# IEEE Electrification Magazine - December 2017 - 60

```Z for V = Vmin to V = Vmax
]
] for ~ e = 0 to ~ e = ~ max
]
Tmin ^i d, i q h = min ^T ^i, Ihh
]
]
^
R p .i q + z d ^i, Ih .~ eh2 + ^R p .i d - z q ^i, Ih .~ eh2 # V 2
]
]
i ! 60, 2.r@
[
I ! 60, I max@
]
]
i d = I cos ^ih
]
]
i q = I sin ^ih
]
end
]
]
\ end.

30

Torque (N . m)

20
220 V

10

350 V

500 V

0
-10

220 V

-20

350 V 500 V

-30
0

200

400

600 800 1,000 1,200 1,400

Torque Maximum (Motor)
Torque Minimum (Generator)

Current (A) (Normalized %)

Figure 13. The torque versus the speed versus Usup.

After the second step, we define a setpoint torque grid
Tcons between the maximum and minimum values. At
each point ^~ e, V, Tconsh, we calculate the vector (i d, i q) minimizing the module of the current I according to the optimization algorithm:

Torque Minimum
(Generator)

Torque Maximum
-100 (Motor)
0
20
40
60
80
100 100
(Normalized %)

50

0

-50

-100

Torque (N . m)
(Normalized %)

Id = f (Speed, Torque)

Current (A) (Normalized %)

Figure 14. The Id versus the torque and the speed.

Torque Maximum
(Motor)

100
50
0
-50
-100
20

0
-20
.
Torque (N m)
(Normalized %)

Torque Minimum
(Generator)
0
500
1,000
(Normalized %)

Figure 15. The Iq versus the torque and the speed.

60

I E E E E l e c t r i f i cati o n M a gaz ine / DECEMBER 2017

The value Tmax ^i d, i q h is also obtained by the same problem
of optimization under the same constraints. In this way,
we obtain the curves of the torque versus the speed for
several power supply levels, as seen in Figure 13.

Step 3: The Lookup Table Construction Giving
Optimal Currents (id, iq) for a Given Voltage,
Speed, and Torque

0
-50

(24)

Z for V = Vmin to V = Vmax
]
] for ~ e = 0 to ~ e = ~ max
]
for Tcons = Tmin to Tcons = Tmax
]
]
^i d, i q h = min _ I = i d2 + i q2 i
]
]
T ^I, ih = Tcons
]
^
R p .i q + z d ^i, I h .~ e h2 + ^R p .i d - z q ^i, I h .~ eh2 # V 2
]]
i ! 60, 2.r@
[
]
I ! 60, I max@
]
]
i d = I cos ^ih
]
i q = I sin ^ih
]
]
end
]
] end
]
\ end.

(25)

The result is a set of surfaces of i d (Tcons, ~ e, U sup) and i q (Tcons,
~ e, U sup) , as illustrated in Figures 14 and 15, respectively.
In Figure 14, the I d current area corresponding to
the upper limit of the torque (red line) is computed by
extrapolation; it hasn't any physical signification, but
it's necessary for the completeness of the lookup table
dimension. The same is the case for the I d current area
corresponding to the lower limit of torque (blue line).
Notice that, because of magnetic saturation, as shown
in Figure 15, the curve i q (Tcons, ~ e, U sup) is not a plan. In the
same way as I d current, the upper and lower areas of the
I q current corresponding, respectively, to the maximum
and minimum torque (the red and blue lines, respectively,
in Figure 15), are obtained by extrapolation.

```

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