Losses estimation,thermal-structure coupled simulation analysis of a magnetic-bearing reaction wheel

Abstract

Magnetic-bearing reaction wheel (MBRW) can be applied to stabilizing satellites with stringent pointing accuracy. Temperature rise subjected to losses is a key issue for a MBRW. The prediction models of iron loss and copper loss for magnetic bearings and BLDCM are established, and the loss coefficients of iron core are determined by measured loss curves at different frequencies. The copper loss distribution and iron loss distribution of the MBRW are investigated. Based on the loss values, the heat generation rates of the different parts in the MBRW are calculated, then the thermal field of the prototype is analyzed by 3-D finite element model. Thermal-structure coupled analysis of rotor is also investigated. Finally, the prediction and analysis of losses are verified by experimental results in a prototype of MBRW with rated speed of 5000 r/min and rated angular momentum of 15 Nms.

Keywords

Magnetic bearing flywheel iron loss magnetic bearing BLDCM finite element model

1. Introduction

Satellite remote sensing of the earth is stabilized by three-axis spacecraft in order to meet the fine pointing accuracy requirements of the payloads [1,2]. Reaction wheel (RW) serves as actuator to stabilize the attitudes of the satellite. The pointing precision of the satellite declines due to the friction torque spike in mechanical bearing RW under the conditions of cross through zero speed. Also, unbalanced vibration in the dynamic friction with rotational speed, temperature, lubrication, and operation history are shortcomings of the mechanical bearing RW. In order to avoid the aforementioned shortcomings for improving pointing performance of satellite, magnetic bearing RW (MBRW) has been the subject of intense research. The outstanding advantages of MB are no lubrication, no wear, and low rotational losses. Other performances obtainable with MBs, are the absence of frictional drag torque effect and can suppress the unbalanced disturbance through the active vibration suppression [3–6]. Therefore, the MBRWs are ideal actuators for the satellites, which enable remarkable improvements in the properties of the attitude control system for the satellite application. But, temperature rise of the MBRW subjected to losses is a key issue for satellite application.

Fig. 1.

Section view of the MBRW.

In the MBRW, the copper loss and iron loss in the MBs and the brushless direct current motor (BLDCM) are the component loss. These losses will produce an increase of temperature rise in the MBRW. The temperature rise will lead to the increase of stress in the MBRW. Another potential problem associated with the temperature rise may also weaken the performance of the permanent magnet material in the MBRW. Therefore, it is very important to calculate the loss and conduct the thermal analysis in the MBRW. The calculation methods of copper and iron loss in electrical machine are used for reference. In Ref. [7], the influence factors of the ac copper losses in an ironless brushless DC motor are analyzed. An eddy current loss analysis method in the retainer of the PM machine with high-power-density is proposed [8]. The calculation model of a doubly salient generator (DSG) is established, and its iron loss distribution of the DSG is investigated [9]. In Ref. [10], the performances of a surface-mounted PM motor (SMPMM) and a hybrid magnetization interior PM motor are compared, with particular reference to losses and temperature rise. The core losses in a high-speed switched-reluctance machine are predicted [11]. The rotor eddy-current losses and iron losses in both motor and generator are calculated by Nannan Zhao, etc. al. [12]. Based on the Bertotti loss model, the rotating loss separation method is proposed [13]. The losses of a high-speed PM synchronous machine with magnetic bearings are predicted with analytical calculation and compared with measured results [14]. The losses prediction and thermal analysis of a high-speed rotor supported by MBs are proposed [15].

The MBRW can be applied to stabilizing satellites with stringent pointing accuracy compared with traditional ball bearing RW, but few literatures systematically predict its losses and conduct an accurate thermal field analysis. Jinji Sun [16] analyzed the eddy current loss in homopolar magnetic bearings for the aerospace applications, but other loss components are not considered. Shuna Zhang [17] conducted thermal and structural coupling analysis of magnetically suspended flywheel rotor by using 2D FEM, however, this paper did not give the loss calculation model. In Refs [18] and [19], a thermal design method combining finite element and thermal network model is presented for magnetically suspended reaction flywheel and control moment gyro, however, the loss calculation model in these two papers are too simple to be accurate. In this paper, an accurate thermal field analysis considering all the losses and the special ways of heat transfer in MBRW is proposed by the authors. Besides, the influence of temperature variation on the maximum equivalent stress is addressed by thermal-structure coupled analysis for thermal stress problems.

The originality of this paper is to systematically deal with the loss estimation, thermal analysis, and thermal-structure coupled analysis in the MBRW. The loss prediction and thermal analysis results are verified by the measured value of the prototype.

2. Scheme design and force analysis of the MBRW

The section view of the MBRW is shown in Fig. 1. The MBRW consists of a BLDCM, two radial/axial sensors, two radial MBs (RMBs), a pair of axial MB (AMB), a cover, a rotor, and a base. To reduce the power consumption of the MBRW, the rotor is driven by an ironless BLDCM, and supported by the MBs with PM bias.

Fig. 2.

Force analysis and torque analysis of the rotor.

Figure 2 shows the force and torque analysis of the rotor supported by MBs. The 6 degrees of freedom of the rotor are controlled by two RMBs, an AMB, and a BLDCM. oxyz is the rotor center-of-mass reference coordinate system, o _u x _u y _u z _u and o _d x _d y _d z _d denote the upper and lower RMB’s coordinate system, respectively. The dynamic equation of the MB-rotor system is given by $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}m\ddot{x}=f_{ux}+f_{dx}=f_{x}=(k_{ui}i_{ux}+k_{u}x_{u})+(k_{di}i_{dx}+k_{d}x_{d})\\[4.0pt] m\ddot{y} =f_{uy}+f_{dy}=f_{y}=(k_{ui}i_{uy}+k_{u}y_{u})+(k_{di}i_{ly}+k_{d}y_{d})\\[4.0pt] m\ddot{z}=f_{z}=(k_{zi}i_{z}+k_{z}z)\\[4.0pt] T_{x}=f_{dy}l_{b}-f_{uy}l_{a}=(k_{di}i_{ly}+k_{d}y_{d})l-(k_{ui}i_{uy}+k_{u}y_{u})l\\[4.0pt] T_{y}=f_{dx}l_{b}-f_{ux}l_{a}=(k_{di}i_{lx}+k_{d}x_{d})l-(k_{ui}i_{ux}+k_{u}x_{u})l\\[4.0pt] \displaystyle J_{z}\dot{{\Omega}}=T_{e}-T_{d}=\frac{E_{a}i_{a}+E_{b}i_{b}+E_{c}i_{c}}{{\Omega}}-T_{d}\end{array}\right. & & \displaystyle\end{eqnarray}$ (1) where m is the mass of the flywheel rotor, f _ux, f _uy, f _dx, f _dy, and f _z are the bearing forces of the upper RMB, lower RMB, and AMB in x-axis, y-axis, and z-axis, respectively; k _ui, k _di, k _u, k _d, k _zi, and k _z are the current stiffness and displacement stiffness of the upper RMB, lower RMB, and AMB, respectively; J _z is the moment of inertia of the rotor around the z-axis, l is the distance from center of the magnetic pole of the RMB to the mass center of the rotor, T _e is torque produced by BLDCM, T _d is the disturb torque; E _a, E _b, and E _c are induced back-EMF in stator windings of phases of the BLDCM; i _a, i _b, and i _c are phase currents of the BLDCM.

3. Loss prediction of BLDCM

The rotation speed is controlled by the BLDCM, MBRW that consume low power are particularly suitable for space applications. Thus, a 3-phase ironless BLDCM is applied in this paper (Fig. 3) [20,21]. The stator consists of winding coils and stator frame for holding winding coils and Hall sensors. The rotor is composed of permanent magnet poles, outer rotor core, and the inner rotor core. SmCo magnet is used in the BLDCM. Table 1 shows its parameters. The rotational speed range is from −5000 r/min to +5000 r/min. Its loss type consists of iron core loss and coil copper loss.

Fig. 3.

The structure scheme of BLDCM with ironless and slotless stator.

3.1. Stator loss estimation

The stator loss mainly consists of copper loss. Copper loss is the loss due to the current going through the armature windings. Considering the influence of temperature on coil copper loss, the BLDCM coil copper loss is calculated by $\begin{eqnarray}\displaystyle P_{coil}^{pm}=mR_{p}i_{p}^{2}=m\frac{N_{c}p_{c}}{A_{c}}{\rho}_{cu}[1+{\delta}(T_{c}-20)]i_{p}^{2} & & \displaystyle\end{eqnarray}$ (2) where m is the number of phases, R _p is phase resistance, i _p is phase current, N _c is number of coil winding turns, p _c is coil winding perimeter in one stator slot, A _c is effective cross section area of coil winding, 𝜌_cu is the resistivity of the coil material (𝜌_cu = 0.01851 Ω ⋅ mm²∕m), δ is temperature coefficient of copper (δ = 0.004 ∕°C), T _c is the test temperature. Figure 4 shows the estimated results of coil loss of the BLDCM affected by temperature, copper loss increases when temperature increases. The total copper losses are 2.4 W at room temperature.

Table 1

Main parameters of the BLDCM

Parameters	Value
Rated speed (r/min)	5000
DC-link voltage (V)	28
Number of stator slots	48
Number of pole-pairs	8
Phase resistance (Ω)	0.1
Torque constant (N ⋅ m∕A)	0.0145
Back EMF constant (V/rpm)	0.00084

Fig. 4.

Winding loss of the BLDCM affected by temperature.

3.2. Rotor loss prediction

In this paper, rotor loss consists of inner rotor core loss, outer rotor core loss, and permanent magnet loss. Iron core loss is generally separated into hysteresis loss, eddy current loss, and excess loss. Permanent magnet loss is generated by induced eddy current in permanent magnets.

The iron core is made up of thin laminations in order to reduce iron core loss. The iron core loss depends on loss factor of iron core material, flux density amplitude, and magnetic field changes in frequency. In this paper, the permalloy with 50% of nickel (1J50) is used as iron core in the BLDCM. The iron loss for the 1J50 consists of eddy current loss, hysteresis loss, and excess loss, and can be calculated from [22,23] $\begin{eqnarray}\displaystyle P & = & \displaystyle P_{h}+P_{c}+P_{e}=K_{h}f\mathop{\sum }_{k=0}^{\infty }k(B_{k\max }^{{\alpha}}+B_{k\min }^{{\alpha}})V_{pm}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +K_{c}f^{2}\mathop{\sum }_{k=0}^{\infty }k^{2}(B_{k\max }^{2}+B_{k\min }^{2})V_{pm}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{K_{e}}{(2{\pi})^{\frac{3}{2}}}\frac{1}{T}\int _{0}^{T}\left(\left|\frac{dB_{r}(t)}{dt}\right|^{1.5}+\left|\frac{dB_{{\theta}}(t)}{dt}\right|^{1.5}\right)dtV_{pm}\end{eqnarray}$ (3) where K _h, K _c, and K _e are the coefficients of hysteresis loss, classical eddy current loss, and excess eddy current loss, respectively; B _kmax and B _kmin are the amplitude of each harmonic flux density; f is the operating frequency of magnetic field changes in frequency, V _pm is the effective volume. The coefficients can be calculated using the curve fitting of the iron loss data from material test data (Fig. 5). The loss coefficients and Steinmetz constant calculated by the direct fitting computed method are shown in Table 2.

Fig. 5.

Loss curves of permalloy with 50% of nickel (1J50).

Table 2

Direct fitting results of parameters for 1J50

K _h	𝛼	K _c	K _e
0.009253	1.6	1.5239 × 10⁻⁵	1.8758 × 10⁻⁷

Fig. 6.

Harmonic components of flux density of inner rotor iron core.

3.2.1. Inner rotor iron core loss prediction

Figure 6 shows the spectra of the flux density variation of inner rotor iron core in the BLDCM when the flywheel rotor rotates at 5000 r/min.

The amplitude of more than 15th harmonic flux density is relatively small, and can be negligible. The hysteresis loss and classical eddy current loss of the inner rotor iron core can be calculated, and the calculated results are 0.42 W and 0.0015 W, respectively. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle P_{in\_h}^{m}=K_{h}f_{m}\mathop{\sum }_{k=0}^{15}k(B_{k\max }^{{\alpha}}+B_{k\min }^{{\alpha}})V_{in}=0.42\\[15.0pt] \displaystyle P_{in\_c}^{m}=K_{c}f_{m}^{2}\mathop{\sum }_{k=0}^{15}k^{2}(B_{k\max }^{2}+B_{k\min }^{2})V_{in}=0.0015\end{array}\right. & & \displaystyle\end{eqnarray}$ (4) where f _m = p ∗ n∕60 Hz, p is the number of pole-pairs, n is rated speed, V _in is the effective volume of the inner rotor iron core in the BLDCM.

Figure 7 shows the first derivative of the flux density of the radial component of the inner rotor iron core versus time. Figure 8 shows the first derivative of the flux density of the circumferential component of the inner rotor iron core versus time.

Fig. 7.

The first derivative of the flux density of the radial component of the inner rotor iron core versus time.

Fig. 8.

The first derivative of the flux density of the circumferential component of the inner rotor iron core versus time.

According to the loss calculation Eq. (3), radial and circumferential components (Figs 7 and 8), the excess loss of the inner rotor iron core can be calculated, and the loss value is 0.0008 W. $\begin{eqnarray}\displaystyle P_{in\_e}^{m}=\frac{K_{e}}{(2{\pi})^{\frac{3}{2}}}\frac{1}{T}\int _{0}^{T}\left(\left|\frac{dB_{r}\left(t\right)}{dt}\right|^{1.5}+\left|\frac{dB_{{\theta}}\left(t\right)}{dt}\right|^{1.5}\right)dtV_{in}=0.0008 & & \displaystyle\end{eqnarray}$ (5) where T = 60∕(p ∗ n) = 0.0015 s.

3.2.2. Outer rotor core loss prediction

In the same way, the spectra of the flux density variation of outer rotor iron core in the BLDCM can be calculated and shown in Fig. 9 when the flywheel rotor rotates at 5000 r/min.

Fig. 9.

Harmonic components of flux density of outer rotor iron core in BLDCM.

The hysteresis loss and classical eddy current loss of the outer rotor iron core can be calculated, and the calculated results are 0.67 W and 0.0039 W, respectively. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle P_{o\_h}^{m}=K_{h}f_{m}\mathop{\sum }_{k=0}^{15}k(B_{k\max }^{{\alpha}}+B_{k\min }^{{\alpha}})V_{o}=0.67\\[15.0pt] \displaystyle P_{o\_c}^{m}=K_{c}f_{m}^{2}\mathop{\sum }_{k=0}^{15}k^{2}(B_{k\max }^{2}+B_{k\min }^{2})V_{o}=0.0039\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where V _o is the effective volume of the outer rotor iron core in the BLDCM.

The first derivative of the flux density of the radial component of the outer rotor iron core versus time is shown in Fig. 10. Figure 11 shows the first derivative of the flux density of the circumferential component of the outer rotor iron core versus time.

Fig. 10.

The first derivative of the flux density of the radial component of the outer rotor iron core versus time.

Fig. 11.

The first derivative of the flux density of the circumferential component of the outer rotor iron core versus time.

In the same way, according to the loss calculation Eq. (3), radial and circumferential components (Figs 10 and 11), the excess loss of the outer rotor iron core can be calculated, and the loss value is 0.0021 W. $\begin{eqnarray}\displaystyle P_{o\_e}^{m}=\frac{K_{e}}{(2{\pi})^{\frac{3}{2}}}\displaystyle \frac{1}{T}\int _{0}^{T}\left(\left|\frac{dB_{r}\left(t\right)}{dt}\right|^{1.5}+\left|\frac{dB_{{\theta}}\left(t\right)}{dt}\right|^{1.5}\right)dtV_{o}=0.0021. & & \displaystyle\end{eqnarray}$ (7)

3.2.3. Permanent magnet loss prediction

Figure 12 shows the spectra of the flux density variation of PM in the BLDCM when the flywheel rotor rotates at 5000 r/min.

Fig. 12.

The spectra of the flux density variation of PM in the BLDCM.

The eddy current loss of the PM in the BLDCM can be calculated by $\begin{eqnarray}\displaystyle P_{ie}=\displaystyle \frac{1}{T}\int _{0}^{T}\int _{0}^{\frac{d_{p}}{2}}\int _{0}^{L}\int _{-\frac{{\tau}_{i}}{2}}^{\frac{{\tau}_{i}}{2}}\frac{J_{i}^{2}\left(x,t\right)}{{\sigma}_{i}}dxdydzdt. & & \displaystyle\end{eqnarray}$ (8) The amplitude of more than 15th harmonic flux density is relatively small, and can be negligible. The eddy current loss can be calculated by $\begin{eqnarray}\displaystyle P_{PM\_loss}=\mathop{\sum }_{n=1}^{15}\displaystyle \frac{{\alpha}_{p}{\sigma}_{p}Ld_{pe}{\tau}_{n}^{3}B_{n}^{2}{\omega}_{en}^{2}}{16}\left(\frac{{\delta}_{n}}{3{\tau}_{n}}\frac{\sinh \displaystyle \frac{{\delta}_{n}}{{\tau}_{n}}-\sin \frac{{\delta}_{n}}{{\tau}_{n}}}{\cosh \displaystyle \frac{{\delta}_{n}}{{\tau}_{n}}-\cos \frac{{\delta}_{n}}{{\tau}_{n}}}\right) & & \displaystyle\end{eqnarray}$ (9) where d _pe is the radial thickness of permanent magnet (d _pe = 3.15 mm), τ_n is pole pitch ( ${\tau}_{n}=\frac{{\pi}D_{r}}{2np}$ , D _r is the inner diameter of permanent magnet), L is the height of permanent magnet (L = 7.6 mm), ω_en is electrical angular velocity (ω_en = 2πf _n), f _n is electrical angular velocity of the PM (f _n = nf _m), factor δ_n can be given as ${\delta}_{n}=\frac{1}{\sqrt{{\pi}f_{n}{\mu}_{p}{\sigma}_{p}}}$ , 𝜇₀ is the magnetic permeability of free space, 𝜇_p is the relative permeability of permanent magnet, σ_p is the conductivity of permanent magnet (σ_p = 1.17 ×10⁶ simens/m), 𝛼_p is pole-arc coefficient (𝛼_p = 0.67).

The values will be substituted for the model of Eq. (9) with the parameters for calculating eddy current loss of the PM in the BLDCM as follow $\begin{eqnarray}\displaystyle P_{PM\_loss}=3.04∼\text{W}. & & \displaystyle\end{eqnarray}$ (10)

4. Losses estimation of the MBs

The losses of the MB consist of copper losses and iron losses, and the iron loss consists of hysteresis loss, eddy-current loss, and the excess loss, which are estimated as follow.

Table 3
Coil copper losses of the MBs

Coils Current Resistance Copper loss

RMB coil 1 ∼ 16 0.052 A 2.9 Ω 0.008 W

TMB coil 1 ∼ 2 0.075 A 8.6 Ω 0.048 W

Total loss 0.224 W

Coils	Current	Resistance	Copper loss
RMB coil 1 ∼ 16	0.052 A	2.9 Ω	0.008 W
TMB coil 1 ∼ 2	0.075 A	8.6 Ω	0.048 W
Total loss	0.224 W

4.1. Cooper losses calculation

In order to save power consumption, the RMB and AMB with PM creating bias flux is used in the CMAMB for the satellite application [24]. There are 16 coil windings for the two RMBs each with 8 coil windings, and 2 coil windings for the AMB. The measured control currents in the MBs are listed in Table 3 when the MBRW rotates at the rotation speed of 5000 r/min. According to Eq. (11), the copper losses of the MBs are calculated and shown in Table 3. The total copper losses of the two RMBs and the AMB are 0.224 W at the room temperature. $\begin{eqnarray}\displaystyle P_{coil}^{AMB}=R_{AMB}i_{AMB}^{2}=\frac{N_{AMB}p_{AMB}}{A_{AMB}}{\rho}_{cu}[1+{\delta}(T_{c}-20)]i_{AMB}^{2} & & \displaystyle\end{eqnarray}$ (11) where R _AMB is the coil resistance of the RMB, i _AMB is the control current in the RMB coil wingding, N _AMB is number of coil winding turns, p _AMB is coil winding perimeter in one stator slot, A _AMB is effective cross section area of coil winding, 𝜌_cu is the resistivity of the coil material (𝜌_cu = 0. 01851 Ω ⋅ mm²∕m), δ is temperature coefficient of copper (δ = 0.004 ∕°C), T _c is the test temperature. Figure 13 shows the coil loss is affected by temperature.

Fig. 13.

Winding loss of the AMB coil affected by temperature.

4.2. Iron loss estimation in the RMB

Figure 14 shows the scheme of the RMB with permanent magnet bias. The RMB consists of rotor core, return ring, stator core, magnetic ring, permanent magnet, and the coil.

Fig. 14.

Section view of the RMB with permanent magnet bias.

The estimation method of the iron loss in Ref. [25] is used to predict the iron loss of the RMB and the AMB with permanent magnet in this paper. $\begin{eqnarray}\displaystyle P_{r}=P_{r\_ed}+P_{r\_hy}+P_{r\_ex} & & \displaystyle\end{eqnarray}$ (12) where P _{r_ed} is eddy current loss, P _{r_hy} is hysteresis loss of the iron core, and P _{r_ex} is the excess loss. The same iron core material of permally as the BLDCM is used for fabricating the RMB, therefore, the loss coefficients and Steinmetz constant are the same.

Fig. 15.

Hysteresis loss curve, classical eddy current loss curve, and excess eddy current loss cure of 1J50.

Figure 15 shows the loss curves of the hysteresis loss, the classical eddy current loss, and excess eddy current loss of the iron core at the frequency of 83.33 Hz. Compared with the other losses, the excess eddy current loss is very small, and can be negligible. The Eq. (12) can be simplified as $\begin{eqnarray}\displaystyle P_{r}\approx P_{r\_ed}+P_{r\_hy} & & \displaystyle\end{eqnarray}$ (13) Eddy current loss of the iron core is calculated as $\begin{eqnarray}\displaystyle P_{r\_ed}=K_{c}f_{r}^{2}B_{r\_m}^{2}V_{r} & & \displaystyle\end{eqnarray}$ (14) where k _{r_c} is the factor of eddy-current loss, f _r is flux change frequency, which is twice bigger than the mechanical frequency because two flux density periods are contained in a geometric period of flywheel rotor, B _{r_m} is the maximum magnetic flux density, 𝜌_r is the density of iron core, and V _r is the affected volume of iron core.

Hysteresis loss of the iron core can be estimated by $\begin{eqnarray}\displaystyle P_{r\_hy}=K_{h}f_{r}B_{r\_m}^{a}V_{r} & & \displaystyle\end{eqnarray}$ (15) where k _{r_h} is the hysteresis losses factor, a is Steinmetz constant. $\begin{eqnarray}\displaystyle P_{r}=(K_{h}f_{r}B_{r\_m}^{a}+K_{c}f_{r}^{2}B_{r\_m}^{2})V_{r}. & & \displaystyle\end{eqnarray}$ (16)

Each component of loss, i.e., hysteresis loss and eddy current loss is given separately both for the stator and rotor. The iron core losses of the two RMB are predicted by Table 4, and the total losses are 1.15 W.

Table 4

Iron loss calculation results of the RMB stator and rotor core

Parts	Frequency	Core volume	Maximum flux density	Iron loss
Stator core 1–4	166.7 Hz	10122 mm³	1.2 T	0.202 W
Rotor core 1–4	166.7 Hz	4301 mm³	1.2 T	0.086 W
Total loss				1.15 W

In the RMB, the magnetic ring (MR), the permanent magnet (PM), and the return ring (RR) are made of solid material. The iron loss in each mesh element can be separated into the hysteresis loss (P _{T_h}), and eddy-current loss (P _{T_e}). The excess loss is very small, and can be negligible. $\begin{eqnarray}\displaystyle P_{T\_iron}=P_{T\_h}+P_{T\_e}=(k_{T\_h}+k_{T\_e}){\sigma}_{T}{\delta}_{T}^{2}f_{T}^{2}B_{T}^{2}V_{T} & & \displaystyle\end{eqnarray}$ (17) where k _{T_s} is hysteresis loss factor, k _{T_e} is eddy current loss factor, σ_T is the conductivity, δ_T is the thickness, B _T is the maximum flux density, f _T is operating frequency of the RMB, V _T is the effective volume. The iron cobalt vanadium soft magnetic alloy 1J22 is adopted in this prototype. The losses of the MR, the PM, and the RR are predicted as Table 5, and the total losses are 0.27 W.

Table 5

The loss estimation of the solid materials in the RMB

Parts	Frequency	Volume	Conductivity	Maximum flux density	Iron loss
MR 1–4	166.7 Hz	2333 mm³	2.5 ×10⁶ S/m	0.85 T	0.047 W
PM 1–2	166.7 Hz	1696 mm³	1.1 ×10⁶ S/m	0.85 T	0.007 W
RR 1–2	166.7 Hz	6710 mm³	2.5 × 10⁶ S/m	0.85 T	0.034 W
Total loss					0.27 W

4.3. The loss estimation of the AMB

Figure 16 shows the scheme of the AMB with permanent magnet bias. The AMB consists of main yoke, sub yoke, permanent magnet, thrust disk, and the coil.

Fig. 16.

Section view of the AMB.

The iron loss of the AMB consists of the sub yoke loss (P _sub), the main yoke loss (P _main), the permanent magnet loss (P _pm), and the thrust disk loss (P _disk), the solid material are used in this paper. The maximum flux density in the air gap is 0.84 T, the maximum flux density in the permanent magnet is 0.91 T. The iron loss of the AMB can be given as $\begin{eqnarray}\displaystyle P_{T\_iron}=P_{main}+P_{sub}+P_{pm}+P_{disk}. & & \displaystyle\end{eqnarray}$ (18)

According to Eq. (16), the iron loss in the AMB can be calculated as Table 6, and the total losses are 0.774 W.

Table 6

Iron losses in the AMB

Parts	Frequency	Volume	Conductivity	Maximum flux density	Iron loss
Main yoke 1–2	83.3 Hz	26280 mm³	2.5 ×10⁶ S/m	0.84 T	0.23 W
Sub yoke 1–2	83.3 Hz	2357 mm³	2.5 × 10⁶ S/m	0.84 T	0.004 W
PMR 1–2	83.3 Hz	2109 mm³	1.11 × 10⁶ S/m	0.91 T	0.003W
Thrust disk 1–2	83.3 Hz	13146 mm³	2.5 × 10⁶ S/m	0.84 T	0.15 W
Total loss					0.774 W

5. Thermal field analysis and measurement

5.1. Calculation of heat generation rate

The losses in the MBRW cause temperature rise and thermal expansion eventually. Losses in the MBRW consist of iron losses and winding losses, and these losses are regarded as the heat generation rate. The transfer of heat occur by means of two main mechanisms, conduction and radiation in the MBRW under vacuum condition.

The heat generation rates of the MBRW are calculated by $\begin{eqnarray}\displaystyle q_{v}=\frac{P_{loss}}{V} & & \displaystyle\end{eqnarray}$ (19) where V is volume, P _loss is various loss in the MBRW.

The heat generation rates of the winding losses are shown in Table 7. The total winding losses of the MBRW is 2.62 W. The heat generation rates of the iron core loss in the BLDCM, the RMBs, and the TMB are also estimated by Eq. (19) and given in Table 8.

Table 7

Heat rate of winding losses

Coils	Copper loss	Volume	Heat rate
BLDCM coil 1–48	0.05 W	794 mm³	62972 W/m³
RMB coil 1–16	0.008 W	3596 mm³	2224 W/m³
TMB coil 1–2	0.048 W	15405 mm³	3116 W/m³
Total loss 2.62 W

Table 8

Heat rate of iron losses

	Parts	Iron core loss	Volume	Heat rate
BLDCM	Inner rotor core	0.422 W	28289 mm³	14917 W/m³
	Outer rotor core	0.676 W	30657 mm³	22050 W/m³
	Permanent magnet	3.04 W	14602 mm³	208191 W/m³
RMB	Stator core 1–4	0.202 W	10122 mm³	19957 W/m³
	Rotor core 1–4	0.086 W	4301 mm³	19995 W/m³
	MR 1–4	0.047 W	2333 mm³	20146 W/m³
	PM 1–2	0.007 W	1696 mm³	4127 W/m³
	RR 1–2	0.034 W	6710 mm³	5067 W/m³
TMB	Main yoke 1–2	0.23 W	26280 mm³	8752 W/m³
	Sub yoke 1–2	0.004 W	2357 mm³	1697 W/m³
	PMR 1–2	0.003 W	2109 mm³	1422 W/m³
	Thrust disk 1–2	0.15 W	13146 mm³	11410 W/m³
Total loss		6.334 W

5.2. Thermal field analysis of the MBRW

The MBRW is applied to stabilizing satellites. Heat can be delivered by radiation and thermal conduction, and the heat transfer is determined by the heat generation rates of the MBs and the BLDCM which are shown in Tables 7 and 8.

The 3-D finite element model (FEM) of the MBRW is built and shown in Fig. 17a. The total number of mesh is 450637. Figure 17b shows the temperature distribution of the MBRW. The calculated maximum temperature is 36 °C, which locates at the windings of the BLDCM.

Fig. 17.

3-D finite element model of the MBRW. (a) Mesh plot. (b) Temperature contour plot.

5.3. Stress analysis of the flywheel rotor

By the action of centrifugal force, the equivalent stress contour plot of the flywheel rotor is calculated and shown in Fig. 18 when the MBRW rotates at its rated rotation speed of 5000 r/min. The maximum equivalent stress is 27.7 MPa, which is located at the outer rotor core of the BLDCM.

Fig. 18.

Equivalent stress contour plot of the flywheel rotor with only the action of centrifugal force.

5.4. Thermal-structure coupled analysis of the MBRW

Equivalent stress analysis is conducted to obtain the stress distribution by the action of centrifugal force and the thermal-structure coupling fields respectively. The calculated results are shown in Fig. 19. The maximum equivalent stress under the influence of temperature variation increases to 52.7 MPa from 27.7 MPa. The temperature rise increases the stress.

Fig. 19.

Equivalent stress contour plot of the MBRW with thermal-structure coupled analysis.

6. Temperature rise experiment

In order to testify the accuracy of the method of the loss estimation for the MBRW proposed in this paper. The prototype of a MBRW is fabricated to measure the temperature rise. Six thermistors are placed in the MBRW, which are shown in Fig. 20. The experiment equipment is designed and shown in Fig. 21, which consists of a MBRW, a cover, two DC electrical sources, and the controllers of the MB and the BLDCM.

Fig. 20.

Positions of six thermistors in the MBRW.

Fig. 21.

Test setup of the MBRW prototype.

The measured data of temperature rise is shown in Fig. 22 at room temperature. Except flywheel rotor, in non-rotating parts of the MBRW, the maximum temperature is located at the coil winding of the BLDCM, and the value is 38.9 °C at the room temperature of 26.1 °C. The error between the estimation and measured one in the coil winding of the BLDCM is 7.6%. Comparing the results from analytical data with those from measurement, the results are shown in Table 9. The maximum error between the analytical and the measured one is 12.1%. The measured data verified the loss analytical method.

Fig. 22.

Measured data of temperature rise of the MBRW at room temperature.

Table 9

Comparing results between analytical and measured data of temperature rise at room temperature

Location	Analytical maximum temperature (°C)	Measured maximum temperature (°C)	Error
BLDCM stator coil	35.96	38.9	7.6%
Upper RMB stator coil	34.01	38.7	12.1%
Lower RMB stator coil	33.95	38.2	11.1%
Upper AMB stator coil	33.94	37.8	10.2%
Lower AMB stator coil	33.75	37.6	10.2%
Inside wall of the base	32.93	36.5	9.8%

7. Conclusion

This paper deals with the prediction models of iron loss and copper loss of MBs and BLDCM in a MBRW for the satellite application, and the loss coefficients of iron core are determined by measured loss curves at different frequencies. The copper loss distribution and iron loss distribution of the MBRW are investigated. The thermal field of the prototype is analyzed by 3-D finite element model based on the heat generation rates. Finally, the prediction and analysis of losses are verified by experimental results in a prototype of MBRW. It is found that the proposed systematic loss model and accurate thermal field analysis method considering all the losses and the special ways of heat transfer are necessary for the design of MBRW. This study may serve as a basis for the structural and thermal design of magnetically suspended flywheel system.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant No. 61573032 and 61721091), the National Key Research and Development Program of China (grant No. 2016YFB0500804), and the Beijing Science and Technology Program (grant No. Z171100002217008).

References

Horiuchi

and Inoue

, Development of magnetic bearing momentum wheel for ultra-precision spacecraft attitude control, in: Seventh International Symposium on Magnetic Bearings, ETH Zurich, 2000, pp. 525–503.

Gao

and Song

, Analysis on unbalance vibration and compensation for maglev flywheel, International Journal of Applied Electromagnetics and Mechanics 50(1) (2016), 177–187.

Hedlund

Abrahamsson

Perez-Loya

J.J.

Lundin

and Bernhoff

, Eddy currents in a passive magnetic axial thrust bearing for a flywheel energy storage system, International Journal of Applied Electromagnetics and Mechanics 54(3) (2017), 389–404.

Peng

Fang

and Wang

, Steering law for a cluster of magnetically suspended momentum wheels with vernier gimballing capability, International Journal of Applied Electromagnetics and Mechanics 47(3) (2015), 691–704.

Auer

, Ball bearing versus magnetic bearing reaction and momentum wheels as momentum actuators, in: International Meeting and Technical Display on Global Technology, 2000, AIAA-80-0911.

Han

Zheng

Wang

and Yuan

, Integral design and analysis of passive magnetic bearing and active radial magnetic bearing for agile satellite application, IEEE Transactions on Magnetics 48(6) (2012), 1959–1966.

Liu

Lin

and Tai

, AC Copper losses analysis of the ironless brushless DC motor used in a flywheel energy storage system, IEEE Transactions on Applied Superconductivity 26(7) (2016), 0611105.

Lee

S.Y.

and Jung

H.K.

, Eddy Current loss analysis in the rotor of permanent magnet traction motor with high power density, in: 2012 IEEE Vehicle Power and Propulsion Conference, Seoul, Korea, 2012, pp. 210–214.

Zhang

Sun

Qian

and Huang

, Iron loss analysis of doubly salient brushless DC generators, IEEE Transactions on Industrial Electronics 63(4) (2015), 2156–2163.

10.

Zhao

and Liu

, Loss calculation and thermal analysis of surface-mounted PM motor and interior PM motor, IEEE Transactions on Magnetics 51(11) (2015), 8112604.

11.

Calverley

S.D.

Jewell

G.W.

and Saunders

R.J.

, Prediction and measurement of core losses in a high-speed switched-reluctance machine, IEEE Transactions on Magnetics 41(11) (2005), 4288–4298.

12.

Zhao

Zhu

and Liu

, Rotor eddy current loss calculation and thermal analysis of permanent magnet motor and generator, IEEE Transactions on Magnetics 47(10) (2011), 4199–4202.

13.

Fang

Wang

Wei

Tang

and Fan

, Homopolar 2-pole radial permanent-magnet biased magnetic bearing with low rotating loss, IEEE Transactions on Magnetics 48(8) (2012), 2293–2303.

14.

Binder

and Sabirin

C.R.

, Loss measurement of a 30 kW High speed permanent magnet synchronous machine with active magnetic bearings, in: 2013 International Conference on Electrical Machines and Systems, Busan, Korea, 2013, pp. 905–910.

15.

Huang

Fang

Liu

and Han

, Loss calculation and thermal analysis of rotors supported by active magnetic bearings for high-speed permanent-magnet electrical machines, IEEE Transactions on Industrial electronics 63(4) (2016), 2027–2035.

16.

Sun

J.J.

and Chen

, Analysis and experiment of eddy current loss in homopolar magnetic bearings with laminated rotor cores, Acta Astronautica 89 (2013), 229–235.

17.

Zhang

S.N.

Han

B.C.

and Fang

J.C.

, Thermal and structural coupling analysis of magnetically suspended flywheel rotor, in: Sixth International Symposium on Instrumentation and Control Technology: Sensors, Automatic Measurement, Control, and Computer Simulation, Beijing, China, 2006.

18.

Wang

C.E.

Fang

J.C.

Tang

J.Q.

and Sun

J.J.

, Thermal design method and experimental research of magnetically suspended reaction flywheel, Acta Aeronautica et Astronautica Sinica 32(4) (2011), 598–607, [in Chinese].

19.

Han

W.T.

Liu

Sun

J.J.

and Tang

J.Q.

, Thermal-structure coupling analysis and research of MSCMG, Journal of Beijing University of Aeronautics and Astronautics 42(2) (2016), 391–399, [in Chinese].

20.

Fang

and Li

, Self-compensation of the commutation angle based on DC-link current for high-speed brushless DC motors with low inductance, IEEE Transactions on Power Electronics 29(1) (2014), 428–439.

21.

Han

Zheng

Wang

and Le

, Design, modeling, fabrication, test of a large scale single-gimbal magnetically suspended control moment gyro, IEEE Transactions on Industrial Electronics 62(12) (2015), 7424–7435.

22.

Binesti

and Ducreux

J.P.

, Core losses and efficiency of electrical motors using new magnetic materials, IEEE Transactions on Magnetic 32(5) (1996), 4887–4889.

23.

Roshen

, Iron loss model for permanent magnet synchronous motors, IEEE Transactions on Magnetic 43(8) (2007), 3428–3434.

24.

Zhilichev

, Analysis of a magnetic bearing pair with a permanent magnet excitation, IEEE Transactions on Magnetics 36(5) (2000), 3690–3692.

25.

Bakay

Dubois

Viarouge

and Ruel

, Losses in hybrid and active magnetic bearings applied to long term flywheel energy storage, in: 5th IET International Conference on Power Electronics, Machines and Drives, Bristol, United Kingdom, 2010.

Losses estimation,thermal-structure coupled simulation analysis of a magnetic-bearing reaction wheel

Abstract

Keywords

1. Introduction

Table 3 Coil copper losses of the MBs Coils Current Resistance Copper loss RMB coil 1 ∼ 16 0.052 A 2.9 Ω 0.008 W TMB coil 1 ∼ 2 0.075 A 8.6 Ω 0.048 W Total loss 0.224 W

5.1. Calculation of heat generation rate

Footnotes

Acknowledgements

References

Table 3
Coil copper losses of the MBs

Coils Current Resistance Copper loss

RMB coil 1 ∼ 16 0.052 A 2.9 Ω 0.008 W

TMB coil 1 ∼ 2 0.075 A 8.6 Ω 0.048 W

Total loss 0.224 W