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FOC: Field Oriented Control
A. Introductory Considerations
Variable speed electric drives are nowadays utilized in almost
every walk of life from the most basic devices such as hand-held tools
and other home appliances to the most sophisticated ones such as
electric propulsion systems in cruise ships and high-precision
manufacturing technologiesDepending on the application the
control variable may be the motor’s torque speed or position of the
rotor shaft In the most demanding applications the requirement is
to be able to control the electric machine’s elec- tromagnetic torque in
order to be able to provide a controlled transition from one operating
speed (posi- tion) to another speed (position) This means that the
control of the drive must be able to achieve desired dynamic response
of the controlled variable in a minimum time interval This can only
be achieved if the motor’s electromagnetic torque can be practically
instantaneously stepped from the previous steady- state value to the
maximum allowed value, which is in turn governed by the allowed
maximum cur- rent Variable speed electric drives that are capable of
achieving such a performance are usually called high-performance
drives, since the control is effective not only in steady state but in
transient as well Common features of all high-performance drives
are that they require information on instantaneous rotor position
(speed), operation is with closed-loop control, and the machine is
supplied from a power electronic converter Applications that
necessitate use of a high-performance drive are numerous and
include robotics, machine tools, elevators, rolling mills, paper
mills, spindles, mine winders, electric traction, electric and hybrid
electric vehicles, and the like
A principal schematic outlay of a high-performance electric
drive is shown in Figure 24 1 and it applies equally to all types of
electric machinery Electromagnetic torque of an electric machine
can be expressed as a product of the flux-producing current and
torque-producing current, so that the control system in Figure 24 1
has two parallel paths Flux-producing current reference is shown as
a constant; however, this may or may not be the case, as discussed
later Torque-producing current is in principle the output of the
torque controller However, torque controller of Figure 24 1 is
usually not present in high-performance drives, since the torqueproducing current reference can be obtained directly from the
reference torque by means of a simple scaling (or the output of the
speed controller can be made to be directly the torque-producing
current reference) This is so since the torque and the torqueproducing current are, when a high-performance control algorithm is
applied, related through a constant The con- trol structure in Figure
24 1 is composed of cascaded controllers (typically of proportional
plus integral [PI] type) An asterisk stands for reference quantities,
while θ, ω, and Te designate further on instan- taneous values of
electrical rotor position, electrical rotor angular speed (speed is
shown in figures as n in rpm; this is not to be confused with phase
number n) and electromagnetic torque developed by the motor,
respectively The cascaded structure is based on the fundamental
equations that govern rotor rotation, which are for a machine with P
pole pairs given with (TL stands for load torque, k is the friction
coefficient, and J is the inertia of rotating masses)
High-performance drives typically involve measurement of the
rotor position (speed) and motor sup- ply currents, as indicated in
Figure 24 1 Since the machine’s torque is governed by currents rather
than voltages, measured currents are used in the block “Drive
control algorithm” to incorporate the closed- loop current control
(CC) algorithm What this means is that the power electronic
converter is current- controlled, so that applied voltages are such as to
minimize the errors in the current tracking
Until the early 1980s of the last century, the separately excited
dc motor was the only available elec- tric machine that could be used
in a high-performance drive A dc motor is by virtue of its construction ideally suited to meeting control specifications for high
performance However, due to numerous shortcomings, dc motor
drives are nowadays replaced with ac drives wherever possible To
explain the requirements on high-performance control, consider a
separately excited dc motor Stator of such a machine can be
equipped with either a winding (excitation winding) or with
permanent magnets The role of the stator is to provide excitation
flux in the machine, which is in the case of permanent mag- nets
constant, while it is controllable if there is an excitation winding For
the sake of explanation, it is assumed that the stator carries
permanent magnets, which provide constant flux, ψm, so that the
upper input into the “Drive control algorithm” block in Figure 24 1
does not exist The permanent magnet flux is stationary in space and
it acts along a magnetic axis, as schematically illustrated in Figure 24
2, where the cross section of the machine is shown Rotor of the
machine carries a winding (armature winding)
access to which is provided by means of stationary brushes
and an assembly on the rotor, called commu- tator The supply is from
a dc source (in principle, a power electronic converter of dc–dc or ac–
dc type, depending on the application), which provides dc armature
current as the input into the rotor winding The brushes are placed in
an axis orthogonal to the permanent magnet flux axis (Figure 24 2)
Since the brushes are stationary, flux and the armature terminal
current are at all times at 90° It is this orthogonal position of the
torque-producing current (armature current ia) and the permanent
magnet flux ψm that enables instantaneous torque control of the
machine by means of instantaneous change of the armature current
This follows from the electromagnetic torque equation of the
machine, which is given by (K is a constructional constant)
It also follows that since the torque-producing (armature)
current and the torque are related through a constant, armature
current reference in Figure 24 1 can be obtained by scaling the torque
reference with the constant (which is normally embedded in the
speed controller PI gains), so that the torque controller is not required
On the basis of these explanations and (24 2) it is obvious that the
machine’s torque can be stepped if armature current can be stepped
This of course requires current-controlled operation of the armature
dc supply, so that the armature voltage is varied in accordance with
the armature current requirements
It is important to remark here that, inside the rotor winding,
the current is actually ac It has a fre- quency equal to the frequency
of rotor rotation, since the commutator converts dc input into ac
output current and therefore performs, together with fixed stationary
brushes, the role of a mechanical inverter (in motoring operation; in
generation it is the other way round, so that the commutator acts as a
recti- fier) As the rotor winding is rotating in the stationary
permanent magnet flux, a rotational electromo- tive force (emf ) is
induced in the rotor winding according to the basic law of
electromagnetic induction,
The machine in Figure 24 2, with constant permanent magnet
excitation, can operate with variable speed in the base speed region
only (i e , up to the rated speed), since operation above base speed
(field weakening region) requires the means for reduction of the flux
in the machine This is so since the arma- ture voltage cannot exceed
the rated voltage of the machine, which corresponds to rated speed,
rated torque operation To operate at a speed higher than rated, one
has to keep the induced emf as for rated speed operation Since speed
goes up flux must come down, something that is not possible if
permanent magnets are used but is achievable if there is an
excitation winding In such a case “flux-producing cur- rent
reference” of Figure 24 1 has a constant rated value up to the rated
speed and is further gradually reduced to achieve operation with
speeds higher than rated (hence the name, field weakening region)
However, due to the orthogonal position of the flux and armature
axes, flux and torque control do not mutually impact on each other
as long as the flux-producing current is kept constant It is hence
said that torque and flux control are decoupled (or independent) and
this is the normal mode of operation in the base speed region Once
when field weakening region is entered, dynamic decoupled flux and
torque control is not possible any more since reduction of the flux
impacts on torque production
The preceding discussion can be summarized as follows:
high-performance operation requires that torque of a motor is
controllable in real time; instantaneous torque of a separately excited
dc motor is directly controllable by armature current as flux and
torque control are inherently decoupled; indepen- dent flux and
torque control are possible in a dc machine due to its specific
construction that involves commutator with brushes whose
position is fixed in space and perpendicular to the flux position;
instantaneous flux and torque control require use of current
controlled dc source(s); current and posi- tion (speed) sensing is
necessary in order to obtain feedback signals for real-time control
Substitution of dc drives with ac drives in high-performance
applications has become possible only relatively recently From the
control point of view, it is necessary to convert an ac machine
into its equivalent dc counterpart so that independent control of two
currents yields decoupled flux and torque control The set of control
schemes that enable achievement of this goal is usually termed “fieldoriented control (FOC)” or “vector control” methods The
principal difficulty that arises in all multiphase machines (with a
phase number n ≥ 3) is that the operating principles are based on
the rotating field (flux) in the machine (note that the machines
customarily called two-phase machines are in essence four-phase
machines, since spatial displacement of phases is 90°; in two-phase
machines phase pairs in spatial opposition are connected into one
phase) As a consequence, the flux that was stationary in a sep- arately
excited dc machine is now rotating in the cross section of the
machine at a synchronous speed, determined with the stator
winding supply frequency Thus, the stationary flux axis of Figure 24
2 now becomes an axis that rotates at synchronous speed Since
decoupled flux and torque control require that flux-producing current
is aligned with the flux axis, while the torque-producing current is
in an axis perpendicular to the flux axis, the control of a multiphase
machine has to be done using a set of orthogo- nal coordinates that
rotates at the synchronous speed (speed of rotation of the flux in the
machine) The situation is further complicated by the fact that, in a
multiphase machine, there are in principle three different fluxes (or
flux linkages, as they will be called further on), stator, air-gap, and
rotor flux linkage While in steady-state operation they all have
synchronous speed of rotation, the instantaneous speeds during
transients differ Hence a decision has to be made with regard to
which flux the control should be performed Basic outlay of the drive
remains as in Figure 24 1 However, while in the case of a dc drive
the block “drive control algorithm” in essence contains only current
controllers, in the case of a mul- tiphase ac machine this block
becomes more complicated The reason is that using design of the
drive control as for a dc machine, where there exist flux and torqueproducing dc current references, means that the control will operate
in a rotating set of coordinates (rotating reference frame) In other
words, current components used in the control (flux- and torqueproducing currents) are not currents that physically exist in the
machine Instead, these are the fictitious current components that are
related to physically existing ac phase currents through a coordinate
transformation This coordinate transforma- tion produces, from dc
current references, ac current references for the supply of the stator
winding of a multiphase machine Thus, what commutator with
brushes does in a dc machine (dc–ac conversion) has to be done in ac
machines using a mathematical transformation in real time
Fundamental principles of FOC (vector control), which
enable mathematical conversion of an ac multiphase machine into
an equivalent dc machine, were laid down in the early 1970s of the
last century for both induction and synchronous machines [1–5]
What is common for both dc and ac high- performance drives is
that the supply sources are current-controlled power electronic
converters, cur- rent feedback and position (speed) feedback are
required, and torque is controlled in real time However, stator
winding of multiphase ac machines is supplied with ac currents,
which are characterized with amplitude, frequency, and phase rather
than just with amplitude as in dc case Thus, an ac machine has to be
fed from a source of variable output voltage, variable output
frequency type Power electronic con- verters of dc–ac type
(inverters) are the most frequent source of power in highperformance ac drives
Application of vector-controlled ac machines in highperformance drives became a reality in the early 1980s and has been
enabled by developments in the areas of power electronics and
microprocessors Control systems that enable realization of
decoupled flux and torque control in ac motor drives are
relatively complex, since they involve a coordinate transformation
that has to be executed in real time Application of microprocessors
or digital signal processors is therefore mandatory
In what follows the basic principles of FOC are summarized
The discussion is restricted to the multi- phase machines with
sinusoidal magnetomotive force distribution The range of available
multiphase ac machine types is huge and includes both singly-fed and
doubly-fed (with or without slip rings) machines The coverage is here
restricted to singly-fed machines, with supply provided at the stator
side The consid- ered machine types are induction machines with a
squirrel-cage rotor winding, permanent magnet syn- chronous
machines (PMSMs) (with surface mounted and interior permanent
magnets and without rotor cage, i e , damper winding), and
synchronous reluctance (Syn-Rel) motors (without damper winding)
This basically encompasses the most important types of ac machines as
far as the servo (high performance) drives are concerned FOC of
synchronous motors with excitation and damper windings (used in
the high-power applications) and of slip ring (wound rotor) induction
machines (used as generators in wind electricity generation) is thus
not covered and the reader is referred to the literature referenced
shortly for more information Considerations here cover the general
case of a multiphase machine with three or more phases on stator (n ≥
3) since the basic field–oriented control principles are valid in the same
manner regardless of the actual number of phases It has to be noted
that the complete theory of vector control has been developed under
the assumption of an ideal variable voltage, variable frequency,
symmetrical and balanced sinusoidal stator winding multiphase
supply Hence, the fact that such a supply does not exist and a
nonideal (power electronic) supply has to be used instead is just a
nuisance, which has no impact on the control principles (this being
in huge contrast with another group of high-performance control
schemes for multiphase electric drives, direct torque control (DTC)
schemes, where the whole idea of the control is based around the
utilization of the nonideal power electronic converter as the supply
source; DTC is beyond the scope of this chapter)
Since the 1980s of the last century, FOC has been
extensively researched and has by now reached a mature stage, so
that it is widely applied in industry when high performance is
required It has also been treated in a number of textbooks [6–25] at
varying levels of complexity and detail Assuming that the machine
is operated as a speed-controlled drive, a generic schematic block
diagram of a field-oriented multiphase singly-fed machine in closedloop speed control mode can be represented, as shown in Figure 24 3
Since the machine is supplied from stator side only, flux- and torqueproducing current references refer now to stator current components
and are designated with indices d and q Here d applies to the flux axis
and q to the axis perpendicular to the d-axis, while index s stands for
stator This scheme is valid for both synchronous and induction
machines and the type of the machine impacts on the setting of the
flux- producing current reference and on the structure of the “vector
controller” block It is assumed in Figure 24 3 that CC algorithm is
applied to the machine’s stator phase currents (so-called current
control in the stationary reference frame; phases are labeled with
numerical indices 1 to n) As indicated in Figure 24 3, blocks “CC
algorithm,” “vector controller,” “Rotational transformation” and
“2/n” are now constituent parts of the block “Drive control
algorithm” of Figure 24 1 Blocks “Rotational transformation” and
“2/n” take up the role of the commutator with brushes in dc machines,
by doing the dc–ac conversion (inversion) of control signals (flux- and
torque-producing stator current references) Vector control schemes for
synchronous machines are, in principle, simpler than the equivalent
ones for an induction machine This is so since the frequency of the
stator-winding supply uniquely determines the speed of rotation of a
synchronous machine If there is excitation, it is provided by permanent
magnets (or dc excitation current in the rotor winding) Rotor
carries with it the excitation flux as it rotates and the instantaneous
spatial position of the rotor flux is always fixed to the rotor Hence, if
rotor position is measured, position of the excitation flux is known
Such a situation leads to relatively simple vector control algorithms for
PMSMs, which are therefore considered first The situation is
somewhat more involved in Syn-Rel machines Rotor is of salient
pole structure but without either magnets or excitation winding, so
that excitation flux stems from the ac supply of the multiphase stator
winding By far the most complex situation results in induction
machines where not only that the excitation flux stems from stator
winding supply, but the rotor rotates asynchronously with the rotating
field This means that, even if the rotor posi- tion is measured, position
of the rotating field in the machine remains unknown Vector control
of induc- tion machines is thus the most complicated case and is
considered last The starting point for derivation of an FOC scheme
is, regardless of the type of the multiphase machine, a
mathematical model obtained using transformations of the general
theory of electrical machines For all synchronous machine types,
such a model is always developed in the common refer- ence frame
firmly fixed to the rotor, while for induction machines the speed of
the common reference frame is arbitrarily selectable All the
standard assumptions of the general theory apply: those that are
the most relevant further on are the assumption of sinusoidal field
(flux) spatial distribution and constancy of all the parameters of the
machine, including magnetizing inductance(s) where applicable
(meaning that the nonlinearity of the ferromagnetic material is
neglected) As noted already, the FOC schemes are developed
assuming ideal sinusoidal supply of the machine If the control
scheme is of the form illustrated in Figure 24 3, where CC is
performed using stator phase cur- rents, then the current-controlled
voltage source (say, an inverter) is treated as an ideal current source
and the machine is said to be current fed In simple words, it is
assumed that the multiphase power supply can deliver any required
stator voltage, such that the actual stator currents perfectly track the
reference currents of Figure 24 3 This greatly simplifies the overall
vector control schemes, since dynamics of the stator (stator voltage
equations) can be omitted from consideration Note that for an n-phase
machine with a single neutral point, the control scheme of Figure 24 3
implies existence of (n−1) current controllers These are typically of
hysteresis or ramp-comparison type and are the same regardless of the
ac machine type CC of the supply is not considered here, nor are the
PWM control schemes that are relevant when CC is not in the stationary
ref- erence frame It is therefore assumed further on that whatever the
machine type and the actual FOC scheme used, the source is capable of
delivering ideal sinusoidal stator currents (or voltages, as discussed
shortly)
B. Field-Oriented Control of Multiphase Permanent Magnet
Synchronous Machines
Consider a multiphase star-connected PMSM, with spatial
shift between any two consecutive phases of 2π/n, and let the phase
number n be an odd number without any loss of generality The
neutral point of the stator winding is isolated Permanent magnets
are on the rotor and they can be surface mounted (surface-mounted
permanent magnet synchronous machine [SPMSM]) or embedded in
the rotor (inte- rior permanent magnet synchronous machine
[IPMSM]) In the former case the air-gap of the machine can be
considered as uniform, while in the latter case the air-gap length is
variable, since permanent magnets have a permeability that is
practically the same as for the air Thus SPMSMs are characterized
with a rather large air gap (which will make operation in the field
weakening region difficult, as dis- cussed later), while the air gap of
the IPMSMs is small, but the magnetic reluctance is variable, due to
the saliency effect produced by the embedded magnets Rotor of the
machine does not carry any windings, regardless of the way in
which the magnets are placed Mathematical model of an IPMSM
can be given in the common reference frame firmly attached to the
rotor with the following equations: where index l stands for leakage
inductance, v, i, and ψ denote voltage, current, and flux linkage,
respec- tively, d and q stand for the components along permanent
magnet flux axis (d) and the axis perpendicular to it (q), and s denotes
stator Inductances Ld and Lq are stator winding self-inductances along
d- and q-axis Voltage and flux linkage equations (24 3) through (24
6) represent an n-phase machine in terms of sets of new n
variables, obtained after transforming the original machine model
in phase-variable
domain by means of a power invariant
transformation matrix that relates original phase variables and new
variables through where f stands for voltage, current, or flux
linkage and [D] and [C] are the rotational transformation matrix
and decoupling transformation matrix (block “2/n” in Figure 24 3)
for stator variables, respec- tively For an n-phase machine with an
odd number of phases, these matrices are Due to the selected powerinvariant form of the transformation matrices, the inverse
transformations are governed with [T]−1 = [T]t, [D]−1 = [D]t, [C]−1
= [C]t Angle of transformation θs in (24 9) is identically equal to the
rotor electrical position, so that As the d-axis of the common
reference frame then coincides with the instantaneous position of
the permanent magnet flux, this means that the given model is
already expressed in the common reference frame firmly attached to
the permanent magnet flux The pairs of d–q equations (24 3) and (24
5) constitute the flux/torque-producing part of the model, as is
evident from torque equation (24 7) Since in a star-connected
winding, with isolated neutral, zero-sequence current cannot flow,
the last equation of (24 4) and (24 6) can be omitted The model
then contains, in addition to the d–q equations, (n − 3)/2 pairs of x–
y component equations in (24 4) and (24 6), which do not
contribute to the torque production and are therefore not
transformed with rotational transformation (24 9) (i e , their form
is the one obtained after application of decoupling transformation
(24 10) only) It has to be noted however, that the reference value
of zero for all of these components (which will exist in the model
for n ≥ 5) is implicitly included in the control scheme of Figure 24
3, since reference phase currents are built from d–q current
references only Equations 24 4 and 24 6 are of the same form for
all the multiphase ac machines considered here (all types of
synchronous and induction machines)
For a SPMSM machine, the set of equations (24 3), (24 5),
and (24 7) further simplifies since the air-gap is regarded as
uniform and hence L s = Ld = Lq Thus (24 3) and (24 5) reduce to
while the torque equation takes the form (24 13) By comparing (24
13) with (24 2), it is obvious that the form of the torque equation is
identical as for a separately excited dc motor The only but important
difference is that the role of the armature current is now taken by the
q-axis stator current component Assuming that the machine is
current-fed (i e , CC is executed in the stationary reference frame),
stator current dynamics of (24 12) are taken care of by the fast CC
loops and the global control scheme of Figure 24 3 becomes as in
Figure 24 4 Since the machine has permanent magnets that provide
excitation flux, there is no need to provide flux from the stator
side and the stator current reference along d-axis is set to zero
According to (24 11), the measured rotor electrical position is the
transformation angle of (24 9)
The control scheme of Figure 24 4 is a direct analog of the
corresponding control scheme of perma- nent magnet excited dc
motors, where the role of the commutator with brushes is now
replaced with the mathematical transformation [T]−1 A few
remarks are due Figure 24 4 includes a limiter after the speed
controller This block is always present in high-performance drives
(although it was not included in Figures 24 1 and 24 3, for
simplicity) and limiting ensures that the maximum allowed stator
current (normally governed by the power electronic converter) is not
exceeded Next, as already noted, a con- stant that relates torque and
stator q-axis current reference according to (24 13) and which is
shown in Figure 24 4 will normally be incorporated into speed
controller gains, so that the limited output of the speed controller
will actually directly be the stator q-axis current reference
The control scheme of Figure 24 4 satisfies for control in the
base speed region If it is required to oper- ate the machine at speeds
higher than rated, it is necessary to weaken the flux so that the voltage
applied to the machine does not exceed the rated value However,
permanent magnet flux cannot be changed and the only way to
achieve operation at speeds higher than rated is to keep the term
ω(Lsids + ψm) of (24 12) is shown in an arbitrary position, as though
it has positive both d- and q-axis components As noted, in the base
speed region stator d-axis current component is zero, meaning that
the complete stator cur- rent space vector of (24 15) is aligned with
the q-axis Stator current is thus at 90° (δ = 90°) with respect to the
flux axis in motoring, while the angle is −90° (δ = −90°) during
braking In the field weakening d-axis current is negative to provide
an artificial effect of the reduction in the flux linkage of the stator
winding, so that δ > 90° in motoring If the machine operates in field
weakening region, simple q-axis current limiting of Figure 24 4 is
not sufficient any more, since the total stator current of (24 15) must
not exceed the prescribed limit, while d-axis current is now not zero
any more Hence, the q-axis current must have a variable limit,
governed by the maximum allowed stator current ismax and the
value of the d-axis current command of (24 14) A more detailed
discussion is available in [19]
In PMSMs, since there is no rotor winding, flux linkage in the
air-gap and rotor is taken as being the same and this is the flux
linkage with which the reference frame has been aligned for FOC
purposes in Figure 24 4 Schematic representation of Figure 24 5 is
the same regardless of the number of stator phases as long as the
CC is implemented, as shown in Figure 24 4 The only thing that
changes is the number of stator winding phases and their spatial
shift An illustration of a three-phase SPMSM performance, obtained
from an experimental rig, is shown next PI speed control algorithm
is implemented in a PC and operation in the base speed region is studied Stator d-axis current reference is thus set to zero at all times, so
that the drive operates in the base speed region only (rated speed of
the motor is 3000 rpm) The output of the speed controller, stator qaxis current command, is after D/A conversion supplied to an
application-specific integrated circuit that per- forms the coordinate
transformation [T]−1 of Figure 24 4 Outputs of the coordinate
transformation chip, stator phase current references, are taken to the
hysteresis current controllers that are used to control a 10 kHz
switching frequency IGBT voltage source inverter Stator currents are
measured using Hall-effect probes Position is measured using a
resolver, whose output is supplied to the resolver to digital converter
(another integrated circuit) One of the outputs of the R/D converter is
the speed signal (in analog form) that is taken to the PC (after A/D
conversion) as the speed feedback signal for the speed control loop
Speed reference is applied in a stepwise manner Speed PI
controller is designed to give an aperiodic speed response to
application of the rated speed reference (3000 rpm) under no-load
conditions, using the inertia of the SPMSM alone Figure 24 6
presents recorded speed responses to step speed references equal to
3000 and 2000 rpm Speed command is always applied at 0 25 s As can
be seen from Figure 24 6, speed response is extremely fast and the set
speed is reached in around 0 25–0 3 s without any overshoot SPMSM
is next mechanically coupled to a permanent magnet dc generator (load),
whose armature termi- nals are left open An effective increase in inertia
is therefore achieved, of the order of 3 to 1 As the dc motor rated speed
is 2000 rpm, testing is performed with this speed reference, Figure 24
7 Operation in the cur- rent limit now takes place for a prolonged
period of time, as can be seen in the accompanying q-axis current
reference and phase a current reference traces included in Figure 24 7
for the 2000 rpm reference speed Due to the increased inertia, duration
of the acceleration transient is now considerably longer, as is obvious
from the general equation of rotor motion (24 1a) In final steady state,
stator q-axis current reference is of con- stant nonzero value, since the
motor must develop some torque (consume some real power) to
overcome the mechanical losses according to (24 1a), as well as the
core losses in the ferromagnetic material of the stator
If a machine’s electromagnetic torque can be
instantaneously stepped from a constant value to the maximum
allowed value, then the speed response will be practically linear, as
follows from (24 1a) Stepping of torque requires stepping of the qaxis current in the machine Due to the very small time constant of
the stator winding (very small inductance) in a SPMSM, stator q-
axis current component changes extremely quickly (although not
instantaneously) and, as a consequence, speed response to step
change of the speed reference is practically linear during operation in
the torque (stator q-axis current) limit This is evident in Figures 24
6 and 24 7 An important property of any high-performance drive is
its load rejection behavior (i e , response to step loading/unloading)
For this purpose, during operation of the SPMSM with constant
speed refer- ence of 1500 rpm the armature terminals of the dc
machine, used as the load, are suddenly connected to a resistance
in the armature circuit, thus creating an effect of step load torque
application
Speed response, recorded during the sudden load
application at 1500 rpm speed reference, is shown in Figure 24 8
Since load torque application is a disturbance, the speed inevitably
drops during the transient How much the speed will dip from the
reference value depends on the design parameters of the speed
controller and on the maximum allowed stator current value, since
this is directly proportional to the maximum electromagnetic torque
value Control scheme of Figure 24 4, which in turns corresponds to the
one of Figure 24 1, assumes that the CC is in the stationary reference
frame, exercised upon machine’s phase currents This was the preferred
solu- tion in the 1980s and early 1990s of the last century, which was
based on utilization of digital electronics for the control part, up to the
creation of stator phase current references The CC algorithm for power
electronic converter (PEC) control was typically implemented using
analog electronics Due to the rapid developments in the speed of
modern microprocessors and DSPs and reduction in their cost, a
completely digital solution
C. Field-Oriented Control
reluctance Machines
of
Multiphase Synchronous
Syn-Rel machines for high-performance variable speed drives
have a salient pole rotor structure without any excitation and without
the cage winding The model of such a machine is obtainable
directly from (24 3) through (24 7) by setting the permanent magnet
flux to zero If there are more than three phases, then stator
equations (24 4) and (24 6) also exist in the model but remain the
same and are hence not repeated Thus, from (24 3), (24 5), and (24
7), one has the model of the Syn-Rel machine, which is again given in
the reference frame firmly attached to the rotor d-axis (axis of the
minimum magnetic reluc- tance or maximum inductance): It follows
from (24 23) that the torque developed by the machine is entirely
dependent on the difference of the inductances along d- and q-axis
Hence constructional maximization of this difference, by mak- ing
Ld/Lq ratio as high as possible, is absolutely necessary in order to
make the Syn-Rel a viable candidate for real-world applications For
this purpose, it has been shown that, by using an axially laminated
rotor rather than a radially laminated rotor structure, this ratio
can be significantly increased From FOC point of view, it is
however irrelevant what the actual rotor construction is (for more
details see [13])
As the machine’s model is again given in the reference frame
firmly attached to the rotor and the real axis of the reference frame
again coincides with the rotor magnetic d-axis, transformation
expressions that relate the actual phase variables with the stator d–q
variables (24 9) through (24 11) are the same as for PMSMs Rotor
position, being measured once more, is the angle required in the
transformation matrix (24 9) Thus one concludes that FOC
schemes for a Syn-Rel will inevitably be very similar to those of an
IPMSM
Since in a Syn-Rel there is no excitation on rotor, excitation
flux must be provided from the stator side and this is the principal
difference, when compared to the PMSM drives Here again a question
arises as to how to subdivide the available stator current into
corresponding d–q axis current references The same idea of MTPA
control is used as with IPMSMs Using (24 19), electromagnetic torque
(24 23) can be written as
By differentiating (24 24) with respect to angle δ, one gets this
time a straightforward solution δ = 45° as the MTPA condition This
means that the MTPA results if at all times stator d-axis and q-axis
cur- rent references are kept equal FOC scheme of Figure 24 4
therefore only changes with respect to the stator d-axis current
reference setting and becomes as illustrated in Figure 24 10 The qaxis current limit is now set as ± is max
2 , since the MTPA
algorithm sets the d- and q-axis current references to the same
values
The same modifications are required in Figure 24 9, where
additionally now the permanent magnet flux needs to be set to zero
in the decoupling voltage calculation (24 18) Otherwise the FOC
scheme is identical as in Figure 24 9 and is therefore not repeated It
should be noted that the simple MTPA solution, obtained above, is
only valid as long as the satura- tion of the machine’s ferromagnetic
material is ignored In reality, however, control is greatly improved
(and also made more complicated) by using an appropriate modified
Syn-Rel model, which accounts for the nonlinear magnetizing
characteristics of the machine in the two axes As an illustration,
some responses collected from a five-phase Syn-Rel experimental
rig are given in what follows To enable sufficient fluxing of the
machine at low load torque values, the MTPA is modified and is
implemented according to Figure 24 11, with a constant d-axis
reference in the initial part The upper limit on the d-axis current
reference is implemented in order to avoid heavy saturation of the
magnetic circuit Phase currents are measured using LEM sensors
and a DSP performs closed- loop inverter phase CC in the stationary
reference frame, using digital form of the ramp-comparison
method Inverter switching frequency is 10 kHz The five-phase
Syn-Rel is 4-pole, 60 Hz with 40 slots on stator It was obtained
from a 7 5 HP, 460 V three-phase induction machine by designing
new stator laminations, a five-phase stator winding, and by cutting
out the original rotor (unskewed, with 28 slots), giving a ratio of the
magnetizing d-axis to q-axis inductances of approximately 2 85
The machine is equipped with a resolver and control operates in the
speed-sensored mode at all times Response of the drive during
reversing transient with step speed reference change from 800
to −800 rpm under no-load conditions is illustrated in Figure 24 12,
where the traces of measured speed, stator q-axis current reference
(which in turn determines the stator current d-axis reference,
according to Figure 24 11), and reference and measured phase
current are shown It can be seen that the quality of Time (s)
the transient speed response is practically the same as with a
SPMSM (Figure 24 6 and 24 7), since the same linearity of the speed
change profile is observable again In final steady-state operation at
−800 rpm the machine operates with q-axis current reference of more
than 1 A rms, although there is no load This is again the
consequence of the mechanical and iron core losses that exist in
the machine but are not accounted for in the vector control scheme
(mechanical loss appears, according to (24 1a), as a certain nonzero
load torque) Measured and reference phase current are in an
excellent agreement, indicating that the CC of the inverter operates
very well
D. Field-Oriented Control of Multiphase Induction Machines
Similar to synchronous machines, FOC schemes for
induction machines are also developed using math- ematical models
obtained by means of general theory of ac machines An n-phase
squirrel cage induc- tion motor can be described in a common
reference frame that rotates at an arbitrary speed of rotation ωa with
the flux–torque-producing part of the model
This is at the same time the complete model of a three-phase
squirrel cage induction machine If stator has more than three
phases, the model also includes the non-flux/torque-producing
equations (24 4) and (24 6), which are of the same form for all nphase machines with sinusoidal magnetomotive force distribution
As the rotor is short-circuited, no x-y voltages of nonzero value
can appear in the rotor (since there is not any coupling between
stator and rotor x-y equations, [26]), so that x-y (as well as zerosequence) equations of the rotor are always redundant and can be
omitted Index l again stands for leakage inductances, indices s and
r denote stator and rotor, and Lm is the magnetizing inductance
Relationship between phase variables and variables in the
common reference frame is once more governed with (24 9) and (24
10) for stator quantities What is however very different is that the
setting of the stator transformation angle according to (24 11) would
be of little use, since rotor speed is different from the synchronous
speed In simple terms, rotor rotates asynchronously with the rotating
field, meaning that rotor position does not coincide with the position
of a rotating flux in the machine The other difference, compared to a
PMSM, is that the rotor does not carry any means for producing the
excitation flux Hence the flux in the machine has to be produced
from the stator supply side, this being similar to a Syn-Rel Torque
equation can be given in different ways, including the two that are the
most relevant for FOC, (24 27), in terms of stator flux and rotor flux
linkage d–q axis components It is obvious from (24 27) that the
torque equation of an induction machine will become identical in form
to a dc machine’s torque equa- tion (24 2) if q-component of either
stator flux or rotor flux is forced to be zero Thus, to convert an induction machine into its dc equivalent, it is necessary to select a reference
frame in which the q-component of either the stator or rotor flux
linkage will be kept at zero value (the third possibility, of very low
practical value, is to choose air-gap [magnetizing] flux instead of
stator or rotor flux, and keep its q-component at zero) Thus, FOC
scheme for an induction machine can be developed by aligning the
reference frame with the d-axis component of the chosen flux linkage
While selection of the stator flux linkage for this purpose does have
certain applications, it results in a more complicated FOC scheme and
is therefore not consid- ered here By far the most frequent selection,
widely utilized in industrial drives, is the FOC scheme that aligns the
d-axis of the common reference frame with the rotor flux linkage As
with synchronous motor drives, CC of the power supply can be
implemented using CC in station- ary or in rotating reference frame
Since with CC in the stationary reference frame one may assume that
the supply is an ideal current source, so that again under no-load
conditions are shown Comparison of Figures 24 16 and 24 17 shows
that the same qual- ity of dynamic response is achievable regardless
of the number of phases on the stator of the machine Load rejection
properties of a three-phase 0 75 kW, 380 V, 4-pole, 50 Hz induction
motor drive with IRFOC are illustrated in Figure 24 18, where at
constant speed reference of 600 rpm rated load torque is at first
applied and then removed The response of the stator q-axis current
reference and rotor speed are shown Once more, speed variation
during sudden loading/unloading is inevitable, as already discussed
in conjunction with Figure 24 8 IRFOC scheme discussed so far
suffices for operation in the base speed region, where rotor flux (stator d-axis current) reference is kept constant If the drive is to operate
above base speed, it is necessary to weaken the field Since flux is
produced from stator side, this now comes to a simple reduction of
the stator d-axis current reference for speeds higher than rated The
necessary reduction of the rotor flux reference is, in the simplest
case, determined in very much the same way as for a PMSM Since
supply voltage of the machine must not exceed the rated value, then
at any speed higher then rated product of rotor flux and speed should
stay the same as at rated speed Hence Since change of rotor speed
takes place at a much slower rate than the change of rotor flux (i e ,
mechani- cal time constant is considerably larger than the
electromagnetic time constant), industrial drives nor- mally base
stator current d-axis setting in the field weakening region on the
steady-state rotor flux relationship, id*s = ψ r* /Lm However, since
modern induction machines are designed to operate around the knee
of the magnetizing characteristic of the machine (i e , in saturated
region), while during opera- tion in the field weakening region flux
reduces and operating point moves toward the linear part of the
magnetizing characteristic, it is necessary to account in the design of
the IRFOC aimed at wide-speed operation for the nonlinearity of the
magnetizing curve (i e , variation of the parameter Lm) One rather
simple and widely used solution is illustrated in Figure 24 19,
where only creation of stator d–q axis current references and the
reference slip speed is shown The rest of the control scheme is the
same as in Figure 24 15 Here again, one can use either stator
current d–q reference currents or d–q axis current components
calculated from the measured phase currents The principal RFOC
scheme, assuming that rotor flux position is again determined
according to the indirect field orientation principle, is shown in
Figure 24 23 (current limiting block is not shown for simplicity)
Operation in the field weakening region can again be realized
by using the stator d-axis current and slip speed reference setting as
in Figure 24 19 Since the rotor flux reference will change slowly, the
rate of change of rotor flux in (24 46) is normally neglected in the
decoupling voltage calculation, so that ed calculation remains as in
(24 47) However, since rotor flux reference reduces with the increase
in speed, eq calculation has to account for the rotor flux (stator d-axis
current) variation As noted in the section on RFOC of PMSMs,
vector control with only two current controllers, as in Figure 24 23,
suffices for three-phase machines While, in theory, this should also be
perfectly sufficient for machines with more than three phases, in
practice various nonideal characteristics of the PEC sup- ply (for
example, inverter dead time) and the machine (any asymmetries in
the stator winding) lead to the situation where the performance with
only two current controllers is not satisfactory [26] To illus- trate
this statement, an experimental result is shown in Figure 24 24 for a
five-phase induction machine (which has already been described in
conjunction with Figure 24 16) Control scheme of Figure 24 23