## 1 Introduction  
Permanent magnet (PM) machines produce high magnetic field, torque density and efficiency due mainly to their structures without coil excitation [1, 2]. Several studies of these machines have investigated developing their performance for many years [3, 4]. In particular, the PM machines, in which the magnet is placed in the stator region, are called stator-PM machines and these have been proposed to be beneficial for efficient wide-speed-range operation because of the lightweight rotor with no winding coil and brushes [5, 6]. Based on this advantage, the stator-PM machines have been developed for many applications, such as renewable energy generators, electric vehicle motors, maglev trains and electrical aircraft [7, 8, 9, 10]. Such machines have been categorized into three popular types according to different PM positions, namely the flux reversal PM machine (FRPM), flux switching PM machine (FSPM), and doubly salient PM machine (DSPM). The FRPM type has the PM addressed at the stator tooth tip [11]. Its research studies have discovered that the structure of the FRPM helps cogging torque reduction due to an improved working airgap flux density harmonic and is appropriate to operate over a high-speed range due to the high number of rotor poles [12]. The FSPM includes the PM installed at the stator tooth. This structure exhibits a bipolar magnetic flux path, resulting in high torque density [13]. Most reviews related to the FSPM operation have identified torque capability improvement for several applications [14]. In addition, the DSPM with the PM mounted in the stator yoke has been of considerable interest because it has many notable merits over other stator-PM machines, such as an uncomplicated structure, effective cooling and a direct drive [15; 16; 17]. Especially, several literature reviews of the DSPM structure have noted that it is suitable for low-speed operation because of its wide-diameter configuration [18; 19]. These properties result in high back-electromotive force (back-EMF), reliability and torque production [20]. However, the DSPM has suffered the trapezoidal waveform of the back-EMF due to asymmetrical magnetic flux path in the machine structure, which severely impacts on torque production [21; 22]. The magnetic field excitation combined between the PM and field coil has been contained in the hybrid-excited (HE) machine. Its structure is helpful in reducing the risk of demagnetization for PM pieces because of the dc current excitation in the field coil [23]. Good flux control is achieved in this machine to improve the flux regulation quality and magnetic flux path [24]. Thus, the HE functions can be an important choice in order to fix the aforementioned non-sinusoidal back-EMF of the DSPM [25; 26]. Recently, a hybrid-excited dual-PM has been utilised in the DSPM structure (HE-DSDPM) to receive a sinusoidal back-EMF with a symmetrical magnetic flux path and better torque production than traditional DSPM [27]. Figure 1a shows the main structural components of the HE-DSDPM, including the stator part with dual-PM and armature winding and the rotor part with the salient lightweight poles. The magnetic field in this machine can be excited by the dual-PM and the armature winding using an innovative dc-biased sinusoidal current [28]. The dual-PM contains the PM at the slot opening and another one at the stator yoke to produce a strong supplemental magnetic flux, as inspired by the combination of a slot-PM machine and a biased-flux PM machine [29; 30]. The innovative dc-biased sinusoidal current injected in the armature winding can be divided into ac-armature current and dc-excited current. The ac-armature current is used to produce the rotating armature field, while the dc-excited current is used for magnetic flux regulation [31]. The dc-biased sinusoidal current utilizes only one winding coil set for copper loss reduction, as introduced by [32].