Since the PM material exhibits poor mechanical characteristics, in SPM, the retaining sleeve is necessary [5N. Bianchi, S. Bolognani, and F. Luise, "Potentials and limits of high-speed PM motors", IEEE Trans. Ind. Appl., vol. 40, no. 6, .
[http://dx.doi.org/10.1109/TIA.2004.836173] ]. Considerable candidates for the sleeves are non-ferromagnetic and have high yield stresses. The electric conductivity of the Inconel 718 is which is 2.5 times lower than 35PN230. Furthermore, due to high yield strength, the Inconel 718 is appropriate to use as sleeve in SPM. Before manufacturing super high speed motor, securing enough safety factor that describes the load carrying capacity of a system beyond the expected loads is important to keep the PM from scattering. The PM diameter and thickness of the sleeve is important to determine the safety factor. The cross section of the rotor is presented in Fig. (3). The pressure when the rotor rotates is proportional to the diameter of the PM and rotation velocity of the motor. The following equation (1) explains relationship among variables.

Fig. (3) 2D cross section of the rotor.

(1)

At the outer radius, r = R₁,

(2)

At the inner radius, r = R₂,

(3)

Where

σ _c: Inner pressure of the motor (Mpa)

σ _c(max): Maximum inner pressure (Mpa)

σ _c(min): Minimum inner pressure (Mpa)

p: Mass density (g/cm³)

ω: Angular speed (rad/s)

g: Gravitational acceleration (kgm/s²)

R₁: Inner radius (mm)

R₂: Outer radius (mm)

r: (mm)

P.S: Poisson’s ratio

The inner pressure of the rotor is proportional to radius of the motor and proportional to the square of rotational speed of the motor. In this paper, initially three rotor models are selected.

Table 2
Constraint & requirement of motor/generator.

Table 3
Mechanical analysis condition.

In Table (2) based on equation (2), maximum inner pressure of the PM which is generated -at the end- of the circle is presented for three models. As presented in Fig. (4). the forces caused by the inner pressure is generated called circumferential stress or tangential stress. As mentioned before, for high speed operation, the retaining sleeve is necessary to the SPM to protect the PM from the forces by the inner pressure (Table 3). The forces caused by pressure is related to the inner pressure and the volume of the magnet as following equation (4).

(4)

Fig. (4) 3D cross section of the motor.

When the sleeve -exists- around the PM, the total tension in the sleeve is expressed as following equation.

(5)

(6)

(7)

From equation (5), (6), (7) and (8), the tangential stress in sleeve is expressed as following equation (9).

(8)

(9)

Furthermore, safety factor is calculated as following equation (10).

(10)

Where

F: The forces generated by the inner pressure (Mpa)

A: Sleeve area (mm²)

D: The diameter of PM (mm)

L: Axial length (mm)

T: Total tension in the sleeve (Mpa)

t: The thickness of the sleeve (mm)

σ_t(max): Tangential stress of the sleeve (Mpa)

S.F: Safety factor

σ_t: Maximum tangential stress of the sleeve (Mpa)

TS;_{inconel 718}: Tensile strength of the inconel 718 (Mpa)

The tangential stress of the sleeve is proportional to the diameter of the magnet and the inner pressure and inversely proportional to the sleeve thickness. Minimum sleeve thickness which satisfies safety factor more than 1.6 according to the diameter of magnet is calculated by equation (10). According to the sleeve thickness, tangential stress and safety factor for PM diameter 20mm are shown in Table 4.

Table 4
Tangential stress & safety factor according to the sleeve thickness (PM diameter: 20mm).

Table 5
Optimized tangential stress & safety factor for three models.

In the previous section 3.1, appropriate sleeve thickness that satisfies safety factor constraint according to the magnet diameter is predicted (Table 5). In this section each models are simulated by FEM with following assumptions [6L. Yi, Y. Pei, P. Liang, and C. Feng, "Analysis of the rotor mechanical strength of interior permanent magnet synchronous in-wheel motor with high speed and large torque", 2014 IEEE Conference and Expo Transportation Electrification Asia-pacific, pp. 1-5.].

• Constant speed in the steady state (maximum speed)
• Thermal effects neglected;
• Yield indicated by planar Von Mises st
• Electromagnetic force considered negligible;
• Rotor eccentricity and vibration neglected

In Figs. (5, 7 and 9), the analysis result of the inner pressure of the magnet and in Figs. (6, 8 and 10) the results of tangential stress in the sleeve according to PM -diameter are shown-.

Fig. (5) PM 20 mm.

Fig. (6) PM 20 mm / sleeve 0.65 mm.

Fig. (7) PM 30 mm.

Fig. (8) PM 30 mm / sleeve 2.4 mm.

Fig. (9) PM 35 mm.

Fig. (10) PM 35 mm / sleeve 4 mm.

1) Copper Loss: The copper loss dealt with this paper concerns the dc loss only. AC loss due to skin and proximity effects is neglected. Analytically, the copper loss for one phase can be determined as equation (11)

(11)

The winding resistance calculation expression is given as equation (12)

(12)

Where

I_rms: Current (rms)

R: Phase resistance (ohm)

l_ω: The average turn length (mm)

N_ω: The number of turns per phase

p_cu: The specific resistance of copper [ohm]

A_C: The slot area (mm²)

In Table 6, copper loss at rated speed is presented. The copper loss which is relatively influential element to the efficiency at low speed is the smallest in model 2 due to low phase resistance and current value Table 7.

Table 6
Copper loss at rated speed.

Table 7
Copper loss at maximum speed.

2) Core Losses: The core loss is composed of the the eddy-current loss and the hysteresis loss, and can be approximately expressed as equation (13) (Table 8).

(13)

Where

f: The frequency [Hz]

K_n: Constant eddy current loss coefficient

K_e: Constant hysteresis loss coefficient

Table 8
Core loss at rated and maximum speed.

Except for the mechanical losses, the overall losses that are determined in the previous sections are considered to drive the machine efficiency. In Table 9, the efficiency of three machine types is compared. For model 2, the machine efficiency is higher than the other models at the rated speed. Furthermore, as rated torque is 2.39, only model 2 satisfies the requirement (Table 10).

Table 9
Performance of the models at rated speed.

Table 10
Performance of the models at maximum speed.