FOC: Field Oriented Control
A. Introductory Considerations
Variable speed electric drives are nowadays utilized in almost every walk of life from the most basic devices such as hand-held tools and other home appliances to the most sophisticated ones such as electric propulsion systems in cruise ships and high-precision manufacturing technologiesDepending on the application the control variable may be the motor’s torque speed or position of the rotor shaft In the most demanding applications the requirement is to be able to control the electric machine’s elec- tromagnetic torque in order to be able to provide a controlled transition from one operating speed (posi- tion) to another speed (position) This means that the control of the drive must be able to achieve desired dynamic response of the controlled variable in a minimum time interval This can only be achieved if the motor’s electromagnetic torque can be practically instantaneously stepped from the previous steady- state value to the maximum allowed value, which is in turn governed by the allowed maximum cur- rent Variable speed electric drives that are capable of achieving such a performance are usually called high-performance drives, since the control is effective not only in steady state but in transient as well Common features of all high-performance drives are that they require information on instantaneous rotor position (speed), operation is with closed-loop control, and the machine is supplied from a power electronic converter Applications that necessitate use of a high-performance drive are numerous and include robotics, machine tools, elevators, rolling mills, paper mills, spindles, mine winders, electric traction, electric and hybrid electric vehicles, and the like
speed controller can be made to be directly the torque-producing current reference) This is so since the torque and the torque-producing current are, when a high-performance control algorithm is applied, related through a constant The con- trol structure in Figure 24 1 is composed of cascaded controllers (typically of proportional plus integral [PI] type) An asterisk stands for reference quantities, while θ, ω, and Te designate further on instan- taneous values of electrical rotor position, electrical rotor angular speed (speed is shown in figures as n in rpm; this is not to be confused with phase number n) and electromagnetic torque developed by the motor, respectively The cascaded structure is based on the fundamental equations that govern rotor rotation, which are for a machine with P pole pairs given with (TL stands for load torque, k is the friction coefficient, and J is the inertia of rotating masses)
High-performance drives typically involve measurement of the rotor position (speed) and motor sup- ply currents, as indicated in Figure 24 1 Since the machine’s torque is governed by currents rather than voltages, measured currents are used in the block “Drive control algorithm” to incorporate the closed- loop current control (CC) algorithm What this means is that the power electronic converter is current- controlled, so that applied voltages are such as to minimize the errors in the current tracking
2, where the cross section of the machine is shown Rotor of the machine carries a winding (armature winding)
access to which is provided by means of stationary brushes and an assembly on the rotor, called commu- tator The supply is from a dc source (in principle, a power electronic converter of dc–dc or ac–dc type, depending on the application), which provides dc armature current as the input into the rotor winding The brushes are placed in an axis orthogonal to the permanent magnet flux axis (Figure 24 2) Since the brushes are stationary, flux and the armature terminal current are at all times at 90° It is this orthogonal position of the torque-producing current (armature current ia) and the permanent magnet flux ψm that enables instantaneous torque control of the machine by means of instantaneous change of the armature current This follows from the electromagnetic torque equation of the machine, which is given by (K is a constructional constant)
It also follows that since the torque-producing (armature) current and the torque are related through a constant, armature current reference in Figure 24 1 can be obtained by scaling the torque reference with the constant (which is normally embedded in the speed controller PI gains), so that the torque controller is not required On the basis of these explanations and (24 2) it is obvious that the machine’s torque can be stepped if armature current can be stepped This of course requires current-controlled operation of the armature dc supply, so that the armature voltage is varied in accordance with the armature current requirements
It is important to remark here that, inside the rotor winding, the current is actually ac It has a fre- quency equal to the frequency of rotor rotation, since the commutator converts dc input into ac output current and therefore performs, together with fixed stationary brushes, the role of a mechanical inverter (in motoring operation; in generation it is the other way round, so that the commutator acts as a recti- fier) As the rotor winding is rotating in the stationary permanent magnet flux, a rotational electromo- tive force (emf ) is induced in the rotor winding according to the basic law of electromagnetic induction,
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
into one phase) As a consequence, the flux that was stationary in a sep- arately excited dc machine is now rotating in the cross section of the machine at a synchronous speed, determined with the stator winding supply frequency Thus, the stationary flux axis of Figure 24 2 now becomes an axis that rotates at synchronous speed Since decoupled flux and torque control require that flux-producing current is aligned with the flux axis, while the torque-producing current is in an axis perpendicular to the flux axis, the control of a multiphase machine has to be done using a set of orthogo- nal coordinates that rotates at the synchronous speed (speed of rotation of the flux in the machine) The situation is further complicated by the fact that, in a multiphase machine, there are in principle three different fluxes (or flux linkages, as they will be called further on), stator, air-gap, and rotor flux linkage While in steady-state operation they all have synchronous speed of rotation, the instantaneous speeds during transients differ Hence a decision has to be made with regard to which flux the control should be performed Basic outlay of the drive remains as in Figure 24 1 However, while in the case of a dc drive the block “drive control algorithm” in essence contains only current controllers, in the case of a mul- tiphase ac machine this block becomes more complicated The reason is that using design of the drive control as for a dc machine, where there exist flux and torque-producing dc current references, means that the control will operate in a rotating set of coordinates (rotating reference frame) In other words, current components used in the control (flux- and torque-producing currents) are not currents that physically exist in the machine Instead, these are the fictitious current components that are related to physically existing ac phase currents through a coordinate transformation This coordinate transforma- tion produces, from dc current references, ac current references for the supply of the stator winding of a multiphase machine Thus, what commutator with brushes does in a dc machine (dc–ac conversion) has to be done in ac machines using a mathematical transformation in real time
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
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
not in the stationary ref- erence frame It is therefore assumed further on that whatever the machine type and the actual FOC scheme used, the source is capable of delivering ideal sinusoidal stator currents (or voltages, as discussed shortly)
B. Field-Oriented Control of Multiphase Permanent Magnet Synchronous Machines
Consider a multiphase star-connected PMSM, with spatial shift between any two consecutive phases of 2π/n, and let the phase number n be an odd number without any loss of generality The neutral point of the stator winding is isolated Permanent magnets are on the rotor and they can be surface mounted (surface-mounted permanent magnet synchronous machine [SPMSM]) or embedded in the rotor (inte- rior permanent magnet synchronous machine [IPMSM]) In the former case the air-gap of the machine can be considered as uniform, while in the latter case the air-gap length is variable, since permanent magnets have a permeability that is practically the same as for the air Thus SPMSMs are characterized with a rather large air gap (which will make operation in the field weakening region difficult, as dis- cussed later), while the air gap of the IPMSMs is small, but the magnetic reluctance is variable, due to the saliency effect produced by the embedded magnets Rotor of the machine does not carry any windings, regardless of the way in which the magnets are placed Mathematical model of an IPMSM can be given in the common reference frame firmly attached to the rotor with the following equations: where index l stands for leakage inductance, v, i, and ψ denote voltage, current, and flux linkage, respec- tively, d and q stand for the components along permanent magnet flux axis (d) and the axis perpendicular to it (q), and s denotes stator Inductances Ld and Lq are stator winding self-inductances along d- and q-axis Voltage and flux linkage equations (24 3) through (24 6) represent an n-phase machine in terms of sets of new n variables, obtained after transforming the original machine model in phase-variable domain by means of a power invariant transformation matrix that relates original phase variables and new variables through where f
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 θs in (24 9) is identically equal to the rotor electrical position, so that As the d-axis of the common reference frame then coincides with the instantaneous position of the permanent magnet flux, this means that the given model is already expressed in the common reference frame firmly attached to the permanent magnet flux The pairs of d–q equations (24 3) and (24 5) constitute the flux/torque-producing part of the model, as is evident from torque equation (24 7) Since in a star-connected winding, with isolated neutral, zero-sequence current cannot flow, the last equation of (24 4) and (24 6) can be omitted The model then contains, in addition to the d–q equations, (n − 3)/2 pairs of x–y component equations in (24 4) and (24 6), which do not contribute to the torque production and are therefore not transformed with rotational transformation (24 9) (i e , their form is the one obtained after application of decoupling transformation (24 10) only) It has to be noted however, that the reference value of zero for all of these components (which will exist in the model for n ≥ 5) is implicitly included in the control scheme of Figure 24 3, since reference phase currents are built from d–q current references only Equations 24 4 and 24 6 are of the same form for all the multiphase ac machines considered here (all types of synchronous and induction machines)
For a SPMSM machine, the set of equations (24 3), (24 5), and (24 7) further simplifies since the air-gap is regarded as uniform and hence 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
provide excitation flux, there is no need to provide flux from the stator side and the stator current reference along d-axis is set to zero According to (24 11), the measured rotor electrical position is the transformation angle of (24 9)
The control scheme of Figure 24 4 is a direct analog of the corresponding control scheme of perma- nent magnet excited dc motors, where the role of the commutator with brushes is now replaced with the mathematical transformation [T]−1 A few remarks are due Figure 24 4 includes a limiter after the speed controller This block is always present in high-performance drives (although it was not included in Figures 24 1 and 24 3, for simplicity) and limiting ensures that the maximum allowed stator current (normally governed by the power electronic converter) is not exceeded Next, as already noted, a con- stant that relates torque and stator q-axis current reference according to (24 13) and which is shown in Figure 24 4 will normally be incorporated into speed controller gains, so that the limited output of the speed controller will actually directly be the stator q-axis current reference
The control scheme of Figure 24 4 satisfies for control in the base speed region If it is required to oper- ate the machine at speeds higher than rated, it is necessary to weaken the flux so that the voltage applied to the machine does not exceed the rated value However, permanent magnet flux cannot be changed and the only way to achieve operation at speeds higher than rated is to keep the term ω(Lsids + ψm) of (24 12) is shown in an arbitrary position, as though it has positive both d- and q-axis components As noted, in the base speed region stator d-axis current component is zero, meaning that the complete stator cur- rent space vector of (24 15) is aligned with the q-axis Stator current is thus at 90° (δ = 90°) with respect to the flux axis in motoring, while the angle is −90° (δ = −90°) during braking In the field weakening d-axis current is negative to provide an artificial effect of the reduction in the flux linkage of the stator winding, so that δ > 90° in motoring If the machine operates in field weakening region, simple q-axis current limiting of Figure 24 4 is not sufficient any more, since the total stator current of (24 15) must not exceed the prescribed limit, while
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
C. 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 d- and q-axis Hence constructional maximization of this difference, by mak- ing Ld/Lq ratio as high as possible, is absolutely necessary in order to make the Syn-Rel a viable candidate for real-world applications For this purpose, it has been shown that, by using an axially laminated rotor rather than a radially laminated rotor structure, this ratio can be significantly increased From FOC point of view, it is however irrelevant what the actual rotor construction is (for more details see [13])
As the machine’s model is again given in the reference frame firmly attached to the rotor and the real axis of the reference frame again coincides with the rotor magnetic d-axis, transformation expressions that relate the actual phase variables with the stator d– q variables (24 9) through (24 11) are the same as for PMSMs Rotor position, being measured once more, is the angle required in the transformation matrix (24 9) Thus one concludes that FOC schemes for a Syn-Rel will inevitably be very similar to those of an IPMSM
Since in a Syn-Rel there is no excitation on rotor, excitation flux must be provided from the stator side and this is the principal difference, when compared to the PMSM drives Here again a question arises as to how to subdivide the available stator current into corresponding d–q axis current references The same idea of MTPA control is used as with IPMSMs Using (24 19), electromagnetic torque (24 23) can be written as
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 transient speed response is practically the same as with a SPMSM (Figure 24 6 and 24 7), since the same linearity of the speed change profile is observable again In final steady-state operation at −800 rpm the machine operates with q-axis current reference of more than 1 A rms, although there is no load This is again the consequence of the mechanical and iron core losses that exist in the machine but are not accounted for in the vector control scheme (mechanical loss appears, according to (24 1a), as a certain nonzero load torque) Measured and reference phase current are in an excellent agreement, indicating that the CC of the inverter operates very well
D. Field-Oriented Control of Multiphase Induction Machines
Similar to synchronous machines, FOC schemes for induction machines are also developed using math- ematical models obtained by means of general theory of ac machines An n-phase squirrel cage induc- tion motor can be described in a common reference frame that rotates at an arbitrary speed of rotation ωa with the flux–torque-producing part of the model
This is at the same time the complete model of a three-phase squirrel cage induction machine If stator has more than three phases, the model also includes the non-flux/torque-producing equations (24 4) and (24 6), which are of the same form for all 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
from stator side, this now comes to a simple reduction of the stator d -axis current reference for speeds higher than rated The necessary reduction of the rotor flux reference is, in the simplest case, determined in very much the same way as for a PMSM Since supply voltage of the machine must not exceed the rated value, then at any speed higher then rated product of rotor flux and speed should stay the same as at rated speed Hence Since change of rotor speed takes place at a much slower rate than the change of rotor flux (i e , mechani- cal time constant is considerably larger than the electromagnetic time constant), industrial drives nor- mally base stator current d-axis setting in the field weakening region on the steady-state rotor flux relationship, id*s = ψ r* /Lm However, since modern induction machines are designed to operate around the knee of the magnetizing characteristic of the machine (i e , in saturated region), while during opera- tion in the field weakening region flux reduces and operating point moves toward the linear part of the magnetizing characteristic, it is necessary to account in the design of the IRFOC aimed at wide-speed operation for the nonlinearity of the magnetizing curve (i e , variation of the parameter
Lm) One rather simple and widely used solution is illustrated in Figure 24 19, where only creation of stator d–q axis current references and the reference slip speed is shown The rest of the control scheme is the same as in Figure 24 15 Here again, one can use either stator current d–q reference currents or d–q axis current components calculated from the measured phase currents The principal RFOC scheme, assuming that rotor flux position is again determined according to the indirect field orientation principle, is shown in Figure 24 23 (current limiting block is not shown for simplicity)
