Torque 100 80 60 40 20 Torque (N . m) (Normalized) Flux d Axis (Wb) (Normalized) Flux d Axis 0 -20 -40 -60 -80 -100 0 45 90 0% Imax (A) 32% Imax (A) 63% Imax (A) 91% Imax (A) 10% Imax (A) 42% Imax (A) 73% Imax (A) 100% Imax (A) 21% Imax (A) 52% Imax (A) 82% Imax (A) 0 45 90 135 180 225 270 315 360 Electrical Position (°) 0% Imax (A) 32% Imax (A) 63% Imax (A) 91% Imax (A) 0 -20 -40 -60 -80 -100 Figure 6. The torque waveform versus the rotor position for the current levels. z d ^ i, I h = z q ^ i, I h = 45 0% Imax (A) 32% Imax (A) 63% Imax (A) 91% Imax (A) 90 135 180 225 270 315 360 Electrical Position (°) 10% Imax (A) 42% Imax (A) 73% Imax (A) 100% Imax (A) 21% Imax (A) 52% Imax (A) 82% Imax (A) Figure 5. The z q waveform versus the rotor position for the current levels. Notice that this flux is only the one generated by the current; the permanent-magnet flux is not included. Therefore, z 0 = 0. But the combined effect of the coil flux and the permanent-magnet flux on saturation is considered. Therefore, the expression of the flux in the d-q axis is given by the Park transformation: R VR V S cos ^ih cos a i - 2.r k cos a i - 4.r k WSz AW zd 3 3 2 S WS W < F= 2.r 4.r z B , (4) 3S zq S- sin ^i h - sin a i - 3 k - sin a i - 3 kWWSz CW T XT X where i is electrical motor position. I E E E E l e c t r i f i cati o n M a gaz ine / DECEMBER 2017 21% Imax (A) 52% Imax (A) 82% Imax (A) In this study, we consider an example of real PMSMs. The obtained flux z d, z q and the torque T are shown in Figures 4, 5, and 6, respectively. The aim of this step is to make an approximated mathematical expression of these waveforms for each current level and electrical position. This approximation is based on a Fourier series decomposition: 100 80 60 40 20 0 10% Imax (A) 42% Imax (A) 73% Imax (A) 100% Imax (A) Case Study Flux q Axis Flux q Axis (Wb) (Normalized) 0 -20 -40 -60 -80 -100 135 180 225 270 315 360 Electrical Position (°) Figure 4. The z d waveform versus the rotor position for the current levels. 56 100 80 60 40 20 T ^ i, I h = ^ I h cos ^n.i h, (5) ^ I h sin ^n.i h, (6) ^ I h sin ^n.i h, (7) h /{ dn n=1 h /{ qn n=1 h /{ cn n=1 where { dn ^ I h, { qn ^ I h, and { cn ^ I h are Fourier series coefficients and h is the maximum number of harmonics that will be fixed according to the waveform quality. Note that z d is the coil flux of the stator only, and, therefore, the permanent-magnet flux, { d0, shall be added to obtain the total flux, z d. In this study, { d0 is considered a constant, even in magnetic saturation. This assumption is based on the case where the ratio between the magnet thickness and the air gap is important. As a result, the total flux in the d-axis can be expressed as follows: z d ^i, I h = { d0 + / { dn ^ I h cos ^n.i h. h (8) n=1 Therefore, for a given current level, starting from (6), (7), and (8), we can estimate all harmonic amplitudes by a mean square equation: 6x@ = ^H T H h-1 H T 6y ^i h@, (9)

IEEE Electrification Magazine - December 2017 - Cover2

IEEE Electrification Magazine - December 2017 - 1

IEEE Electrification Magazine - December 2017 - 2

IEEE Electrification Magazine - December 2017 - 3

IEEE Electrification Magazine - December 2017 - 4

IEEE Electrification Magazine - December 2017 - 5

IEEE Electrification Magazine - December 2017 - 6

IEEE Electrification Magazine - December 2017 - 7

IEEE Electrification Magazine - December 2017 - 8

IEEE Electrification Magazine - December 2017 - 9

IEEE Electrification Magazine - December 2017 - 10

IEEE Electrification Magazine - December 2017 - 11

IEEE Electrification Magazine - December 2017 - 12

IEEE Electrification Magazine - December 2017 - 13

IEEE Electrification Magazine - December 2017 - 14

IEEE Electrification Magazine - December 2017 - 15

IEEE Electrification Magazine - December 2017 - 16

IEEE Electrification Magazine - December 2017 - 17

IEEE Electrification Magazine - December 2017 - 18

IEEE Electrification Magazine - December 2017 - 19

IEEE Electrification Magazine - December 2017 - 20

IEEE Electrification Magazine - December 2017 - 21

IEEE Electrification Magazine - December 2017 - 22

IEEE Electrification Magazine - December 2017 - 23

IEEE Electrification Magazine - December 2017 - 24

IEEE Electrification Magazine - December 2017 - 25

IEEE Electrification Magazine - December 2017 - 26

IEEE Electrification Magazine - December 2017 - 27

IEEE Electrification Magazine - December 2017 - 28

IEEE Electrification Magazine - December 2017 - 29

IEEE Electrification Magazine - December 2017 - 30

IEEE Electrification Magazine - December 2017 - 31

IEEE Electrification Magazine - December 2017 - 32

IEEE Electrification Magazine - December 2017 - 33

IEEE Electrification Magazine - December 2017 - 34

IEEE Electrification Magazine - December 2017 - 35

IEEE Electrification Magazine - December 2017 - 36

IEEE Electrification Magazine - December 2017 - 37

IEEE Electrification Magazine - December 2017 - 38

IEEE Electrification Magazine - December 2017 - 39

IEEE Electrification Magazine - December 2017 - 40

IEEE Electrification Magazine - December 2017 - 41

IEEE Electrification Magazine - December 2017 - 42

IEEE Electrification Magazine - December 2017 - 43

IEEE Electrification Magazine - December 2017 - 44

IEEE Electrification Magazine - December 2017 - 45

IEEE Electrification Magazine - December 2017 - 46

IEEE Electrification Magazine - December 2017 - 47

IEEE Electrification Magazine - December 2017 - 48

IEEE Electrification Magazine - December 2017 - 49

IEEE Electrification Magazine - December 2017 - 50

IEEE Electrification Magazine - December 2017 - 51

IEEE Electrification Magazine - December 2017 - 52

IEEE Electrification Magazine - December 2017 - 53

IEEE Electrification Magazine - December 2017 - 54

IEEE Electrification Magazine - December 2017 - 55

IEEE Electrification Magazine - December 2017 - 56

IEEE Electrification Magazine - December 2017 - 57

IEEE Electrification Magazine - December 2017 - 58

IEEE Electrification Magazine - December 2017 - 59

IEEE Electrification Magazine - December 2017 - 60

IEEE Electrification Magazine - December 2017 - 61

IEEE Electrification Magazine - December 2017 - 62

IEEE Electrification Magazine - December 2017 - 63

IEEE Electrification Magazine - December 2017 - 64

IEEE Electrification Magazine - December 2017 - 65

IEEE Electrification Magazine - December 2017 - 66

IEEE Electrification Magazine - December 2017 - 67

IEEE Electrification Magazine - December 2017 - 68

IEEE Electrification Magazine - December 2017 - 69

IEEE Electrification Magazine - December 2017 - 70

IEEE Electrification Magazine - December 2017 - 71

IEEE Electrification Magazine - December 2017 - 72

IEEE Electrification Magazine - December 2017 - 73

IEEE Electrification Magazine - December 2017 - 74

IEEE Electrification Magazine - December 2017 - 75

IEEE Electrification Magazine - December 2017 - 76

IEEE Electrification Magazine - December 2017 - 77

IEEE Electrification Magazine - December 2017 - 78

IEEE Electrification Magazine - December 2017 - 79

IEEE Electrification Magazine - December 2017 - 80

IEEE Electrification Magazine - December 2017 - 81

IEEE Electrification Magazine - December 2017 - 82

IEEE Electrification Magazine - December 2017 - 83

IEEE Electrification Magazine - December 2017 - 84

IEEE Electrification Magazine - December 2017 - 85

IEEE Electrification Magazine - December 2017 - 86

IEEE Electrification Magazine - December 2017 - 87

IEEE Electrification Magazine - December 2017 - 88

IEEE Electrification Magazine - December 2017 - 89

IEEE Electrification Magazine - December 2017 - 90

IEEE Electrification Magazine - December 2017 - 91

IEEE Electrification Magazine - December 2017 - 92

IEEE Electrification Magazine - December 2017 - 93

IEEE Electrification Magazine - December 2017 - 94

IEEE Electrification Magazine - December 2017 - 95

IEEE Electrification Magazine - December 2017 - 96

IEEE Electrification Magazine - December 2017 - 97

IEEE Electrification Magazine - December 2017 - 98

IEEE Electrification Magazine - December 2017 - 99

IEEE Electrification Magazine - December 2017 - 100

IEEE Electrification Magazine - December 2017 - Cover3

IEEE Electrification Magazine - December 2017 - Cover4

http://www.nxtbook.com/nxtbooks/pes/electrification_december2019

http://www.nxtbook.com/nxtbooks/pes/electrification_september2019

http://www.nxtbook.com/nxtbooks/pes/electrification_june2019

http://www.nxtbook.com/nxtbooks/pes/electrification_march2019

http://www.nxtbook.com/nxtbooks/pes/electrification_december2018

http://www.nxtbook.com/nxtbooks/pes/electrification_september2018

http://www.nxtbook.com/nxtbooks/pes/electrification_june2018

http://www.nxtbook.com/nxtbooks/pes/electrification_december2017

http://www.nxtbook.com/nxtbooks/pes/electrification_september2017

http://www.nxtbook.com/nxtbooks/pes/electrification_march2018

http://www.nxtbook.com/nxtbooks/pes/electrification_june2017

http://www.nxtbook.com/nxtbooks/pes/electrification_march2017

http://www.nxtbook.com/nxtbooks/pes/electrification_june2016

http://www.nxtbook.com/nxtbooks/pes/electrification_december2016

http://www.nxtbook.com/nxtbooks/pes/electrification_september2016

http://www.nxtbook.com/nxtbooks/pes/electrification_december2015

http://www.nxtbook.com/nxtbooks/pes/electrification_march2016

http://www.nxtbook.com/nxtbooks/pes/electrification_march2015

http://www.nxtbook.com/nxtbooks/pes/electrification_june2015

http://www.nxtbook.com/nxtbooks/pes/electrification_september2015

http://www.nxtbook.com/nxtbooks/pes/electrification_march2014

http://www.nxtbook.com/nxtbooks/pes/electrification_june2014

http://www.nxtbook.com/nxtbooks/pes/electrification_september2014

http://www.nxtbook.com/nxtbooks/pes/electrification_december2014

http://www.nxtbook.com/nxtbooks/pes/electrification_december2013

http://www.nxtbook.com/nxtbooks/pes/electrification_september2013

http://www.nxtbookMEDIA.com