FOC: Field Oriented Control

  • Introductory Considerations

Variable speed electric drives are nowadays utilized in almost every walk of life from the most basic devices such as hand-held tools and other home appliances to the most sophisticated ones such as electric propulsion systems in cruise ships and high-precision manufacturing technologiesDepending on the application the  control variable may be the motor’s torque speed or position of the rotor shaft  In the most demanding applications  the requirement is to be able to control the electric machine’s elec- tromagnetic torque in order to be able to provide a controlled transition from one operating speed (posi- tion) to another speed (position)  This means that the control of the drive must be able to achieve desired  dynamic response of the controlled variable in a minimum time interval  This can only be achieved if  the motor’s electromagnetic torque can be practically instantaneously stepped from the previous steady- state  value  to  the  maximum  allowed  value,  which  is  in  turn  governed  by  the  allowed  maximum  cur- rent  Variable speed electric drives that are capable of achieving such a performance are usually called  high-performance drives, since the control is effective not only in steady state but in transient as well   Common  features  of  all  high-performance  drives  are  that  they  require  information  on  instantaneous  rotor position (speed), operation is with closed-loop control, and the machine is supplied from a power  electronic converter  Applications that necessitate use of a high-performance drive are numerous and  include  robotics,  machine  tools,  elevators,  rolling  mills,  paper  mills,  spindles,  mine  winders,  electric  traction, electric and hybrid electric vehicles, and the like

A  principal  schematic  outlay  of  a  high-performance  electric  drive  is  shown  in  Figure  24 1  and  it  applies equally to all types of electric machinery  Electromagnetic torque of an electric machine can be  expressed as a product of the flux-producing current and torque-producing current, so that the control  system in Figure 24 1 has two parallel paths  Flux-producing current reference is shown as a constant;  however, this may or may not be the case, as discussed later  Torque-producing current is in principle  the output of the torque controller  However, torque controller of Figure 24 1 is usually not present in  high-performance drives, since the torque-producing current reference can be obtained directly from the reference torque by means of a simple scaling (or the output of the speed controller can be made to  be directly the torque-producing current reference)  This is so since the torque and the torque-producing  current are, when a high-performance control algorithm is applied, related through a constant  The con- trol structure in Figure 24 1 is composed of cascaded controllers (typically of proportional plus integral  [PI]  type)   An  asterisk  stands  for  reference  quantities,  while  θ,  ω,  and  Te  designate  further  on  instan- taneous values of electrical rotor position, electrical rotor angular speed (speed is shown in figures as  n in rpm; this is not to be confused with phase number n) and electromagnetic torque developed by the  motor,  respectively   The  cascaded  structure  is  based  on  the  fundamental  equations  that  govern  rotor  rotation, which are for a machine with P pole pairs given with (TL stands for load torque, k is the friction  coefficient, and J is the inertia of rotating masses)

High-performance drives typically involve measurement of the rotor position (speed) and motor sup- ply currents, as indicated in Figure 24 1  Since the machine’s torque is governed by currents rather than  voltages, measured currents are used in the block “Drive control algorithm” to incorporate the closed- loop current control (CC) algorithm  What this means is that the power electronic converter is current- controlled, so that applied voltages are such as to minimize the errors in the current tracking

Until the early 1980s of the last century, the separately excited dc motor was the only available elec- tric machine that could be used in a high-performance drive  A dc motor is by virtue of its construc- tion ideally suited to meeting control specifications for high performance  However, due to numerous  shortcomings, dc motor drives are nowadays replaced with ac drives wherever possible  To explain the  requirements  on  high-performance  control,  consider  a  separately  excited  dc  motor   Stator  of  such  a  machine can be equipped with either a winding (excitation winding) or with permanent magnets  The  role of the stator is to provide excitation flux in the machine, which is in the case of permanent mag- nets constant, while it is controllable if there is an excitation winding  For the sake of explanation, it is  assumed that the stator carries permanent magnets, which provide constant flux, ψm, so that the upper  input into the “Drive control algorithm” block in Figure 24 1 does not exist  The permanent magnet flux  is stationary in space and it acts along a magnetic axis, as schematically illustrated in Figure 24 2, where  the cross section of the machine is shown  Rotor of the machine carries a winding (armature winding)

access to which is provided by means of stationary brushes and an assembly on the rotor, called commu- tator  The supply is from a dc source (in principle, a power electronic converter of dc–dc or ac–dc type,  depending on the application), which provides dc armature current as the input into the rotor winding   The brushes are placed in an axis orthogonal to the permanent magnet flux axis (Figure 24 2)  Since the  brushes are stationary, flux and the armature terminal current are at all times at 90°  It is this orthogonal  position of the torque-producing current (armature current ia) and the permanent magnet flux ψm that  enables instantaneous torque control of the machine by means of instantaneous change of the armature  current  This follows from the electromagnetic torque equation of the machine, which is given by (K is  a constructional constant)

It also follows that since the torque-producing (armature) current and the torque are related through a constant, armature current reference in Figure 24 1 can be obtained by scaling the torque reference with  the constant (which is normally embedded in the speed controller PI gains), so that the torque controller  is not required  On the basis of these explanations and (24 2) it is obvious that the machine’s torque can  be stepped if armature current can be stepped  This of course requires current-controlled operation of  the armature dc supply, so that the armature voltage is varied in accordance with the armature current  requirements

It is important to remark here that, inside the rotor winding, the current is actually ac  It has a fre- quency equal to the frequency of rotor rotation, since the commutator converts dc input into ac output  current and therefore performs, together with fixed stationary brushes, the role of a mechanical inverter  (in motoring operation; in generation it is the other way round, so that the commutator acts as a recti- fier)  As the rotor winding is rotating in the stationary permanent magnet flux, a rotational electromo- tive force (emf ) is induced in the rotor winding according to the basic law of electromagnetic induction,

The machine in Figure 24 2, with constant permanent magnet excitation, can operate with variable speed in the base speed region only (i e , up to the rated speed), since operation above base speed (field  weakening region) requires the means for reduction of the flux in the machine  This is so since the arma- ture  voltage  cannot  exceed  the  rated  voltage  of  the  machine,  which  corresponds  to  rated  speed,  rated  torque operation  To operate at a speed higher than rated, one has to keep the induced emf as for rated  speed operation  Since speed goes up flux must come down, something that is not possible if permanent  magnets are used but is achievable if there is an excitation winding  In such a case “flux-producing cur- rent reference” of Figure 24 1 has a constant rated value up to the rated speed and is further gradually  reduced to achieve operation with speeds higher than rated (hence the name, field weakening region)   However, due to the orthogonal position of the flux and armature axes, flux and torque control do not mutually impact on each other as long as the flux-producing current is kept constant  It is hence said  that torque and flux control are decoupled (or independent) and this is the normal mode of operation in  the base speed region  Once when field weakening region is entered, dynamic decoupled flux and torque  control is not possible any more since reduction of the flux impacts on torque production

The preceding discussion can be summarized as follows: high-performance operation requires that  torque of a motor is controllable in real time; instantaneous torque of a separately excited dc motor is  directly controllable by armature current as flux and torque control are inherently decoupled; indepen- dent flux and torque control are possible in a dc machine due to its specific construction that involves  commutator  with  brushes  whose  position  is  fixed  in  space  and  perpendicular  to  the  flux  position;  instantaneous flux and torque control require use of current controlled dc 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- 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 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)

  • FieldOriented Control  of Multiphase Permanent Magnet Synchronous Machines

Consider a multiphase star-connected PMSM, with spatial shift between any two consecutive phases of 2π/n, and let the phase number n be an odd number without any loss of generality  The neutral point  of the stator winding is isolated  Permanent magnets are on the rotor and they can be surface mounted  (surface-mounted permanent magnet synchronous machine [SPMSM]) or embedded in the rotor (inte- rior permanent magnet synchronous machine [IPMSM])  In the former case the air-gap of the machine can  be  considered  as  uniform,  while  in  the  latter  case  the  air-gap  length  is  variable,  since  permanent  magnets have a permeability that is practically the same as for the air  Thus SPMSMs are characterized  with  a  rather  large  air  gap  (which  will  make  operation  in  the  field  weakening  region  difficult,  as  dis- cussed later), while the air gap of the IPMSMs is small, but the magnetic reluctance is variable, due to the  saliency effect produced by the embedded magnets  Rotor of the machine does not carry any windings,  regardless of the way in which the magnets are placed Mathematical model of an IPMSM can be given in the common reference frame firmly attached to  the rotor with the following equations: where index l stands for leakage inductance, v, i, and ψ denote voltage, current, and flux linkage, respec- tively, d and q stand for the components along permanent magnet flux axis (d) and the axis perpendicular  to it (q), and s denotes stator  Inductances Ld and Lq are stator winding self-inductances along d– and q-axis Voltage  and  flux  linkage  equations  (24 3)  through  (24 6)  represent  an  n-phase  machine  in  terms of  sets  of  new  n  variables,  obtained  after  transforming  the  original  machine  model  in  phase-variable  domain by means of a power invariant transformation matrix that relates original phase variables and  new variables through where  f  stands  for  voltage,  current,  or  flux  linkage  and  [D]  and  [C]  are  the  rotational  transformation  matrix and decoupling transformation matrix (block “2/n” in Figure 24 3) for stator variables, respec- tively  For an n-phase machine with an odd number of phases, these matrices are Due to the selected 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 xy 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 ω(Lsids + ψm) of (24 12) is shown in an arbitrary position, as though it has positive both d– and q-axis components  As noted,  in the base speed region stator d-axis current component is zero, meaning that the complete stator cur- rent space vector of (24 15) is aligned with the q-axis  Stator current is thus at 90° (δ = 90°) with respect  to the flux axis in motoring, while the angle is −90° (δ = −90°) during braking  In the field weakening  d-axis current is negative to provide an artificial effect of the reduction in the flux linkage of the stator  winding, so that δ > 90° in motoring  If the machine operates in field weakening region, simple q-axis  current limiting of Figure 24 4 is not sufficient any more, since the total stator current of (24 15) must  not exceed the prescribed limit, while d-axis current is now not zero any more  Hence, the q-axis current  must have a variable limit, governed by the maximum allowed stator current ismax and the value of the  d-axis current command of (24 14)  A more detailed discussion is available in [19]

In PMSMs, since there is no rotor winding, flux linkage in the air-gap and rotor is taken as being the  same  and  this  is  the  flux  linkage  with  which  the  reference  frame  has  been  aligned  for  FOC  purposes  in Figure 24 4  Schematic representation of Figure 24 5 is the same regardless of the number of stator  phases as  long  as  the  CC  is implemented, as  shown  in  Figure  24 4   The only  thing  that  changes  is the  number of stator winding phases and their spatial shift An illustration of a three-phase SPMSM performance, obtained from an experimental rig, is shown  next  PI speed control algorithm is implemented in a PC and operation in the base speed region is 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  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

  • FieldOriented 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

By differentiating (24 24) with respect to angle δ, one gets this time a straightforward solution δ = 45°  as the MTPA condition  This means that the MTPA results if at all times stator d-axis and q-axis cur- rent references are kept equal  FOC scheme of Figure 24 4 therefore only changes with respect to the  stator d-axis current reference setting and becomes as illustrated in Figure 24 10  The 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

  • FieldOriented 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

Relationship between phase variables and variables in the common reference frame is once more governed with (24 9) and (24 10) for stator quantities  What is however very different is that the setting of the  stator transformation angle according to (24 11) would be of little use, since rotor speed is different from  the synchronous speed  In simple terms, rotor rotates asynchronously with the rotating field, meaning that  rotor position does not coincide with the position of a rotating flux in the machine  The other difference,  compared to a PMSM, is that the rotor does not carry any means for producing the excitation flux  Hence  the flux in the machine has to be produced from the stator supply side, this being similar to a Syn-Rel Torque equation can be given in different ways, including the two that are the most relevant for FOC,  (24 27), in terms of stator flux and rotor flux linkage d–q axis components  It is obvious from (24 27) that  the torque equation of an induction machine will become identical in form to a dc machine’s torque equa- tion (24 2) if q-component of either stator flux or rotor flux is forced to be zero  Thus, to convert an 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   =  ψ r* /Lm  However, since modern induction machines are designed to operate around  the knee of the magnetizing characteristic of the machine (i e , in saturated region), while during opera- tion in the field weakening region flux reduces and operating point moves toward the linear part of the  magnetizing characteristic, it is necessary to account in the design of the IRFOC aimed at wide-speed  operation for the nonlinearity of the magnetizing curve (i e , variation of the parameter Lm)  One rather  simple  and  widely  used  solution  is  illustrated  in  Figure  24 19,  where  only  creation  of  stator  d–q  axis  current references and the reference slip speed is shown  The rest of the control scheme is the same as  in Figure 24 15 Here  again,  one  can  use  either  stator  current  d–q  reference  currents  or  d–q  axis  current  components  calculated  from  the  measured  phase  currents   The  principal  RFOC  scheme,  assuming  that  rotor  flux  position  is  again  determined  according  to  the  indirect  field  orientation  principle,  is  shown  in  Figure 24 23 (current limiting block is not shown for simplicity)

Operation in the field weakening region can again be realized by using the stator d-axis current and  slip speed reference setting as in Figure 24 19  Since the rotor flux reference will change slowly, the rate  of change of rotor flux in (24 46) is normally neglected in the decoupling voltage calculation, so that ed  calculation remains as in (24 47)  However, since rotor flux reference reduces with the increase in speed,  eq calculation has to account for the rotor flux (stator d-axis current) variation As noted in the section on RFOC of PMSMs, vector control with only two current controllers, as in Figure 24 23, suffices for three-phase machines  While, in theory, this should also be perfectly sufficient  for machines with more than three phases, in practice various nonideal characteristics of the PEC sup- ply (for example, inverter dead time) and the machine (any asymmetries in the stator winding) lead to  the situation where the performance with only two current controllers is not satisfactory [26]  To illus- trate this statement, an experimental result is shown in Figure 24 24 for a five-phase induction machine  (which has already been described in conjunction with Figure 24 16)  Control scheme of Figure 24 23