Dynamic Modeling Approach in View of Vector Control and Behavior Analysis of a Multi-Three-Phase Star Permanent Magnet Synchronous Motor Drive

Abstract

In order to fully utilize the control degrees of freedom of a multi-three-phase-star smooth-pole permanent magnet synchronous motor (PMSM), this paper first develops a modeling approach using a new matrix transformation method. The proposed transformation produces decoupled and independent star windings, removing the inductive couplings and preserving the model and torque control’s consistency as the number of phases increases. The model, together with a new vector control scheme, is superior for studying the effect of the winding phase angle shift on motor performance. Based on a numerical simulation, this paper focuses on the quality analysis of phase currents, non-sequential currents, and torque ripple with different phase angles for double- and triple-star PMSM drives. The control of a triple-star PMSM is validated, and the behavior analysis is investigated by OPAL-RT.

1. Introduction

In the current energy transition, multi-phase machines play a crucial role by offering a unique combination of advantages. Their power segmentation and high energy density, combined with their high tolerance to faults due to the elevated number of phases, make them indispensable choices for applications, such as naval and railway propulsion, as well as wind energy systems. By meeting the growing energy needs while reducing our environmental footprint, they significantly contribute to the transition towards more sustainable and efficient energy solutions.

The basis of multi-phase motor control is the development of its relatively accurate mathematical model. The stator phase number of multi-phase machines can be a prime integer or a composite number. Furthermore, if the winding sets have three phases, the machine is a multi-three-phase machine. This kind is particularly appealing since it can be powered by regular three-phase inverters. Recently, there has been a considerable rise in the research on this specific sort of multi-phase machine [1,2,3]. Due to the specificity of this type of structure, different coordinate transformations can be chosen: one is the coordinate transformation of the dual three-phase motor considered as a whole, i.e., based on the harmonic-based vector space decoupling (VSD) technique, proposed by Lipo [4] in 1995, and the other is the dual dq decoupling method proposed by Radu in 2003. The dual three-phase motor is considered as two three-phase motors for coordinate transformations; it is based on a dual synchronous rotational coordinate transformation (dual dq) [5].

The vector space decomposition (VSD) method is a technique used to perform machine modeling [4]. The machine is decoupled into orthogonal subspaces using this method: a single flux/torque-producing (α-β) subspace and numerous non-flux/torque-producing (x–y) subspaces [6,7,8,9]. Low-order harmonics and phase variable imbalances map onto the x–y subspaces and can be manipulated individually [10,11,12,13]. However, the transformation matrix calculation for motor modeling is complicated and requires the orthogonal computation of multi-dimensional vectors [14].

Another approach for modeling the machine can be by applying the well-known three-phase transformation by Clarke to each three-phase winding set [15,16], followed by the standard three-phase rotational transformation. By carrying this out, the machine is separated into several flux/torque-producing subspaces, and each subspace can be controlled using well-known methods created for three-phase machines [17,18]. Since this technique decomposes the multi-star motor into N mutually independent subspaces, it is conducive to the realization of fault-tolerant control. However, its motor model is not fully decoupled with the presence of weakly coupled mutual inductance [14].

Modeling is the basis of motor performance analysis, and one of the significant features of multi-phase motors is that the phase angle shift of the windings affects the performance of the motor, and the current research objective targets double three-phase motors. Many researchers have conducted a detailed analysis of the torque performance of surface-mounted permanent magnet synchronous motors [19,20], and the results of the studies demonstrated that the 30° phase-shift structure not only improved the air-gap magnetic fundamental content but also effectively eliminated the harmonic component, thus achieving the improvement of the motor’s permanent magnet torque performance. Another study conducted an in-depth comparative analysis of the torque performance of built-in dual three-phase permanent magnet motors [21]. Compared with the conventional three-phase winding method, the average torque of the motor increased by 4.8% and the torque pulsation reduced by 9% from 13.3% with the dual three-phase 30° phase shift structure. It can be seen that the optimal phase-shift structure is universal in its effectiveness in improving torque performance and is not affected by the rotor structure. However, the law of phase angle shift has not been further promoted, and there are few studies on the performance analysis of triple-three-phase motors.

The goal of this article is to develop a novel matrix transformation for removing the inductive couplings of a multi-three-phase star PMSM. It addresses the issue of increasing dimensionality with the VSD method as the number of phases increases and the problem of inductance coupling in the dq method. The modeling approach maintains the same form for the model matrix in the dq coordinate system, ensuring modeling consistency. Based on the unified model, the impact of phase angle shift in the windings of multi-three-phase star PMSM is analyzed and a generalized law is obtained. Finally, the real-time simulation of a triple-three-phase star PMSM has been carried out, which provides a new reference for the design of multi-three-phase star PMSM parameters.

2. Dynamical Model of the Multi-Three-Star PMSM

Figure 1 shows the electrical drives under consideration. The motor is a multi-three-phase star permanent magnet synchronous machine. These windings are coupled into q three-phase stars. The neutrals of all stars are connected together. Each three-phase star is supplied by each three-phase pulse width modulation (PWM) inverter. Each inverter consists of six insulated gate bipolar transistors (IGBTs). The PWM inverters supplied by the same DC voltage source will be assimilated to an N × 3-phase PWM inverter. Speed and torque control are ensured thanks to generalized vector control.

2.1. Assumptions

The PMSM in smoothed pole machine. The study is based on the following assumptions.

(1)

The multi-phase winding consists of q × 3 identical phases.

(2)

Variable reluctance effects and saturation phenomena are neglected.

(3)

Only the first space harmonic is taken into account.

(4)

The temperature effects are neglected.

(5)

The capacitive effect between the windings is neglected.

(6)

The semiconductor components constituting the inverters are supposed to be perfect.

(7)

The DC voltage source is assumed to be perfect.

2.2. Electrical Model in the Natural Basis

The multi-three-phase star permanent magnet synchronous motors under consideration in our study are composed of q three-phase windings, and the angle between each star winding is not uniformly distributed in space. The general electrical equation in the natural basis can be written as follows:

$$\left[V_{\mathrm{s}}\right]=\left[R_s\right]\left[I_s\right]+\left[L_s\right]\frac{d}{dt}\left[I_s\right]+\left[E_s\right]$$

(1)

[V_s] represents the supply voltage vector of the stator windings. It is defined as follows:

$$\left[V_s\right]={\left[\begin{array}{lllll}{\left[V_{s1}\right]}^t\hfill & {\left[V_{s2}\right]}^t\hfill & \cdots \hfill & \cdots \hfill & {\left[V_{sq}\right]}^t\hfill \end{array}\right]}^t$$

(2)

where $\left[V_{si}\right]={\left[\begin{array}{ccc}{V}_{ai}& V_{bi}& V_{ci}\end{array}\right]}^t$, i = 1, q and V_ai, V_bi, V_ci are the supply voltage of the star i.

[I_s] represents the supply current vector of the stator windings. It is defined as follows:

$$\left[I_s\right]={\left[\begin{array}{lllll}{\left[I_{s1}\right]}^t\hfill & {\left[I_{s2}\right]}^t\hfill & \cdots \hfill & \cdots \hfill & {\left[I_{sq}\right]}^t\hfill \end{array}\right]}^t$$

(3)

where $\left[I_{si}\right]={\left[\begin{array}{ccc}{I}_{ai}& I_{bi}& I_{ci}\end{array}\right]}^t$, i = 1, q and I_ai, I_bi, I_ci are the supply current of the star i.

[E_s] represents the EMF voltage vector. It is defined as follows:

$$\left[E_s\right]={\left[\begin{array}{lllll}{\left[E_{s1}\right]}^t\hfill & {\left[E_{s2}\right]}^t\hfill & \cdots \hfill & \cdots \hfill & {\left[E_{sq}\right]}^t\hfill \end{array}\right]}^t$$

(4)

where $\left[E_{si}\right]={\left[\begin{array}{ccc}{E}_{ai}& E_{bi}& E_{ci}\end{array}\right]}^t$, i = 1, q and E_ai, E_bi, E_ci are the supply voltage of the star i. It is given by the following relation:

$$\left[E_{si}\right]=\sqrt{2}\omega {\phi }_f{\left[\sin (\theta -(q-1)\gamma )\hspace{1em}\sin \left(\theta -\frac{2\pi }{3}-(q-1)\gamma \right)\hspace{1em}\sin \left(\theta +\frac{2\pi }{3}-(q-1)\gamma \right)\right]}^t$$

(5)

The resistance matrix [R_s] is a diagonal matrix:

$$\left[R_s\right]=r_s{[I]}_{3q\times 3q}$$

(6)

where r_s is the resistance of each winding and [I]_3q×3q represents the identity matrix with both rows and columns of size 3q.

[L_s] is the inductance matrix and it is given by the following relation:

$$[Ls]=\left[\begin{array}{ccccc}L& M_{s12}& \cdots & \cdots & M_{s13\mathrm{q}}\\ M_{s12}& \ddots & \dots & \dots & ⋮\\ ⋮& \dots & \ddots & \dots & ⋮\\ ⋮& ⋮& \dots & \ddots & ⋮\\ M_{s3\mathrm{q}1}& \cdots & \cdots & \cdots & L\end{array}\right]$$

(7)

where L represents the inductance of each winding. It can be written as follows:

$$L=l_{fs}+M_{ss}$$

(8)

where l_fs is leakage inductance and M_ss is mutual inductance.

M_sij represents the mutual inductance matrix between the phases i and j. It is defined as follows:

$$\left[M_{sij}\right]=M_{ss}\cos ({\theta }_{ij})$$

(9)

where θ_ij is the angle difference between the two phase windings i and j.

From the inductance matrix it can be easily shown that the matrix is fully coupled, so the control variables of the motor are complicated.

2.3. Electrical Model in the (α_iβ_io_i)_i=1,q Frame

In the case of polyphase systems, the windings are regular [6]. The inductance matrix is cyclic and it can be diagonalized. Hence, there is an orthogonal basis for the eigenvectors. Therefore, we can identify a coordinate frame such that the related variables are decoupled. The advantage of this is that it simplifies the matrix calculation during the simulation and enables the design of a decoupled control system. However, in the case of multi-star winding, generally, the inductance matrix is not cyclic.

In this section, we present an approach to reduce coupling terms based on the classical Concordia transformation. This process involves four steps to reach the electrical equations of the generalized model of the synchronous machine in the (alpha, beta) reference frame.

• STEP 1:

Hence, as shown in Figure 2 each star winding is a traditional star winding. According to the classical Concordia transformation, the axes of each star (a_ib_ic_i)_i=1,q will be moved to their new own axis system (α_iβ_io_i)_i=1,q.

In order to achieve this operation, the following transformation will be applied:

$$\left[\begin{array}{c}{X}_{\alpha i}\\ X_{\beta i}\\ X_{oi}\end{array}\right]={\left[T_{33}\right]}^t\left[\begin{array}{c}{X}_{ai}\\ X_{bi}\\ X_{ci}\end{array}\right] i=1,q$$

(10)

$${\left[T_{33}\right]}^t={\left[\begin{array}{cc}\left[T_{32}\right]& \left[T_{31}\right]\end{array}\right]}^t$$

(11)

$$\begin{array}{l}\left[T_{32}\right]=\sqrt{\frac{2}{3}}{\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ 0& -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]}^{\mathrm{t}}\\ \hspace{1em}\left[T_{31}\right]={\left[\begin{array}{lll}\frac{1}{\sqrt{3}}\hfill & \frac{1}{\sqrt{3}}\hfill & \frac{1}{\sqrt{3}}\hfill \end{array}\right]}^{\mathrm{t}}\end{array}$$

(12)

• STEP 2:

In order to reduce the magnetic coupling effect between the windings, as shown in Figure 3, a new rotation of the (i − 1) × γ angle is applied to each (α_iβ_io_i)_i=1,q system.

Thus, we define a new basis transformation for the above transformation:

$$\left[\begin{array}{l}{X}_{\alpha i}^{\prime }\hfill \\ X_{\beta i}^{\prime }\hfill \\ X_{oi}\hfill \end{array}\right]={\left[T_{3si}\right]}^t\left[\begin{array}{l}{X}_{asi}\hfill \\ X_{bsi}\hfill \\ X_{csi}\hfill \end{array}\right] i=1,q$$

(13)

The transformation matrix to convert the (α_sqβ_sq) plane to the (α^′_sq β^′_sq) plane is as follows:

$${\left[T_{3si}\right]}^t=\left[\begin{array}{c}{\left[\left[T_{32}\right]\cdot P(-\gamma )\right]}^t\\ {\left[T_{31}\right]}^t\end{array}\right] i=1,q$$

(14)

$$P\left(\gamma \right)=\left[\begin{array}{cc}\cos (\left(q-1\right)\gamma )& \sin (\left(q-1\right)\gamma )\\ -\sin (\left(q-1\right)\gamma )& \cos (\left(q-1\right)\gamma )\end{array}\right] i=1,q$$

(15)

where q is the serial number of the star winding.

Finally, the global transformation matrix which allows to write the all-physical variables in the new frame is:

$$\begin{array}{l}{\left[\left[\begin{array}{ccc}{X}_{\alpha 1}& X_{\beta 1}& X_{o1}\end{array}\right]\left[\begin{array}{ccc}{X}_{\alpha 2}^{\prime }& X_{\beta 2}^{\prime }& X_{o2}\end{array}\right]\cdots \left[\begin{array}{ccc}{X}_{\alpha q}^{\prime }& X_{\beta g}^{\prime }& X_{oq}\end{array}\right]\right]}^t\\ =\left[T_{3q3q}\right]{\left[\left[\begin{array}{ccc}{X}_{as1}& X_{bs1}& X_{cs1}\end{array}\right]\left[\begin{array}{ccc}{X}_{as2}& X_{bs2}& X_{cs2}\end{array}\right]\cdots \left[\begin{array}{ccc}{X}_{asq}& X_{bsq}& X_{csq}\end{array}\right]\right]}^t\end{array}$$

(16)

$$\left[T_{3q3q}\right]=\left[\begin{array}{cccc}{\left[T_{3s1}\right]}^t& {[0]}_{3\times 3}& \dots & {[0]}_{3\times 3}\\ {[0]}_{3\times 3}& {\left[T_{3s2}\right]}^t& \ddots & ⋮\\ ⋮& \ddots & \ddots & {[0]}_{3\times 3}^t\\ {[0]}_{3\times 3}& \cdots & {[0]}_{3\times 3}& {\left[T_{3sq}\right]}^t\end{array}\right]$$

(17)

By applying this transformation, the stator’s inductance matrix [L_s] in the new frame can be deduced:

$$\left[{L_s}^{\prime }\right]={\left[T_{3q3q}\right]}^t\left[L_s\right]\left[T_{3q3q}\right]$$

(18)

Thus, after the development of the calculation, we obtain

$$\left[{L_s}^{\prime }\right]=\left[\begin{array}{cccc}{\left[L_{cd}\right]}_{3\times 3}& {\left[M_c\right]}_{3\times 3}& \cdots & {\left[M_c\right]}_{3\times 3}\\ {\left[M_c\right]}_{3\times 3}& {\left[L_{cd}\right]}_{3\times 3}& \ddots & ⋮\\ ⋮& \ddots & \ddots & {\left[M_c\right]}_{3\times 3}\\ {\left[M_c\right]}_{3\times 3}& \cdots & {\left[M_c\right]}_{3\times 3}& {\left[L_{cd}\right]}_{3\times 3}\end{array}\right]$$

(19)

where the submatrices [L_cd] and [M_c] are given by the following relations:

$$\left[L_{cd}\right]=\left[\begin{array}{ccc}{l}_{fs}+\frac{3}{2}{M}_{ss}& 0& 0\\ 0& l_{fs}+\frac{3}{2}{M}_{ss}& 0\\ 0& 0& l_{fs}\end{array}\right] \left[M_c\right]=\left[\begin{array}{ccc}\frac{3}{2}{M}_{ss}& 0& 0\\ 0& \frac{3}{2}{M}_{ss}& 0\\ 0& 0& 0\end{array}\right]$$

(20)

• STEP 3:

In this new basis, the inductance matrix [L_s^’] is not completely diagonal. There is a coupling between the different stars represented by the matrix [M_c]. To solve this problem, the general torque expression in the (α^′_iβ^′_io^′_i)_{i=1, q} frame is analyzed. It can be written as follows:

$$\mathrm{\Gamma }=\frac{e_{a_1}{i}_{{\alpha }_1}+e_{{\beta }_1}{i}_{{\beta }_1}+e_{{a^{\prime }}_2}{i}_{{{\alpha }^{\prime }}_2}+e_{{{\beta }^{\prime }}_2}{i}_{{{\beta }^{\prime }}_2}+\cdots e_{a^{\prime }q}{i}_{{\alpha }^{\prime }q}+e_{{\beta }^{\prime }q{\beta }^{\prime }q}}{\mathrm{\Omega }}$$

(21)

It is important to notice that the transformation P(γ) defined above only introduces a rotation of the Emf vector and does not modify its module. We have the following:

$$\begin{array}{l}{e}_{{\alpha }^{\prime }i}=e_{\alpha 1}=e_{\alpha }\\ e_{{\beta }^{\prime }i}=e_{\beta 1}=e_{\beta }i=2,q\end{array}$$

(22)

Then, the torque expression can be simplified and it becomes the following:

$$\mathrm{\Gamma }=\frac{e_{\alpha }\left(i_{\alpha 1}+{{\displaystyle \sum }}_{i=2}^q{i}_{\alpha i}^{\prime }\right)+e_{\beta }\left(i_{\beta 1}+{{\displaystyle \sum }}_{i=2}^q{i}_{\beta i}^{\prime }\right)}{\mathrm{\Omega }}$$

(23)

This final formulation of the torque demonstrates that it depends on the sum of currents in the (α β o) axes. We then define equivalent currents in the (α β o) subspace:

$$\begin{array}{l}{i}_{\alpha }=i_{\alpha 1}+i_{\alpha 2}^{\prime }+\cdots +i_{\alpha q}^{\prime }\\ i_{\beta }=i_{\beta 1}+i_{\beta 2}^{\prime }+\cdots +i_{\beta q}^{\prime }\\ i_O=i_{01}+i_{02}^{\prime }+\cdots +i_{0q}^{\prime }\end{array}$$

(24)

By applying this transformation to all physical quantities and ensuring the preservation of vector norms, we then define the change-of-basis matrix from (α^′_iβ^′_io^′_i)_i=1,q to (α β o) and we obtain the following:

$$\left[\begin{array}{c}{X}_{\alpha }\\ X_{\beta }\\ X_o\end{array}\right]=\frac{1}{\sqrt{q}}\left[\begin{array}{c}{X}_{\alpha 1}+X_{\alpha 2}^{\prime }+\cdots +X_{\alpha q}^{\prime }\\ X_{\beta 1}+X_{\beta 2}^{\prime }+\cdots +X_{\beta q}^{\prime }\\ X_{o1}+X_{o2}^{\prime }+\cdots X_{oq}^{\prime }\end{array}\right]=\left[T_p\right]\left[\begin{array}{c}\left[\begin{array}{c}{X}_{\alpha 1}\\ X_{\beta 1}\\ X_{o1}\end{array}\right]\\ \left[\begin{array}{c}{X}_{\alpha 2}^{\prime }\\ X_{\beta 2}^{\prime }\\ X_{o2}^{\prime }\end{array}\right]\\ ⋮\\ ⋮\\ \left[\begin{array}{c}{X}_{\alpha q}^{\prime }\\ X_{\beta q}^{\prime }\\ X_{oq}^{\prime }\end{array}\right]\end{array}\right]$$

(25)

where the matrix [T_P] of dimension (3 × 3q) is given by the following relationship:

$$\left[T_p\right]=\frac{1}{\sqrt{q}}\left[{[I]}_{3\times 3}\hspace{1em}\left[\begin{array}{llll}I{]}_{3\times 3}\hfill & \cdots \hfill & \cdots \hfill & {[I]}_{3\times 3}\hfill \end{array}\right]\right.$$

(26)

Now, in order to preserve the order of the system (3q) and ensure the bijectivity of the transformation from the (α^′_iβ^′_io^′_i)_i=1,q frame to the new frame (α β o), this transformation also has the advantage of eliminating the coupling present between the elements of the inductance matrix. The currents resulting from this transformation do not produce torque and will now be referred to as non-sequential currents (i_zi)_i=1,(3q−2).

$$\begin{array}{l}{\left[\begin{array}{cccc}{i}_{z1}& i_{z2}& \cdots & i_{z(3q-2)}\end{array}\right]}^t=\\ \left[T_s\right]{\left[\begin{array}{cccc}{\left[\begin{array}{ccc}{i}_{\alpha 1}& i_{\beta 1}& i_{o1}\end{array}\right]}^t& {\left[\begin{array}{ccc}{i}_{\alpha 2}^{\prime }& i_{\beta 2}^{\prime }& i_{o2}^{\prime }\end{array}\right]}^t& \cdots & {\left[\begin{array}{ccc}{i}_{\alpha q}^{\prime }& i_{\beta q}^{\prime }& i_{oq}^{\prime }\end{array}\right]}^t\end{array}\right]}^t\end{array}$$

(27)

where the matrix [T_s] is a normalized matrix given by:

$$\left[T_s\right]=\frac{1}{\sqrt{q(q-1)}}\left[\begin{array}{ccccc}(q-1){[I]}_{3\times 3}& -{[I]}_{3\times 3}& \cdots & \cdots & -{[I]}_{3\times 3}\\ -{[I]}_{3\times 3}& (q-1){[I]}_{3\times 3}& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ -{[I]}_{3\times 3}& \cdots & -{[I]}_{3\times 3}& (q-1){[I]}_{3\times 3}& -{[I]}_{3\times 3}\end{array}\right]$$

(28)

Hence, to transform the physical variables of the (α^′_iβ^′_io^′_i)_i=1, frame to this new frame called (α β z_i)_i=1,(3q−2), the following matrix is defined:

$$\left[T_t\right]=\left[\begin{array}{c}\left[T_{\mathrm{p}}\right]\\ \left[T_s\right]\end{array}\right]$$

(29)

Finally, the transformation from the natural basis (α_iβ_io_i)_i=1,q to the new basis (α β z_i)_i=1,(3q−2) is obtained by making the product of the two matrices and taking into account the criterion of power conversion:

$${\left[T_{3q}\right]}^t=\left[T_t\right]\left[\begin{array}{cccc}{\left[T_{3s1}\right]}^t& {[0]}_{3\times 3}& \dots & {[0]}_{3\times 3}\\ {[0]}_{3\times 3}& {\left[T_{3s2}\right]}^t& \ddots & ⋮\\ ⋮& \ddots & \ddots & {[0]}_{3\times 3}\\ {[0]}_{3\times 3}& \dots & {[0]}_{3\times 3}& \left[T_{3s\mathrm{q}}\right]\end{array}\right]$$

(30)

• STEP 4:

By applying this transformation, the stator’s inductance matrix in the new frame can be deduced:

$$\left[L_{\mathrm{dia}g}\right]={\left[T_{3q}\right]}^t\left[L_s\right]\left[T_{3q}\right]$$

(31)

Thus after the development of the calculation, we obtain the following:

$$\left[L_{\mathrm{diag} }\right]=\left[\begin{array}{cc}{L}_c{[I]}_{2\times 2}& {[0]}_{2\times (3q-2)}\\ {[0]}_{(3q-2)\times (3q-2)}& l_{fs}{[I]}_{(3q-2)\times (3q-2)}\end{array}\right]$$

(32)

where

$$L_c=l_{ss}+\frac{3q}{2}{M}_{ss}$$

(33)

Now the electrical equation in this (α β z_i)_i=1,(3q−2) can be easily deduced:

$$\left\{\begin{array}{l}{V}_a=Ria+L_c\frac{d}{dt}\left(ia\right)-\sqrt{3q}\omega {\phi }_f\sin (\theta )\\ V_{\beta }=R{i}_{\beta }+L_c\frac{d}{dt}\left(i_{\beta }\right)+\sqrt{3q}\omega {\phi }_f\cos (\theta )\\ V_{zi}=R{i}_{zi}+l_{fs}\frac{d}{dt}\left(i_{zi}\right)\\ i=1,3q-2\end{array}\right.$$

(34)

2.4. Electrical Equation in Park’s Frame

By applying the classical Park transformation only to αβ component the dynamical electrical model in the (dqz_i)_i=1,3q−2 frame can be established:

$$\left\{\begin{array}{l}{V}_d=R{i}_d+L_c\frac{d}{dt}\left(i_d\right)-\omega L{i}_q\\ V_q=R{i}_q+L_c\frac{d}{dt}\left(i_q\right)+\omega L_c{i}_d+\sqrt{3q}\omega {\phi }_f\\ V_{zi}=R{i}_{zi}+l_{fs}\frac{d}{dt}\left(i_{zi}\right)\\ i=1,3q-2\end{array}\right.$$

(35)

The torque equation is:

$$\mathrm{\Gamma }=\frac{e_d{i}_d+e_q{i}_q}{\mathrm{\Omega }}=\frac{e_q{i}_q}{\mathrm{\Omega }}=p\frac{e_q{i}_q}{\omega }=\sqrt{3q}p{\phi }_f{i}_q$$

(36)

So, we deduce on three-phase motors, double-star motors and three-star motors, and the results obtained are regular, as shown in

Table 1. Summary of machine equations based on phase number.

	Diagonalized Inductance Matrix	Electrical Equation in dq Frame	Torque Equation
Three-phase PMSM	$\left[\begin{array}{cc}{L}_c{[I]}_{2\times 2}& {[0]}_{2\times 1}\\ {[0]}_{1\times 2}& l_{fs}\end{array}\right]$ $L_c=l_{fs}+\frac{3}{2}{M}_{ss}$	$\left\{\begin{array}{l}{V}_d=R{i}_d+L_c\frac{d}{dt}\left(i_d\right)-\omega L_c{i}_q\\ V_q=R{i}_q+L_c\frac{d}{dt}\left(i_q\right)+\omega L_c{i}_d+\sqrt{3}\omega {\phi }_f\\ V_o=R{i}_o+l_{fs}\frac{d}{dt}\left(i_o\right)\end{array}\right.$	$\mathrm{\Gamma }=\sqrt{3}p{\phi }_f{i}_q$
Double three-phase PMSM	$\left[\begin{array}{cc}{L}_c{[I]}_{2\times 2}& {[0]}_{2\times 4}\\ {\left[0\right]}_{4\times 2}& l_{fs}{[I]}_{4\times 4}\end{array}\right]$ $L_c=l_{fs}+3M_{ss}$	$\left\{\begin{array}{l}{V}_d=R{i}_d+L_c\frac{d}{dt}\left(i_d\right)-\omega L_c{i}_q\\ V_q=R{i}_q+L_c\frac{d}{dt}\left(i_q\right)+\omega L_c{i}_d+\sqrt{6}\omega {\phi }_f\\ V_{zi}=R{i}_{zi}+l_{fs}\frac{d}{dt}\left(i_{zi}\right)\\ i=1,4\end{array}\right.$	$\mathrm{\Gamma }=\sqrt{6}p{\phi }_f{i}_q$
Triple three-phase PMSM	$\left[\begin{array}{cc}{L}_c{[I]}_{2\times 2}& {[0]}_{2\times 7}\\ {\left[0\right]}_{7\times 2}& l_{fs}{[I]}_{7\times 7}\end{array}\right]$ $L_c=l_{fs}+\frac{9}{2}{M}_{ss}$	$\left\{\begin{array}{l}{V}_d=R{i}_d+L_c\frac{d}{dt}\left(i_d\right)-\omega L_c{i}_q\\ V_q=R{i}_q+L_c\frac{d}{dt}\left(i_q\right)+\omega L_c{i}_d+\sqrt{9}\omega {\phi }_f\\ V_{zi}=R{i}_{zi}+l_{fs}\frac{d}{dt}\left(i_{zi}\right)\\ i=1,7\end{array}\right.$	$\mathrm{\Gamma }=\sqrt{9}p{\phi }_f{i}_q$
Multi-three-phase PMSM	$\left[\begin{array}{cc}{L}_c{[I]}_{2\times 2}& {[0]}_{2\times (3q-2)}\\ {[0]}_{(3q-2)\times 2}& l_{fs}{[I]}_{(3q-2)\times (3q-2)}\end{array}\right]$ $L_c=l_{ss}+\frac{3q}{2}{M}_{ss}$	$\left\{\begin{array}{l}{V}_d=R{i}_d+L_c\frac{d}{dt}\left(i_d\right)-\omega L_c{i}_q\\ V_q=R{i}_q+L_c\frac{d}{dt}\left(i_q\right)+\omega L_c{i}_d+\sqrt{3q}\omega {\phi }_f\\ V_{zi}=R{i}_{zi}+l_{fs}\frac{d}{dt}\left(i_{zi}\right)\\ i=1,3q-2\end{array}\right.$	$\mathrm{\Gamma }=\sqrt{3q}p{\phi }_f{i}_q$

.

This proves that the method is effective when extended to multi-star winding motors. We derived the transformation matrix applicable to all multi-star windings. The transformation matrix can diagonalize the inductance matrix and simplify the electrical equation of the motor.

2.5. Calculation of the Voltage Vector [V_s]

The phase-to-phase voltages between phase a₁ and all other (3q − 1) phases can be easily written in the following form:

$$\left\{\begin{array}{l}{U}_{a1b1}=V_{a1n}-V_{b1n}=V_{a1m}-V_{b1m}\\ U_{a1c1}=V_{a1n}-V_{b1n}=V_{a1m}-V_{b1m}\\ U_{a1a1}=V_{a1n}-V_{a2n}=V_{a1m}-V_{a2m}\\ U_{a1b2}=V_{a1n}-V_{b2n}=V_{a1m}-V_{b2m}\\ U_{a1c2}=V_{a1n}-V_{c2n}=V_{a1m}-V_{c2m}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⋮\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⋮\\ U_{a1aq}=V_{a1n}-V_{aqn}=V_{a1m}-V_{aqm}\\ U_{a1bq}=V_{a1n}-V_{bqn}=V_{a1m}-V_{bqm}\\ U_{a1cq}=V_{a1n}-V_{cqn}=V_{a1m}-V_{cqm}\end{array}\right.$$

(37)

where n is the neutral of the multi-phase stator and m is the midpoint of the continuous bus. As the neutrals of all stars are connected and form a single isolated neutral and the phases are balanced, the sum of all phase voltages is zero:

$${\displaystyle \sum }_{i=1}^q\left(V_{ain}+V_{bin}+V_{cin}\right)=0$$

(38)

By combining Equations (32) and (33) we can deduce the following relation:

$$\left[V_s\right]=\left[T_q\right]\left[V_m\right]$$

(39)

where

$$\left[V_s\right]={\left[\begin{array}{ccc}\begin{array}{ccc}{V}_{a1n}& V_{b1n}& V_{c1n}\end{array}& \begin{array}{ccc}\cdots & \cdots & \cdots \end{array}& \begin{array}{ccc}{V}_{aqn}& V_{bqn}& V_{cqn}\end{array}\end{array}\right]}^t$$

(40)

$$\left[V_m\right]={\left[\begin{array}{ccc}\begin{array}{ccc}{V}_{a1m}& V_{b1m}& V_{c1m}\end{array}& \begin{array}{ccc}\cdots & \cdots & \cdots \end{array}& \begin{array}{ccc}{V}_{aqm}& V_{bqm}& V_{cqm}\end{array}\end{array}\right]}^t$$

(41)

$$\left[T_q\right]=\frac{1}{3q}\left[\begin{array}{ccc}\begin{array}{cc}3q-1& -1\\ -1& 3q-1\end{array}& \cdots & \begin{array}{cc}-1& -1\\ -1& -1\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}-1& -1\\ -1& -1\end{array}& \cdots & \begin{array}{cc}3q-1& -1\\ & 3q-1\end{array}\end{array}\right]$$

(42)

Each three-phase inverter is modeled by a connection matrix that connects the DC source to the phases of a three-phase winding of a star constituting the stator:

$$\left[C{M}_i\right]=\left[\begin{array}{ccc}{S}_{1i}& S_{2i}& S_{3i}\\ S_{i4}& S_{i5}& S_{i6}\end{array}\right] i=1,q$$

(43)

where S_{ji(j=1, 6)} represents the conduction state of the equivalent switch in the inverter i.

The voltage vector [V_m] depends on the conduction state of the inverter switches, by substituting Equation (38) into Equation (34), we deduce the voltage vector [V_m] as a function of the conduction states of the q three-phase inverters:

$$\left[V_m\right]=\left[\begin{array}{ccc}\begin{array}{cc}{\left[C{M}_1\right]}^t& {\left[0\right]}_{3\times 2}\\ {\left[0\right]}_{3\times 2}& {\left[C{M}_2\right]}^t\end{array}& \cdots & \begin{array}{cc}{\left[0\right]}_{3\times 2}& {\left[0\right]}_{3\times 2}\\ {\left[0\right]}_{3\times 2}& {\left[0\right]}_{3\times 2}\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}{\left[0\right]}_{3\times 2}& {\left[0\right]}_{3\times 2}\\ {\left[0\right]}_{3\times 2}& {\left[0\right]}_{3\times 2}\end{array}& \cdots & \begin{array}{cc}{\left[C{M}_{q-1}\right]}^t& {\left[0\right]}_{3\times 2}\\ {\left[0\right]}_{3\times 2}& {\left[C{M}_q\right]}^t\end{array}\end{array}\right]\left[\begin{array}{c}\frac{V_{DC}}{2}\\ \frac{-V_{DC}}{2}\end{array}\right]$$

(44)

3. Torque Control Strategy-Vector Control

Traditional vector control adopts double closed-loop control of speed outer loop and current inner loop. The reference value of the q-current is calculated by calculating the difference between the the reference torque and the actual torque, the current reference value of the d-current sets the zero-sequence component to 0, and sets The difference between the reference value and the actual value of the dq current and the zero-sequence current component is used as the input signal of the pi controller, and finally the control signal passes through the PWM converter to output the control signal of the motor.

Firstly, the torque control of the three-phase motor is established based on the PWM instantaneous model and then extrapolated to the three-phase double-star motor, and finally, the three-phase triple-star motor is studied.

Vector control transforms the three-phase stator current of the permanent magnet synchronous generator into decoupled quadrature direct-axis current i_d and torque current i_q through coordinate transformation, which is equivalent to the control of a DC motor.

In order to control the torque well, we need to determine the factors that affect the torque. From the first section, we obtain the torque equation:

$$\mathrm{\Gamma }=\frac{e_d{i}_d+e_q{i}_q+e_{zi}{{\displaystyle \sum }}_{i=1}^{3q-2}izi}{\mathrm{\Omega }}=\sqrt{3q}p{\phi }_f{i}_q$$

(45)

As the zero-sequence voltage e_zi = 0, the zero-sequence currents i_zi do not contribute to the development of the torque. In order to limit the losses by the Joule effect, i_d is imposed at 0. For a smooth pole motor, vector control torque Γ has a linear relationship with current i_q, and the given value of current i_q under the running state of the generator can be obtained by setting the reference value of the torque.

The magnetic torque Γ is determined by the number of pole pairs p of the motor, the rotor flux linkage φ_f, and the torque current i_q. For permanent magnet synchronous motors, after the motor is manufactured, the flux linkages φ_f of the permanent magnets will basically not change, and the number of pole pairs is also a constant. Therefore, the electromagnetic torque of the permanent magnet synchronous motor is only related to the torque current i_q, as long as the size of the torque current is controlled, the electromagnetic torque can be controlled, and the motor can be controlled under the condition of constant speed output power.

The schematic diagram of the vector control principle is shown in Figure 4. The direct current control method is adopted, and the feedback link is added to realize the current closed loop. Compared with the indirect current control, the sensitivity to parameters is reduced, and the static and dynamic performance of the system is optimized. The robustness of the system is enhanced.

As non-sequential currents do not contribute to torque production, and in order to minimize losses due to Joule effects, the instruction values of currents i_z1*, i_z2*, …, i_z(3q−2)* are set to zero.

The difference between the reference value and the actual value of the dq plane and the zero-sequence component passes through the PI control link to generate the dq plane and the zero-sequence component voltage. Then, the dq voltage passes through the decoupling module and then passes through the conversion module with other zero-sequence components to obtain the n-phase voltage signal input to the PWM converter. Finally, the inverter outputs the voltage signal of the multi-phase motor capable of controlling the torque.

It can be seen from the above formulas that the vector control of multi-phase motors is basically equivalent to that of three-phase motors, and the currents of the dq axes are all controlled. The difference is the number of zero-sequence components, such as six-phase output. The output is a six-phase signal, and the nine-phase output is a nine-phase signal.

4. Simulation Results

The section dedicated to simulating this study holds crucial importance in evaluating the performance and reliability of our generalized machine model. This study focuses on two specific machine configurations: one with a double-star winding and another with a triple-star winding. It is noteworthy that the program remains general, provided that the correct expressions for inductance (L_c) and torque (Γ) are specified (

Table 2. Expressions of inductance torque.

	Nine-Phases	Six-Phases
Lc	$L_c=l_{fs}+\frac{9}{2}{M}_{ss}$	$L_c=l_{fs}+3M_{ss}$
Γ	$\mathsf{\Gamma }=\sqrt{9}p{\phi }_f{i}_q$	$\mathsf{\Gamma }=\sqrt{6}p{\phi }_f{i}_q$

).

In this study, emphasis was placed on investigating the impact of the phase shift angle between the star windings. Additionally, we explored scenarios where the number of stars increases. Specifically, we examined both the case of two stars and the case of three stars. For the phase shift angles, we initially considered the scenario where the stars are in phase ($\gamma$ = 0). The second case involved equally spaced phases (for the double star, $\gamma$ = 60, and for the triple star, $\gamma$ = 40). Finally, we examined the case where the stars are shifted by $\gamma$ = 30, a scenario widely discussed in the literature due to its significant advantages in the context of the double star machine. Throughout these simulations on the Matlab/Simulink platform V2018A, we meticulously examined the machine’s behavior under the previously specified operating conditions.

During the simulation, we defined specific operating conditions for the machine. Between 0 and 3 s, a speed of 300 rpm was maintained, followed by an increase to 400 rpm from 3 to 10 s. As for the torque, a reference value of 10 N·m was applied from 0 to 5 s and then increased to 20 N·m from 5 to 10 s.

Table 3. Machine parameters.

Parameters	Values
Stator resistance Rs (Ohm)	2
Leakage inductance $l_{fs}$ (H)	0.562 × 10−3
Maximum mutual inductance between two stator’s windings $M_{ss}$ (H)	3.373 × 10−3
Rotor flux (Wb)	0.42
Moment of inertia (Kg·m2)	0.025
Number of pole pairs	6
Coefficient of viscose friction (N·m·s/rd)	0.01

presents the parameters used in the simulation.

Below are curves illustrating the performance of the two machine configurations with different angular offsets. Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 depict results for the double-star machine, while Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 represent simulation results for the triple-star machine. The results obtained for the double-star machine are presented below:

The results obtained for the triple-star machine are presented below:

Figure 10. Currents i_q and i_d of triple star PMSM.

Figure 10. Currents i_q and i_d of triple star PMSM.

Figure 11. Currents i_z1 to i_z4 of triple star PMSM.

Figure 11. Currents i_z1 to i_z4 of triple star PMSM.

Figure 12. Currents i_a1, i_a2 and i_a3 of triple star PMSM.

Figure 12. Currents i_a1, i_a2 and i_a3 of triple star PMSM.

Figure 13. Torque and speed of triple star PMSM.

Figure 13. Torque and speed of triple star PMSM.

Figure 14. RMS voltage of triple star PMSM.

Figure 14. RMS voltage of triple star PMSM.

From a global perspective, the simulation results for the various configurations discussed in this document indicate that the controlled variables, including torque, speed, and currents, closely follow their respective references, as shown in Figure 5 and Figure 10 for the currents (i_d) and (i_q), as well as in Figure 8 and Figure 13 for speed and torque.

Figure 6 and Figure 11 confirm our expectations by showing that the current i_z1 is zero. This observation is crucial as it demonstrates that, as anticipated, unconnected neutrals result in a null homopolar current.

In the scenario where γ ≠ 0, a non-sequential current flows between the stars through a loop formed by leakage inductances and winding resistances of the machine. This current does not affect the magnetizing part of the machine and, therefore, does not impact the quality of the generated torque.

However, when γ = 0, this same current passes through both the leakage inductances and the magnetizing part, representing a filtering of currents compared to the previous case. Figure 6 and Figure 11 demonstrate that in the case of γ = 0, the values of currents (i_z1) to (i_z4) (for the double-star machine) and (i_z1) to (i_z7) (for the triple-star machine) are of the order of (10⁻¹⁴). Conversely, when γ ≠ 0, currents (i_z2) to (i_z4) and (i_z2) to (i_z7) take non-zero values.

For the double-star machine, the current amplitudes reach 1 A for γ=π/6 and 0.3 A for γ = 2π/6. In the case of the triple-star machine, the current amplitudes reach 1.1 A for γ = π/6 and 0.8 A for γ = 2π/9.

Figure 5 and Figure 10 highlight a more significant ripple amplitude in the (i_q) current when (γ = 0) compared to the other cases at γ=π/6 and configurations where phases are evenly distributed ((γ = 2π/6) for double-star and (γ=2π/9) for triple-star machines). Note that ($∆i_q$) for (γ = 0) is 0.2 A for double-star and 0.14 A for triple-star machines. When (γ = π/6), the ripple amplitude is 0.1 A for the double-star and 7.78 × 10⁻² A for (i_q) ripples in the triple-star machine. With a (γ = 2π/6) for the double star machine, the ripple is 1.26 × 10⁻¹ A. For a (γ = 2π/9) angle in a triple-star machine, the ripple is 5.18 × 10⁻² A.

These results highlight the influence of the γ value on non-sequential currents and the magnitude of ripples in the i_q currents, which directly correlates with the torque. This analysis underscores the paramount importance of this parameter in the design, control, and efficient operation of electrical machines.

According to the formula $\mathsf{\Gamma }=\sqrt{3q}p{\phi }_f{i}_{qn}$, since the reference torque is the same for both double-star and triple-star machines, we have the following:

$$\sqrt{6}p{\phi }_f{i}_{q6}=\sqrt{9}p{\phi }_f{i}_{q9}$$

(46)

The average values of the two currents i_q6 and i_q9 in a steady state between t = 5 s and t = 10 s are 3.3 A and 2.7 A, respectively. The ratio between these two values is $\sqrt{6}/3$ confirming that our model is consistent with the equations of the machine stated earlier.

The phase currents of the two machines, namely the double-star and the triple-star machines, are, respectively, represented in Figure 7 and Figure 12, for which we zoomed into a region of the steady-state regime, specifically between t = 7 s and t = 7.1 s.

The results for the phase currents, as illustrated in Figure 7 and Figure 12, are consistent with the nature of non-sequential currents. In the case where (γ = 0), the phase currents take sinusoidal forms, but disturbances occur when (γ ≠ 0). These disturbances are due to the presence of low-order harmonics close to the operating frequency. We also observe that, for an equivalent power, increasing the number of phases leads to a reduction in the amplitudes of the phase currents. This can be explained by one of the advantages of multi-phase machines, which is power segmentation.

We also observe, as shown in Figure 9 and Figure 14, that as the load torque increases, the effective voltage value also increases. The same phenomenon is observed in the phase currents.

5. Real-Time Simulation of a Triple-Star PMSM Drive

In the preceding section of this paper, we underscored the significance of augmenting the number of phases, specifically emphasizing the superiority of three stars over two stars. In this vein, to authenticate the vector control methodology applied to the triple-star machine, propelled by three three-phase PWM inverters, we executed the control algorithm on a rapid prototyping platform. For this validation, we harnessed the capabilities of the OP4510 (target station) rapid prototyping platform by OPAL-RT. This platform, characterized as a real-time PC/FPGA-based simulator, facilitates hardware-in-the-loop (HIL) setups and rapid control prototyping (RCP) systems. Its utilization extends across a spectrum of industries including power grids, power electronics, electric motors, automotive, trains, aircraft, and beyond. Its robust architecture integrates high-end INTEL multicore processors with a powerful Xilinx® Kintex® 7 FPGA from Xilinx employed by the company OPAL RT in their OP4510 equipment, endowing it with enhanced simulation power and sub-microsecond simulation time steps. This combination ensures optimum accuracy and efficiency, particularly crucial for modeling fast power electronic systems.

Figure 15 illustrates the test bench used for validation purposes. It consists of a “target station”, which is the OP4510 platform, a “host station” representing the computer equipped with Matlab/Simulink software used to develop the model of the converter machine assembly, a digital and analog input/output panel used to interface between the program executed on the FPGA and the physical elements, and finally, two oscilloscopes for tracing the curves presented thereafter.

During the testing phase, three scenarios were implemented to thoroughly study the machine’s behavior in different situations. The three scenarios are as follows:

-

For 0 < t < 2 s: The reference speed is set to 300 rpm, representing 75% of its nominal value, and the reference electromagnetic torque is 80 Nm, representing 56% of its nominal value.

-

For 2 s < t < 5 s: The reference torque is maintained at its setpoint, while the reference speed transitions to its nominal value of 400 rpm.

-

In the last case, for 5 s < t < 10 s: The reference values for speed and torque transition to their respective nominal values of 400 rpm and 143 Nm. Therefore, in each curve presented hereafter, the plot between t = 5 s and t = 10 s represents the nominal values. To show the phase current quality in the steady state, we included the current curve at the nominal operating point.

The results obtained for the triple stars machine are presented below:

Overall, the simulation results obtained for the specific nine-phase machine configuration discussed in this document reveal a remarkable alignment between the controlled variables—namely torque, speed, and currents—and their respective references. This coherence is prominently highlighted in Figure 16, depicting speed and torque, a close convergence between the simulated values and the predefined references is evident. Similarly, Figure 17 is dedicated to illustrating currents (i_d) and (i_q), where a precise adherence to the reference values is observed.

It is important to emphasize that these optimal performances are maintained regardless of the value of the phase shift angle between the three windings. The angles studied in this document are γ = 0, γ = π/6, and γ = 2π/9. This diversity of angles demonstrates the robustness of the control model, capable of maintaining a precise and stable response in different configurations. This underscores the versatility and reliability of the vector control adopted for the nine-phase permanent magnet synchronous machine.

Figure 18 depicts the current of the first phase of each winding in the machine, namely i_a1, i_a2, and i_a3, for three values of the phase shift angle between the stator windings mentioned earlier. Upon comparing the curves obtained for each angle value, it is evident that, for an angle γ = 0, the stator currents are purely sinusoidal and exhibit no waveform distortion compared to those obtained for γ = π/6 and γ = 2π/9. This can be explained by the fact that, for the γ = 0 configuration, the sequential currents and non-sequential currents are in phase and traverse the same inductances, unlike the other two configurations where sequential currents and non-sequential currents follow different paths and are not in phase.

Figure 19A–C depict the evolution of the φ_β as a function of the φ_α, forming a circle in all three different cases. This is explained by the fact that the two components of the stator flux are sinusoidal components shifted by 90° from each other.

The plot of the components of the machine’s supply voltage in the (alpha, beta) coordinate system (V_β as a function of V_α) takes on three different forms. In the case of γ = 0, the clear formation of six sectors in the (alpha, beta) plane is evident. This is explained by the superposition of the supply to the three stator windings, functioning equivalently to a three-phase machine. In the case of γ = π/6, each three-phase winding is supplied by a power system consisting of six sectors. As the phase shift between the three windings is π/6, this results in an overlap between the sectors of the three windings. Hence, we observe the appearance of multiple sectors, but with different angles. In the final case where γ equals 2π/9, we observe a uniform distribution among the machine phases in the (alpha, beta) plane. This results in the emergence of 18 sectors, each with a width of π/18. Consequently, we have 18 different voltage vectors supplying the machine while adhering to the direction in the trigonometric circle.

The plot of the current i_z3 as a function of i_z2 is presented in Figure 19G–I, and we observe a minimal amplitude compared to the stator currents. These currents do not contribute to torque production but manifest as losses due to Joule effects. In the case of γ = 0 radians, the iz currents are truly negligible, in the order of millivolts. This simulation confirms the occurrence of non-sequential currents when the phase shift angle between windings is not zero.

6. Conclusions

This paper proposed a new modeling approach for a multi-three-phase star PMSM. The devised modeling approach uses a new decoupling transformation to remove the multi-three-phase star PMSM couplings of a generic modular configuration. This transformation matrix has a universality and regularity that enables the modeling of a multi-three-phase star PMSM that has the same structure as the number of phases increases.

On the basis of this unified model, the effect of phase angle shift during vector control of a multi-three-phase star PMSM is investigated. The results are as follows:

(1)

When all the stars are in same the phase, the value of the non-sequential current is close to 0. However, its value is not 0 when there is an angle shift. The phase current has the same law as the non-sequential current, and the waveform is close to sinusoidal when all the stars are in the same phase.

(2)

The ripple amplitude of the i_q of the multi-three-phase PMSM decreases as the angle of each star windings increases.

(3)

For an equivalent power, increasing the number of phases leads to a reduction in the amplitudes of the phase currents. An increase in load torque will make both the effective voltage amplitude and phase current amplitude increase.

The numerical simulation and real-time simulation results demonstrate the feasibility of the new model for phase angle shift performance analysis of a triple-three-phase star PMSM. Moreover, the proposed method can be extended to multi-three-phase star PMSM applications.