IEEE Electrification Magazine - December 2017 - 56

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