The cross-coupling problem caused by the structure of a combined radial-axial magnetic bearing for DC motors

Abstract

Different from separated magnetic bearing units, combined radial-axial magnetic bearing (CRAMB) has cross coupling problem caused by structure itself. This paper focuses on the study of cross coupling problem for a CRAMB caused by structure itself. An equivalent magnetic circuit model (EMCM) considering the leakage and coupling effects was presented to analyze the mechanism of cross coupling. The analysis results show that due to the influence of magnetic flux leakage and common bias magnetic circuit, cross coupling between axial and radial direction caused by the structure itself of a CRAMB cannot be ignored. The coupling degrees of factors that cause cross coupling are analyzed based on the combination of EMCM and finite element method (FEM). The analysis results show that the coupling effect caused by axial magnetic flux leakage was most serious. To solve the coupling problem caused by the structure itself, the coupled transfer function is derived. This coupled transfer function has been added in the original controller. Experimental results show that the cross coupling effect was reduced effectively and CRAMB can levitate the rotor at 48000 r/min stably with the improved controller.

Keywords

Cross coupling combined radial-axial magnetic bearing decoupling design magnetic flux leakage

1. Introduction

High-speed permanent magnet (PM) motors have many advantages including high efficiency, high power density, small size, and light weight, so they have been applied in many areas [1–3], such as compressors, machine tools, electric vehicles, vacuum pumps, and so on. Magnetic bearings (MBs) support the rotor by the magnetic force. Compared to mechanic bearings, MBs have many merits including no lubrication, no friction, active vibration control, and so on [4–7]. Generally, a magnetic suspension rotor needs to be equipped with two radial magnetic bearings and one axial magnetic bearing to realize the stable suspension of five degrees of freedom. However, three separated magnetic bearings require more space and longer axial length. For high speed electric machines, a larger volume can decrease energy density; a longer axial length can decrease bending critical speed of the rotor. For reducing the cost, size, and saving energy, increasing power density and enhancing bending critical speed of the rotor, combined radial-axial magnetic bearing (CRAMB) is considered to be an appropriate choice for a high speed motor.

The combined radial-axial magnetic bearing with conical rotor was first proposed in the 1990s [8]. In recent years, CRAMB has attracted the interests of many scholars. Literature [9] proposed a novel structure. Literature [10] provided a comparison of the overall performance of compressor systems designed with CRAMB and separated magnetic bearings. With the compact structure, the coupling problem of CRAMB is more serious than that of separated magnetic bearings. In [11,12], dynamic modeling and design method were presented for a CRAMB. But coupling effect was not taken into consideration. In [13], the force coupling of a CRAMB was designed and analyzed with a coupling factor. However, the coupling model and experimental results were not provided. In [14], the author proposed a 3-DOFs hybrid magnetic bearing with the secondary air-gap. The 3-DOFs magnetic bearings without a large disk [15,16] rotor were proposed. But those structures were not suitable for producing the large axial force application. In [17], magnetic flux leakage of a CRAMB was analyzed. The structure parameters were optimized to decrease flux leakage. In [18], the dynamic stiffness models for a CRAMB were studied. The coupling effects on stiffness were analyzed, too. However, in these literatures, fewer researchers discussed decouple measurement of CRAMB.

There are two main structures for CRAMB. One is symmetry about the middle plane, as shown in [9]. The other is the structure whose radial and axial control magnetic circuits are arranged side by side along the actuator axis, as shown in [19]. Here, we will study the second structure as shown in Fig. 1(a). Due to the lack of the symmetry structure in this design, for axial control flux path, there exists a leakage magnetic flux. Since the leakage flux adds to or subtracts from the bias flux in the radial direction, it results in the axial-to-radial cross-coupling. In [19], the author proposed a method to solve the coupling problem by adding compensation coils on the radial magnetic poles. The compensation flux flows along the same path as the leakage flux but in the opposite direction. However, compensation coils need extra space. In this paper, a decoupling transform function was added to the original controller. This method doesn’t need to add coils. It realizes real-time decoupling on the control stage.

Fig. 1.

Structure and work flux path of CRAMB (a) Configuration (a) Bias flux path (b) Radial control flux path (c) Axial control flux path.

The aim of the paper is to study the coupling problem caused by the structure itself of CRAMB. The structure of this paper is organized as the research proceeds. Firstly, structure of this CRAMB was briefly introduced. Secondly, the mechanism of cross coupling was analyzed. Thirdly, the coupling degrees caused by various structural factors were evaluated. Fourthly, decoupling transfer function was deduced. The corresponding decoupling module was added to the original controller. Finally, the relative experiments are conducted on the CRAMB applied in magnetically suspended high-speed electromotor with the power 30 kW.

2. Structure of CRAMB

Structure of this CRAMB is shown in Fig. 1(a). Figure 1(b), (c) and (d) show the work flux path of CRAMB.

From Fig. 1, it can be seen that the CRAMB consists of a magnetic conductor ring, solid axial stator, solid thrust disk, laminated radial stator, laminated radial rotor, axial coils and radial coils and an axially magnetized permanent magnet (PM) ring. The PM ring assembled between the magnetic conductor ring and axial stators is used to provide both of the radial and axial bias flux.

3. Analysis of cross coupling mechanism

In order to simplify the analysis, it is assumed that compared with the permeability of air-gap, the permeability of material of rotor and stator can be considered as infinite. And the fringing effect was ignored. This section illustrates how structural cross coupling occurs based on equivalent magnetic circuit model. As shown in Fig. 2(a), (b) and (c), the corresponding equivalent magnetic circuit models of CRAMB can be obtained, according to Fig. 1(b), (c) and (d), respectively.

Fig. 2.

Plot of equivalent magnetic circuit model (a) Bias equivalent magnetic circuit (b) Radial control equivalent magnetic circuit (c) Axial control equivalent magnetic circuit.

3.1. Magnetic circuit mode

In Fig. 2(a), F _pm is the magneto motive force (MMF) of the PM; R _rgx+, R _rgx−, R _rgy+, R _rgy− are magnetic reluctances of air gap in x+, x−, y+ and y− direction, respectively; R _zg+ and R _zg− are magnetic reluctances of air gap in z+ and z− direction, respectively; 𝜙_pm, 𝜙_rpm and 𝜙_zpm are the magnetic flux flows through the PM, each radial magnetic pole and each axial magnetic pole, respectively. Flux leakage between the radial and axial stators is denoted by LF _pm1 in Fig. 1(b) and represented by R _lpm1 in Fig. 2(a). Flux leakage between axial magnetic poles and rotor is denoted by LF _pm2 and LF _pm3 in Fig. 1(b) and represented by R _lpm2 and R _lpm3 in Fig. 2(a).

According to KCL and KVL applied in magnetic circuit, each magnetic flux in Fig. 2(a) can be calculated as $\begin{eqnarray}\displaystyle & \displaystyle {\phi}_{pm}=\frac{F_{pm}}{R_{ptotal}} & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle & \displaystyle {\phi}_{rpm}=\frac{1}{4}\frac{R_{lpm1}}{R_{lpm1}+\left(\displaystyle\frac{R_{rg}}{4}+\displaystyle\frac{R_{zg}}{2}\bigg/\!\!\bigg/\frac{R_{lpm2}}{2}\right)}{\phi}_{pm} & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \displaystyle {\phi}_{zpm}=\frac{2{\phi}_{rpm}R_{lpm2}}{R_{gz}+R_{lpm2}} & \displaystyle\end{eqnarray}$ (3) where $\begin{eqnarray}\displaystyle R_{ptotal}=R_{pm}+R_{lpm1}\left/\!\!\left/\left(\frac{R_{rg}}{4}+\left.\frac{R_{zg}}{2}\right/\!\!\!\left/\frac{R_{lpm2}}{2}\right.\right)\right.\right.. & & \displaystyle \nonumber\end{eqnarray}$ The magnetic reluctance of each leak magnetic circuit can be calculated as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}R_{lpm1}=\displaystyle \frac{l_{rta}}{{\mu}_{0}{\pi}({r_{pmi}}^{2}-{r_{zpi}}^{2})}\\[15.0pt] R_{lpm2}=R_{lpm3}=\displaystyle \frac{\ln \left(\displaystyle \frac{r_{zpi}}{r_{rout}}\right)}{{\mu}_{0}2{\pi}l_{zs}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (4) in which r _pmi is the inner radius of PM ring; r _zpi is the inner radius of axial stators; l _zs is the radial length of each axial magnetic pole; r _rout is the outer radius of the rotor; l _rta is the distance between the radial stator and the axial stator.

In Fig. 2(b), G _grx+, G _grx−, G _gry+ and G _gry− are magnetic conductance of air-gap in each corresponding direction; G _rp is radial total magnetic conductance. Each magnetic flux in Fig. 2(b) can be calculated as $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}{\phi}_{cy+}\\ {\phi}_{cx+}\\ {\phi}_{cy-}\\ {\phi}_{cx-}\end{array}\right)=\frac{1}{G_{rp}}\boldsymbol{R}_{x}N_{x}i_{x}+\frac{1}{G_{rp}}\boldsymbol{R}_{y}N_{y}i_{y} & & \displaystyle\end{eqnarray}$ (5) where $\begin{eqnarray}\displaystyle \boldsymbol{R}_{x}=\left(\begin{array}{@{}c@{}}G_{y+}(G_{x-}-G_{x+})\\ G_{x+}(G_{y+}+G_{y-}+2G_{x-})\\ G_{y-}(G_{x+}-G_{x-})\\ G_{x-}(G_{y+}+G_{y-}+2G_{x+})\end{array}\right),\quad \mathbf{R}_{y}=\left(\begin{array}{@{}c@{}}G_{y+}(G_{x-}+G_{x+}+2G_{y-})\\ G_{x+}(G_{y-}-G_{y+})\\ G_{y-}(G_{x+}+G_{x-}+2G_{y+})\\ G_{x-}(G_{y+}-G_{y-})\end{array}\right). & & \displaystyle\end{eqnarray}$ (6) When the rotor is located at the intermediate position with stable equilibrium state and only y channel has current, the equation ((5)) can be written as $\begin{eqnarray}\displaystyle {\phi}_{cy+}={\phi}_{cy-}=\frac{{\mu}_{0}A_{rp}}{g_{r0}}N_{y}i_{y},\quad {\phi}_{cx+}={\phi}_{cx-}=0 & & \displaystyle\end{eqnarray}$ (7) where A _rp is the cross area of radial magnetic poles.

Combining the ((2)) and ((7)), the total magnetic flux density of radial y direction without coupling can be expressed as $\begin{eqnarray}\displaystyle B_{y+}=\frac{{\phi}_{py+}+{\phi}_{cy+}}{A_{rp}},\quad B_{y-}=\frac{{\phi}_{py-}-{\phi}_{cy-}}{A_{rp}} & & \displaystyle\end{eqnarray}$ (8) In Fig. 2(c), G _LFZ denotes the flux leakage in Fig. 1(d). Each magnetic flux in Fig. 2(c) can be calculated as $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}{\phi}_{cz+}\\ {\phi}_{cz-}\\ {\phi}_{LFz}\end{array}\right)=\left(\begin{array}{@{}c@{}}N_{z}i_{z}\displaystyle\frac{G_{gz+}(G_{gz-}+G_{LFz})}{G_{gz+}+G_{gz-}+G_{LFz}}\\[13.0pt] N_{z}i_{z}\displaystyle\frac{G_{gz+}G_{gz-}}{G_{gz+}+G_{gz-}+G_{LFz}}\\[13.0pt] N_{z}i_{z}\displaystyle\frac{G_{gz+}G_{LFz}}{G_{gz+}+G_{gz-}+G_{LFz}}\end{array}\right) & & \displaystyle\end{eqnarray}$ (9) where $\begin{eqnarray}\displaystyle G_{gz+}=\frac{1}{R_{gz+}},\quad G_{gz-}=\frac{1}{R_{gz-}},\quad G_{pm}=\frac{1}{R_{pm}},\quad G_{LFz}=\frac{G_{rp}.G_{pm}}{G_{rp}+G_{pm}}. & & \displaystyle \nonumber\end{eqnarray}$ The magnetic flux in x and y direction caused by axial control current are denoted by 𝜙_LFzx and 𝜙_LFzy, respectively. Due to the radial symmetrical structure, it can be obtained $\begin{eqnarray}\displaystyle {\phi}_{LFzx}={\phi}_{LFzy}=\frac{N_{z}i_{z}}{4}.\frac{G_{gz+}G_{LFz}}{G_{gz+}+G_{gz-}+G_{LFz}}. & & \displaystyle\end{eqnarray}$ (10) Combining the ((2)) and ((9)), the total magnetic flux density of axial z direction without coupling can be expressed as $\begin{eqnarray}\displaystyle B_{z+}=\frac{{\phi}_{pz+}+{\phi}_{cz+}}{A_{zp}},\quad B_{z-}=\frac{{\phi}_{pz-}-{\phi}_{cz-}}{A_{zp}}. & & \displaystyle\end{eqnarray}$ (11) Based on the theory of virtual work, the magnetic force in y direction can be written as $\begin{eqnarray}\displaystyle F_{y}=\frac{(({\phi}_{rpm}+{\phi}_{cy+}-{\phi}_{LFzy})^{2}-({\phi}_{rpm}-{\phi}_{cy+}-{\phi}_{LFzy})^{2})}{2{\mu}_{0}A_{rp}}. & & \displaystyle\end{eqnarray}$ (12) According to the structure and magnetic circuit analysis above, the magnetic flux in radial stator generated by control current flows only inside of the x or y channel. Here, assume that radial current has little influence on the axial magnetic flux. Therefore, magnetic force in axial z direction can be written as $\begin{eqnarray}\displaystyle F_{z}=\frac{1}{2{\mu}_{0}A_{zp}}(({\phi}_{zpm}+{\phi}_{cz+})^{2}-({\phi}_{zpm}-{\phi}_{cz-})^{2}). & & \displaystyle\end{eqnarray}$ (13)

3.2. Analysis of coupling factor

There are two main reasons for coupling of CRAMB. One reason is that the bias magnetic flux of radial and axial direction is provided by the same bias magnetic circuit. The other reason is flux leakage.

(1) Coupling caused by bias magnetic circuit

Due to common bias magnetic circuit in radial and axial direction, the displacement of one direction not only causes the change of the magnetic density itself but also causes the change of the magnetic density in the other direction.

The couple factor k _zy is used to denote the coupling degree in z direction caused by displacement in y direction. k _zy is defined as follows $\begin{eqnarray}\displaystyle k_{zy}=\frac{|B_{zy}-B_{z0}|}{|B_{z0}|} & & \displaystyle\end{eqnarray}$ (14) where B _zy is the magnetic flux density of axial air-gap when displacement happens in y direction. B _z0 is the magnetic flux density of axial air-gap at the central position without any displacement and current in radial y direction.

Figure 3(a) shows that axial air-gap magnetic flux density is affected by displacement in axial z direction and radial y direction. Figure 3(b) shows that the axial coupling factor with different displacement in radial y direction. The blue solid lines and red dashed lines denoted the magnetic flux density coupling factor and magnetic force coupling factor, respectively. It can be seen that coupling factor with displacement in y direction is very small.

Fig. 3.

Coupling caused by radial y displacement (a) Relation of radial air-gap magnetic flux density with displacement in axial z direction and radial y direction (b) Coupling factor with displacement in y direction.

(2) Coupling caused by axial displacement

The couple factor k _yzd and k _yzc are used to denoted the coupling degree in radial y direction caused by the displacement and current of z channels, respectively. k _yzd and k _yzc are defined as follows $\begin{eqnarray}\displaystyle k_{yzd}=\frac{|B_{yzd}-B_{y0}|}{|B_{y0}|} & & \displaystyle\end{eqnarray}$ (15) where B _yzd is the magnetic flux density of radial air-gap in y direction when displacement happens in axial z direction; B _y0 is the magnetic flux density of radial air-gap in y direction at the intermediate position without any displacement in axial z direction.

Figure 4(a) shows the relation of radial air-gap magnetic flux density with displacement in axial z direction and radial y direction. Figure 4(b) shows the radial coupling factor with different displacement in axial z direction. The blue solid lines and red dashed lines are denoted magnetic flux density coupling factor and magnetic force coupling factor, respectively. It can be seen coupling caused by axial z displacement is very small.

Fig. 4.

Coupling caused by axial z displacement (a) Relation of radial air-gap magnetic flux density with displacement in axial z direction and radial y direction (b) Radial coupling factor with different displacement in axial z direction.

(3) Coupling caused by flux leakage

According to the analysis above, radial current has little influence on the magnetic flux in axial direction. Hence, we assumed the coupling factor of z direction caused by the current in radial stators remains zero when the current is not very large. But due to the leakage circuit, the axial current control flux will affect radial magnetic flux density.

The couple factor k _yzc are used to denote the coupling degree in radial y direction caused by the current of z channels. k _yzc was defined as $\begin{eqnarray}\displaystyle k_{yzc}=\frac{|B_{yzc}-B_{y0}|}{|B_{y0}|} & & \displaystyle\end{eqnarray}$ (16) where B _yzC is the the magnetic flux density of radial air-gap in y direction when there is control current in axial z direction; B _y0 is the magnetic flux density of radial air-gap in y direction at the intermediate position without any displacement and current in axial z direction.

Figure 5(a) shows that the value of radial air-gap magnetic flux density is affected by both of axial and radial control current. Figure 5(b) shows the radial coupling factor with different current in axial z direction. The value of magnetic force coupling factor achieves the maximum value 8.84% when current is ±1.2 A. The value of magnetic flux density coupling factor achieves the maximum value 4.53% when current is ±1.2 A.

Fig. 5.

Coupling caused by axial z current (a) Relation of radial air-gap magnetic flux density with current in axial z direction and radial y direction (b) Radial coupling factor with different current in axial z direction.

4. Analysis of coupling based FEM

4.1. Simulation model

This CRAMB is designed for magnetically suspended brushless DC motor, with rated speed 48000 r/min, output power 30 kW. The main parameters of the CRAMB are listed in Table 1 below.

Table 1
Parameters of CRAMB

Parameter Design value

Axial air gap length, mm 0.4

Radial air gap length, mm 0.4

Outer radius of the rotor, mm 27.5

Inner radius of the disk, mm 20

Outer radius of the disk, mm 37.5

Height of the disk, mm 9

Outer radius of axial stator, mm 58

Inner radius of axial stator, mm 32.5

Height of axial stator, mm 20

Outer radius of magnetic conductor ring, mm 58

Height of magnetic conductor ring, mm 20

Outer radius of radial stator, mm 53

Height of radial stator, mm 20

Outer radius of pm ring, mm 58

Inner radius of pm ring, mm 52

Length of PM ring magnetization, mm 8

Width of radial magnetic pole, mm 34

Relative permeability of PM 1.05

Coercive force of PM, kA/m 796

Radial maximum number of ampere turns 200

Axial maximum number of ampere turns 350

Parameter	Design value
Axial air gap length, mm	0.4
Radial air gap length, mm	0.4
Outer radius of the rotor, mm	27.5
Inner radius of the disk, mm	20
Outer radius of the disk, mm	37.5
Height of the disk, mm	9
Outer radius of axial stator, mm	58
Inner radius of axial stator, mm	32.5
Height of axial stator, mm	20
Outer radius of magnetic conductor ring, mm	58
Height of magnetic conductor ring, mm	20
Outer radius of radial stator, mm	53
Height of radial stator, mm	20
Outer radius of pm ring, mm	58
Inner radius of pm ring, mm	52
Length of PM ring magnetization, mm	8
Width of radial magnetic pole, mm	34
Relative permeability of PM	1.05
Coercive force of PM, kA/m	796
Radial maximum number of ampere turns	200
Axial maximum number of ampere turns	350

The materials of main elemental parts of the CRAMB are shown in Table 2.

Table 2

Materials of main elemental parts of the CRAMB

Main elements	Materials
Shaft	40CrNiMo
Radial stators	Silicon steel
Rotor	Silicon steel
Axial stators	DT4C
Magnetic conductor ring	DT4C
PM	SmCo
Radial coils	Copper
Axial coils	Copper

4.2. Simulation of magnetic flux density

Based on the analysis above, the CRAMB is analyzed by 3-D FEM with Ansoft 15.0. Figure 6 shows the magnetic flux density of radial air-gaps. The red dashed lines denote the waveform diagram of bias magnetic flux density of radial air-gap without axial current. The blue solid lines denote the waveform diagram of bias magnetic flux density of radial air-gap with axial current 1.2 A.

Fig. 6.

Simulation result of radial air-gap magnetic flux density.

It can be seen from Fig. 6 that radial magnetic flux density rises to 0.597 T from 0.55 T. It can be calculated maximum coupling factor caused by axial current is $\begin{eqnarray}\displaystyle k_{yzc}=\frac{0.597-0.55}{0.55}=\frac{0.047}{1.55}=8.55\%. & & \displaystyle\end{eqnarray}$ (17) The coupling factor deduced from analytical model was shown in Fig. 5(b) with Matlab software. The coupling factor from finite element model with Ansoft software has been calculated, which was shown in Eq. (17). Comparing the results gotten from these two methods, it can be seen that error between them is small. Therefore, the analytical coupling model can be thought as sufficiently accurate for design purposes.

5. Improvements of controller

Generally, decentralized control method was applied to separate MBs. The control of each degree of freedom is considered to be independent. That method is applicable because separated magnetic bearings do not have structural interdependence and coupling caused by structure itself. As for CRAMB, if the structure coupling can be predicted and preprocessed, it will greatly improve the stability of the MB control. Figure 7 shows the scheme of improved control with decoupling module.

Fig. 7.

The scheme of control with improved decoupling module.

Based on the analysis above, the radial force generated by the axial current is $\begin{eqnarray}\displaystyle F_{zy} & = & \displaystyle \frac{1}{2{\mu}_{0}A_{rp}}(({\phi}_{rpm}+{\phi}_{cy+}-{\phi}_{LFzy})^{2}-({\phi}_{rpm}-{\phi}_{cy+}-{\phi}_{LFzy})^{2})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼\frac{1}{2{\mu}_{0}A_{rp}}(({\phi}_{rpm}+{\phi}_{cy+})^{2}-({\phi}_{rpm}-{\phi}_{cy+})^{2}).\end{eqnarray}$ (18) When the rotor is at the equilibrium position, we can calculated $\begin{eqnarray}\displaystyle F_{zy}=\frac{-2{\phi}_{LFzy}{\phi}_{cy+}}{{\mu}_{0}A_{rp}}=\frac{-N_{z}i_{z}}{2g_{r0}}\frac{G_{z+}G_{LFz}}{G_{z+}+G_{z-}+G_{LFz}}N_{y}i_{y}. & & \displaystyle\end{eqnarray}$ (19) According to Newton’s law, it can be obtained $\begin{eqnarray}\displaystyle F_{zy}=m\ddot{y} _{z} & & \displaystyle\end{eqnarray}$ (20) where m is mass of rotor, y _z is displacement in y direction caused by axial control current.

Combining (19) and (20), it can be obtained based on the Laplace transform $\begin{eqnarray}\displaystyle \frac{Y_{Z}(s)}{I_{Z}(s)}=\frac{-N_{z}N_{y}i_{y}}{2g_{r0}}\frac{G_{z+}G_{LFz}}{G_{z+}+G_{z-}+G_{LFz}}\frac{1}{ms^{2}}. & & \displaystyle\end{eqnarray}$ (21) As an input of the comparator, the displacement signal should be converted into a voltage of the sensor. Therefore, the decoupling transform function by Laplace transform can be expressed as $\begin{eqnarray}\displaystyle K_{cpl}(s)=\frac{Y_{Z}(s)}{I_{Z}(s)}K_{sen}(s) & & \displaystyle\end{eqnarray}$ (22) where K _sen(s) is the gain of sensor.

Fig. 8.

The photograph of the CRAMB.

According to the Fig. 7, radial control current can be obtained $\begin{eqnarray}\displaystyle I_{y}(s)=\frac{(K_{p}+K_{D}s)K_{am}(s)[V_{f}(s)+I_{z}(s)K_{cpl}(s)]}{1+K_{sen}(s)\frac{k_{i}}{ms^{2}- k_{s}}(K_{p}+K_{D}s)K_{am}(s)} & & \displaystyle\end{eqnarray}$ (23) where V _f(s) denotes the reference displacement sensor value at central position.

6. Experiments

The CRAMB has been applied in magnetically suspended high-speed brushless DC motor for compressor. The prototype has been manufactured. To verify the analysis of couple effect, the relative experiments has been conducted. Figure 8 shows the photograph and test curves of the CRAMB. Front and back faces of the CRAMB were shown in Fig. 8.

Fig. 9.

Waveform of the displacement signal without and with decoupling (a) Displacement signal graph without decoupling (b) Displacement signal graph with decoupling.

6.1. The improved effect of controller

In order to verify the decoupling effect, the experiments were conducted on two cases. However, it is not convenient to test the change of air gap magnetic field directly. Therefore, we test the mutual influence of control current which can indirectly reflect coupling. The experimental steps are described as follows. Firstly, a tensiometer was used to push the thrust on the rotor in z direction. Secondly, axial displacement sensor detects axial displacement signal, and axial control current adjusts the axial force according to the signal of the axial displacement sensor. Meanwhile, in parallel with the previous step, the changes of the radial displacement sensor and the radial control current were recorded. Under the same experimental excitation and steps, the experiments were carried out under two different control conditions. One case is without the decoupling module on each axis. The other is with the decoupling module on each axis. For both of the two cases, the rotor is in static suspension state.

Fig. 10.

Waveform of the current signal without and with decoupling (a) Current signal graph without decoupling (b) Current signal graph with decoupling.

Figure 9 is the waveform diagram of the displacement signal without and with decoupling. The detection precision of displacement sensor is 6.25 V/mm. Figure 10 is the waveform of the current signal without and with decoupling. It can be seen from Fig. 9(a), without decoupling, the displacement of X direction changes to 64 μm, and the displacement of Y direction changes to 48 μm; It can be seen from Fig. 9(b), with decoupling, the displacement of X direction and Y direction is about 10 μm, which is reduced by 84.4% and 79.2% respectively compared to the situation without decoupling. It can be seen from Fig. 9(a), without decoupling, the current change in the X direction is 0.2 A, and the current in the Y direction is 0.15 A; It can be seen from Fig. 10(b), with decoupling, the current variation in the Y direction of the X direction is about 0.04 A, which is reduced by 80% and 73.3%, respectively, compared to the situation without decoupling.

6.2. Test of performance of CRAMB

When the rotor operates at a speed of 48000 r/min, the signals from eddy-current displacement sensors for CRAMB were shown in the Fig. 11. It can be calculated that the radial vibrations of CRAMB are within 25 μm and the axial vibration of CRAMB is within 11.2 μm. Both of the length of radial and axial protective air gaps are 0.2 mm. It can be obtained that radial vibration didn’t exceed 12.5% of the protective air-gap. The axial vibration didn’t exceed 5.6% of the protective air-gap. Therefore, it can be seen from Fig. 11 that the designed CRAMB and its control systems have a good and stable performance in the high-speed PM machine

Fig. 11.

Displacement signals of the suspended rotor with a speed of 48000 r/min.

7. Conclusion

The coupling problems in different degrees of freedom will make it difficult to design the controller and influence dynamic characteristics of the magnetic bearing system. In this paper, study on the cross coupling analysis caused by the structure of CRAMB are conducted. Coupling mechanism was analyzed based on magnetic circuit model considering flux leakage. Coupling factors caused by the structure were analyzed. Both analytical and experimental results show that the coupling caused by displacement is small. The coupling effect caused by axial flux leakage is most serious. Therefore, axial magnetic flux leakage should be paid special attention in the design stage.

Coupling degree was evaluated by the coupling factor. The magnetic flux density coupling factor is defined as the ratio of magnetic flux density variation caused by coupling to the original magnetic flux density without coupling effect. Based on the EMCM and FEM, structure coupling transfer function was deduced and added to the controller to improve the performance of CRAMB. Experimental results show the radial current coupling variation was decreased by 37.9% after the addition of decoupling module. With the improved controller, the designed CRAMB and its control systems have a good and stable performance at a speed of 48000 r/min in the high-speed PM machine.

Footnotes

Acknowledgements

This work was supported by the National Nature Science Foundation of China under Grant 61573032, Fundamental Research Funds for the Central Universities under Grant FRF-TP-18-058A1 and in part by Chinese Postdoctoral Science Foundation under Grant 2019M650483, Beijing Municipal Education Commission Science and Technology Program under Grant KM201911232018.

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