}_\mathrm{M})\), produces the force
Rotation of the modes	Same electrical angular speed	\(2\omega_\mathrm{M}\) in both planes (steady-state force)