## 2 Rotor Structure for Initial Motor and Proposed Motor  
### Rotor Structure of the Initial Motor  
Generally speaking, the size of the stator of the SynRM is the same as that of the three-phase asynchronous motor with the same capacity. Therefore, the stator core and winding of the motor in this paper refer to the three-phase asynchronous motor. The difference between the two is only in the rotor structure. Considering the shortcomings of the axially laminated rotors, such as weak mechanical strength, high manufacturing process, and high cost, this paper chooses the laterally laminated rotors.  
The rotor structure of the initial motor is derived from the magnetic field line distribution of a finite element simulation of a four-pole motor with a virtual solid rotor [25]. In addition, ABB has also produced synchronous reluctance motors with fluid-shaped flux-barriers for industrial applications.  
The equation of the magnetic field potential lines in the rotor in polar coordinates can be expressed as:  
\[r(\theta,C)=\frac{d}{2}\cdot\sqrt[p]{\frac{C+\sqrt{C^{2}+4\sin^{2}(p\theta)}}{2 \sin(p\theta)}} \tag{1}\]  
where \(r\) and \(\theta\) are the radius and polar angles of each point in the polar coordinate system, \(d\) is the shaft diameter, and \(p\) is the number of pole pairs of the motor. \(C\) is used to represent one of the field potential lines. In other words, each magnetic field line corresponds to a certain value of \(C\).  
The design method of initial rotor structure in literature [26; 27] are referred and the rotor structure of the referenced design motor with fluid-shaped flux-barriers is shown in Figure 1. \(\alpha_{i}\) is the angle between the end of the adjacent two layers of the barriers, and \(\beta\) is the angle between the \(q\)-axis and the imaginary extra point in the last segment. To determine the location of the end of the flux-barriers, it is assumed that the angle of the virtual point near the \(q\)-axis \(\beta=\alpha_{m}/2\) and so that the angle between the end of the flux-barriers is constant that \(\alpha_{m}=\pi/[2p(k+1)]\), as shown in Figure 1, where \(t_{i}\) and \(h_{i}\) are the thicknesses of the flux-barriers and the segments, respectively.  
In this paper, the thickness of segments and flux-barriers are optimized following the principle of sine distribution of magnetic motive force [28] to determine the value of the thickness of segments and flux-barriers.  
The parameters of the initial SynRM are provided in Table 1. The slot type of stator used in this paper is piriform slot, and the winding connection adopts delta connection.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Parameter** & **Symbol** & **Value** \\ \hline Rated phase voltage & \(U_{p}\) & 220 V & Stack length & \(L\) & 155 mm \\ Rated phase current & \(I_{p}\) & 21.79 A & Number of phases & \(m\) & 3 \\ Rated speed & \(n_{s}\) & 1500 rpm & Number of slots per pole per phase & \(q\) & 3 \\ Frequency & \(f\) & 50 Hz & Number of pole pairs & \(p\) & 2 \\ Air-gap length & \(g\) & 0.5 mm & Number of parallel branches & \(a_{s}\) & 1 \\ Outer stator diameter & \(D_{s}\) & 260 mm & Number of strands & \(b_{s}\) & 2 \\ Inner stator diameter & \(D_{si}\) & 170 mm & Wingding layers & \(b_{l}\) & 1 \\ Outer rotor diameter & \(D_{r}\) & 169 mm & Winding factor & \(K_{w}\) & 0.95 \\ Inner rotor diameter & \(d\) & 60 mm & Stacking factor & \(K_{fe}\) & 0.95 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the initial motor.  
Figure 1: Rotor structure and design variables of the initial motor.