IEEE Electrification Magazine - December 2017 - 58

Flux-Weakening Control

Ωméca

θ Correction

ic

a, b, c

ib

Clarke
iβ Transform

α, β

Park
Transform

d, q

Ωméca

Vq∗

Iq
Id

Iq

+−

Iq Field
Weakening Table

Γ∗

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

Speed Loop

Ωméca

Ω∗méca

+
−

P.I

Γ∗∗

Ce_min

Ωméca

q

Id∗
Γ∗
Ωméca I Field
d
Weakening Table
Vdc
I∗

+−

Vdc
Saturation

Vdc

Vdc
Ce_max

58

Figure 10. The principle of flux weakening using lookup tables.

Current Loop

The Principle of the Flux-Weakening Control
Under Saturation
Id

P.I

Vq∗∗

P.I

Vd∗∗

Decoupling

Vd∗

d, q

α, β

θelec

Vα∗
Park-1
Transform Vβ∗

iα

α, β

Duty 1
Duty 2
SVM
Duty 3

ia

Inverter

θméca

V1
V2
V3

Position and
Speed
Processing

MSAP

For a given torque value, the flux-weakening control
using lookup tables must deliver a vector ^i d, i q h as a
function of speed and the available power supply voltage. To use the tables, the first operation is to verify if
the torque setpoint (the output of the speed loop regulator) is in the allowed range of torque (a function of the
speed and the voltage). If the torque value is out of
range, the torque shall be set at a saturated value. The
second operation is to define the vector ^i d, i q h, which
leads to the torque value in the admissible range. This
current vector shall respect several constraints. The calculation of these lookup tables is detailed in the following paragraphs. Once lookup tables are computed
offline, they should be implemented, as illustrated in
Figure 10, which describes the flux-weakening control
strategy in this article.
In this way, we propose a method of building optimal lookup tables for the flux-weakening control under
magnetic saturation. The computation of these lookup
tables is based on the analytics expressions of z d ^i, I h,
z q ^i, I h, and T ^i, I h found in the previous section. Before
describing the procedure of the lookup tables computation, we present the principle of the flux-weakening
control by the current vector orientation for an operating point and under magnetic saturation. Then, this
principle will be generalized in the next subsections.

The curves in Figure 6 give, for each current level, the
torque waveforms according to the rotor position. Therefore, above a level of current (42% in this example), these
curves are not perfectly sinusoidal because of the magnetic saturation. Figure 11 presents the torque curve for
100% of the current level. In the case of no magnetic saturation, the torque will be propositional to the current. For
example, the dashed curve in Figure 11 gives the torque
at 100% of current without magnetic saturation. This
torque is, however, maximum (respectively minimum) at
position 170° (respectively 90°). In the case of magnetic
saturation, this torque setpoint is not reachable with conventional controls. Indeed, the optimal torque will be
reached by applying some negative current in the direct
axis and by keeping the same level of current norm, as
presented in Figure 12.
This current orientation is applied according to the same
angle (24°) between torques with and without saturation
cases [Figure 11(b)]. Therefore, despite the magnetic saturation, we gained up to 9.3% of torque in this operating point
thanks to the optimization of the current vector orientation.
To generalize this method on all operating points, let's
remember the fundamental PMSM equations. The norm
of voltage is given by
V = V 2d + V 2q .

(19)



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