# High-Frequency Signal Injection-Based Sensorless Control for Dual-Armature Flux-Switching Permanent Magnet Machine

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling of the Machine

#### 2.1. The Fundamental Frequency Model of the Machine

**I**is the decoupled currents as in (7); the

_{s_dq}**u**and

_{s_feedforward}**u**are shown in (8).

_{r_feedforward}#### 2.2. The High-Frequency Model of the Machine

**u**and

_{sh_dq}**u**are the injected high-frequency voltages in the rotational coordinate system;

_{rh_d1q1d3q3}**i**and

_{sh_dq}**i**are the high-frequency response currents in the rotational coordinate system;

_{rh_d1q1d3q3}**L**,

_{sh_dq}**L**, and

_{rh_d1q1d3q3}**M**are the high-frequency inductance matrices which are same as the fundamental frequency inductances, respectively.

_{srhdq}## 3. High-Frequency Injection

#### 3.1. The High-Frequency Current Response

_{se}, and the same is true for the rotor windings.

**A**is the coefficient matrix that relates to the position error and inductances of the DA-FSPM, which is given in detail in Appendix A. The injection in the third harmonic plane of the five-phase rotor windings is not considered in this paper.

**A**contains δ

_{se}and δ

_{re}, some conclusions can be drawn from (12). When a high-frequency voltage signal is injected into the stator or the rotor side, the position angle signal will be carried in the current response of the two-sided winding. The current response on the same side of the injected high-frequency voltage side only carries the position information of that side, while the current response on the opposite side carries the position information of both sides. However, as the initial position of both windings are identified, the position of the stator windings and rotor winding has a certain relationship. Thus, the position can be estimated through both sides regardless of which side the high-frequency voltage is injected.

#### 3.2. Injected through the Stator Side

#### 3.3. Injected through the Rotor Side

## 4. Position Estimation

#### 4.1. Estimate from the Same Side

#### 4.2. Estimate from the Opposite Side

_{rqh}

_{1}is the high-frequency current response of the rotor windings when the high-frequency voltage is injected through the stator windings, and i’

_{sqh}is the high-frequency current response of the stator windings when the high-frequency voltage is injected through the rotor windings.

#### 4.3. The Position Observer

#### 4.3.1. The Position Error Function

#### 4.3.2. The Phase-Locked Loop

_{err}represents the slope of the position error function.

_{c}is the cut-off frequency of the low-pass filter, k

_{p}is the proportional coefficient, and k

_{i}is the integral coefficient of the PI controller.

_{p}and k

_{i}are calculated as

#### 4.3.3. The Effect of Current Sample Error on Position Observation

_{err}. Based on the above analysis, the relationship between the error of the speed estimation and the position estimation, and the current sampling disturbance is affected by the error function coefficient k

_{err}. The smaller the k

_{err}is, the error generated in the estimated value is larger. Thus, when the current sampling accuracy is the same, and the PI controller parameters of the position observer are the same, the position estimation method with a smaller k

_{err}should be adopted to improve the accuracy of the estimation.

#### 4.4. The Initial Position Detection

#### 4.4.1. The Initial Position of the Stator Windings

#### 4.4.2. The Initial Position of the Rotor Windings

#### 4.4.3. The Control Scheme of the Initial Position Detection

#### 4.5. The Overall Control Scheme

## 5. Experiment Results

#### 5.1. The Experiment Platform

#### 5.2. Initial Position Detection Results

#### 5.3. The Position Estimation Results

#### 5.3.1. Startup Results

#### 5.3.2. Speed Step Results

#### 5.3.3. Load Step Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

back-EMF | Back-electromagnet force. |

DA-FSPM | Dual-armature flux-switching permanent magnet machine. |

EKF | Extended Kalman Filter. |

FSPM | Flux switching permanent magnet. |

HFI | High-frequency injection. |

PLL | Phase lock loop. |

PMSM | Permanent magnet synchronize machine. |

PI | Proportional-integral. |

SMO | Sliding mode observer. |

u, i, ψ | Matrixes of voltages, currents, and flux linkages. |

R, L, M | Matrixes of resistances, self-inductances, and mutual inductances. |

W | Matrixes of cross-coupling coefficients. |

^{s}, ^{r} | Parameters of the stator windings and rotor windings. |

^{s_dq}, ^{r_d1q1d3q3} | Parameters of the d-q axes of the stator windings and d1-q1-d3-q3 axes of the rotor windings. |

^{srdq} | Coupling component of the stator windings and rotor windings. |

p | Differential operator |

ω_{se}, ω_{re} | Stator and rotor electrical angular velocities. |

L_{sd}, L_{sq} | Self-inductances of the stator windings. |

L_{rd}_{1}, L_{rq}_{1}, L_{rd}_{3}, L_{rq}_{3} | Self-inductances of the rotor windings. |

M_{srd}_{1} | Mutual inductances between the stator d-axis windings and the rotor d1-axis windings. |

M_{srq}_{1} | Mutual inductances between the stator q-axis windings and the rotor q1-axis windings. |

M_{srd}_{3} | Mutual inductances between the stator d-axis windings and the rotor d3-axis windings. |

M_{srq}_{3} | Mutual inductances between the stator q-axis windings and the rotor q3-axis windings. |

M_{rd}_{13} | Mutual inductances between the rotor d1-axis windings and d3-axis windings. |

M_{rq}_{13} | Mutual inductances between the rotor d1-axis windings and d3-axis windings. |

f, f’ | Variable in the real and estimated rotational coordinate system. |

δ_{se}, δ_{re} | Errors of the estimated angle and the real angle of the stator windings and rotor windings. |

ω_{h}, θ_{h}, V_{c} | Electrical angular speed, electrical angle, and amplitude of the injected voltage. |

## Appendix A

**A**in this paper is

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**Figure 7.**The initial position detection scheme based on the stator side high-frequency voltage injection.

**Figure 10.**The initial position detection results. (

**a**) The actual initial is 0.4 rad. (

**b**) The actual initial is 0.8 rad. (

**c**) The actual initial is 1.2 rad. (

**d**) The actual initial is 1.5 rad.

**Figure 11.**The startup results of estimating from the same side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

**Figure 12.**The startup results of estimating from the opposite side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

**Figure 13.**The speed step results of estimating from the same side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

**Figure 14.**The speed step results of estimating from the opposite side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

**Figure 15.**The load step results of estimating from the same side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

**Figure 16.**The load step results of estimating from the opposite side. (

**a**) Speed. (

**b**) Stator position. (

**c**) Rotor position.

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

n_{ps} | 10 | n_{pr} | 6 |

R_{s} (Ω) | 0.7 | R_{r} (Ω) | 0.66 |

L_{sd} (mH) | 4.9605 | L_{rd}_{1} (mH) | 6.3961 |

L_{sq} (mH) | 5.4473 | L_{rq}_{1} (mH) | 7.1630 |

M_{srd}_{1} (mH) | 1.7208 | L_{rd}_{3} (mH) | 3.6681 |

M_{srq}_{1} (mH) | 1.6906 | L_{rq}_{3} (mH) | 3.6974 |

M_{srd}_{3} (mH) | 0.4332 | M_{srq}_{3} (mH) | 0.4768 |

ψ_{fsd} (Wb) | 0.058 | ψ_{frd}_{1} (Wb) | 0.0923 |

ψ_{frd}_{3} (Wb) | −0.0065 |

Estimating from the Same Side | Estimating from the Opposite side | |||
---|---|---|---|---|

Stator Windings | Rotor Windings | Stator Windings | Rotor Windings | |

Response time (s) | 0.5 | 0.5 | ||

Steady speed ripple (rpm) | $\pm 3.4$ | $\pm 1.8$ | ||

Steady position ripple (rad) | $\pm 0.03$ | $\pm 0.02$ | $\pm 0.01$ | $\pm 0.008$ |

Maximum speed error (rpm) | 10.54 | 12.16 | ||

Maximum position error (rad) | 0.39 | 0.23 | 0.52 | 0.31 |

Estimating from the Same Side | Estimating from the Opposite side | |||
---|---|---|---|---|

Stator Windings | Rotor Windings | Stator Windings | Rotor Windings | |

Response time (s) | 0.5 | 0.2 | ||

Steady speed ripple (rpm) | $\pm 4$ | $\pm 2$ | ||

Steady position ripple (rad) | $\pm 0.11$ | $\pm 0.3$ | $\pm 0.01$ | $\pm 0.006$ |

Maximum speed error (rpm) | 20.8 | 10.75 | ||

Maximum position error (rad) | 1.23 | 0.74 | 0.31 | 0.18 |

Estimating from the Same Side | Estimating from the Opposite side | |||
---|---|---|---|---|

Stator Windings | Rotor Windings | Stator Windings | Rotor Windings | |

Response time (s) | 0.95 | 0.7 | ||

Steady speed ripple (rpm) | $\pm 2$ | $\pm 1.5$ | ||

Steady position ripple (rad) | $\pm 0.05$ | $\pm 0.03$ | $\pm 0.01$ | $\pm 0.01$ |

Maximum speed error (rpm) | 6.6 | 4.1 | ||

Maximum position error (rad) | 0.12 | 0.07 | 0.1 | 0.06 |

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## Share and Cite

**MDPI and ACS Style**

Wu, L.; Yi, J.; Lyu, Z.; Zhang, Z.; Hu, S. High-Frequency Signal Injection-Based Sensorless Control for Dual-Armature Flux-Switching Permanent Magnet Machine. *World Electr. Veh. J.* **2023**, *14*, 85.
https://doi.org/10.3390/wevj14040085

**AMA Style**

Wu L, Yi J, Lyu Z, Zhang Z, Hu S. High-Frequency Signal Injection-Based Sensorless Control for Dual-Armature Flux-Switching Permanent Magnet Machine. *World Electric Vehicle Journal*. 2023; 14(4):85.
https://doi.org/10.3390/wevj14040085

**Chicago/Turabian Style**

Wu, Lijian, Jiali Yi, Zekai Lyu, Zhengxiang Zhang, and Sideng Hu. 2023. "High-Frequency Signal Injection-Based Sensorless Control for Dual-Armature Flux-Switching Permanent Magnet Machine" *World Electric Vehicle Journal* 14, no. 4: 85.
https://doi.org/10.3390/wevj14040085