# Current Harmonic Suppression Algorithm for Asymmetric Dual Three-Phase PMSM
## 3 Analysis of Dead-Time Effects  
In order to facilitate the analysis, Figure 3 shows one phase leg of the widely used six-phase PWM inverter during the dead-time period. If the phase current flows into the stator, the direction is defined as positive and vice versa. Figure 3: Phase leg of the converter during the dead-time period. In Figure 3, the turn-on time is usually shorter than the turn-off time. In order to avoid the simultaneous conduction of both switching devices Q1 and Q2, fixed dead-time insertion is necessary to ensure the safety of the inverter system. The dead time could cause the desired reference voltages to distort and affect the control performance. During the dead-time period, both switches are turned off, and the current can only flow through the free-wheeling diode D1/D2, if the polarity of phase current is negative/positive. So, the output voltage relies on the direction of phase current during the dead-time period from the above analysis. Taking the PWM waveforms of one phase leg for example, Figure 4a,b show the situations when the phase current is positive or negative, respectively. The average voltage error in one PWM period can be obtained as  
\[U_{err}=U_{d}*sign(i),\hskip 5.690551 ptsign(i)=\left\{ \begin{array}{cc}-1,&i<0 \\ 1,&i>0\end{array} \right. \tag{4}\]  
where \(sign(x)\) is a sign function and \(U_{d}\) represents the magnitude of voltage error. \(U_{d}\) can be expressed as  
\[U_{d}=\frac{T_{d}+T_{ond}-T_{offd}}{T_{s}}(U_{dc}-U_{sat}+U_{dicide})+U_{v} \tag{5}\]  
\[U_{v}=\left\{ \begin{array}{cc}\frac{T_{an}}{T_{s}}U_{sat}+\frac{T_{off}}{T _{s}}U_{dicide}&i>0\\ \frac{T_{off}}{T_{s}}U_{sat}+\frac{T_{on}}{T_{s}}U_{dicide}&i<0\end{array} \right. \tag{6}\]  
where \(T_{d}\) is the dead time, \(T_{s}\) is the PWM carrier period, \(T_{on}\) and \(T_{off}\) are the on-period and off-period of the switching device of the inverter leg, while \(T_{ond}\) and \(T_{offd}\) define the turn-on time delay and turn-off time delay of the switching device, respectively. \(U_{sat}\) and \(U_{dicide}\) denote the forward voltage drop of the  
Figure 4: Switching patterns and output voltages. (**a**) Phase current is positive. (**b**) Phase current is negative. switching device and the diode. \(U_{v}\) represents the average voltage drop in one PWM cycle. The error voltages of six phases can be transformed into the \(x\)-\(y\) coordinate axis in the \(x\)-\(y\) subspace as follows:  
\[\left[ \begin{array}{c}U_{xd}\\ U_{yd}\end{array} \right]=\frac{1}{3}\!\!\left[ \begin{array}{cccccc}1&-\frac{1 }{2}&-\frac{1}{2}&-\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}&0\\ 0&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&\frac{1}{2}&\frac{1}{2}&-1\end{array}  \right]\!\!\left[ \begin{array}{cccccc}U_{Ad}&U_{Bd}&U_{Cd}&U_{Dd}&U_{Ed}&U_{Fd }\end{array} \right]^{\!T} \tag{7}\]  
where \(U_{xd}\) and \(U_{yd}\) are the error voltages in the \(x\)-\(y\) coordinate axis in the \(x\)-\(y\) subspace, while \(U_{Ad}\), \(U_{Bd}\), \(U_{Cd}\), \(U_{Dd}\), \(U_{Ed}\), and \(U_{Fd}\) are the error voltages of phase A, B, C, D, E, and F, respectively. The error voltages in the \(x\)-\(y\) subspace can be obtained by applying the Fourier series as follows:  
\[\left[ \begin{array}{c}U_{xd}\\ U_{yd}\end{array} \right]=\frac{4U_{d}}{\pi}\left[ \begin{array}{c}\sum\limits \limits_{n=6k-1}^{\infty}\frac{\sin n(w_{e}t+\delta)}{n}+\sum\limits_{n=6k+1} ^{\infty}\frac{\sin n(w_{e}t+\delta)}{n}\\ -\sum\limits_{n=6k-1}^{\infty}\frac{\cos n(w_{e}t+\delta)}{n}+\sum\limits_{n=6k +1}^{\infty}\frac{\cos n(w_{e}t+\delta)}{n}\end{array} \right]\qquad k=1,3,5... \tag{8}\]  
where \(\delta\) denotes the angle between the current vector \(I_{s}\) and the \(q\)-axis, as depicted in Figure 5. Here, the anticlockwise rotation is defined as the positive direction. It is noteworthy that the fifth and seventh current harmonics will appear as the \(+5\)th and \(-7\)th in the \(x\)-\(y\) coordinate axis in the \(x\)-\(y\) subspace.