Chapter 11 - first part

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ELECTRIC DRIVES
Ion Boldea S.A.Nasar 1998
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PM AND RELUCTANCE SYNCHRONOUS MOTOR DRIVES 11.1.
INTRODUCTION PM and reluctance synchronous
motors are associated with PWM voltage - source
inverters in variable speed drives. While PM - SM
drives enjoy by now world - wide markets,
reluctance synchronous motor (RSM) drives still
have a small share of the market in low power
region, though recently high saliency (and
performance) RSMs with q axis PMs have been
successfully introduced for wide constant power
speed range applications (such as spindle
drives). High performance applications are in
general provided with PM - SM drives. RSM drives
may be used for general applications as the costs
of the latter are lower than of the former and,
in general, close to the costs of similar IM
drives. In what follows we will deal first with
PM - SM drives and after that with RSM drives.
Constant power operation will be discussed at the
end of the chapter for both types of synchronous
motors.
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• 11.2. PM - SM DRIVES CLASSIFICATIONS
• Basically we may distinguish three ways to
  classify PM - SM drives with respect to current
  waveform, voltage - frequency correlation, motion
  sensor presence.
• From the point of view of current waveform we
  distinguish
• rectangular current control - figure 11.1.a - the
  so called brushless d.c. motor drive - q 1 slot
  per pole per phase, surface PM rotor
• sinusoidal current control - figure 11.1.b - the
  so called brushless a.c. drive - q ? 2 slots per
  pole per phase.
• From the point of view of motion sensor presence
  there are
• drives with motion sensors
• drives without motion sensors (sensorless).
• Finally sinusoidal current (brushless a.c.)
  drives may have
• scalar (V / f) control - a damper cage on the
  rotor is required
• vector control (current or current and voltage)
• direct torque and flux control (DTFC).
• The stator current waveforms - rectangular or
  sinusoidal (figure 11.1) - have to be
  synchronized with the rotor position.

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Figure 11.1. Rectangular or sinusoidal current
control aa - advance angle
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Scalar control (V / f) is related to sinusoidal
current control without motion sensors
(sensorless) (figure 11.2).
Figure 11.2. V / f (scalar) control for PM - SM
(and for RSM) with torque angle increment
compensation
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For faster dynamics applications vector control
is used (figure 11.3).
Figure 11.3. Basic vector control of PM - SM (or
for RSM) 1 - with motion sensor 2 - sensorless
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Figure 11.4. Direct torque and flux control
(DTFC) of PM - SMs (and RSMs)
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To simplify the motor control, the direct torque
and flux control (DTFC for IMs) has been extended
to PM - SM (and to RSMs) as torque vector control
(TVC) in 10. The stator flux and torque direct
control leads to a table of voltage switchings
(voltage vector sequence). Vector rotation has
been dropped but flux and torque observers are
required. While speed is to be observed, rotor
position estimation is not required in sensorless
driving. Again, fast flux and torque control may
be obtained even in sensorless driving. Rectangul
ar current control and sinusoidal current control
(through vector control or DTFC) are going to be
detailed in what follows for motion sensor and
sensorless driving.
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11.3. RECTANGULAR CURRENT CONTROL (BRUSHLESS D.C.
MOTOR DRIVES) Rectangular current control is
applied to nonsinusoidal (trapezoidal) PM -
e.m.f. waveform - typical to concentrated coil
stator windings (three slots per pole q 1). To
reduce torque pulsations rectangular current is
needed. With rectangular current, however, the
reluctance torque production is not so efficient
and thus nonsalient pole (surface PM) rotors is
preferred. The motor phase inductance La is
1 (11.1) (11.2) where W1 -
turns per phase, L - stack length, t - pole
pitch, g - airgap, Kc - Carter coefficient, hPM -
PM radial thickness, p - pole pairs, Lsl -
leakage inductance. The mutual inductance between
phases, Lab, is (11.3) Thus the cycling
inductance Ls (per phase) for two phase
conduction is (11.4)
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11.3.1. Ideal brushless d.c. motor waveforms In
principle the surface PM extends over an angle
aPM less than 1800 (which represents the pole
pitch, figure 11.1). The two limits of aPM are
2p/3 and p. Let us suppose that the PM produces a
rectangular airgap flux distribution over aPM p
(1800) (figure 11.5.a). The stator phase m.m.f.
is supposed to be rectangular, a case
corresponding to q 1 (three slots per pole).
Consequently the PM flux linkage in the stator
winding lPM(qer) varies linearly with rotor
position (figure 11.5.b). Finally the phase
e.m.f. Ea is rectangular with respect to rotor
position (figure 11.5.c). For zero advance angle
aa 0 (figure 11.1) - the phase currents are in
phase with the e.m.f. (ia, Ea) for motoring
(figure 11.5.d). At any instant, due to assumed
instantaneous current commutation, only two
phases are in conduction. Consequently only two
thirds of the PMs are utilized. So the back core
flux is 50 larger than necessary. On the other
hand, if the currents would extend over 1800
(with three phases conducting at any time) for
1200 wide PMs, it would mean 50 more than
necessary copper losses for a given torque.
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Figure 11.5. Ideal waveforms for BLDC (brushless
d.c. motor) a.) PM airgap flux density b.) PM
flux per phase a c.) e.m.f. in phase a d.) ideal
currents for motoring e.) ideal currents for
generating
For 1200 wide magnets delta connection is used
while 1800 wide magnets require star connection.
To reduce the fringing (leakage) flux between
neighboring magnets their span is 1500 - 1600.
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Returning to the linear phase flux linkage
variation with rotor position (figure 11.5.b) we
have (11.5) The maximum flux linkage per
phase lPM is (11.6) The phase e.m.f.,
Ea, is (11.7) where wr is the
electrical angular speed. As two phases conduct
at any time the ideal torque Te is
constant (11.8) Between any two current
instantaneous commutation the phase current is
constant (id idc) and thus the voltage equation
is (11.9)
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(11.10) With (11.11) The ideal
speed - torque curve is linear (figure 11.6) like
for a d.c. brush PM motor.
Figure 11.6. Ideal speed - torque curves of BLDC
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11.3.2. The rectangular current control
system In general a rectangular current control
system contains the BLDC motor, the PWM inverter,
the speed and current controllers and the
position (speed) sensors (or estimators, for
sensorless control) and the current sensors
(figure 11.7).
Figure 11.7. Rectangular current control of BLDC
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The currents sequence, produced through inverter
adequate control, with 1200 current waveforms in
figure 11.8 shows also the position of the 6
elements of the proximity sensors with respect to
the axis of the phase a for a zero advance angle
aa 0.
Figure 11.8. a.) Current sequencing b.) phase
connection
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With two phases conducting the stator active
m.m.f. is on from 600 to 1200 with respect to the
rotor position. The ideal voltage vector (figure
11.8) also jumps 600 for any phase commutation in
the inverter. Each phase is on 1200 out of 1800
for the 1200 conducting strategy. To reverse the
speed the addresses (IGBTs) of the proximity
sensor elements action are shifted by 1800 (P(a)
? P(a-) P(b) ? P(b-) P(c) ? P(c-)). The
proximity sensor has been located for zero
advance angle to provide similar performance for
direct and reverse motion. However, through
electronic means, the advance angle may be
increased as speed increases to reduce the peak
PM flux in the stator phase and thus produce more
torque, for limited voltage, at high
speeds. Using the same hardware we may also
provide for 1800 conduction conditions, at high
speeds, when all three phases conduct at any time.
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11.3.3. The hysteresis current controller
Figure 11.9. Current chopping
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Figure 11.10. Conduction of phases a and b a.) on
- time b.) off - time
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During the on - time, figure 11.10.a, the ab-
equation is (11.12) (11.13) Note
that if the turning on is advanced by aa part of
the on time Ea - Eb 0 and thus a faster current
increase is possible. The solution of equation
(11.12) is (11.14) To allow for
current rising Vd gt E0. Above a certain speed Vd
lt E0 and thus current chopping is not feasable
any more. The current waveform contains in this
case a single on - off pulse triggered by the
proximity sensor (estimator).
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During the off time (diodes D1 and D4 conducting,
in figure 11.10.b) the voltage equation
is (11.15) Vc0 is the capacitor voltage
at the end of on time (11.15), or at the
beginning of off - time. The solution of (11.15)
with t t - ton is (11.16) with
(11.17) The torque Te(t) expression
is (11.18) So if the e.m.f. is constant
in time the electromagnetic torque reproduces the
current pulsations between imin and imax.
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Example 11.1. A BLDC motor is fed through a PWM
inverter from a 300V d.c. source (Vd 300V).
Rectangular current control is performed at an
electrical speed wr 2p10rad/s. The no load line
voltage at wr is E0 48V const. The cyclic
inductance Ls 0.5mH, rs 0.1W and the stator
winding has q 1 slot per pole per phase and two
poles (2p 2). The filter capacitor Cf 10mF,
the current chopping frequency fc 1.25kHz and
ton / toff 5/3. Determine a.) the minimum and
maximum values of current (imin, imax) during
current chopping b.) calculate the torque
expression and plot it. Solution a.) To find
imin and imax we have to use equation (11.14) for
i(t) imax and t ton and (11.16) for i(t)
imin and t toff. (11.19) Notice that
ton 0.5ms and toff 0.3ms (11.20)
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Also from (11.17) (11.21) (11.22
) (11.23) (11.24) From (11.19)
and (11.24) we may calculate Imax and Imin
(11.25) The average current . b.)
The average torque Tav is (11.26)
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The instantaneous torque (11.18) includes current
pulsations (as from (11.14) and (11.16), figure
11.11).
Figure 11.11. Torque pulsations due to current
chopping only
It should be noticed that the chopping frequency
is low for the chosen speed (E0 ltlt Vd) and thus
the current and torque pulsations are
large. Increasing the chopping frequency will
reduce these pulsations. To keep the current
error band 2(Imax - Imin) whithin reasonable
limits the chopping frequency should vary with
speed (higher at lower speeds and lower at medium
speeds) Though the high frequency torque
pulsations due to current chopping are not
followed by the motor speed, due to the much
larger mechanical time constant, they produce
flux density pulsations and, thus, notable
additional core and copper losses (only the
average current I0 is, in fact, useful).
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11.3.4. Practical performance So far the phase
commutation transients - current overlapping -
have been neglected. They however introduce
notable torque pulsations at 6wr frequency
(figure 11.12) much lower than those due to
current chopping. To account for them complete
simulation or testing is required 3.
Figure 11.12. Torque pulsations due to phase
commutation
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Special measures are required to reduce the
cogging torque to less than 2 - 5 of rated
torque for high performance drives. While at low
speeds current chopping is feasable at high
speeds one current pulse remains (figure 11.13).
The current controller gets saturated and the
required current is not reached.
Figure 11.13. Current waveform at high speeds
As the advance angle is zero (aa 0) there is a
delay in installing the current and thus, as
the e.m.f. is in phase with the reference
current, a further reduction in torque occurs.
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11.3.5. Extending the torque - speed
domain Extending the torque - speed domain may
be obtained (for a given drive) by advancing the
phase commutation time by an angle aa dependent
on speed. This phase advancing allows fast
current rise before the occurence of the e.m.f.
(assuming a PM span angle aPM lt 1500 - 1600) An
approximate way to estimate the advance angle
required aa, for 1200 conduction, may be based on
linear current rise 4 to the value
I (11.27) where n - is the rotor speed
in rps. Torque at even higher speeds may be
obtained by switching from 1200 to 1800 current
conduction (three phase working at any time). The
current waveform changes, especially with
advancing angle (figure 11.14). This time the
e.m.f. is considered trapezoidal, that is close
to reality. The advancing angle aa may be, for
high speeds, calculated assuming sinusoidal
e.m.f. 4 and current variation (11.28)
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Figure 11.14. 1800 conducting with advancing
angle at high speed
It has been demonstrated 3,4 that 1200
conduction is profitable at low to base speeds
while 1800 conduction with advancing angle is
profitable for high speeds (figure 11.15).
Figure 11.15. Torque - speed curves for various
advancing angle aa
A smooth transition between 1200 and 1800
conduction is required to fully exploit the
torque - speed capabilities of brushless d.c.
motor drives.
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Example 11.2. Digital simulation a brushless
d.c. motor drive We will present here the
simulation results on a permanent magnet
brushless DC motor drive (BLDC). The motor
equations (see equation (10.6) in chapter 10)
are (11.29) with (11.30) La, M
self and mutual inductances. Introducing the null
point of d.c. link (0), the phase voltages
are (11.31) For star connection of
phases (11.32) Here n represents the
star connection point of stator windings. Va0,
Vb0, Vc0 could easily be related to the d.c. link
voltage and inverter switching state.
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The simulation of this drive was implemented in
MATLAB - SIMULINK. The motor model was integrated
in a block (PM_SM). The changing of motor
parameters for different simulations is as simple
as possible. After clicking on this block, a
dialog box appears and you can change them by
modifying their default values. The drive system
consists of a PI speed controller (Ki 20, Ti
0.05s), a reference current calculation block,
hysteresis controller and motor blocks. The study
examines the system behavior for starting, load
perturbation and speed reversal. The motor
operates at desired speed and the phase currents
are regulated within a hysteresis band around the
reference currents (as functions of rotor
position). The integration step (50ms) can be
modified from the Simulinks Simulation /
Parameters. To find out the structure of each
block presented above unmask it (Options/Unmask).
Each masked block contains a short help
describing that block (inputs / outputs /
parameters).
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Figure 11.16. The BLDC rectangular current -
drive controller
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Figure 11.17. The BLDC motor block diagram
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The drive and motor used for this simulation have
the following parameters Vdc 220V, 2p 2, Rs
1W, Ls 0.02H, M -0.006667H, J 0.005kgm2,
K 0.763, hb (hysteresis band) 0.2. The
following figures represent the speed (figure
11.18), torque (figure 11.19) and current (figure
11.20) responses, and e.m.f. waveform (figure
11.21), for a starting process, loading (6Nm) at
0.2s, unloading at 0.4s, reversal at 0.5s and
loading (6Nm) again at 0.8s.
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Figure 11.18. Speed transient response
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Figure 11.19. Torque response
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Figure 11.20.Current waveform (ia)
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Figure 11.21.Induced voltage waveform (ea)
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11.4. VECTOR (SINUSOIDAL) CONTROL The vector
(sinusoidal) control is applied both for surface
PM and interior PM rotors and distributed stator
windings (q ? 2). According to chapter 10 the
torque Te expression is (11.33) The
space vector equation (chapter 10)
is (11.34) (11.35) (11.36
)
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In general Ld ? Lq and thus the second (reactive)
torque component is positive only if id ? 0. For
id ? 0 the PM flux is diminished by the id m.m.f.
(figure 11.22) but always the PM prevails to
avoid complete PM demagnetization. For vector
control, in general, the rotor does not have a
damper cage and thus there is no rotor circuit,
and, as a direct consequence, no current
decoupling network is necessary, in contrast to
IM case.
Figure 11.22. Space vector diagram of PM - SM
with negative id (id lt 0)
In vector control the PM - SM is controlled in d
- q coordinates and then the reference d - q
currents (or voltages) are transformed into
stator coordinates through the vector rotator
(inverse Park transformation) to be then realized
by PWM in the inverter.
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11.4.1. Optimum id - iq relationship Since the
current decoupling network is missing and the
drive command variable is the reference torque
Te as required by a speed (speed and position)
controller, we should now choose id and iq from
the torque equation (11.33). Evidently we need
one more equation. This additional information
may be obtained through an optimisation criterion
such as maximum torque per current, maximum
torque per flux, maximum efficiency etc. As at
low speeds the drive is current limited and above
base speed wb it is flux limited, we may use
these two criteria combined for a high
performance drive. The maximum torque / ampere
criterion is applied by using (11.34)
and (11.37) (11.38) to
find (11.39) For Ld Lq, starting with
(11.33), it follows that (11.40)
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On the other hand, for maximum torque per
flux (11.41) Proceeding as above we
obtain a new relationship between and
(11.42) where (11.43) For Ld
Lq (11.41) becomes (starting all over with
(11.33)) (11.44) Equation (11.44)
signifies complete cancellation of PM flux. The
PM is still not completely demagnetized as part
of the Ldid is leakage flux which does not flow
through the PM. Also notice that constant current
is means a circle in the id - iq plane and
constant stator flux ls means an ellipse.
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We may now represent (11.38) and (11.40) and
(11.41) and (11.42) as in figure 11.23.
Figure 11.23. id - iq a.) for given current is
b.) for given flux ls
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So, for each value of reference torque Te,
according to one of the two optimisation
criteria, unique values of id and iq are
obtained provided the current limit is and the
flux limit ls are not surpassed. Notice that
the flux limit is related to speed wr (through
11.43). So, in fact, the torque Te is limited
with respect to stator current and flux (speed)
(figure 11.23.a). (11.45) (11.46)
Te limit is dependent on speed (figure 11.24.b).
Figure 11.24. id - iq optimum relationship and
torque and flux limits with speed
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Example 11.3. A PM - SM with Vsn 120 V, isn
20A, ld 0.6p.u., no load voltage e0
0.7p.u., lq 1.2p.u., nn 15rps, p 1 (pole
pairs), rs 0, is driven at a current of 10A up
to base speed nb according to the criterion of
maximum torque per current. Calculate the torque
Tei up to base speed and the base speed
wb. Solution First we have to calculate ld, lq,
e0 in absolute units Ld, Lq, Eo (11.47)
(11.48) (11.49) The PM flux lPM
is (11.50)
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Making use of (11.39) with (11.50) (11.51
) (11.52) (11.53) and the
torque (11.54)
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Now we have to find the maximum speed for which
this torque may be produced (11.55)
(11.56) Consequently for rated voltage
(maximum inverter voltage) the base speed wb
is (11.57) (11.58) In a similar
way we may proceed above base speed using the
maximum torque / flux criterion.
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11.4.2. The indirect vector current
control Indirect vector current control means,
in fact, making use of precalculated and
((11.45 - (11.46)) to produce the reference d -q
currents id, iq. Then, with Park
transformation, the reference phase current
controllers are used to produce PWM in the
inverter. A rotor position sensor is required for
position and speed feedback (11.25).
Figure 11.25. Indirect vector current control of
PM - SM
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As for the IMs the a.c. controllers may be
replaced by d.c. current controllers (in rotor
coordinates) to improve the performance at high
speeds, especially. Though, in principle, direct
vector control is possible, it is hardly
practical unless the drive is sensorless.
11.4.3. Indirect voltage and current vector
control As expected, the vector current control
does not account for the e.m.f. effect of slowing
down the current transients with increasing
speed. This problem may be solved by using the
stator voltage equation for voltage
decoupling (11.59) (11.60)
The d.c. current controllers may replace the
first terms in (11.59) - (11.60) and thus only
the motion induced voltages are feedforwarded.
Then PWM is performed open loop to produce the
phase voltages at the motor terminals through the
PWM inverter (figure 11.26).
48
Figure 11.26. Indirect voltage and current vector
control of PMSM with dq (dc) current controllers
As expected the open loop (voltage) PWM has to
observe the voltage limit (11.61)
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At low speeds the current controllers prevail
while at high speeds the voltage decoupler takes
over. Results obtained with such a method 5,
based only on fdi, fqi (maximum torque per
current criterion) proved a remarkable
enlargement of the torque - speed envelope
(figure 11.27).
Figure 11.27. Torque - speed envelope for id 0
and for voltage and current control
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11.4.4. Fast response PM - SM drives surface PM
rotor motors with predictive control Interior PM
- SM drives have been so far treated for vector
control. As Lq is rather high in such motors, at
least iq response is rather slow in comparison
with IM drives. However for surface PM rotor PM -
SMs, Ld Lq Ls is small and thus fast current
(and torque) response may be obtained. The
current increment Dis is (11.62) This
classical method 7 has produced spectacular
results with position sensors (encoders) - figure
11.28.
The low speed / time linearity 7 is obtained
also by using special measures to reduce all
torque pulsations to less than 1.5 of rated
torque.
Figure 11.28. Speed responses at low speed
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Example 11.4. Digital simulation Indirect vector
a.c. current control For the PM - SM motor in
example 11.3, making use of the indirect vector
current control system of figure 11.25, perform
digital simulations. The simulation of this drive
was implemented in MATLAB - SIMULINK. The motor
model was integrated in a block (PM_SM) (figures
11.29, 11.30, 11.31).
Figure 11.29. id, iq referencers
52
Figure 11.30. A.c. current controllers
53
Figure 11.31. PM - SM block diagram
54
The motor used for this simulation has the
following parameters Pn 900W, Un 220V, 2p
4, n 1700rpm In 3A, lPM 0.272Wb, Rs
4.3W, Ld 0.027H, Lq 0.067H, J
0.00179kgm2.The following figures represent the
speed (figure 11.32), torque (figure 11.33) and
current waveform (figure 11.34), for a starting
process and load torque (2Nm) applied at 0.4s.
Figure 11.32. Speed transient response
55
Figure 11.33. Torque response
56
Figure 11.34.Current waveform (ia) under steady
state
57
The d - q reference current relationships to
torque for maximum torque per current are given
in figure 11.35.
Figure 11.35.
58
Example 11.5. Digital simulation indirect vector
d.c. current control of PM - SM The simulation of
this drive (figure 11.36) was implemented in
MATLAB - SIMULINK simulation program. The motor
model was integrated in the PM_SM block (figure
11.37).
The motor used for this simulation has the
following parameters Pn 900W, Un 220V, 2p
4, n 1700rpm In 3A, lPM 0.272Wb, Rs
4.3W, Ld 0.027H, Lq 0.067H, J 0.000179kgm2.
59
Figure 11.36. The indirect vector d.c. current
controller
60
Figure 11.37. PM - SM model
61
Figure 11.38. id- iq reference currents versus
torque for maximum torque / current criterion
62
The following figures represent the speed (figure
11.39), torque (figure 11.40) and d - q current
responses (figure 11.41, 11.42), for a starting
process and load torque (2Nm) applied at 0.4s.
Figure 11.39. Speed transient response
63
Figure 11.40. Torque response
64
Figure 11.41.Current waveform (id) under steady
state
65
Figure 11.42.Current waveform (iq) under steady
state