## 1 Introduction  
In recent decades, with energy shortages and environmental degradation becoming the focus of global attention, proton-exchange membrane fuel cell (PEMFC) systems for the automobile industry have become an interesting and promising solution, and the research and development of PEMFC systems have dramatically increased [1]. As the core technology of PEMFC systems, the power density, efficiency, and water balance characteristics of fuel-cell systems can be improved by using air compression technology. However, there is currently no guideline for choosing a compressor and a suitable electrical machine topology. The main requirements of fuel-cell systems for electrical machines are high rotational speed, compact size, high efficiency, high power density, and low noise. The maximal speed of the targeted machine is almost over 100,000 rpm. Induction machines (IM) [2], permanent-magnet synchronous machines (PMSMs) [3, 4], switched-reluctance machines (SRM) [5], and synchronous-reluctance machines (SynRM) [6] exist for air the compressors of fuel cell-systems. Among them, PMSMs are currently the preferred option in the literature when compared with other high-speed (HS) electrical machines because they present high efficiency, high power density, and low acoustic noise. However, few works show how to design an electrical machine regarding electromagnetic, mechanical, rotor-dynamics, and thermal performance; rotor design is a very important issue for HS PMSMs. A major challenge in PMSMs is the trade-off between integrated rotor and high efficiency. Rotordiameter and length are limited by the maximal rotation speed and minimal frequency of bending critical speed, respectively. In order to ensure the high rotational speed of the rotor, maximal rotor diameter and length must be as small as possible. It is also beneficial to reduce windage losses. [PERSON] et al. [7] proposed a critical design flow of HS PMSMs regarding electromagnetics, mechanical stresses, rotor dynamics, thermal analysis, and manufacturing and assembly. However, the design methodology was inadequate in terms of mechanical and thermal performance, so it could not adapted for other rotor topologies. Numerous studies were performed to overcome these constraints. To ensure PM rotor integrity, a high-strength sleeve is required for the PM rotor, and Inconel, titanium alloys, stainless steel, and carbon fiber composites (CFCs) can be used for retaining the sleeve [8; 9; 10; 11]. [PERSON] et al. [11] proposed an integrated design for rotor topologies considering electromagnetic and mechanical issues. Inconel and titanium alloys have good thermal conductivity, but their eddy current loss is high, as its conductivity is high if compared with that of CFC. Though CFC can achieve very high strength and low eddy current loss, its thermal conductivity is very low, which leads to the PM rotor facing the risk of overheating and demagnetization. Therefore, to achieve overall good performance, the above aspects need to be considered. In the design of high-power machines, [PERSON] et al. [9] developed a hybrid sleeve composed of a titanium alloy and carbon fiber. Moreover, [PERSON] et al. [12] proposed an HS sleeve-free PM rotor using amorphous magnetic materials; thus, sleeves were avoided. It can trade off core losses and mechanical stress, but rotor heat was high. To resolve this challenge, reducing losses and improving the cooling method are both effective. In order to achieve accurate thermal distribution, HSPMSM losses need to be accurately calculated [13]. The rotor eddy current loss of HSPMSMs is induced by space harmonics caused by the slotting effect and time harmonics, caused by the winding current [14]. In [15], a new method was proposed to predict rotor losses considering the slotting effect and its interaction with stator winding, and it could be extended for predicting whole rotor loss, including retaining sleeve and shaft. However, it is difficult to obtain actual power losses when an inverter exists. To solve this problem, [PERSON] et al.