**1. Introduction**

The last decade has witnessed rapid developments of magnetic gears (MGs) and electric machines that utilize the magnetic-gearing effect, which are also called magnetic-geared machines (MGMs) [1–3]. Ever since their invention in 2001 [4], MGs have become a research hotspot due to their high efficiency and self-protection characteristics [5–7].

The concept of MGMs is derived from MGs. By substituting stator windings with AC current for one rotating permanent magnet (PM) component, MGMs change one mechanical port of MGs into an electrical port. Thus, the two rotating components of the MGM and its stator windings can be regarded as a combination of a magnetic gear and an electric machine [8–10]. Indeed, with the introduction of another rotating component and the ability to alternate the speed ratio and torque ratio between two rotating components, MGMs have broadened the application scenarios of electric machines [11–13]. A good example is that MGMs can serve as the power split component (PSC) in hybrid electric vehicles (HEVs) to realize energy exchange among the internal combustion engine (ICE), wheels, and battery [14–16]. The ICE and electric machine can provide traction for the wheels independently. The electric machine can work as a generator and a motor. When the electric machine serves as a generator, it can absorb power from the ICE or wheels (depending on working modes) to get the battery charged. When the electric machine serves as a motor, the power flows from the battery to the electric machine to drive the wheels. Hence, the ICE can always work at its highest efficiency to save fuel by alternating the working modes of the electric machine. This application scenario has drawn more and more attention as environmental problems become severe [17]. HEVs do offer a chance to alleviate the exhaust gas emission problem caused by fuel vehicles [18]. Moreover, compared to its counterpart, namely the mechanical gearbox with an electric machine, MGMs not only save space, but also improve efficiency and reduce noise and vibration by eliminating the physical contact of two gear sets [19,20].

Just like permanent magnet synchronous machines (PMSMs), MGMs utilize permanent magnets as the magnetic sources instead of using the electrical excitation method. Thus, the carbon brush structure can be eliminated and the durability of electric machines can be enhanced. However, the rare earth elements make the price of PMs extremely expensive [21]. To solve this problem, a consequent-pole structure can be adopted. The consequent-pole structure can not only reduce the flux linkage, but also improve the torque density [22–24]. Two different topologies of consequent-pole MGMs, i.e., single consequent-pole magnetic-geared machines (SCP-MGM) and dual consequent-pole magnetic-geared machines (DCP-MGM) have been proposed [25], but their mathematical modeling has not been well studied.

Although the MGMs offer many new possibilities for electric machines, their magnetic field distribution is much more complex compared to traditional electric machines with one rotor. Many scholars have focused on the magnetic field distribution calculation of MGMs [26–28]. Yet, to the best of author's knowledge, no literature has studied the magnetic field distribution of consequent-pole MGMs. The introduction of soft magnetic material (SMM) to replace the PM part will make the magnetic field distribution of consequent-pole MGMs even more complicated. Research [29] has solved the magnetic field distribution of a PMSM with PMs inserted into the SMM part, but did not consider of the saturation of SMM. Additionally, the subdomain division method [30] is not suitable for MGMs, since too many subdomains increase the calculation time rapidly. Research [31] has proposed a new harmonic modeling method (HMM) to calculate the magnetic distribution of electric machines. By introducing complex Fourier series and a convolution matrix of permeability, HMM can reduce the number of subdomains to within ten. This is because the total number of these ring-like subdomains will not increase with the increase of modulator pieces and slots.

In this paper, two consequent-pole MGMs were studied using HMM. The paper is organized as follows. Section 2 discusses the configurations and operating principles of consequent-pole MGMs. Mathematical models of SCP-MGM and DCP-MGM considering iron saturation are then proposed and elaborated in Section 3. Finally, the effectiveness of proposed HMM is validated by using finite element analysis (FEA) in Section 4.

#### **2. Configurations and Operating Principles of SCP-MGM and DCP-MGM**

When the MGM (either SCP-MGM or DCP-MGM) is applied in HEV, its inner rotor can be connected to the ICE, while the outer rotor can be connected to a permanent magnet synchronous machine (PMSM), which will be further connected to the differential to drive the wheels; the battery provides energy to the windings of both the MGM and PMSM via an inverter. The whole system configuration can be seen in Figure 1. The MGM together with the PMSM can be regarded as the E-CVT in a Toyota Prius. They can cooperate with each other according to different working conditions of HEVs [32]. Briefly speaking, either the torque from the ICE or the torque on the outer rotor driven by AC current can be the prime power to drive the HEV, and they can also work together to enhance the output power. Additionally, the battery can be charged under a regenerative braking state. The concept that a PMSM is added after the CP-MGM is derived from that in E-CVT [33]. The PMSM in Figure 1 is used to regulate the performances of the outer rotor. For instance, it can be used to drag the outer rotor of CP-MGM to a synchronous state (the rated rotating speed) at startup state. In addition, it

can deliver extra output torque to the outer rotor shaft if the output torque of the CP-MGM cannot meet the requirement. Since this paper mainly focuses on the operating modes of the CP-MGM, it is reasonable to assume that there is no power flow between the PMSM and the outer rotor shaft at the four steady states mentioned in this paper. In fact, power exchange between the PMSM and the wheels would not affect the conclusion obtained in this paper.

The working principle of MGMs is similar to that of magnetic gears. By adopting a modulator layer, the magnetic field distribution can be changed. Assuming that the pole pair number of the original magnetic field is *Pi*, and the modulator number is *Q*, then a novel magnetic field will have a component that has (*Q* − *Pi*) pole pairs. Thus, the fundamental structural requirement of an MGM is [4]:

$$P\_i + P\_s = Q \tag{1}$$

where *Ps* is the pole pair number of stator windings.

Under steady working conditions, the rotating speed of two rotating rotors and the current frequency *f* within stator windings should then satisfy:

$$P\_i \omega\_i - Q \omega\_o = P\_s w\_s = 60f \tag{2}$$

where ω*i*, ω*o*, and ω*<sup>s</sup>* are the rotating speed of the inner rotor, outer rotor, and the equivalent rotating speed of stator windings.

**Figure 1.** Propulsion system configuration of the consequent-pole magnetic-geared machine (MGM) applied in hybrid electric vehicles (HEVs).

Since the ICE reaches its highest efficiency at the range of ~2000 r/min–3000 r/min, the rotating speed of outer rotor and the current frequency of stator winding must cooperate with the rotating speed of the inner rotor to ensure the highest efficiency of the ICE. However, if the stator windings need to provide energy for the HEV, the rotating speed of the inner rotor must be smaller than that of the outer rotor. Thus, a gearbox must come into service under hybrid mode to reduce the rotating speed of the inner rotor. Therefore, the operation modes of the proposed HEV propulsion system can be divided into four kinds, and their typical operating parameters are shown in Table 1. The rotating speed of the outer rotor is calculated according to the different driving speeds of the HEV, and the current frequency of stator winding is obtained via Equation (2).



The topologies of SCP-MGM and DCP-MGM are shown in Figure 2. By substituting SMM for PMs with the same polarity, a consequent-pole structure is obtained. The name "consequent-pole" is due to SMM, and PM appears alternately on the circumferential direction. Although SMM cannot generate a magnetic field itself, it can be easily magnetized to conduct flux lines. Hence, SMM in a consequent-pole structure can be regarded as a magnetic source to some degree. The greatest advantage of using the consequent-pole structure is saving PM material, which is the most expensive material in an electric machine. Both SCP-MGM and DCP-MGM utilize a consequent-pole structure to save PM material. The SMM part in the outer rotor of a DCP-MGM not only works as a consequent-pole structure for the PMs inserted in the outer rotor, it also modulates the magnetic field of the inner rotor. Thus, the *P*s-th harmonic component within the DCP-MGM is larger than that of the SCP-MGM. Additionally, the saturation of the DCP-MGM is more severe than that of the SCP-MGM.

**Figure 2.** Proposed machine topologies: (**a**) single consequent-pole magnetic-geared machines (SCP-MGMs); (**b**) dual consequent-pole magnetic-geared machines (DCP-MGMs).
