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 electromagnetic torque in order to be able to provide a controlled transition from one operating speed (position) 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 current Variable speed electric drives that are capable of achieving such a performance are usually called high-performance drives, since the control is effective not only in steady state but in transient as well Common features of all high-performance drives are that they require information on instantaneous rotor position (speed), operation is with closed-loop control, and the machine is supplied from a power electronic converter Applications that necessitate use of a high-performance drive are numerous and include robotics, machine tools, elevators, rolling mills, paper mills, spindles, mine winders, electric traction, electric and hybrid electric vehicles, and the like

A principal schematic outlay of a high-performance electric drive is shown in Figure 24 1 and it applies equally to all types of electric machinery Electromagnetic torque of an electric machine can be expressed as a product of the flux-producing current and torque-producing current, so that the control system in Figure 24 1 has two parallel paths Flux-producing current reference is shown as a constant; however, this may or may not be the case, as discussed later Torque-producing current is in principle the output of the torque controller However, torque controller of Figure 24 1 is usually not present in high-performance drives, since the torque-producing current reference can be obtained directly from the reference torque by means of a simple scaling (or the output of the speed controller can be made to be directly the torque-producing current reference) This is so since the torque and the torque-producing current are, when a high-performance control algorithm is applied, related through a constant The con-trol structure in Figure 24 1 is composed of cascaded concon-trollers (typically of proportional plus integral [PI] type) An asterisk stands for reference quantities, while ?, ?, and Te designate further on instantaneous 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 supply 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 electric machine that could be used in a high-performance drive A dc motor is by virtue of its construction ideally suited to meeting control specifications for high performance However, due to numerous shortcomings, dc motor drives are nowadays replaced with ac drives wherever possible To explain the requirements on high-performance control, consider a separately excited dc motor Stator of such a machine can be equipped with either a winding (excitation winding) or with permanent magnets The role of the stator is to provide

excitation flux in the machine, which is in the case of permanent magnets 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)

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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 frequency 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 armature 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 current 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; independent 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 position (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 separately 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

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mul-tiphase ac machine this block becomes more complicated The reason is that using design of the drive control as for a dc machine, where there exist flux and torque-producing dc current references, means that the control will operate in a rotating set of coordinates (rotating reference frame) In other words, current components used in the control (flux- and torque-producing currents) are not currents that physically exist in the machine Instead, these are the fictitious current components that are related to physically existing ac phase currents through a coordinate transformation This coordinate transformation 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, current 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 converters 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 multiphase 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 considered machine types are induction machines with a squirrel-cage rotor winding, permanent magnet synchronous 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)

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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 position is measured, position of the rotating field in the machine remains unknown Vector control of induction 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 reference 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 currents, 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 reference 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

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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 permanent 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 constant 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 operate 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 current 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 performs 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

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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 terminals 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 current 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 constant 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 reference 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 solution 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 reluctance or maximum inductance): It follows from (24 23) that the torque developed by the machine is entirely dependent on the difference of the inductances along d-and q-axis Hence constructional maximization of this difference, by mak- ing Ld/Lq ratio as high as possible, is absolutely necessary in order to make the Syn-Rel a viable candidate for real-world applications For this purpose, it has been shown that, by using an axially laminated rotor rather than a radially laminated rotor structure, this ratio can be significantly increased From FOC point of view, it is however irrelevant what the actual rotor construction is (for more details see [13])

As the machine's model is again given in the reference frame firmly attached to the rotor and the real axis of the reference frame again coincides with the rotor magnetic d-axis, transformation expressions that relate the actual phase variables with the stator d?q variables (24 9) through (24 11) are the same as for PMSMs Rotor position, being measured once more, is the angle required in the transformation matrix (24 9) Thus one concludes that FOC schemes for a Syn-Rel will inevitably be very similar to those of an IPMSM

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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 current 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 saturation 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 mathematical models obtained by means of general theory of ac machines An n-phase squirrel cage induction 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

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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 equation (24 2) if q-component of either stator flux or rotor flux is forced to be zero Thus, to convert an induction machine into its dc equivalent, it is necessary to select a reference frame in which the q-component of either the stator or rotor flux linkage will be kept at zero value (the third possibility, of very low practical value, is to choose air-gap [magnetizing] flux instead of stator or rotor flux, and keep its q-component at zero) Thus, FOC scheme for an induction machine can be developed by aligning the reference frame with the d-axis component of the chosen flux linkage While selection of the stator flux linkage for this purpose does have certain applications, it results in a more complicated FOC scheme and is therefore not considered 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 stationary 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 quality of dynamic response is achievable regardless of the number of phases on the stator of the machine Load rejection properties of a three-phase 0 75 kW, 380 V, 4-pole, 50 Hz induction motor drive with IRFOC are illustrated in Figure 24 18, where at constant speed reference of 600 rpm rated load torque is at first applied and then removed The response of the stator q-axis current reference and rotor speed are shown Once more, speed variation during sudden loading/unloading is inevitable, as already discussed in conjunction with Figure 24 8 IRFOC scheme discussed so far suffices for operation in the base speed region, where rotor flux (stator d-axis current) reference is kept constant If the drive is to operate above base speed, it is necessary to weaken the field Since flux is produced from stator side, this now comes to a simple reduction of the stator d-axis current reference for speeds higher than rated The necessary reduction of the rotor flux reference is, in the simplest case, determined in very much the same way as for a PMSM Since supply voltage of the machine must not exceed the rated value, then at any speed higher then rated product of rotor flux and speed should stay the same as at rated speed Hence Since change of rotor speed takes place at a much slower rate than the change of rotor flux (i e , mechanical time constant is considerably larger than the electromagnetic time constant), industrial drives normally 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 operation 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)