FOC: Field Oriented Control | Karya Tulis Ilmiah
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FOC: Field Oriented Control
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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 control structure in Figure 24 1 is composed of cascaded controllers (typically of proportional plus integral [PI] type) An asterisk
stands for reference quantities, while ?, ?, 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, ?m, 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
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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 ?m 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 ?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-
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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 ?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
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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)
Field-Oriented Control of Multiphase Permanent Magnet Synchronous Machines
Consider a multiphase star-connected PMSM, with spatial shift between any two consecutive phases of 2?/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 ? 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 power-invariant 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 ?s 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
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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 Ls = 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 ?(Lsids + ?m) 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° (? = 90°) with respect to the flux axis in motoring, while the angle is
?90° (? = ?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 ? > 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 stud- ied 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 q-axis 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
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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
q-axis 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
Field-Oriented Control of Multiphase Synchronous reluctance Machines
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 dand 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
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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 ?, one gets this time a straightforward solution ? = 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
q-axis 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
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 ?a 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
n-phase 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 zero-sequence) 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,
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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 induc- tion 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 (sta- tor 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 = ? 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
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FOC: Field Oriented Control
LINK DOWNLOAD [266.62 KB]
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 control structure in Figure 24 1 is composed of cascaded controllers (typically of proportional plus integral [PI] type) An asterisk
stands for reference quantities, while ?, ?, 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, ?m, 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
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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 ?m 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 ?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-
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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 ?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
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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)
Field-Oriented Control of Multiphase Permanent Magnet Synchronous Machines
Consider a multiphase star-connected PMSM, with spatial shift between any two consecutive phases of 2?/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 ? 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 power-invariant 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 ?s 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
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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 Ls = 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 ?(Lsids + ?m) 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° (? = 90°) with respect to the flux axis in motoring, while the angle is
?90° (? = ?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 ? > 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 stud- ied 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 q-axis 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
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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
q-axis 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
Field-Oriented Control of Multiphase Synchronous reluctance Machines
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 dand 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
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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 ?, one gets this time a straightforward solution ? = 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
q-axis 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
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 ?a 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
n-phase 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 zero-sequence) 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,
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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 induc- tion 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 (sta- tor 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 = ? 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
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