本論文主要目的為建立12/8極磁阻馬達之徑向力控制法則，且將所提出之法則應用於降低馬達震動與建立自軸承控制系統。透過有限元素分析軟體分析並建立馬達之數學模式，其中馬達定子與轉子凸極間之吸引力模式為轉子角度與激磁電流之函數，並且在模式中考慮互感之影響。分析此吸引力可分解出馬達之轉矩與徑向力，最後得到轉矩與徑向力模式。
本論文提出三種徑向力控制法則，研究最初首先提出利用具有相移之單相弦波激磁法則控制磁阻馬達之轉矩及徑向力，之後再提出兩相弦波激磁法則用以提昇單相弦波激磁法則徑向力之操作能力。另外，亦研究六極激磁之徑向力控制法則。此三種徑向力控制法則皆可以有效將馬達轉矩及徑向力解耦，並產生期望之徑向力。其中單相弦波激磁法則之驗證結果顯示可以應用於降低馬達振動。兩相弦波激磁法則則用以實現單軸自軸承控制系統。此自軸承馬達之轉子只需要一個軸承用以維持馬達轉動與軸向位置，而另一端則可以自由地徑向移動，但由馬達所產生的徑向力平衡轉子並維持在氣隙的中心。

The main objective of this research investigates the control of radial force of 12/8-pole SRM, and the applications of this scheme to motor vibration reduction and self-bearing control. The mathematical models of the motor were analyzed and developed through finite-element analysis. The attraction force between the stator and rotor pole teeth and its orientation was modeled as a function of the rotor angle and the excitation current. Mutual inductance was included in the modeling. The attraction force is decomposed into motor torque and radial force, and consequently, the torque and radial force produced by the motor is analyzed.
Three radial force control schemes were investigated in this research. In the beginning, single-phase sinusoidal currents with phase shift was proposed to control the SRM. Then, a two-phase sinusoidal excitation scheme was presented to extend the force producing capability of the single-phase scheme. In addition, a six-pole excitation scheme for radial fore control was also investigated. All of these schemes have been verified to be able to produce radial force and torque effectively, and independently. The single-phase sinusoidal current scheme has been applied to show that motor vibration can be reduced by appropriate radial force control. The two-phase sinusoidal excitation scheme has been applied to realize self-bearing control of the SRM. In this one-axis self-bearing motor, the rotor needs only one bearing for rotation and constrain of axial movement; the other end can move freely in radial direction and is balanced at the center of the air gap with the radial force produced by the motor.

CONTENTS

CHINESE ABSTRACT	I
ABSTRACT	II
ACKNOWLEDGEMENTS	IV
CONTENTS	V
LIST OF FIGURES	VIII
LIST OF TABLES	XIV
LIST OF SYMBOLS	XV
CHAPTER 1 INTRODUCTION	1
1.1 Motivation	1
1.2 Review of Prior Work	3
1.3 Contributions of the Dissertation	5
1.4 Outline of the Contents	6
CHAPTER 2 MATHEMATICAL MODELS OF SWITCHED RELUCTANCE MOTOR	8
2.1 Introduction	8
2.2 Single Pole Model	9
2.3 Multi-Pole Model	12
2.4 Simulation Results	15
CHAPTER 3 RADIAL FORCE CONTROL WITH SINGLE- PHASE SINUSOIDAL EXCITATIONS	20
3.1 Introduction	20
3.2 Radial Force Control with Single-Phase Sinusoidal Excitations	21
3.3 Radial force and torque limitations	27
3.4 Phase Commutation	31
3.5 Experimental Results	33
3.6 Discussions	42
CHAPTER 4 RADIAL FORCE CONTROL WITH SIX-POLE EXCITATIONS	44
4.1 Introduction	44
4.2 Radial Force Control with Six-Pole Excitations	44
4.3 Simulation Results	49
4.4 Control System	52
4.5 Experimental Results	54
4.6 Discussions	60
CHAPTER 5 RADIAL FORCE CONTROL WITH TWO-PHASE SINUSOIDAL EXCITATIONS	61
5.1 Introduction	61
5.2 Single-Phase Sinusoidal Excitations with Consideration of Mutual  Inductance	62
5.3 Radial Force Control with Two-Phase Sinusoidal Excitations	64
5.4 Simulation Results	66
5.5 Control System	68
5.6 Experimental Results	70
5.7 Discussions	75
CHAPTER 6 SELF-BEARING CONTROL	77
6.1 Introduction	77
6.2 Control System	77
6.3 Experimental Results	79
6.4 Discussions	87
CHAPTER 7 CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK	88
7.1 CONCLUTIONS	88
7.2 FUTURE WORK	90
REFERENCE	91
APPENDIX	97

 
LIST OF FIGURES

Fig. 2.1. Schematic and coordinate systems of the 12/8 pole SRM.	10
Fig. 2.2. Attraction force produced by pole A1.	10
Fig. 2.3. The relationship between qp and qr.	12
Fig. 2.4. Comparison of qf vs. qr calculated by FEA and by Eq.(2.3).	15
Fig. 2.5. Comparison of torque and radial force calculated with FE software and by Eqs.(2.4)-(2.6), (a) torque, (b) radial force.	16
Fig. 2.6. Comparison of torque and radial force calculated with FE software and by Eqs.(2.7)-(2.9), (a) torque, (b) radial force.	16
Fig. 2.7. Comparison of torque and radial force calculated with FE software and by Eqs.(2.10)-(2.14), when three poles of phase A are excited, (a) torque, (b) radial force.	17
Fig. 2.8. Comparison of torque and radial force calculated with FE software and by Eqs.(2.10)-(2.14), when four poles of phase A are excited, (a) torque, (b) radial force.	17
Fig. 2.9. Comparison of torque and radial force calculated with FE software and by Eqs.(2.7)-(2.8), Eqs.(2.10)-(2.13), (a) iA1=1A, iA2=1A, 2A, and 3A, respectively, (b) iA1=1A, iA2=2A, iA3=1A, 2A, and 3A, respectively, (c) iA1=1A, iA2=2A, iA3=1A, iA4=1A, 2A, 3A, and 4A, respectively.	18
Fig. 3.1. Control block diagram for torque and radial force control with single-phase sinusoidal excitations.	25
Fig. 3.2. FEA calculated waveforms when iT = 2A, iF =2 A, qr = 5o, and qf varied from 0 to 360o,(a) phase A currents, (b) torque, (c) radial force.	26
Fig. 3.3. FE analysis calculated radial force waveforms when qr varied from 0 o to 15o and qf varied from 0 to 360o.	26
Fig. 3.4. Radial force and toque calculated with FE under single-phase excitations, iT=iT_rated, iF= imax, qf varied from 0 to 360o, qr=0o, -7o, and -14o, respectively, (a)iA1~iA4 , (b)torque, (c)radial force.	28
Fig. 3.5. Radial force calculated with FE under single-phase excitations, iT= 33%, 66%, and 100% iT_rated, respectively, qf varied from 0 to 360o, qr = 0o, (a)iA1, (b)radial force.	29
Fig. 3.6. Operating range of the SRM used in this paper, (a) iF vs. iT, (b) radial force vs. motor torque.	30
Fig. 3.7. Phase A and phase C currents when commutating from Phase A to C.	32
Fig. 3.8. Comparison of FE calculated force and torque waveforms for (a) conventional commutation, (b) proposed commutation scheme.	33
Fig. 3.9. Experimental setup.	34
Fig. 3.10. Radial force and phase A currents when the SRM was at standstill, |Fr*| =15N, load torque = 0.7Nm, qr = 0 o, -7 o, -14o, respectively, and rotating at 1Hz, single-phase excitation, (a) radial force, (b) iA1~iA4, (c)(d) radial force.	35
Fig. 3.11. Radial force and phase A currents when the SRM was at standstill, qr = -14o, load torque = 1Nm, |Fr*| = 30N and rotating at 1Hz, single-phase excitation, (a) radial force, (b) pole A1~A4 currents.	36
Fig. 3.12. Block diagram of the experimental system.	37
Fig. 3.13. Motor currents and vibration spectrum with the square current control, motor speed=1000 rpm(17Hz), (a) iA1, iB1, and iC1, (b) vibration spectrum.	39
Fig. 3.14. Currents and vibration spectrum with the sinusoidal currents, motor speed=1000 rpm, and Fr* =35N, 25 Hz: (a) iA1, iB1, and iC1; (b) vibration spectrum.	40
Fig. 3.15. Same conditions as in Fig. 3.14, (a) frequency of Fr*=17 Hz, and (b) frequency of Fr*=34 Hz.	40
Fig. 3.16. Comparison of motor vibration spectrum when an eccentric inertia was attached at the rotor, motor speed=1000 rpm, (a) Fr* = 0, (b) Fr* =35N, 17 Hz.	41
Fig. 3.17. Currents and vibration spectrum with the conventional commutation, motor speed=1000 rpm, (a) iA1, iB1, and iC1, (b) vibration spectrum.	41
Fig. 3.18. Currents and vibration spectrum with the proposed commutation, motor speed=1000 rpm, (a) iA1, iB1, and iC1, (b) vibration spectrum.	42
Fig. 4.1. Idealized inductance of 12/8-pole SRM.	46
Fig. 4.2. Schematic of the SRM, conduction phase: phase A, and radial force producing poles: B1, B2.	46
Fig. 4.3. Motor radial force calculated with FE,  qr = 0 o, |Fr*|=10N, 20N, 30N, respectively, no load, Fr* varied from 0 to 360o, (a) radial force vector, (b) iA1~iA4 , when |Fr*| = 10N, (c) iA3 for various |Fr*|.	50
Fig. 4.4. Motor radial force calculated with FE, qr = 0 o, 7 o, 14 o, respectively, no load, Fr* varied from 0 to 360o, (a) radial force vector, (b) iA3 for various qr.	51
Fig. 4.5. (a) Compensation torque calculated with the same conditions as shown in Fig.4.4, (b) motor torque after compensation.	51
Fig. 4.6. Radial force calculated with FE, |Fr*|=15N, Fr* varied from 0 to 360o, qr = 7o, and iT* =1A, 2A and 3A, respectively.	52
Fig. 4.7. Calculate iF1*, iF2* and dT from FX*, FY* and qr.	53
Fig. 4.8. Block diagram of the control system.	53
Fig. 4.9. Radial force and pole A1~A4 currents, standstill, qr = 0 oM, |Fr*| = 10N, no load, and Fr* varied from 0 to 360o, (a) radial force vector, (b) pole A1~A4 currents.	56
Fig. 4.10. Radial force, standstill, qr = 0 oM, |Fr*| = 15N, no load, and Fr* varied from 0 to 360o, (a) included, and (b) not included the mutual inductance.	57
Fig. 4.11. Radial force and iA3 current, standstill, |Fr*| = 15N, Fr* varied from 0 to 360o, (a) radial force when qr = 7oM, (b) radial force when qr = 14oM, (c) iA3 for various qr.	57
Fig. 4.12. Radial force vector and pole A1~A4 currents, 100 rpm under 0.5 Nm load torque and |Fr*| set to: (a) 0N, (b)10N and rotating synchronously with the rotor, (c)10N and rotating at 1 Hz.	58
Fig. 4.13. Radial force vector, 600 rpm under 0.5Nm load torque and |Fr*| set to: (a) 0N, (b)10N and rotating synchronously with the rotor, (c)10N and rotating at 1 Hz.	59
Fig. 4.14. Radial force vector, 1000 rpm under 0.5Nm load torque and |Fr*| set to: (a) 0N, (b)10N and rotating synchronously with the rotor, (c)10N and rotating at 1 Hz.	59
Fig. 5.1. Force and torque for two-phase excitation.	66
Fig. 5.2. Radial force and toque calculated with FE under two-phase excitations, qr = -14o in phase A, T* = 0.1Nm, Fr*=15N, 30N, and 45N, respectively, (a) iA3 , (b) iB3 , (c) torque, (d) radial force.	67
Fig. 5.3. Operating range (Fr vs. T) of the SRM used two-phase sinusoidal excitations for various qr.	68
Fig. 5.4. Radial force control system.	69
Fig. 5.5. Selection of excitation strategy.	69
Fig. 5.6. Calculate current commands for two-phase excitations.	69
Fig. 5.7. Radial force and phase A, B currents, standstill, qr = -14o, load torque = 0.1Nm, |Fr*| =15N, 30N, 45N, respectively, and rotating at 1Hz, two-phase excitation, (a) radial force, (b) iA1, (c) iB1.	71
Fig. 5.8. Radial force and phase A currents, 100 rpm under 1.0Nm load torque and |Fr*| set to (a) 0N, (b)15N, (c)45N, and rotating synchronously with the rotor.	72
Fig. 5.9. The same operating conditions as Fig.5.8 except motor speed = 500rpm, |Fr*| set to (a) 0N, (b)15N, (c)45N.	73
Fig. 5.10. The same operating conditions as Fig.5.8 except motor speed = 1000rpm, |Fr*| set to (a) 0N, (b)15N, (c)45N.	73
Fig. 5.11. |Fr*| rotating synchronously with the rotor and stepped from 15N to 45N at time=0.5sec, 500rpm under 0.3Nm load torque, (a)Fx vs. Fy, (b) Fx, Fy vs. time (c) iA1, (d) motor speed.	74
Fig. 5.12. Radial force vector, 500 rpm under 0.3Nm load torque, and |Fr*| =45N and the frequency is (a)0Hz at 45o, (b)4Hz, (c)8Hz, (d)16Hz.	75
Fig. 6.1. Schematic of the self-bearing SRM.	78
Fig. 6.2. Block diagram of the self-bearing control system.	79
Fig. 6.3. Radial position responses, standstill and subjected to 0.03Nm load torque, qr = 0o, |x*| = |y*| = 0.1mm, 10Hz, (a) position responses vs. time, (b) x vs. y.	80
Fig. 6.4. Radial position responses, same running conditions as Fig.6.3, (a) iT = 1A, (b)iT = 2A, (c)iT = 3A.	80
Fig. 6.5. Control responses, standstill and subjected to 0.1Nm load torque, qr = 0o, |x*| = |y*| = 0, (a) X-, Y-axis radial position feedback, X-, Y-axis radial force commands, and motor speed, (b) phase A currents.	81
Fig. 6.6. Control responses, 100rpm and subjected to 0.1Nm load torque, |x*| = |y*| = 0, (a) X-, Y-axis radial position feedback, X-, Y-axis radial force commands, motor speed, and torque command, (b) position feedback in XY form, (c) A1, B1, C1 currents.	82
Fig. 6.7. Control responses, 100 rpm and subjected to 0.7Nm load torque, |x*| = |y*| = 0, (a) X-, Y-axis radial position feedback, X-, Y-axis radial force commands, motor speed, and torque command, (b) position feedback in XY form, (c) A1, B1, C1 currents.	84
Fig. 6.8. Control responses, 1500rpm and subjected to 0.25Nm load torque, |x*| = |y*| = 0, (a) X-, Y-axis radial position feedback, X-, Y-axis radial force commands, motor speed, and torque command, (b) position feedback in XY form, (c) A1, B1, C1 currents.	85
Fig. 6.9. Control responses, 1500rpm and subjected to 0.7Nm load torque, |x*| = |y*| = 0, (a) X-, Y-axis radial position feedback, X-, Y-axis radial force commands, motor speed, and torque command, (b) position feedback in XY form, (c) A1, B1, C1 currents.	86
Fig. 6.10. Comparisons of radial position errors at various motor speed and load, (a)500rpm, load torque = 0.1Nm, (b) 500rpm, load torque = 0.7Nm, (c)1000rpm, load torque = 0.1Nm, (d) 1000rpm, load torque = 0.7Nm.	87
 
LIST OF TABLES

Table 4.1 Selection of force control poles.	47

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