Table of Contents

About the Author xv

Preface xvii

1 A Brief History of Reference Frame Theory 1

References 3

2 Tesla’s Rotating Magnetic Field 5

2.1 Introduction 5

2.2 Rotating Magnetic Field for Symmetrical Two-Phase Stator Windings 5

2.3 Rotating Magnetic Field for Symmetrical Three-Phase Stator Windings 11

2.4 Rotating Magnetic Field for Symmetrical Two-Phase Rotor Windings 13

2.5 Rotating Magnetic Field for Symmetrical Three-Phase Rotor Windings 15

2.6 Closing Comments 17

References 17

3 Tesla’s Rotating Magnetic Field and Reference Frame Theory 19

3.1 Introduction 19

3.2 Transformation of Two-Phase Symmetrical Stator Variables to the Arbitrary Reference Frame 20

3.3 Transformation of Two-Phase Symmetrical Rotor Variables to the Arbitrary Reference Frame 24

3.4 Transformation of Three-Phase Stator and Rotor Variables to the Arbitrary Reference Frame 26

3.5 Balanced Steady-State Stator Variables Viewed from Any Reference Frame 31

3.6 Closing Comments 35

References 35

4 Equivalent Circuits for the Symmetrical Machine 37

4.1 Introduction 37

4.2 Flux-Linkage Equations for a Magnetically Linear Two-Phase Symmetrical Machine 37

4.3 Flux-Linkage Equations in the Arbitrary Reference Frame 39

4.4 Torque Expression in Arbitrary Reference Frame 41

4.5 Instantaneous and Steady-State Phasors 42

4.6 Flux-Linkage Equations for a Magnetically Linear Three-Phase Symmetrical Machine and Equivalent Circuit 45

4.7 Closing Comments 49

References 50

5 Synchronous Machines 51

5.1 Introduction 51

5.2 Synchronous Machine 51

5.3 Equivalent Circuit For Three-Phase Synchronous Generator 53

5.4 Closing Comment 57

Reference 57

6 Brushless dc Drive with Field Orientation 59

6.1 Introduction 59

6.2 The Permanent-Magnet ac Machine 59

6.3 Instantaneous and Steady-State Phasors 62

6.4 Field Orientation of a Brushless dc Drive 65

6.5 Torque Control of a Brushless dc Drive 75

6.6 Closing Comments 78

References 79

7 Field Orientation of Induction Machine Drives 81

7.1 Introduction 81

7.2 Field Orientation of a Symmetrical Machine 81

7.3 Torque Control of Field-Orientated Symmetrical Machine 86

7.4 Closing Comments 89

References 89

8 Additional Applications of Reference Frame Theory 91

8.1 Introduction 91

8.2 Neglecting Stator Transients 91

8.3 Symmetrical Components Derived by Reference Frame Theory 93

8.4 Multiple Reference Frames 97

8.5 Closing Comments 97

References 97

Index 99

About the Author

PAUL C. KRAUSE, PHD, started PC Krause and Associates, Inc. in 1983. He was a Professor in the School of Electrical and Computer Engineering at Purdue University for 39 years and before that he taught at the University of Wisconsin as well as the University of Kansas. He is a Life Fellow of the IEEE and has authored or co-authored over 100 technical papers and three textbooks on electric machines. He was the recipient of the IEEE Nikola Tesla Award in 2010.