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 torque-
producing 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 torque-
producing 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 construc- tion 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 sources; 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 “field-
oriented 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 torque-producing 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 torque-producing
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 high- performance ac drives
Application of vector-controlled ac machines in high- performance 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 closed-loop speed control mode can be represented, as
shown in Figure 24 3 Since the machine is supplied from stator side only, flux- and torque-producing 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 “2n” are now constituent parts of the block “Drive control algorithm” of Figure 24 1 Blocks
“Rotational transformation” and “2n” 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 inductances 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
Kategori