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 2πn, 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 “2n” 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 fluxtorque-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 − 32
pairs of x–y component equations in 24 4 and 24 6, which do not contribute to the torque production and are therefore not
transformed with rotational transformation 24 9 i e , their form is the one obtained after application of decoupling
transformation 24 10 only It has to be noted however, that the reference value of zero for all of these components which
will exist in the model for n ≥ 5 is implicitly included in the control scheme of Figure 24 3, since reference phase currents are built
from d–q current references only Equations 24 4 and 24 6 are of the same form for all the multiphase ac machines
considered here all types of synchronous and induction machines
For a SPMSM machine, the set of equations 24 3, 24 5, and 24 7 further simplifies since the air-gap is regarded as
uniform and hence Ls = Ld = Lq Thus 24 3 and 24 5 reduce to while the torque equation takes the form
24 13 By comparing 24 13 with 24 2, it is obvious that the form of the torque equation is
identical as for a separately excited dc motor The only but important difference is that the role of the armature current is now taken by the
q-axis stator current component Assuming that the machine is current-fed i e , CC is executed in the stationary reference frame,
stator current dynamics of 24 12 are taken care of by the fast CC loops and the global control scheme of Figure 24 3 becomes as in
Figure 24 4 Since the machine has permanent magnets that
provide excitation flux, there is no need to provide flux from the stator side and the stator current reference along d-axis is set to
zero According to 24 11, the measured rotor electrical position is the transformation angle of 24 9
The control scheme of Figure 24 4 is a direct analog of the corresponding control scheme of perma- nent magnet excited dc
motors, where the role of the commutator with brushes is now replaced with the mathematical transformation [T]−1 A few
remarks are due Figure 24 4 includes a limiter after the speed controller This block is always present in high-performance drives
although it was not included in Figures 24 1 and 24 3, for simplicity and limiting ensures that the maximum allowed stator
current normally governed by the power electronic converter is not exceeded Next, as already noted, a con- stant that relates torque
and stator q-axis current reference according to 24 13 and which is shown in Figure 24 4 will normally be incorporated into speed
controller gains, so that the limited output of the speed controller will actually directly be the stator q-axis current reference
The control scheme of Figure 24 4 satisfies for control in the base speed region If it is required to oper- ate the machine at speeds
higher than rated, it is necessary to weaken the flux so that the voltage applied to the machine does not exceed the rated value However,
permanent magnet flux cannot be changed and the only way to achieve operation at speeds higher than rated is to keep the term
ωLsids + 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 DA
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 RD converter is the speed signal in analog form that is taken to the PC after AD 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 loadingunloading 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
C. Field-Oriented Control of Multiphase Synchronous reluctance Machines
