energies

Review
A Review about Flux-Weakening Operating Limits and Control
Techniques for Synchronous Motor Drives
Nicola Bianchi *,† , Paolo Gherardo Carlet † , Luca Cinti † and Ludovico Ortombina †

Department of Industrial Engineering, University of Padova, Via Gradenigo 6A, 35131 Padova, Italy;
paologherardo.carlet@phd.unipd.it (P.G.C.); luca.cinti@phd.unipd.it (L.C.); ludovico.ortombina@unipd.it (L.O.)
* Correspondence: nicola.bianchi@unipd.it
† These authors contributed equally to this work.

Abstract: This paper deals with motor design aspects and control strategies for the flux-weakening
(FW) operation of synchronous motors. The theory of FW is described by taking into account differ-
ent control schemes. The advantages and drawbacks of each one are discussed, as well. Moreover,
some motor design considerations for achieving an effective FW operation are illustrated for per-
manent magnet (PM), wound rotor (WR) and reluctance (REL) synchronous machines, using the
per unit approach. The distinguishing characteristic of this review provides two-fold attention on
both machine design and control strategies obtained by several collaborations with industrial and
commercial companies.

Keywords: permanent magnet (PM) machines; permanent magnet (PM) motor control; interior
permanent magnet (IPM) motor; hybrid excitation (HE); per-unit system; flux-weakening (FW)
operation; magnetic analysis






Citation: Bianchi, N.; Carlet, P.G.;
Cinti, L.; Ortombina, L. A Review
1. Introduction
about Flux-Weakening Operating
Limits and Control Techniques for
In recent years, mobility is experiencing a disruptive revolution due to the world-
Synchronous Motor Drives. Energies
wide diffusion of electric vehicles (EVs). EVs represent a highly demanding application
2022, 15, 1930. https://doi.org/ for electric motor technologies, requiring high torque and power for a very wide speed
10.3390/en15051930 range. E-motor technology is populated by several configurations [1], which are shown in
Figure 1. Wound rotor synchronous machines (WRSM), fed by load commutated inverters,
Academic Editor: Lieven
are suitable for high-power applications. On the other hand, interior permanent magnet
Vandevelde
(IPM), surface permanent magnet (SPM) [2] and reluctance (REL) synchronous motors are
Received: 31 January 2022 preferable in the low and medium power range which are slightly replacing induction
Accepted: 2 March 2022 motors (IM)s. A non-standard configuration that exhibits promising performance is repre-
Published: 7 March 2022 sented by the Normal-Saliency PM (NSPM) motor [3]. Finally, an interesting compromise
Publisher’s Note: MDPI stays neutral
is represented by hybrid excited permanent magnet (HEPM) motors, which combine the
with regard to jurisdictional claims in
benefit of wound rotor machines and permanent magnet synchronous motor (PMSM) [4].
published maps and institutional affil-
Synchronous Motors
iations.

WRSM PMSM REL

Copyright: © 2022 by the authors. CR SP HEPM IPM SPM NSPM

Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/ Figure 1. Classification of synchronous motors (CR means Continuous Rotor and SP stands for
4.0/). Salient Pole).

Energies 2022, 15, 1930. https://doi.org/10.3390/en15051930 https://www.mdpi.com/journal/energies

Energies 2022, 15, 1930 2 of 18

A maximum torque per ampere (MTPA) strategy is commonly implemented for

PMSM control up to the nominal speed. This strategy allows for a constant maximum
available torque up to the nominal speed, thus maximizing the available power. In addition,
an efficient operation of the drive is assured by minimizing the joule losses. Achieving
high torque in a wide speed range is an essential feature. However, most of PMSMs are
characterized by a limited speed range since the back-electromotive force (BEMF) of such
machines increases linearly with the speed. Thus, the voltage limit of the converter is
quickly reached [5].
The feasible speed working region of e-motors can be extended by means of several
solutions. First, it is possible by acting on the design of the overall mechanical system.
Considering EV applications, the introduction of multiple gear-ratios can address the
issue. However, this solution involves significant costs, increasing weights, and it may
reduce the reliability of the system. As an alternative, the motor control strategy can be
modified, leaving the MTPA operation above the nominal speed. In this condition, the flux-
weakening (FW) mode represents the most widespread solution [6,7]. Such strategy allows
for a constant deliverable maximum power and it is achieved by decreasing the motor
stator current component aligned with the PM flux or by acting on the rotor windings
excitation. This solution has a limited impact on the overall system where the electric
motor is mounted. Depending on the PM flux linkage and on the direct axis inductance,
PMSM can have a finite or infinite theoretical maximum operating speed. In the latter
case, a constant available maximum power can not be guaranteed in the whole speed
range. Above a defined speed, a new strategy must be implemented for the motor control,
namely, the maximum torque per voltage (MTPV) [8,9]. It allows all the current and voltage
constraints of the system to be met, at the cost of a reduction in the delivered power.
An effective FW operation of PMSM drives requires particular care both during the
electric machine design stage and the control scheme desing one. Machine designers
are required to enhance the flux-weakening capability of e-motors by changing the motor
structures [10] or by using excitation rotor windings. For example, annular iron mounted on
the surface of the PM and flux barriers can be used to reduce the direct flux demagnetization
and to increase the machine operation speed range, as in [10].
A proper control strategy is needed to extract the maximum output torque given
the converter voltage constraint [11], once a PMSM with satisfactory FW capabilities is
designed. Several challenges need to be overcome in the selection of the most effective
motor control architecture [12–14]. Different FW techniques are available in the literature
depending on the PMSM controller type. Synchronous PID current controllers, DTCs, MPCs
and flux-based controls are considered in this review. Robustness against motor parameters
uncertainties, computational burden, anti-windup handling, and maximum deployment of
the motor feeding converter are some of the aspects that the control designer needs to take
into account. The most widespread architectures in the literature are presented, including
feed-forward, feedback and mixed FW control structures. For sake of completeness, some
cutting-edge solutions which couples, for instance sensorless and FW operations, are
included in this review, as well.
To cover most of the topics related to FW control of electric drives, the paper is
organized as follows:
• Section 2 defines the steady-state operation loci of PMSMs, distinguishing between
MTPA and FW operation mode.
• Section 3 resumes some considerations concerning the design of SPM, IPM, REL and
HEPM motors for given FW performances. A per unit (p.u.) analysis is proposed to
generalize the design guidelines.
• Section 4 presents FW control architectures, discussing advantages and drawbacks.
The widespread feed-back, feed-forward and hybrid architectures are presented,
together with other promising configurations.
Energies 2022, 15, 1930 3 of 18

2. Motor Model and Operating Condition

The steady-state voltage equations of a synchronous motor in the dq rotating reference
frame are:
Vd = Rs Id − ωLq Iq Vq = Rs Iq + ω (Λm + Ld Id ), (1)
where Vd , Vq are the stator voltages, Id , Iq are the stator currents, Ld , Lq are the motor
apparent inductances, Rs is the stator resistance, Λm is the PM flux linkage and ω = pωm
are the electromechanical speed, pole pairs and mechanical speed, respectively. To describe
all synchronous motors by (1), the d-axis of the rotating reference frame is aligned to PM
flux linkage. To simplify the notation and the dissertation, magnetic saturation, iron losses
and PM demagnetisation effects are neglected. However, the aforementioned effects can
be included as in [15,16]. Finally, the steady-state torque of a synchronous motor can be
computed as follows:
3
T = p[Λm Iq + ( Ld − Lq ) Id Iq ], (2)
2
where two terms can be recognized, namely, the PM and the reluctance torque components.
Operating motor conditions are studied by drawing Equations (1) and (2) in the dq
current plane as in Figure 2a,b [17,18]. Before analyzing the motor operating regions, some
relevant curves are introduced in the dq plane, in particular the current limit locus, the
voltage limit loci and the constant torque loci. The motor current limit describes a circle
centered in the origin in the dq plane, given by:
2
IN = Id2 + Iq2 , (3)

where IN is the nominal current of the motor. The current limit is represented by a red solid
line in Figure 2a,b.

Iq Iq
Tb T Tb
p

B B
I I
P Id
i Id i

p - - m max - - m
m m
b Ld Ld-Lq Ld Ld-Lq
b
(a) (b)
T
T

Tb Tb

Tp

b p b max
(c) (d)

Figure 2. Circle diagrams and torque versus speed characteristic when electric motors have
Λm > Ld IN or Λm < Ld IN . (a) Circle diagram with Λm < Ld IN . (b) Circle diagram with Λm > Ld IN .
(c) Torque vs. speed Λm < Ld IN . (d) Torque vs. speed with Λm > Ld IN .
Energies 2022, 15, 1930 4 of 18

The voltage limit is retrieved from (1), by imposing a maximum voltage magnitude
equal to the nominal value VN . The resulting equation describes elliptical trajectories in the
dq current plane, which depend on the actual motor speed as:

VN2 = ω 2 [(Λm + Ld Id )2 + ( Lq Iq )2 ]. (4)

The curve ellipticity is equal to the motor saliency ratio ξ = Lq /Ld . Moreover, the ellipses
are centered in (−Λm /Ld , 0), where the ratio Λm /Ld is equal to the magnitude of the
steady-state three-phase short-circuit current. Furthermore, the ellipse major semi-axis
length is equal to VN /(ωLd ), thus it is inversely proportional to the operating speed. As
limit cases, the voltage limit curves of SPM machines are circular, having an unitary saliency
ratio, whereas voltage ellipses of REL motors are centered in the origin, having a zero
steady-state short-circuit current. Voltage limit curves are depicted by blue solid line in
Figure 2a,b. It is remarked that (4) holds at high speeds and for medium-high power
machines. In fact, the resistive voltage drop is not negligible for low power motors and it
has to be accounted in the machine description, as in [19].
The constant torque loci shape is obtained by inspecting (2). In particular, constant
torque curves are described by hyperbola, whose asymptotes are the d axis and the vertical
straight line defined by the equation Id = −Λm /( Ld − Lq ). Since the d axis is assumed to
be aligned with the PM flux, the vertical asymptote lies in the positive Id semiplane, indeed
Ld < Lq .
Constant torque loci are the black solid line hyperbolas in Figure 2a,b. As particular
case, SPM motors are characterized by horizontal straight lines constant loci as in Figure 3a,
having an unitary saliency ratio, whereas REL machines are characterized by hyperbolic
constant torque curves centered in the origin, mounting no PMs (Figure 3b.)

Iq Iq
b
Tb T
B B I
P
Tb
max i Id Tp i Id
- m
p

Ld I b

(a) (b)

Figure 3. Circle diagrams of the SPM and REL motors. (a) SPM circle diagram. (b) REL circle diagram
(Lq > Ld ).

2.1. Constant Maximum Available Torque Region

At the standstill condition, the only constraint that limits the motor torque capabilities
is the current limit circle (3). The voltage ellipse constraint described by (4) expands,
covering the entire dq current plane. Indeed, the ellipse size is inversely proportional to
the speed. In this condition, the motor is controlled along the MTPA strategy. In particular,
given a desired torque T, the dq currents magnitude I and phase αi are computed such that
the desired torque is guaranteed and the Joule losses in the machine are minimized. For
any current amplitude, the current angle αi should be chosen equal to:
q
−Λm + Λ2m + 8( Ld − Lq )2 I 2
cos αi = . (5)
4( L d − L q ) I
Energies 2022, 15, 1930 5 of 18

Considering Figure 2a,b, the MTPA trajectory is obtained as the tangent points between
current circles and torque hyperbola. The maximum available torque is retrieved from (5) by
substituting the nominal current magnitude IN , and it represents the nominal motor torque.
The nominal torque remains the maximum available one until the voltage constraint
ellipse, which shrinks for increasing speeds, crosses the MTPA current locus at the nominal
current circle. This condition occurs in the points denoted as B in Figure 2a,b. Above such a
speed, known as nominal speed and denoted as ωb , the motor is not longer able to deliver
its nominal torque, since the voltage ellipse constraint forces the working point to lie on a
lower torque hyperbola, given the nominal current [20].

2.2. Flux-Weakening: Constant Maximum Available Volt-Ampere Region

FW operation begins above the nominal speed [21]. Since the MTPA strategy can not
be implemented, the nominal torque can not be obtained. The maximum available torque
for a given speed and a rated current is retrieved by the intersection of the current limit
circle with the voltage limit ellipse. This intersection starts moving the working points
from the base ones B (Figure 2a,b) along the limit circle towards the point ( IN , 0), keeping
a constant Volt-Ampere rating.
Two main possibilities can occur while moving along the current limit circle, depend-
ing on the position of the voltage ellipse center with respect to the nominal motor current.
If the ratio Λm /Ld is greater than the nominal current IN , the ellipse center lies outside
the current limit circle. In this condition, the movement of the motor current along the
current limit circle can proceed till the point (− IN , 0), where the motor exhibits zero torque.
From a geometrical point of view, it implies that the ellipse voltage limit is tangent to
the current limit circle. This condition is depicted in Figure 2b and it corresponds to the
smallest voltage ellipse. Moreover, in this condition the voltage limit corresponds to a
higher achievable speed:
VN
ωmax = . (6)
Λm − Ld IN
The motor is not able to operate above such speed, since there are no longer intersections
between the current limit and voltage limit locus. In other words, the electric drive has a
maximum operating speed. This behavior is often retrieved in SPM machines, characterized
by small synchronous inductances and, consequently, high Λm /Ld ratios, as shown in
Figure 3a.
On the contrary, a different behavior characterizes machines with the ellipse voltage
limit center placed inside the current limit circle. The behavior is shown in Figure 2a. In this
case, the movement of the current point along the limit circle reaches the point P in the same
figure, where the torque hyperbola is tangent to the voltage limit ellipse. In this operating
point P, the drive exhibits its maximum torque-to-voltage ratio and the corresponding
electrical speed ω p represents the maximum speed of the FW constant volt-ampere region.
Above such speed the control strategy needs to be changed to guarantee a feasible working
condition.

2.3. Flux-Weakening: Decreasing Volt-Ampere Region

The third operating region exists only for those machines whose ellipse center lies
within the current circle, as in Figure 2a. This is also the case of REL motor [18,22] whose
circle diagram is reported in Figure 3b. SPM machines achieve rarely such operating mode,
if not supported by external inductances. Above the speed ω p , torque performances are
limited by the voltage limit ellipse. In particular, it is not possible to proceed moving the
current along the current limit circle. The maximum drive current must decreased in order
to respect the voltage limit ellipse, describing the Maximum Torque per Voltage (MTPV)
trajectory as reported in Figure 2c. Since the ellipse axis is inversely proportional to the
speed, the drive can theoretically reach an infinite maximum speed, i.e., till the ellipse
collapses in a single point. Of course, mechanical effects limit the maximum achievable
speed operation.
Energies 2022, 15, 1930 6 of 18

3. Design of Electric Motors for Given FW Requirements

Synchronous motor type and inverter Volt-Ampere ratings can be selected to meet
a desired torque versus speed characteristic. Alternatively, the motor drive design can
be optimized for a given nominal and FW requirement as proposed in [23]. This allows
both the energy losses and the constraints on the power converter to be reduced. A proper
combination of machine parameters of rotor PM and excitation flux linkage can be derived,
in particular d- and q-axis inductances. The inverter current rating can be obtained, as well.
For the sake of generality, the design is carried out using normalized motor data, as
defined in Appendix A. Normalized quantities are denoted by lower case letters whereas
uppercase letters refers to actual quantities.
The flux linkage, the dq-axis inductances and the inverter Volt-Ampere ratings are
selected such that the drive exhibits the desired nominal torque Tb at the desired base
speed Ωb . Moreover, the maximum p.u. speed and p.u. torque achievable in FW operating
condition are defined, namely ω f w and t f w . They represent the maximum FW speed
and torque for which the motor is designed, respectively. The motor has to guarantee all
requirements in its operating conditions.
The design procedure works as follows:
1. Set the desired values of maximum FW speed ω f w and torque t f w ;
2. A suitable couple of rotor flux linkage λr and saliency ratio ξ must be selected;
3. In order to fulfill the specification at nominal point of base torque tb = 1, voltage
v = 1, and speed ωb = 1, only one value of direct inductance ld and current i can
assure the desired performance once λr and ξ are set;
4. Finally, the defined p.u. parameters can be reported to the absolute magnitude value
and then the machine can be designed. These values must be realized with a proper
motor design by taking into account practical limitation.
An example is reported in Figure 4, considering a desired FW speed ω f w = 4 p.u. The
requirement on FW torque t f w at speed ω f w = 4 p.u. is 0.28 p.u. If ξ = 2 is defined, λr has
to be chosen equal to 0.65, accordingly to the green line in Figure 4. Once values of λr and
ξ are selected, the values of ld and i can be grabbed by exploiting Figures 5 and 6. With the
selected values, ld = 0.4 p.u. and i = 1.3 p.u. It is worth noting that Figure 4 shows that the
FW torque t f w does not exist at a desired speed for all couple of the rotor flux linkage λr
and saliency ratio ξ. Moreover, there is only one peak for each saliency ratio ξ.

r
Figure 4. FW torque as a function of the saliency ratio ξ with a FW speed of 4 p.u. [24].
Energies 2022, 15, 1930 7 of 18

Figure 5. Normalized inductance as a function of the saliency ratio ξ [24].

Figure 6. Normalized current as a function of the saliency ratio ξ [24].

3.1. PM Motors
In PM motors, the rotor flux linkage λr is equal to λm and it can change in the range
between 0 and 1. Comparing Figures 4–6, the maximum torque occurs with λm = ld i,
or when the voltage limit ellipse center is exactly placed√on the current limit circle. The
corresponding maximum power is approximately equal to 2 p.u., i.e., the theoretical value
when ω f w = ∞. A wide FW speed range can also be obtained with a lower ξ, provided
that high inductances or additional external inductances are used. Moreover, comparing
the FW torque in Figure 4 with the current values in Figure 6, it is always preferable to
design synchronous motors with λr > ld i to minimize the losses. As far as joule losses are
concerned, PM motors should be preferred rather than REL motors. Other examples of
IPM design are reported in [25].
In case of SPM motors, the problem can be solved analytically, namely, ld and i can
be computed in closed form. This kind of motors have been studied in [26–29]. A SPM
motor is characterized by a unitary saliency ratio ξ. In addition, it is worth reminding
that the MTPA locus corresponds to a current angle αi = 90 degrees. By fixing tb = 1 p.u.,
v = 1 p.u., ωb = 1 p.u., and αi = 90 degrees, the motor inductance and the drive current
result as [24]:
1
q
ld = lq = λm 1 − λ2m , i= . (7)
λm
Then, the maximum FW speed can be evaluated as:

1
ωmax = p . (8)
λm − 1 − λ2m

All speeds ω f w can be reached, even if high values of current or inductances may be
required for the highest speed. Since high inductances are not obtained with an SPM motor
configuration, a wide FW speed range requires the use of external inductances.
Energies 2022, 15, 1930 8 of 18

3.2. Pure Reluctance Motor

In REL motors, the rotor flux linkage λr is zero. Thus, the MTPA angle is αi = 135 degrees
(with the adopted convention lq > ld ). The normalized values of the dq-axis inductances
and drive current can be expressed analytically as a function of the motor saliency ratio ξ
as follow: p
ξ −1 2( ξ 2 + 1)
ld = 2 i= (9)
ξ +1 ξ −1
√
which is always greater than 2.
As regards the FW performance, the torque is always greater than zero at any speed
ω f w [30–32]. However, both the maximum torque and power decrease with ω f w , while for
√
an IPM motor the maximum power can be kept constant, close to 2.

3.3. Motor with Rotor Excitation Windings

Power generation and traction systems use motors with excitation windings. In these
machines, the rotor flux linkage λr is given by the sum of PM λm and excitation λe flux
linkages. The control of excitation current gives an additional degree of freedom that can
improve the motor performance at high speeds. The formulation proposed in [33]:

ξ ( ω f w l d i )2 + v2
λr∗ = q (10)
ω f w ( ω f w ξ l d i )2 + v2

computes the rotor flux linkage λr∗ that maximizes the delivered motor torque at each motor
speed. In order to keep the power constant, its value increases as the motor speed decreases.
However, to guarantee the nominal motor behavior, the actual rotor flux linkage λr must
be limited to its base point value. Figure 7 shows an example of the rotor flux linkage
trend as a function of motor speed with a rotor flux linkage at base point of λr = 0.85 p.u.
The rotor flux linkage is constant and equal to its base value for motor speed smaller
than ωth . For higher speed, the excitation flux linkage λe decreases to assure a total rotor
flux linkage equal to λr∗ . The speed ωth is the threshold speed at which excitation must
decreases. It is worth noting that ωth is greater than unity indeed stator currents are already
flux-weakening the machine when the excitation flux linkage λe start decreasing. During
that FW operations, the rotor flux λhe and the corresponding stator current are regulated so
as to achieve a torque as high as possible according to that speed ω f w . For different values
of λr , ξ, ld , i N and v N , it is possible to verify that the reduction of rotor flux is convenient
only if λr > ld id as described in [33,34]. Similarly to Figure 4, Figure 8 shows the FW torque
at ω f w = 4 p.u. for different values of λr in a HEPM motor, considering the strategy of
rotor flux linkage reduction as in (10). The maximum delivered torque by the machine at
speed ω f w = 4 p.u. is represented as a function of the rotor flux linkage at base point that
changes from 0 to 1 p.u. Motor with an excited rotor allows for operating at high speed,
e.g., ω f w = 4 p.u., even if the base rotor flux linkage is high as in the aforementioned
example. This is a important difference with respect to classical motor configuration where
a trade-off between rotor flux linkage magnitude and maximum achievable speed must be
found (see Figure 4). This difference allows for reducing the base motor current and, in
turn, the motor losses.
The motor design procedure is the same already described in Section 3. The ratio
between λe and λm must be defined and it can be chosen according to the machine applica-
tion. The described strategy can be applied in HEPM and WRSM motor configurations.
It was compared to conventional IPM motors in terms of torque, speed capabilities and
efficiency in [35–39].
Energies 2022, 15, 1930 9 of 18

1
*
r r

0.8

[p.u.]
0.6
*
r = r

0.4
r

0.2

0
0 1 th 2 3 4 5
e
[p.u.]

Figure 7. Rotor flux linkages as a function of various electrical speed.

0.4
=6 =4
=2
0.3
t fw [p.u.]

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
r [p.u.]
Figure 8. FW torque as a function of excitation control for a FW speed of 4 p.u. [33].

4. Flux-Weakening Control Strategies

FW operation of synchronous motors has been investigated for all the most widespread
control architectures, including current controllers, flux-based controllers and model pre-
dictive control schemes. All these three structures are considered in this review. More
attention is paid on FW operation with synchronous PI current regulators, since it is by far
the most common solution. FW control strategies are clustered according to the presence
or absence of a feedback of the motor voltage. Actually, the feedback is obtained by using
the inverter reference voltage, since the actual motor voltage is not available in industrial
drives. According to the voltage feedback criterion, three main categories of FW strategies
are identified:
• feed-forward architectures, which do not implement any voltage feedback;
• feedback schemes, where only a voltage feedback provides the FW operation;
• hybrid methods, which couple both a voltage feedback and feed-forward action.
All these topologies have been deeply investigated in the literature and each of them
has its own benefits. In the following, each category is reviewed, considering first drives
which implement synchronous current PIs.

4.1. Feed-Forward Schemes

Feed-forward schemes are known also as model-based methods. In fact, this technique
consists of computing the FW d-axis current reference by exploiting an accurate analytical
model of the motor, e.g., the voltage balance equation as in [17]. A block scheme repre-
sentation of feed-forward structures is reported in Figure 9, including some aspects that
may be taken into consideration when the feed-forward term is computed. The required
inputs by the algorithm are the torque reference, which comes from a speed regulator, the
motor speed and all the electric motor parameters, i.e., the stator windings resistance and,
possibly, the current to flux linkages relationship [22]. All the asterisks in Figure 9 denote
reference variables. The same notation is used hereinafter.
Energies 2022, 15, 1930 10 of 18

Online parameters
Feed-forward Linear motor tracking
calculation Constant parameters
Nonlinear motor
(look-up tables)
Motor model i∗d
∗
Speed loop τ Neglect
Inverter model Feed-forward Stator resistance
calculation Account
ω
Extra features i∗q Neglect
Inverter
id iq nonlinearities Account

Figure 9. Scheme of a standard feed-forward FW control architecture and an overview of the most
common features available in the literature for the feed-forward term calculation.

The most relevant advantage of this approach is a superior behavior during fast
transients. Indeed, the dynamic performances of feedback type schemes are slow down
by the closed-loop dynamic of the voltage loop. Moreover, as further merits, feed-forward
methods are not affected by significant stability problems [40] and no tuning parameters
are necessary.
However, a pure open-loop computation of the FW current reference can be badly
affected by any parameter mismatch. Since FW operation is often required in demanding
applications, such as the automotive, the parameter sensitivity issue needs to be addressed.
Most of the works on feed-forward schemes of the last two decades aim to overcome this
problem. In particular, the temperature effect on the stator resistance and the nonlinear
magnetic characteristic have been deeply analyzed.
In [41], the iron saturation effect is addressed by an online estimation of the electric
motor inductive parameters. The paper focuses on an IPM machine, but the method can be
easily implemented for other highly saturated motor topologies, e.g., pure REL machines
or PM assisted REL motors. However, the work neglects the effect of the stator resistance.
The latter is studied in [42], where the stator resistance is included in the computation
of the current set points during the FW operation. Moreover, the computational burden of
the scheme is reduced by approximating the elliptical voltage constraint in a piece-wise
linear manner. The latter method appears indeed as the benchmark for feed-forward
solutions. Look-up-tables may be too small because of hardware limitations. In this case,
advanced computational tools are available to expand small tables to larger ones, such as
the second-order bilinear interpolation method [43]. More and more detailed model [44]
may be implemented in order to improve the current reference generation. For example,
the inverter nonlinear voltage drop are included in [9], too. However, the computation
burden increases with the complexity of the model. Thus, the main advantages of the
feed-forward architecture is its simplicity. When the model becomes too cumbersome, it is
convenient to prefer feedback-based solutions. Despite all the improvements, it is worth
reminding also that the available DC-bus of the inverter is rarely entirely exploited by
feed-forward algorithms, since a small derating needs to be introduced to deal with the
parameter sensitivity issue.

4.2. Feedback Schemes

Feedback schemes [45,46] implement a feedback on the inverter reference voltages to
reach the FW operation. This topology is also known as robust, to further highlight the
key benefit with respect to feed-forward architectures. The plant model is not explicitly
exploited by feedback algorithms. However, the electric model is required for tuning the
voltage loop regulator. In addition, an accurate model knowledge results mandatory to
linearise the loop, as will be clear in the following. The two merits of the feedback solutions
are the enhanced parametric robustness and a higher exploitation of the available inverter
DC-bus voltage. However, the voltage control loop introduces a delay in the FW response.
Thus, dynamic performances are slightly penalized. Moreover, the additional control loop
poses many challenges, e.g., tuning, linearisation and selection of a reasonable bandwidth.
Energies 2022, 15, 1930 11 of 18

Feedback topologies are subdivided in two subcategories, depending on how the loop
acts on the current references coming from the speed loop. In particular, it is possible to
distinguish between solutions acting on the angle of the MTPA current references [47,48]
and solutions acting on the d-current reference [49,50]. The control schemes are reported
in Figure 10 whereas their operating principle is depicted in Figure 11. The two different
approaches affect both control effectiveness and regulator tuning. It is worth noting that
the voltage loop linearization deeply differs between the two schemes.

anti-windup i∗d
upper cos(·) ×
u ∗ + + limit
u αi
Cu (s) π |·|
− +
∗
i∗q
u αi,M TPA sin(·) ×

Speed loop τ∗ MTPA

(a)
anti-windup

u∗ + ϵu + i∗d i∗d
Cu (s) −IN
− +
lower
u i∗d,M T P A limit
∗ i∗q
Speed loop τ τ ∗ → i∗d,M T P A 3
2
p(Λmg + (Ld − Lq )i∗d ) ÷×

(b)
Figure 10. Feedback flux-weakening control architectures. (a) Flux-weakening voltage loop with
angle control. (b) Flux-weakening voltage loop with id control.

Tb Tb
ωb T ∗ < Tb ωb T ∗ < Tb
iq iq
B B
ω > ωb ω > ωb

αi

α∗i,M T P A

id i∗d i∗d,M T P A id
(a) (b)

Figure 11. Principle of angle and current correction in feedback FW schemes. (a) Angle current vector
diagram control scheme. (b) Direct current vector diagram control scheme.

An interesting configuration was proposed in [51] where the voltage error generates
two auxiliary control variables. The former acts on the MTPA current reference angle,
increasing the FW current component. The latter is used when the current amplitude needs
to be limited if the MTPV operation is reached. The voltage loop is often designed in a
Energies 2022, 15, 1930 12 of 18

model-based fashion to accomplish the desired specifications in terms of bandwidth and

phase margin. Alternatively, the tuning can be performed using a modified relay feedback
tuning, as proposed in [52]. This method represents a promising solution for general
purpose applications, where motor parameters are partially or completely unknown and
an online auto-tuning procedure is needed.
Unfortunately, the dynamic of the current angle is strongly nonlinear. Moreover, a
linearization of the angle loop would require the computation of a non trivial linearizing
gain. From this point of view, the d-axis current strategy appears more convenient [53]. The
loop linearization requires an accurate knowledge of the electric motor model and system
parameters and the operating speed as auxiliary input of the voltage loop.
Feedback based strategies are adopted for HEPM motors [54], as well. However, the
voltage loop is used to regulate the rotor excitation current, instead of the d-axis stator
windings current. This is still an open research topic since new motor topologies combining
PM and current winding to generate the rotor flux linkage have been proposed. Different
hybridization ratio leads to an additional degree of freedom in the control that it can be
exploited. Moreover, in HEPM motors, the excitation current affects the motor voltage
equation than the linearization approach [53] is no longer valid and a different dissertation
must be studied.
Particular attention needs to be paid to avoid the wind-up phenomenon in the voltage
loop. Thus, a back-calculation anti-windup strategy is proposed in [55]. As a further
relevant contribution, the same paper proposes an adaptive velocity particle swarm opti-
mization algorithm to optimize the control parameters of the anti-windup proportional
and integral controller. Once the desired architecture is selected, a proper bandwidth of the
voltage loop needs to be chosen. In case of d-axis current compensation schemes, shown in
Figure 10b, three different approaches are proposed in [40]. In a nutshell, the bandwidth
can be selected based on specifications of torque disturbance transients, fast acceleration
transients or quick variation of the grid voltage.

4.3. Hybrid FW Schemes

Finally, mixed or hybrid FW schemes are shortly reviewed. Hybrid architectures
include both a voltage control loop and a feed-forward term. As feed-forward schemes, the
feed-forward contribute is computed by means of the motor model. As feedback structures,
the reference d-axis current or current angle is then modified accordingly to the output of
the voltage loop.
This allows us to reinforce the feed-forward action. Not surprisingly, mixed schemes
aim to guarantee the benefits of both aforementioned structures, namely, fast dynamic and
robustness against parameter variations. Examples of hybrid FW schemes can be found
in [56–58].

4.4. DC-Bus Voltage Use

Before moving towards FW operation of motors which implement flux controllers, it is
worth mentioning another key research topic, i.e., the effective usage of the inverter DC-bus
voltage [59,60]. In fact, when the motor operates at high speed, the inverter works close to
its intrinsic
√ voltage constraints. In particular, voltage magnitudes are clamped to the value
VDC / 3 if the space vector modulation (SVM) method is implemented in the converter.
However, it is possible to go beyond this limit, approaching the six-step operation [61]. Sul
et al. proposes a FW method able to achieve a quasi six-step operation [62], synthesizing
voltage vectors with an amplitude up to 2VπDC . This FW method requires a proper design of
the current regulators. Nevertheless, this technique allows for a higher torque and power
performances, together with a lower current ripple.

4.5. Direct Flux Control

FW operation has been investigated for drives which directly control the motor flux
linkages [63], instead of the d and q axis current. These architectures require a stator flux
Energies 2022, 15, 1930 13 of 18

observer, too. Among these methods, the direct flux FOC and the DTC [64] are considered.
Concerning the first topology, a FW strategy is presented in [11], which implements a simple
feed-forward strategy. The IPM motor flux is decreased linearly with the operating speed.
Since the drive is fed by a battery on an electric scooter, the flux is regulated proportionally
to the feeding battery voltage. DTC schemes behave in a more robust manner with respect
to parametric uncertainties, as described in [65]. In the aforementioned work, the DTC
is coupled with a model-based feed-forward FW strategy. A simplified control scheme
is reported in Figure 12. The resulting scheme is compared with the benchmark current
regulators coupled with feed-forward and feedback FW strategies. The DTC exhibits
promising results even in presence of parameter variation, simplicity of calculation, and
stable control.
FW control
Flux ref. Torque
Voltage ref. PWM
and flux inverter PMSM
MTPA control
Saturation
T∗ +
−
T Torque and flux
Flux estimators Currents
Speed

Figure 12. FW scheme of a DTC drive.

For sake of completeness, it is reminded that the DTC can be coupled even with
pseudo-feedback FW methods, as in [66]. The proposed technique reduces the flux linkage
reference and adjusts the torque reference when the required torque is not achieved. Even
if the voltage loop is not present, the flux linkage reference is adjusted based on a measured
error rather than the computation of a pure open-loop reference. Furthermore, the FW
strategy does not require explicitly motor model parameters. For these two reasons, the
technique is categorized as a feedback-type.

4.6. Model Predictive Control

An emerging and promising control structure in electric motor drives is represented by
the model predictive control. Few preliminary works are found in the literature concerning
the FW operation, e.g., [67], resulting an appealing topic for academic research. Model
predictive control is mostly used to replace the d- and q-axis current regulators, rather
than acting on the generation of the FW current references as in [68]. The d-axis FW
current reference is generated by a standard feed–forward algorithm, whereas the model
predictive control takes care of the current reference tracking, as shown in Figure 13. A
model predictive control is designed for the voltage control loop of a feedback strategy
in [69]. The authors believe that the complexity and non linearity of the problems have not
entirely been addressed yet.

FW control MPC controller

∆ud ud uα
i∗d selection Improved
dq
ωme < Ωb linear model z PWM
∆uq z−1 uq αβ uβ inverter PMSM
ωme > Ωb Constraints
id ia
Speed ref. dq ib
iq ic
abc
ωme d θme
dt

Figure 13. FW scheme of a MPC drive.

4.7. Sensorless Control

Nowadays, a crucial requirements for industrial drives is the sensorless operation
capability. In other words, the motor needs to be controlled without the position sensor
to increase the reliability of the system and to reduce the product costs. Interesting and
challenging issues occur when a drive operates in the FW region without position sensor.
Energies 2022, 15, 1930 14 of 18

For example, the delay introduced by position estimation algorithms could influence FW
loops. Indeed, a Luenberger observer is proposed in [70] to replace the low-pass filter in
the rotor position estimation. Thanks to the proposed observer, the motor speed estimation
delay was reduced. The delay can be further reduced by means of even more sophisticated
position estimators, such as the extended Kalman filter [71]. However, attention must be
taken to keep the computation burden at bay when coupling a voltage loop, a position
observer and, possibly, current-flux linkage look-up tables.
Instability problems of sensorless FW operation are analyzed recently in [72], consider-
ing both feedback and feed-forward strategies. In particular, it has been analytically proven
that the limited bandwidth of the position estimation can induce speed oscillations, which
increase when approaching the FW operation. The system may even become unstable. This
is mainly due to the position estimation error and it is worsened in fast transients. Thus,
from this point of view, low-inertia drives result to be particularly challenging.

5. Conclusions
The flux-weakening operation of synchronous motor is used for guaranteeing a con-
stant power range and for increasing the operation region. Machine design and control
strategies have to take into account the possibility to work in that condition.
The p.u. analysis has been reported considering the design of all the possible PMSMs,
focusing the attention on the HEPM and WRSM motors that has been rediscovered in
the recent years. These machines have been investigated in the literature. They show a
wide speed range with constant unit power factor along the FW region. Three kinds of
control algorithms have been proposed with their own advantages and disadvantages.
The presented architectures include: feed-forward architectures, which do not implement
any voltage feedback; feedback schemes, where only a voltage feedback provides the
FW operation; hybrid methods, which couple both a voltage feedback and feed-forward
action. These controls have been compared in term of robustness, computational cost and
dynamic response.
Robustness to parameter variation and model uncertainties is a flaw in the feed-
forward method as it works in open-loop manner. It assumes a perfect motor model and
parameters knowledge, thus any parameter variation or inaccuracy in the motor model
affects the controller performance. The other two methods, which are feed-back based, are
inherently robuster to such uncertainties.
The perspectives and judgments on that field are of considerable interest. An effective
exploitation of the constant power region is increasingly required in various applications, so
FW operation is mandatory. As illustrated in this review, the problem has to be addressed
from both the machine design and the control point of view. Hybrid excited and wounded
rotor machine exhibit outstanding FW performance. Their rediscovery for this application
is quite recent and it is still an open research topic. On the other hand, FW schemes still need
in-depth study. The process is highly nonlinear; therefore, its description and control are not
trivial as well as the controller tuning. The inherent feature of MPC in handling multi-input
multi-output nonlinear systems makes this control strategy an attractive solution. Finally,
the interaction of several control loops poses several control challenges and this is an open
issue that characterize FW and sensorless control schemes.

Author Contributions: Equal contribution by the Authors. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Normalized Parameters

For the sake of generality, it is convenient to normalize all motor data with respect to
the base quantities defined in the following. Normalised data are denoted by means of small
Energies 2022, 15, 1930 15 of 18

letters. This permits to easily extend the considerations to motors of any power. Moreover,
interesting comparisons can be carried out among different motor types, requiring fixed
FW performances.
Torque, speed, and voltage under full load at the maximum speed of the constant
torque region have been defined as base torque Tb , base angular frequency Ωb and base
voltage VN , respectively. The base values of the motor parameters and current are retrieved
by using the power balance:
Ω 3
Tb b = VN IN . (A1)
p 2
Then, the base current, inductance and flux linkage are computed as:

2Tb Ωb 3pVN2 VN
IN = , Lb = , Λb = Lb IN = . (A2)
3pVN 2Tb Ω2b Ωb

All the normalised motor data are resumed in Table A1.

Table A1. Normalised quantities.

P.U. QUANTITIES
Torque t = T/Tb Electrical speed ω = Ω/Ωb
Phase current i = I/IN Rotor flux linkage λrb = (Λm + Λe )/Λb
Phase voltage v = V/VN Synchronous inductance l = L/Lb

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