# Current Harmonic Suppression Algorithm for Asymmetric Dual Three-Phase PMSM
## 2 Machine Model of Asymmetric Dual Three-phase Permanent Magnet Synchronous Motor (ADT-PMSM)  
As depicted in Figure 2, the drive system of ADT_PMSM in this paper has two sets of three-phase voltage source inverters (VSI) sharing the same DC-link voltage, which can supply two sets of three-phase windings.  
According to the VSD theory, the variables including the current, flux, and voltage of the machine can be mapped into three orthogonal subspaces, i.e., the \(\alpha\)-\(\beta\), \(x\)-\(y\), and zero-sequence subspaces. This approach gives an alternative description of the machine, which is useful for machine control and the development of pulse width modulation (PWM) techniques. Harmonics of different orders are mapped into different subspaces. The fundamental component and harmonics with the order 12n \(\pm\) 1 (n = 1, 2, 3...) are mapped into the \(\alpha\)-\(\beta\) subspace. The harmonics with the order 6n \(\pm\) 1(n = 1, 3, 5...) are mapped into the \(x\)-\(y\) subspace. The harmonics with the order 6n \(\pm\) 3(n = 1, 3, 5...) are mapped into the \(O_{1}\)-\(O_{2}\) subspace, which is also called the zero sequence subspace. With neutral points being isolated, no current components in the \(O_{1}\)-\(O_{2}\) subspace would exist. Therefore, the ADT-PMSM with isolated neutrals is a four-order system. The mathematical model and the VSD modeling process of the dual three-phase motor in more detail are introduced in Reference [9]. According to the magnitude invariant principle, the static decoupling transformation matrix is employed as  
Figure 2: Drive system of asymmetric dual three-phase permanent magnet synchronous motor (ADT-PMSM).  
\[T_{VSD}=\frac{1}{3}\left[\begin{array}{cccccc}1&-\frac{1}{2}&-\frac{1}{2}&\frac{ \sqrt{3}}{2}&-\frac{\sqrt{3}}{2}&0\\ 0&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&\frac{1}{2}&\frac{1}{2}&-1\\ 1&-\frac{1}{2}&-\frac{1}{2}&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&0\\ 0&-\frac{\sqrt{5}}{2}&\frac{\sqrt{5}}{2}&\frac{1}{2}&\frac{1}{2}&-1\\ \end{array}\right]. \tag{1}\]  
In order to achieve the vector control, a transformation is used to transform the \(\alpha\)-\(\beta\) variables into a synchronous reference frame rotating with the fundamental frequency, i.e.  
\[T_{dq}=\left[\begin{array}{cccccc}\cos\theta&\sin\theta&0&0&0&0\\ -\sin\theta&\cos\theta&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&1&0&0\\ \end{array}\right] \tag{2}\]  
where \(\theta\) is the electrical rotor position.  
On the condition that the back electromotive force (EMF) is sinusoidal, two sets of windings are symmetric, and mutual leakage inductances are ignored. The voltage equations in the synchronous rotating \(d\)-\(q\) coordinate axis in the \(\alpha\)-\(\beta\) subspace and in the stationary \(x\)-\(y\) coordinate axis in the \(x\)-\(y\) subspace are  
\[\left\{\begin{array}{c}u_{d}=i_{d}R+L_{d}\frac{di_{d}}{dt}-w_{e}L_{q}i_{q}\\ u_{q}=i_{q}R+L_{q}\frac{di_{q}}{dt}+w_{e}(L_{d}i_{d}+\psi_{f})\\ u_{x}=i_{x}R+L_{z}\frac{di_{x}}{dt}\\ u_{y}=i_{y}R+L_{z}\frac{di_{y}}{dt}\\ \end{array}\right. \tag{3}\]  
where \(R\) is the stator resistance, \(L_{d}\) and \(L_{d}\) are the \(d\)- and \(q\)-axes inductances respectively, \(L_{z}\) is the leakage inductance, and \(u_{k}(k=d\), \(q\), \(x\) and \(y)\) and \(i_{k}(k=d\), \(q\), \(x\) and \(y)\) are the voltage and current components, respectively. \(w_{e}\) is the electrical angular speed, and \(\psi_{f}\) is the permanent magnet flux.