IEEE Electrification Magazine - December 2017 - 57

Flux q Axis (Wb) (Normalized %)

50
0
-50

-100
0

20

100
50
0
-50

-100
0

5

22

5

0

0

27

31

60
80
Current (A)
(Normalized %)

Electrical
Position (°)

45
90
5
13
0
18

40

0

20

36

40 60
80
Current (A)
(Normalized %)

0
45
90
5
13
0
18
5
22
0
27
5
31
0
36

Flux d Axis (Wb) (Normalized %)

100

Electrical
Position (°)

Measured Flux d Axis
Measured Flux q Axis

Estimated Flux d Axis

Estimated Flux q Axis
Figure 8. The mesh grid for z q ^i, Ih.

Figure 7. The mesh grid for z d ^i, Ih.

R
Scos ^i 1 h cos ^2.i 1 h
Scos ^i 2 h cos ^2.i 2 h
H=S
...
S ...
S cos ^i lh cos ^2.i lh
T
x = 6{ d1 { d2 f { dh@T,

...
...
...
...

V
cos ^h.i 1 hW
cos ^h.i 2 hW
W,
...
W
cos ^h.i l hW
X

(10)

(11)

y ^i h = 6z d ^i 1 h z d ^i 2 h f z d ^i l h@T;

(12)

for z q ^ih matrices

-50
-100
0

0

45

90

5

13

5

0

0

18

22

5

50
Current (A)
(Normalized %)

27

y ^i h = 6z q ^i 1 h z q ^i 2 h f z q ^i l h@T;

0

0

x = 6{ q1 { q2 f { qh@T,

50

31

...
...
...
...

V
sin ^h.i 1 hW
sin ^h.i 2 hW
W,
...
W
sin ^h.i l hW
X

100

36

R
Ssin ^i 1 h sin ^2.i 1 h
Ssin ^i 2 h sin ^2.i 2 h
H=S
...
S ...
S sin ^i l h sin ^2.i lh
T

Torque (N. m) (Normalized %)

with the following matrices for z d ^i h:

Electrical Position (°)

(13)

Measured Torque
Estimated Torque

(14)
(15)

Figure 9. The mesh grid for T ^i, Ih.

and for torque matrices
x = 6{ c1 { c2 f { ch@T,

y ^i h = 6T ^i 1 h T ^i 2 h f T ^i lh@T.

p n ^ I h = / a n,j I j,
d

(16)
(17)

Notice that the matrix H for T ^ih is the same as z q ^ih.
After computing harmonics amplitudes for given current levels, the aim now is to find a numerical function to
approximate harmonic amplitudes in the current range
60, I max@. These functions form a supposed polynomial with
a degree d, depending on the current level, as

(18)

j=0

where p n ^ I h is the amplitude of the harmonic n ^n ! 61, h@h,
defined in (6), (7), and (8) with p n ^ I h = " { dn, { qn, { cn ,. The polynomial coefficients a n,j can be evaluated with a meansquare algorithm or another usual optimization algorithm.
Figures 7, 8, and 9 show the difference between the
measured and the estimated flux by using approximated
analytical functions.
IEEE Elec trific ation Magazine / D EC EM BE R 2 0 1 7

57



Table of Contents for the Digital Edition of IEEE Electrification Magazine - December 2017

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