## 2 Study System Definition and Operating Principle
Figure 2: Typical configurations of a star- or delta-connected AC machine. Considering the circuits \(AA^{\prime}O^{\prime}O\), \(BB^{\prime}O^{\prime}O\), and CC\({}^{\prime}O^{\prime}O\) in Figure 3, the three-phase voltages \(V_{AA^{\prime}}\), \(V_{BB^{\prime}}\) and \(V_{CC^{\prime}}\) can be expressed using Kirchhoff's law, as shown in Equation (3):  
\[ \begin{cases}V_{AA^{\prime}}=V_{AO}-V_{A^{\prime}O^{\prime}}+V_{OO^{\prime}}\\ V_{BB^{\prime}}=V_{BO}-V_{B^{\prime}O^{\prime}}+V_{OO^{\prime}}\\ V_{CC^{\prime}}=V_{CO}-V_{C^{\prime}O^{\prime}}+V_{OO^{\prime}}\end{cases}  \tag{3}\]  
where \(V_{AO}\), \(V_{BO}\), \(V_{CO}\) and \(V_{A^{\prime}O^{\prime}}\), \(V_{B^{\prime}O^{\prime}}\), \(V_{CC^{\prime}O^{\prime}}\) represent the three-phase pole voltages of the master inverter and the slave inverter, respectively. Assuming that the three-phase loads are symmetrical, the sum of the load phase voltages must be zero, as expressed in Equation (3). \[V_{AA^{\prime}}+V_{BB^{\prime}}+V_{CC^{\prime}}=0 \tag{4}\]  
Thus, \(V_{OO^{\prime}}\) can be obtained by summing the three equations in (3), leading to Equation (5). \[V_{OO^{\prime}}=\frac{V_{A^{\prime}O^{\prime}}+V_{B^{\prime}O^{\prime}}+V_{C^{ \prime}O^{\prime}}}{3}-\frac{V_{AO}+V_{BO}+V_{CO}}{3} \tag{5}\]  
By substituting Equation (5) into Equation (3), the phase voltage expression is updated as follows:  
\[ \begin{cases}V_{AA^{\prime}}=\left(\frac{2}{3}V_{AO}-\frac{1}{3}V_{BO}-\frac{1 }{3}V_{CO}\right)-\left(\frac{2}{3}V_{A^{\prime}O^{\prime}}-\frac{1}{3}V_{B^{ \prime}O^{\prime}}-\frac{1}{3}V_{C^{\prime}O^{\prime}}\right)\\ V_{BB^{\prime}}=\left(\frac{2}{3}V_{BO}-\frac{1}{3}V_{CO}-\frac{1}{3}V_{AO} \right)-\left(\frac{2}{3}V_{B^{\prime}O^{\prime}}-\frac{1}{3}V_{C^{\prime}O^{ \prime}}-\frac{1}{3}V_{A^{\prime}O^{\prime}}\right)\\ V_{CC^{\prime}}=\left(\frac{2}{3}V_{CO}-\frac{1}{3}V_{AO}-\frac{1}{3}V_{BO} \right)-\left(\frac{2}{3}V_{A^{\prime}O^{\prime}}-\frac{1}{3}V_{C^{\prime}O^{ \prime}}-\frac{1}{3}V_{B^{\prime}O^{\prime}}\right)\end{cases}  \tag{6}\]  
From Equation (6), it can be observed that the phase voltage \(V_{ij}\) applied by the DTLI to the OEWM in Figure 2 corresponds to the voltage difference between the two inverters, \(V_{i}\) and \(V_{j}\). Thus, \(V_{ij}\) can be expressed by Equation (7) [96]. \[V_{ij}=V_{i}-V_{j} \tag{7}\]  
In the power circuit configuration of the three-phase inverters shown in Figure 3, the two switches in each inverter leg operate in a complementary manner, meaning that, when  
Figure 4: Vectorial diagram of the voltage space of each inverter: (**a**) Master inverter. (**b**) Slave inverter. one switch is turned on, the other is turned off. By utilizing this switching behavior, the phase output voltages of each inverter, referenced to the negative terminal of the DCB, can be expressed in matrix form as follows:  
\[ \begin{bmatrix}v_{i,A}\\ v_{i,B}\\ v_{i,C}\end{bmatrix} =\frac{V_{dc}}{3} \begin{bmatrix}2&-1&-1\\ -1&2&-1\\ -1&-1&2\end{bmatrix} \cdot \begin{bmatrix}S_{11}\\ S_{S3}\\ S_{S5}\end{bmatrix}  \tag{8}\]  
\[ \begin{bmatrix}v_{j,A^{\prime}}\\ v_{j,B^{\prime}}\\ v_{j,C^{\prime}}\end{bmatrix} =\frac{V_{dc}}{3} \begin{bmatrix}2&-1&-1\\ -1&2&-1\\ -1&-1&2\end{bmatrix} \cdot \begin{bmatrix}S_{j1}\\ S_{j3}\\ S_{j5}\end{bmatrix}  \tag{9}\]  
As presented in Equation (7), the phase voltages applied by the DTLI to the OEWM are obtained by computing the difference between the output voltages of the two inverters. This relationship can therefore be modeled by the following equation, which yield  
\[ \begin{bmatrix}v_{ij}\\ v_{ij}\end{bmatrix} = \begin{bmatrix}v_{i,A}-v_{j,A^{\prime}}\\ v_{i,B}-v_{j,B^{\prime}}\\ v_{i,C}-v_{j,C^{\prime}}\end{bmatrix} =\frac{V_{dc}}{3} \begin{bmatrix}2&-1&-1 \\ -1&2&-1\\ -1&-1&2\end{bmatrix} \cdot \begin{bmatrix}S_{i1}-S_{j1}\\ S_{S3}-S_{j3}\\ S_{S5}-S_{j5}\end{bmatrix}  \tag{10}\]  
Based on this principle, the space vector diagram of a DTLI can be represented in Figure 5, where the hexagonal space vector set generated by the slave inverter 'orbits' around the hexagon of the master inverter.