# Gain-Scheduled Control of Asymmetric Thrust Magnetic Bearing

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Cold Compressor

#### 2.2. Non-Linear Electromagnetic Force

_{a}generated by the single magnetic pole is related to the coil current i and the distance s between the stator and rotor [2]:

_{0}is the vacuum permeability, N denotes the coil turns, A is the magnetic pole area, and k is the AMB structure parameter, which does not change with the instantaneous state of the system.

_{u}and

_{d}to denote the upper and lower thrust AMB, respectively, and introduce the structural parameters k

_{u}and k

_{d}of the thrust AMB:

_{m}and x

_{p}denote the unilateral air gap of the thrust AMB and the auxiliary bearing, respectively, the specified balanced position is labeled x

_{0}(0 < x

_{0}< 2x

_{p}), the bias current is labeled i

_{0}, the instantaneous distance of the rotor from the bottom end of the auxiliary bearing is indicated by x, and i indicates the instantaneous control current.

_{0}, i = 0) for the Formula (3) is presented:

_{i}and k

_{x}are the current stiffness and displacement stiffness, respectively. They are the two critical parameters related to the AMB geometry, the bias current, and the reference position.

_{a}includes a constant force part f

_{a0}, not varying with the instantaneous state of the system. Even when the rotor is balanced at the physical center, and even if the thrust AMB is asymmetric, the term is not zero. According to the design ideas mentioned above, when the rotor is suspended at the center, f

_{a0}should be exactly the same as the gravity of the rotor.

#### 2.3. Model Analysis

_{i}and k

_{x}characterizing the electromagnetic force model are illustrated in Figure 3 and Figure 4, respectively. It should be noted that the magnitudes of k

_{i}, k

_{x}, and f

_{a0}are also affected by the bias current, according to Formula (4). However, to highlight the variation characteristics with different positions, the bias current remains unchanged, with a value of 0.85 A, as shown in Table 1.

_{i}and k

_{x}are not symmetrical at the center of 300 μm. The change near the upper AMB is much more dramatic than that near the lower AMB due to the larger geometry of the upper AMB. These figures depict that the two parameters remain almost unchanged within the range of 100 μm, deviating downward from the center. However, k

_{i}and k

_{x}increase rapidly, and their slopes increase notably when the deviation increases. When the balanced position deviates from the scope of (200, 300) by 100 μm, the percentage change in k

_{i}and k

_{x}reaches 55% and 85%, respectively. It can be inferred from Figure 4 that the parameter-fixed controller may maintain the system as stable within the range of 200 to 300 μm. Still, beyond this range, the response performance is likely to deteriorate rapidly. The results are confirmed in the following simulation experiment.

_{a0}with the balanced position. As shown in Formula (4), f

_{a0}is a constant force generated by the asymmetrical AMB. When the rotor is balanced at the center, f

_{a0}is not zero but is supposed to be equal to the rotor gravity, according to the previous design expectations. At this equilibrium point, the coil current of the upper and lower AMB should be close. When the balanced position deviates from the center, the resultant force of the electromagnetic force and the gravity is not zero. Thus, additional current is needed to offset the resultant force represented by the gray curve. The figure shows that the resultant force increases with the deviation. The increasing rate of the force close to the upper AMB is more significant than the lower.

_{0}moves away from the AMB center. However, it does not prevent us from analyzing the qualitative law of the problem.

## 3. Controller Design

_{c}in the following form:

_{p}, K

_{i}, and K

_{d}are the proportional gain, integral gain, and derivative gain, respectively. The derivative time constant T

_{d}is also introduced to prevent the controller from infinitely amplifying high-frequency noise signals.

_{p}, K

_{i}, and K

_{d}.

_{a}is the power amplifier’s transfer functions and G

_{s}is the sensor’s. The sweep frequency test shown in Figure 6 displays that the gain of the power amplifier is 0.79 and that the bandwidth is approximately 2.3 kHz. The sensor gain is 20,000 V/m, and the bandwidth of the anti-aliasing filter in the signal processing circuit is 3.3 kHz. The bandwidth of both the power amplifier and the sensor is quite large, so their transfer functions can be simplified into a constant to highlight clearly the relationship between the control parameters and the system supporting performance.

_{e}denotes the stiffness and d

_{e}denotes the damping, the Fourier transform of the dynamic equation of the mass-spring-damping system is:

_{i}and K

_{d}in Formula (11) have little effect on the stiffness and thus can be ignored during the initial estimation stage when the working frequency is below 200 Hz,

_{e}/k

_{x}, and the stiffness coefficient should be greater than 1 to stabilize the rotor. However, excessive stiffness requires a high-bandwidth power amplifier and may cause magnetic field saturation. Therefore, according to (13):

#### 3.1. Parameter-Fixed PID Controller

_{p}, K

_{d}, and K

_{i}can be calculated. These parameters are listed in Table 3.

_{e}begins to decay to a negative value. Negative stiffness means that the characteristic root of the closed-loop system will move from the left half-plane to the real axis, and the rotor cannot remain stable anymore. When the balanced position moves from the center to the magnetic poles at both ends, the rotor is more likely to fall into the unstable area close to the larger upper axial magnetic bearing.

#### 3.2. Gain-Scheduled PID Controller

_{p}and K

_{d}with the balanced position is shown in Figure 8.

_{p}increases rapidly. K

_{p}increases approximately three times at a distance of around 300 μm from the physical center and increases approximately two times at approximately 200 μm. In general, the variation of the balanced position in the asymmetric AMB causes K

_{p}to vary greater, whereas K

_{d}displays more obvious asymmetric features.

## 4. Simulation and Experiment

#### 4.1. Simulation Result Analysis

#### 4.2. Experiment Result Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Step response simulation with parameter-fixed controller. (

**a**) x

_{0}= 50 μm; (

**b**) x

_{0}= 100 μm; (

**c**) x

_{0}= 200 μm; (

**d**) x

_{0}= 300 μm; (

**e**) x

_{0}= 400 μm; (

**f**) x

_{0}= 500 μm.

**Figure 12.**Step response simulation with gain-scheduled controller. (

**a**) x

_{0}= 50 μm; (

**b**) x

_{0}= 100 μm; (

**c**) x

_{0}= 200 μm; (

**d**) x

_{0}= 300 μm; (

**e**) x

_{0}= 400 μm; (

**f**) x

_{0}= 500 μm.

**Figure 14.**Experiment of step response from 0. (

**a**) x

_{0}= 50 μm; (

**b**) x

_{0}= 100 μm; (

**c**) x

_{0}= 150 μm; (

**d**) x

_{0}= 250 μm; (

**e**) x

_{0}= 300 μm; (

**f**) x

_{0}= 350 μm.

**Figure 15.**Experiment of stair-like response with a fixed step size of 50 μm. (

**a**) from x

_{0}= 50 to 100 μm; (

**b**) from x

_{0}=100 to 150 μm; (

**c**) from x

_{0}= 150 to 200 μm; (

**d**) from x

_{0}= 200 to 250 μm; (

**e**) from x

_{0}= 250 to 300 μm; (

**f**) from x

_{0}= 300 to 350 μm.

Name | Value |
---|---|

Bias current, i_{0} (A) | 0.85 |

Unilateral air gap of the thrust AMB, x_{m} (μm) | 400 |

Unilateral air gap of the auxiliary bearing, x_{p} (μm) | 300 |

Upper AMB inner magnetic pole inner/outer diameter, a/b (mm) | 50/58.5 |

Upper AMB outer magnetic pole inner/outer diameter, c/d (mm) | 74/80 |

Lower AMB inner magnetic pole inner/outer diameter, e/f (mm) | 545/85 |

Lower AMB outer magnetic pole inner/outer diameter, g/h (mm) | 76/80 |

Number of coil turns of upper AMB, N_{u} | 225 |

Number of coil turns of lower AMB, N_{d} | 175 |

Structure parameter of upper AMB, k_{u} | 15.062 |

Structure parameter of lower AMB, k_{d} | 13.662 |

Name | Value |
---|---|

Bias current of radial AMB (A) | 1.1 |

Air gap of radial AMB (μm) | 350 |

Air gap of radial auxiliary bearing (μm) | 150 |

Magnetic pole area of radial AMB (mm^{2}) | 263 |

Number of coil turns of radial AMB | 97 |

Number of magnetic poles of radial AMB | 8 |

Total length of rotor (mm) | 412 |

Rotor mass, m (kg) | 6.2 |

Diameter of middle part of rotor (mm) | 63 |

Name | Value |
---|---|

Proportional gain K_{p0} | 0.32 |

Integral gain K_{i0} | 40 |

Differential gain K_{d0} | 7.5 × 10^{−4} |

Differential time constant T_{d} | 1/400π |

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**MDPI and ACS Style**

Zhang, S.; Wu, J.
Gain-Scheduled Control of Asymmetric Thrust Magnetic Bearing. *Actuators* **2021**, *10*, 329.
https://doi.org/10.3390/act10120329

**AMA Style**

Zhang S, Wu J.
Gain-Scheduled Control of Asymmetric Thrust Magnetic Bearing. *Actuators*. 2021; 10(12):329.
https://doi.org/10.3390/act10120329

**Chicago/Turabian Style**

Zhang, Shuyue, and Jihao Wu.
2021. "Gain-Scheduled Control of Asymmetric Thrust Magnetic Bearing" *Actuators* 10, no. 12: 329.
https://doi.org/10.3390/act10120329