# IEEE Electrification Magazine - December 2017 - 59

```Torque

37

20

36

10

35

0

33

-10
Magnetic
Saturation Area

32
31

-30
-40

34

∆T = 3.65 N . m

30

-20

Torque

38

(N . m)

(N . m)

40

0

50

∆θ = 24°
30
210 220 230 240 250 260 270 280 290 300 310
100 150 200 250 300 350 400
(°)
(°)
(b)
(a)
Torque with Saturation
Torque Without Saturation

Figure 11. (a) The torques for 100% of the current level. (b) A zoomed view.

Let's consider U sup the power supply voltage. In the case of
space vector modulation, we have
U sup
.
3

(20)

The norm of the d-q axis current is given by
I = i 2d + i 2q .

(21)

The flux dynamic equations in the d-q axis are given by
Z
] Vd = R p i d +
]
[
]] Vq = R p i q +
\

_
dz d
- ~ e .z q bb
dt
`.
dz q
+ ~ e .z d bb
dt
a

Id = 40.67

V=

00

I=1

∆θ = 24°
Iq = 91.35

(22)

Notice that z d is the total flux that includes permanentmagnet flux.
To compute the lookup tables, we have to define
three separate grids for the speed range [0, ~ max], power
voltage range [Vmin, Vmax], and torque range [Tmin, Tmax].
In the first step, we shall define the range of the
torque under the maximum current norm. In the second
step, we define the torque limit curve depending on the
voltage and speed. And, finally, in the third step, we
define the (i d, i q) vector for each point of the mesh (voltage, speed, and torque) when a solution exists. In the
case of out-of-limit torque, the vector (i d, i q) is computed
by extrapolation.

Step 1: The Calculation of the Minimum
and Maximum Torques under the Maximum
Current Allowed
The first step consists of computing the maximum and
minimum torques available, Tmax and Tmin, that correspond,

Figure 12. The current vector orientation.

respectively, to the maximum motor torque and the maximum generator torque, under the maximum current
norm. This can be formulated according to the following
optimization algorithm:
Z
]] Tmin = min ^T ^i, I hh
i ! 60, 2.r@
.
[
]
I = I max
\

(23)

The value Tmax is also obtained by the same problem of
maximization under the same constraints. So Tmax and Tmin
will be used to build the torque grid used in the third step.

Step 2: The Calculation of the Minimum and
Maximum Torque for a Given Voltage and Speed
The second step is to compute the torque limit for each
voltage value of the grid and each speed value of the grid.
In other words, the aim is to define a set of curves,
Tmin (~ e, I max, U sup) and Tmax (~ e, I max, U sup), according to the following optimization algorithm:
IEEE Elec trific ation Magazine / D EC EM BE R 2 0 1 7

59

```

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

IEEE Electrification Magazine - December 2017 - Cover1
IEEE Electrification Magazine - December 2017 - Cover2
IEEE Electrification Magazine - December 2017 - 1
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IEEE Electrification Magazine - December 2017 - 100
IEEE Electrification Magazine - December 2017 - Cover3
IEEE Electrification Magazine - December 2017 - Cover4
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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
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