# Suspension Flux Internal Model Control of Single-Winding Bearingless Flux-Switching Permanent Magnet Motor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topology and Mathematical Model of BFSPMM

#### 2.1. Topology

_{1}-k

_{2}-k

_{3}-k

_{4}in series, k = A-F. The current i

_{A}-i

_{F}flows through the phase winding A-F, respectively. They contain the torque current components i

_{AT}-i

_{FT}that control the motor’s rotation and the suspension current components i

_{AS}-i

_{FS}that control the motor’s suspension, and i

_{AS}= i

_{DS}, i

_{ES}= i

_{BS}, i

_{CS}= i

_{FS}, i

_{AT}= −i

_{DT}, i

_{ET}= −i

_{BT}, and i

_{CT}= −i

_{FT}. Through the suspension current, which is spatially symmetrical and in the same direction, the suspension force in three directions with a spatial difference of 120° can be generated to produce a controlled suspension operation. While the suspension current which is spatially symmetrical and in the opposite direction can be regarded as a three-phase FSPMM, which can generate stable electromagnetic torque. This current mode allows for the decoupling of rotation and suspension control.

_{R}, the mechanical angle is θ

_{R}, and the electrical angle is θ

_{r}= 10 θ

_{R}.

_{αT}i

_{βT}, suspension-plane current components i

_{αS}i

_{βS}, and zero-sequence currents i

_{o}

_{1}i

_{o}

_{2}by constant power transformation matrix

**T**[6]:

_{6}_{dT}= 0. Thus, the electromagnetic torque is:

_{f}| is the amplitude of the PM flux linkage, and i

_{qT}and i

_{dT}can be obtained by the following transformation:

#### 2.2. Rotor Dynamics Model of BFSPMM

_{r}j

_{r}k

_{r}-axis as the synchronous rotating coordinate system. Due to the unbalanced moment N

_{x}N

_{y}, there is precession along the z-axis when the rotor rotates tangentially along the k

_{r}-axis. The angles of inclination of the k

_{r}-axis to the y- and x-axes are θ

_{x}and θ

_{y}, respectively. As the inclination of the shaft is very small, θ

_{x}and θ

_{y}are proportional to the x- and y-direction displacements of the rotor.

_{r}and I

_{R}are the moment of inertia of the rotor radial and tangential, respectively. According to the moment formula, the rotor dynamics model can be written as follows:

_{x}and F

_{y}are the radial electromagnetic pulls in the x-and y-directions, respectively. According to [10], the linear relationships between F

_{x}F

_{y}and suspension current components i

_{dS}i

_{qS}are as follows:

_{PM}and k

_{qT}are the amplitude of the fundamental component of the suspension force generated by the coactions of unit suspension current with the PMs’ MMF and i

_{qT}, respectively. The suspension current i

_{dS}i

_{qS}can be obtained by the following equation:

#### 2.3. Suspension Force-Flux Mathematical Model

**T**, the stator flux of the stationary coordinate system can be deduced as follows:

_{6}_{ii}(i = A~F) represents the self-inductance of phase winding i, M

_{ij}(i = A~F, j = A~F, I ≠ j) represents the mutual inductance of phase winding i and j, and ψ

_{fi}(i = A~F) represents the PM flux of phase winding i.

#### 2.3.1. Expression of No-Load PM Flux

_{PM}represents the MMF of the PMs, M

_{S}represents the stator modulation function, M

_{R}Represents the rotor modulation function, and W

_{i}(i = A~F) represents the winding function.

_{g}is the air-gap radius, μ

_{0}is the air permeability, and Δg is the air-gap length. W

_{A}represents the winding function of phase A, and it is generated by that conductor distribution function C

_{i}(θ) modulated by stator modulation function M

_{s}(θ), i.e., W

_{A}= M

_{s}·C

_{A}. As the winding of each phase is made up of two coils with symmetrical spatial axes in series, C

_{A}= C

_{A}

_{1}+ C

_{A}

_{2}, C

_{A}

_{1}, and C

_{A}

_{2}are conductor distribution functions of coils A1-A2 and coils A3-A4, respectively, whose Fourier series expansion is as follows:

_{w}is the number of coil turns,

_{S}= M

_{SW}·M

_{SF}, M

_{SW}is the modulation function of the tooth slot at windings, M

_{SF}is the modulation function of the tooth slot at PMs, and the Fourier series expansion is as follows:

_{ts}is the distance from the center of each U-core of the stator, l

_{sp}is the width of the PM, and a

_{s}and b

_{s}are constants.

_{R}is as follows:

_{tr}and l

_{sr}are the widths of the tooth and slot of the rotor, respectively. a

_{r}and b

_{r}are constants.

_{m}is the amplitude of the MMF, and a

_{f}is a Fourier series.

_{m}is a constant. b

_{m}

_{1}, b

_{m}

_{2}, and b

_{nm}are Fourier series.

_{e}. The modulation operator M

_{e}is a sinusoidal function with the eccentric angle φ as the positive direction, i.e.,

_{em}= ψ

_{fm}·m

_{e}, m

_{e}is a constant.

#### 2.3.2. Inductance Expression

_{i}(θ), and the flux induced by another winding, whose winding function is W

_{j}(θ), the mutual inductance between the phase winding is represented by i and j. Thus, the mutual inductance expression without rotor eccentricity is as follows:

_{ji}is the self-inductance of phase winding i when i = j, denoted as L

_{ii}

_{0}. Taking phase winding A as an example, and substituting (14)~(19) into (25), it is deduced that the self-inductance of phase winding A without rotor eccentricity is:

_{0}~L

_{4}are constants, and L

_{0}>> L

_{2}> L

_{3}> L

_{1}> L

_{4}. Thus, it can be obtained that L

_{ii}

_{0}≈ L

_{0}.

_{e}. The inductance changes in phase winding i can be deduced as

_{e}= L

_{0}·l

_{e}, and l

_{e}is a constant.

_{ji}

_{0}is the DC component of the mutual inductance between phase winding i and j without the rotor eccentricity, and M

_{ji}

_{e}is the change in the mutual inductance caused by the eccentric rotor. When i = j, M

_{ji}= L

_{ii}. According to [29], there is a fixed proportional coefficient of the mutual inductance and self-inductance of single-winding BFSPMM. Thus, (28) can be rewritten as

#### 2.3.3. Mathematical Model of Suspension Flux

_{A}~ψ

_{F}can be obtained. By substituting (22), (24), and (29) into (9), the suspension-plane flux of the stationary coordinate system α

_{S}β

_{S}can be deduced as

_{1}~v

_{3}are constants.

_{αS}ψ

_{βS}of the stationary coordinate system can be transformed into a d

_{S}q

_{S}coordinate system, and the flux in the d

_{S}q

_{S}coordinate system is as follows:

_{dS}and i

_{qS}, and the latter two terms can be regarded as the disturbance components ψ

_{dSd}and ψ

_{qSd}.

_{dS}and i

_{qS}in (32) into (6), the linear relationships between F

_{x}F

_{y}and suspension-plane flux ψ

_{dS}ψ

_{qS}are as follows:

## 3. Suspension Flux Internal Model Control Strategy

#### 3.1. Suspension Flux-Dynamics Internal Model of BFSPMM

_{n}is the natural oscillation frequency, and ζ

_{F}is the damping coefficient of the filter.

_{f}≈ x − x*. Thus, the control problem that rotor radial displacement tracks a given signal becomes the control problem that x

_{f}= 0. Similarly, the derivation of the y-direction is not detailed here.

#### 3.2. Feedback Linearization of Auxiliary Function

_{f}= 0, setting ${x}_{f1}={x}_{f},{x}_{f2}={\dot{x}}_{f1}$, it can be deduced that

#### 3.3. Control Block Diagram

_{dS}ψ

_{qS}into equation (34), the internal model value of the rotor radial displacement is obtained. The internal model parameters are shown in Table 1. The given value u

_{x}*u

_{y}* of the output of the internal model controller is calculated according to (36)~(42), and the following equation calculates the suspension-plane flux error:

_{s}is the winding resistance, and T

_{s}is the control period.

_{dT}= 0, according to (3)~(4). The zero-sequence current component i

_{o}

_{2}is controlled to zero by a PI controller. Thus, the torque-plane expected voltage vector u

_{dT}*u

_{qT}*, which controls the motor rotation, the suspension-plane expected voltage vector u

_{dS}*u

_{qS}*, which controls the motor suspension, and the zero-sequence-plane expected voltage vector u

_{o}

_{2}* can be obtained. Finally, the control strategy proposed in this paper can be realized by PWM control.

## 4. Verification

#### 4.1. Test Platform

^{®}controller(Nanjing Rtunit Information Technology Co., Ltd., Nanjing, China) with TMS320F28377D as the core, a building block power module, and a high-power density power, as shown in Figure 6b.

#### 4.2. Simulation Results

#### 4.3. Steady-State Experiment

#### 4.4. Dynamic-State Experiment

## 5. Discussion

## 6. Conclusions

- (1)
- The high-performance decoupling control of a single-winding BFSPMM is realized, and the radial displacement of the rotor is always suspended with a small radial displacement in the geometric center axis.
- (2)
- Compared with the traditional PID control, the proposed method has better steady-state performance, and the maximum radial displacement ripple of the rotor is reduced by 53%, which effectively improves the antijamming and robustness of the system.
- (3)
- Compared with the traditional PID control, the proposed method has better dynamic-state performance and reduces the radial vibration of the rotor in the process of speed regulation under the load-speed step condition.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Simulation results: (

**a**) Rotor displacement under PID control; (

**b**) Suspension current under PID control; (

**c**) Rotor displacement under proposed control; (

**d**) Suspension current under proposed control.

**Figure 8.**Comparison of the steady-state experiment: (

**a**) Traditional PID method; (

**b**) Proposed method.

**Figure 9.**Comparison of the dynamic-state experiment: (

**a**) Rotor displacement under PID control; (

**b**) Suspension current under PID control; (

**c**) Rotor displacement under proposed control; (

**d**) Suspension current under proposed control.

Parameter | Symbol | Value |
---|---|---|

Rotor gravity | G | 98 N |

Shaft length | l_{r} | 135 mm |

Moment of inertia of the rotor radial | I_{r} | 0.16123 kg·m^{2} |

Moment of inertia of the rotor tangential | I_{R} | 0.00986 kg·m^{2} |

Suspension-plane inductance | L_{ss} | 0.004553 H |

Self-inductance variation coefficient | L_{e} | 1.4556 H/m |

Amplitude of PM flux | ψ_{fm} | 0.0294 Wb |

PM flux variation coefficient | ψ_{em} | 11 Wb/m |

Performance | PID Method | Proposed Method |
---|---|---|

Stable suspension time in simulation | 200 ms | 100 ms |

Steady-state rotor displacement | ±0.15 mm | ±0.07 mm |

Maximum rotor displacement in step speed | ±0.2 mm | ±0.1 mm |

Parameter dependence | Low | High |

Complexity | Low | High |

Loss | Low | High |

Control accuracy | Low | High |

Dynamic response | Slow | Fast |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Yu, W.; Yang, R.; Cui, B.
Suspension Flux Internal Model Control of Single-Winding Bearingless Flux-Switching Permanent Magnet Motor. *Actuators* **2023**, *12*, 404.
https://doi.org/10.3390/act12110404

**AMA Style**

Chen Y, Yu W, Yang R, Cui B.
Suspension Flux Internal Model Control of Single-Winding Bearingless Flux-Switching Permanent Magnet Motor. *Actuators*. 2023; 12(11):404.
https://doi.org/10.3390/act12110404

**Chicago/Turabian Style**

Chen, Yao, Wanneng Yu, Rongfeng Yang, and Bowen Cui.
2023. "Suspension Flux Internal Model Control of Single-Winding Bearingless Flux-Switching Permanent Magnet Motor" *Actuators* 12, no. 11: 404.
https://doi.org/10.3390/act12110404