Performance study on newly developed AC magnetic suspension system using magnetic resonance coupling

Abstract

A newly developed AC magnetic suspension system is designed and fabricated to investigate the performances. A new concept of design is revealed for operating the apparatus differentially where the floator is kept at a constant position despite changing the parameters of the upper stator electromagnets. An extensive finite element analysis is conducted to estimate the basic characteristics of the system. A permanent magnet is incorporated in this system to reduce the required supply energy to the stator electromagnets. Magnetic suspension with a maximum gap of 2.0 mm without any control with the upper electromagnet in cumulative coupling mode and with a gap of 3.0 mm with indirect damping in the differentially operated mode is achieved. The indirect damping is achieved by applying PD control to the stator. The individual force, current and phase for variable frequency and gap between primary and secondary electromagnet are measured to examine the basic characteristics and performances.

Keywords

Magnetic suspension magnetic resonance coupling self-stabilization magnetic force indirect damping

1. Introduction

Numerous methods have been proposed depending on the type and combination of the source of magnetic force and the object (floator) in magnetic levitation [1]. Usually, direct-current (DC) electromagnets are used for suspension. In several applications, the power supply to the floator is necessary. However, when electric wires are used for transferring power to the floator, the non-contact property is lost. Several researches prove that alternating current (AC) electromagnet can generate levitation force and simultaneously transfer power to the floator [2]. However, the power transmission efficiency decreases remarkably as the gap increases [3]. We have proposed the solution of this problem by applying the method of AC magnetic levitation using magnetic resonance coupling where magnetic suspension and wireless power transfer both can be achieved. In the previous researches, the floator was levitated by the upward attraction force generated by the upper electromagnet. However, when any parameter of the upper electromagnet like the amplitude of the applied voltage or the applied frequency is changed, the floator position also changes. It may lead to some difficulties in estimating the effects of such parameters experimentally.

In this research, a new design is proposed to levitate the floator by attracting from the upper and lower side electromagnets to maintain the position of the floator. Due to the usage of the lower side electromagnets in the floator, however, the mass of the floator increases. Therefore, the required supply energy to the primary electromagnets increases. A weight support suspension with a permanent magnet (PM) is introduced to reduce the supply energy.

Fig. 1.

Basic model of magnetic suspension.

2. Principles of magnetic resonance coupling

Figure 1(a) shows a basic model of magnetic suspension where electromagnets are placed to each other with a certain gap. A primary electromagnet is fixed to the stator and a secondary electromagnet is fixed on the floator. The gravitational force mg and the attractive force F act on the floator. Figure 1(b) depicts the equivalent circuit of the basic model where primary electromagnet includes a series capacitor C₁ and an AC power source. The circuit of the secondary electromagnet includes a series capacitor C₂. Mutual inductance L_m denotes the interaction between magnetic field of the primary and secondary circuits that induces a voltage in the secondary circuit. There are two resonant circuits which are adjusted to have a common resonant frequency.

2.1. Generation of force

The electrical dynamics of the equivalent circuit are expressed by Eqs. ((1)) and ((2)). $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle R_{1}i_{1}+L_{1}{\displaystyle \frac{di_{1}}{dt}}-L_{m}{\displaystyle \frac{di_{2}}{dt}}+{\displaystyle \frac{1}{C_{1}}}\int i_{1}dt=e \end{eqnarray}$ (1) $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle R_{2}i_{2}+L_{2}{\displaystyle \frac{di_{2}}{dt}}-L_{m}{\displaystyle \frac{di_{1}}{dt}}+{\displaystyle \frac{1}{C_{2}}}\int i_{2}dt=0 \end{eqnarray}$ (2) where i₁ and i₂ are the current flowing through the primary and secondary circuit respectively and e is the supply voltage. It is assumed that both circuits have the same self-inductance L and capacitance C with no loss (R₁ = R₂ =0). The mutual inductance L_m is given by kL where k is the coupling coefficient. To obtain steady-state solutions, the variables are expressed as $\begin{eqnarray}\displaystyle e=Ee^{j{\omega}t},\quad i_{1}=I_{1}e^{j{\omega}t},\quad i_{2}=I_{2}e^{j{\omega}t} & & \displaystyle\end{eqnarray}$ (3) where ω is the angular frequency of the supply source and E, I₁ and I₂ are complex constant that are referred to as phasors. Substituting Eq. ((3)) into Eqs. ((1)) and ((2)) gives $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle j\left({\omega}L-{\displaystyle \frac{1}{{\omega}C}}\right)I_{1}-j{\omega}kLI_{2}=E\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle j\left({\omega}L-{\displaystyle \frac{1}{{\omega}C}}\right)I_{2}-j{\omega}kLI_{1}=0.\end{eqnarray}$ (5)

The flux generated by the stator electromagnet is reinforced with the flux generated by the floator electromagnet so that the force between them is attractive force in the cumulative coupling [4]. In this experiment, cumulative coupling has been introduced that operated by differential mode.

2.2. Self-stabilization

In this principle, self-stabilization can be achieved due to the dependency of coupling coefficient k on the gap between primary and secondary electromagnets. The coupling coefficient is inversely proportional to the gap. The resonant condition of the circuit is expressed in the Eq. (6). $\begin{eqnarray}\displaystyle k=1-{\displaystyle \frac{1}{{\omega}^{2}LC}}. & & \displaystyle\end{eqnarray}$ (6)

For a fixed given ω₀, this value is assigned by k₀. Suppose that k₀ < k and the coupling is cumulative. In this condition, as the gap increases, k decreases and tends to k₀. As a result, the current and the attraction force increase. Contrarily, as the gap decreases, the attractive force decreases. It is expected that this suspension system can achieve self-stabilization characteristic.

Fig. 2.

Schematic drawing of magnetic coupling between the stator and floator.

Fig. 3.

Attractive force versus frequency: analyzed with JMAG.

3. Design and analysis

3.1. Design concept of apparatus

Figure 2 illustrates the schematic drawing of the magnetic coupling between the stator and the floator in the upper and lower side. The force F generated by the upper electromagnet (EM) can balance the gravitational force mg acting on the floator at a definite gap. The attractive force at the specific gap depends on several parameters such as frequency, current and amplitude of the supply source. If any parameter is changed, the floating condition at the specific gap is lost. It may be undesirable to estimate the effects of the parameter purely.

To solve this problem, the lower stator is incorporated to the apparatus to add some force F to maintain the gap. The proposed apparatus is designed in such a way where the floator can be kept at a constant gap even when the parameters of the upper electromagnets are varied. Figure 2(a) shows the stator is fixed with the base structure which may lead to the generation of vibration at higher gap. So, damping is necessary to reduce the inherent vibration to the floator. Full floating condition will be lost if direct damping is applied to the floator. Therefore, damping is applied between base and the stator herein called as indirect damping as shown in Fig. 2(b).

3.2. FEM analysis for force measurement

To design the apparatus, finite element method (FEM) analysis has been done for specifying the electromagnets. The vertical force generated by the one pair such as one primary and one secondary electromagnet is calculated with JMAG. The result is shown by Fig. 3. It has been found from frequency analysis that the force at resonance frequency of 587 Hz is about 20 N where the supply amplitude was 11 V. This result will be verified later in the experimental analysis.

4. Experimental apparatus

Figure 4(a) shows a schematic drawing of the experimental apparatus. There are three parts in this apparatus such as an upper stator, a floator and a lower stator. Each stator part comprises three primary electromagnets, three voice coil motors (VCMs) and passive spring elements. The floator has three secondary electromagnets each on the upper and lower sides. A permanent magnet is added to the upper stator and a target ferromagnetic plate is added to the floator to reduce supply energy to the upper primary electromagnets. Figure 4(b) shows a top view of the experimental apparatus where three primary electromagnets and VCMs are placed on the upper stator. This position is maintained to the lower stator also. Figure 5 shows a photograph of the fabricated experimental apparatus. The core of electromagnets is made of laminated silicon steel plates and a coil with 220 turns of copper wires of 0.8 mm diameter. The VCM is made of an iron core and permanent magnets and a coil with 150 turns of copper wire of 0.8 mm diameter. When the coils of the primary electromagnets are energized with AC supply, magnetic resonance coupling occurs between the primary and secondary electromagnets. As the magnetic suspension intrinsically vibrates, indirect damping is added by the VCM. Three displacement sensors are used to measure the displacement of each of the upper stator plate, the floator plate and the lower stator plate. These sensors are placed 120 degree away from each other.

Fig. 4.

Schematic drawing of experimental apparatus.

Fig. 5.

Photograph of experimental apparatus.

Table 1

Circuit parameters of one pair of electromagnets

Parameters	Symbol	Value
Self-Inductance of primary electromagnet	L ₁	9.84 mH
Capacitance of primary circuit	C ₁	2.01 μF
Resistance of primary coil	R ₁	2.5 Ω
Self-Inductance of secondary electromagnet	L ₂	9.74 mH
Capacitance of secondary circuit	C ₂	2.02 μF
Resistance of secondary coil	R ₂	2.43 Ω

5. Experimental results

The characteristics of each pair of the circuit of primary and secondary electromagnets are investigated. Table 1 shows the circuit parameters of the upper one pair of the primary and secondary electromagnets at the position number 1 as shown by Fig. 4(b). The isolated primary and secondary circuit has a resonance frequency about 1 kHz. However, the resonance frequency changes when the circuit is magnetically coupled.

5.1. Frequency dependency

When the amplitude of the applied voltage is 11 V, the resonance frequency is 632 Hz at a gap of 2.0 mm between the primary and secondary electromagnets is observed as shown in Fig. 6(a). The attractive force is measured with a load cell. The maximum force of 18.7 N is measured at the resonance frequency of about 632 Hz. This result has a good agreement with the FEM analysis result that is shown in Fig. 3 for the magnitude of the force. The measured characteristics of the primary and secondary currents at the resonant frequency are shown by Fig. 6(b). Figure 6(c) shows the measured characteristics of the phase difference between the primary and secondary currents at the resonant frequency. The current is maximum and the phase difference is almost zero when the applied frequency is adjusted to the resonance frequency of 632 Hz. The latter result demonstrates that the cumulative coupling occurs.

Fig. 6.

Characteristics of force, currents and phase difference between coil currents current when the gap is 2.0 mm.

5.2. Gap dependency

Figure 7(a) shows the measured characteristics of force to the gap when the amplitude of the applied voltage is 11 V and the angular frequency, ω = 2π ×650 rad/s. The force is almost zero when the gap is less than 1.5 mm or greater than 2.8 mm. The force increases as the gap increases from 1.5 mm to 2.2 mm. It can be said that self-stabilization is possible in this range. Another important point is that the maximum force at frequency of 632 Hz when the gap is 2.0 mm as shown by Fig. 6(a). It indicates that the resonance frequency increases as the gap becomes larger. Figure 7(b) shows the characteristics of the primary and secondary currents with the gap when the applied frequency is 650 Hz and the amplitude is 11 V. The relation of the phase difference between the primary and secondary currents is shown by Fig. 7(c). The current is maximum and the phase difference is almost zero at the gap of 2.2 mm. So, circuit resonant here at the gap of 2.2 mm.

Fig. 7.

Characteristics of force, currents and phase difference between coil currents current when ω =2π ×650 rad/s.

5.3. Amplitude dependency

The gap between the primary and secondary electromagnets is kept 2 mm. The force is measured for various amplitudes of the applied voltage when the applied frequency is set to be 620 Hz. Figure 8(a) shows the characteristics of the force to the amplitude of the applied voltage. The force increases as the amplitude of the applied voltage increases. The primary and secondary currents also increase when the applied voltage increases as shown in Fig. 8(b). Figure 8(c) shows the characteristics of the phase difference between the primary and secondary currents to the applied voltage at the same applied frequency. The phase difference is very small near the resonance frequency.

Fig. 8.

Characteristics of force, currents and phase difference between coil currents current when frequency is 620 Hz.

Fig. 9.

Displacement of the upper stator (US) and floator (F) during levitation.

Table 2

Variation of frequencies keeping the floator at constant position

Applied frequency of the upper electromagnet [Hz]	765	760	755	750	745	740
Applied Frequency of the lower electromagnet [Hz]	300	411	456	483	504	532

Fig. 10.

Displacement of the upper stator (US), floator (F) and lower stator (LS) during levitation in differential mode.

5.4. Levitation with self-stabilization

Self-stabilization is achieved when the amplitude of the applied voltage is 46 V with a resonance frequency of 725 Hz between the upper stator and floator. Figure 9 shows the results of self-stabilization without applying indirect damping. The gap is kept almost at 2.0 mm without oscillating. The stator plate is suspended by three helical spring with a stiffness of 3.81 N/mm. Stable condition is achieved here without any feedback control.

5.5. Differential operation with indirect damping

The floator is levitated with the gap of 3.0 mm between the upper and lower stator. Figure 10 shows the displacement of the upper stator (US), floator (F) and lower stator (LS) where the data has been taken using three sensors for each. To adjust the downward force, the frequency of the lower stator electromagnets is varied. The levitation starts from applying the frequency to the upper stator electromagnets of 765 Hz and the lower stator electromagnets of 300 Hz with the applied voltage of 46 Volt. The frequency of the upper electromagnets is gradually decreased up to 740 Hz and the lower stator electromagnets is gradually increased up to 532 Hz manually to keep the floator at constant position as mentioned in Table 2. A PD control is applied to the upper stator VCMs to generate indirect damping. The applied damping is 123 Ns/m and the stiffness is 201 N/m. A bias current of 0.9 A is also supplied to the upper VCMs to keep the position of the upper stator. In the lower stator, the applied damping is 78 Ns/m. A bias current of 1.0 A is also supplied to the lower stator VCMs to keep the position of the lower stator. Thus, magnetic suspension is achieved in the differential mode.

6. Conclusions

The performances of newly developed AC magnetic suspension system using magnetic resonance coupling were investigated. The fundamental characteristics of the apparatus were studied theoretically and experimentally. The self-stabilization characteristic was also observed to levitate the floator when the gap is 2.0 mm from the upper stator without control. In addition, the floator was kept at a constant position while changing the frequency of the upper and lower electromagnets. Then, differentially operated AC magnetic stable suspension was achieved.

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