## 2 Sideband Component Feature Identification Based on DPWM
### Analysis of Sideband Current Harmonics and Radial Electromagnetic Force Characteristics  
Sideband current harmonics can be investigated by constructing the double Fourier series in the frame of the stator coordinate system with the modulating wave frequency \(\omega_{0}\) and the carrier frequency \(\omega_{c}\) as variable; the sideband harmonic components in the rotor coordinate system are then directly resolved by rearranging them through coordinate  
Figure 1: Diagram of the voltage space vector. Figure 2: DPWM switching sequence in sector I.  
transformations [6]. Taking the first carrier frequency domain as an example, when \(n=1\), ignoring the higher harmonics with smaller amplitudes, the sideband current harmonics in the DPWM can be shown as  
\[ \begin{array}{c}i_{h}(t)\approx\varepsilon_{1}i_{a_{1}}\cos[2\pi(\omega_{c} \pm\alpha_{1}\omega_{0})t]+\\ \varepsilon_{2}i_{a_{2}}\cos[2\pi(\omega_{c}\pm\alpha_{2}\omega_{0})t]+\\ \varepsilon_{3}i_{a_{3}}\cos[2\pi(\omega_{c}\pm\alpha_{3}\omega_{0})t]\end{array}  \tag{3}\]  
where \(i_{a1}\), \(i_{a2}\) and \(i_{a3}\) denote the current harmonic amplitudes with eigenfrequencies \(\omega_{c}\)\(\pm\)\(\alpha_{1}\omega_{0}\), \(\omega_{c}\pm\alpha_{2}\omega_{0}\) and \(\omega_{c}\pm\alpha_{3}\omega_{0}\), respectively. The specific values of \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\) are shown in Table 1.  
The amplitude of the sideband current harmonics depends on the modulation ratio and speed of the motor [29; 30]. The distribution of current harmonics under different DPWM modes can be derived from Equation (3) and Table 1.  
The vibro-acoustic response of PMSM is mainly caused by the electromagnetic force acting on the stator, and the radial electromagnetic force density can be defined by [PERSON]'s stress tensor method as  
\[f(\theta,t)=\frac{1}{2\mu_{0}}\left[B_{r}(\theta,t)^{2}-B_{t}(\theta,t)^{2} \right]\approx\frac{1}{2\mu_{0}}B_{r}(\theta,t)^{2} \tag{4}\]  
where \(B_{r}\) denotes the radial air-gap flux density. \(B_{t}\) denotes the tangential air-gap flux density. \(\mu_{0}\) denotes the vacuum permeability. Since \(B_{r}\) is much greater than \(B_{t}\), the electromagnetic force effect generated by the tangential air-gap flux density is ignored. Neglecting magnetic saturation effects and considering the principle of magnetic field superposition, as shown in Equation (5), the radial air-gap flux density \(B_{r}\) can be decomposed into the \(v\)-th stator armate magnetic field \(B_{arm\_v}\) and the \(\mu\)-th rotor permanent magnet magnetic field \(B_{mag\_\mu}\). \[B_{r}(\theta,t)=B_{arm_{v}}(\theta,t)+B_{mag_{u}}(\theta,t) \tag{5}\]  
\[B_{arm\_v}(\theta,t)=\sum_{v=1}^{\infty}F_{arm\_v}\cos(vp\theta\pm\omega t)= \sum_{v=1}^{\infty}\frac{3\sqrt{2}NkI_{d/q}}{\pi vp}\cos(vp\theta\pm\omega t) \tag{6}\]  
\[B_{mag\_u}(\theta,t)=\sum_{\mu=1}^{\infty}F_{mag\_u}\cos(\mu p\theta\pm \omega t+\varphi)=\sum_{\mu=1}^{\infty}\frac{3\sqrt{2}NkI_{d/q}}{\pi\mu p} \cos(\mu p\theta\pm\omega t+\varphi) \tag{7}\]  
where \(F_{arm\_v}\) and \(F_{mag\_\mu}\) denote the stator armatromotive force and permanent magnet electromotive force, respectively. \(N\) denotes the number of coil turns. \(p\) denotes the number of pole pairs. \(\mu\) and \(v\) denote the harmonic orders of the permanent magnet magnetic field and the armatropic magnetic field, respectively. \(\varphi\) denotes the angle between the current vector and the \(\alpha\) axis. Given the relatively modest amplitude of both the fundamental armature field and the harmonic armature field, the calculation is simplified by only considering the sideband electromagnetic force that arises from the interaction between the permanent magnet field  
\begin{table}
\begin{tabular}{c c c c c c c} \hline DPWM Mode & \(\varepsilon_{1}\) & \(\varepsilon_{2}\) & \(\varepsilon_{3}\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) \\ \hline
0 & 1 & 1 & 0 & 2 & 4 & / \\
1 & 1 & 1 & 1 & 4 & 6 & 8 \\
2 & 1 & 1 & 0 & 2 & 4 & / \\
3 & 1 & 1 & 1 & 4 & 6 & 8 \\ MAX &and the harmonic armrate field.