## 1 Introduction  
Permanent magnet motors have the advantages of small size, high output torque, high efficiency, good speed regulation performance, and strong reliability, and are extensively used in different fields such as automobiles, aerospace, and ships [1, 2]. The performance of permanent magnets closely affects the performance of motors. At the moment, by applying high-performance permanent magnet material, NdFeB, the torque and efficiency of permanent magnet motors have been significantly improved [3, 4]. But due to the significant fluctuations and high prices of NdFeB, and NdFeB materials being non-renewable resources, their reserves are decreasing year by year with continuous mining [5, 6]. Permanent magnet-assisted synchronous reluctance motors (PMA-SynRMs) typically have fewer permanent magnets for the same volume than conventional permanent magnet synchronous motors. As a result, PMA-SynRMs have better economics. PMA-SynRMs have gradually entered the public sphere and become a research direction for many scholars. PMA-SynRMs, by integrating the advantages of permanent magnet motors and reluctance motors [7], have gradually gained widespread attention due to their advantages such as low permanent magnet usage, high torque and power density, and a wide weak magnetic speed regulation range [8; 9]. Nowadays, the design of permanent magnets, magnetic circuit analysis, and magnetic resistance design are the research hotspots of PMA-SynRM [10; 11; 12]. By analyzing the quantity, volume, geometric shape, and mixing of multiple materials of permanent magnets [13; 14; 15; 16], PMA-SynRM achieved better torque performance. Because of the need for large-scale production of permanent magnets and the more complex and difficult installation of long arc-shaped permanent magnets, this article uses segmented rectangular permanent magnets for the convenience of manufacturing and assembly [17]. In addition, the arrangement of permanent magnets also directly affects the performance of PMA-SynRM. The magnetic pole-biased PMA-SynRM [18] biases the permanent magnet to one side of the magnetic circuit and changes the \(q\)-axis phase by changing the magnetic flux direction, maximizing both permanent-magnet torque and reluctance torque at the same current phase angle to increase the total superimposed torque. But the magnetic pole-biased PMA-SynRM will cause the loss of other magnetic circuits, causing the rotor magnetic circuit not to reach saturation. Moreover, because the arrangement of permanent magnets is irrational, the magnetic force of permanent magnets is not well utilized. However, the Halbach array has been widely used as a permanent magnet arrangement that achieves high magnetic field strength in specific directions [19; 20]. In recent years, the application of Halbach arrays in permanent magnet motors has become a research hotspot, such as in permanent disk motors [21], permanent magnet linear motors [22], permanent magnet vernier motors [23], and so on. However, there is little research on the application of Halbach arrays in PMA-SynRM. Therefore, this paper proposes two models based on the basic model of a permanent magnet arrangement with a magnetic flux focusing effect, combined with the permanent magnet arrangement of Halbach array. The magnetic flux focusing effect is obvious in the magnetic circuit on both sides of the rotor monopole of the proposed model, and the air gap magnetic density is significantly improved. By increasing the excitation, the average torque and efficiency of the proposed model are effectively improved. In this paper, the model of reference [18] is optimized with the aim of improving the average torque. This article proposes two PMA-SynRM models that combine Halbach arrays and exhibit magnetic focusing effects.