Tài liệu High Performance Driver P6b pdf

HIGH PERFORMANCE DRIVES

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Summarising the procedure: speed loop is held open, d-axis current command is set to rated and q-axis
current command of the alternating square waveform is imposed. If the rotor time constant value in the
controller is correctly set, speed response has to be triangular function. Figure 3.21a is valid for the
correct (rated) value of the rotor time constant and speed response is a triangular function in this case.
However, if the rotor time constant significantly deviates from the correct value (1.7 times rated value in
Fig. 3.21b) the speed response is unsatisfactory and far from required triangular waveform.
3.6. DESIGN OF THE CONTROL SYSTEM FOR AN INDIRECT FEED-FORWARD
CURRENT-FED ROTOR FLUX ORIENTED INDUCTION MACHINE
Calculation of all the necessary values, required for indirect rotor flux oriented control, will be
illustrated using an example. Consider a three-phase four-pole star connected squirrel-cage induction
machine, whose parameters at 50 Hz are
RRXXX
srsrm
=====10 6 3 135 12.6 132ΩΩΩΩΩ
γγ
.
Rated current and voltage equal 2.1 A and 380 V. The machine is to be operated as an indirect rotor
flux oriented current-fed induction machine. Current control is performed with phase current controllers
so that actual and reference phase currents can be assumed to be the same for the purpose of
calculation. The speed is to be controlled from zero up to its rated value, using a constant, rated value of
rotor flux. The rated torque and inertia of the machine are 5.07 Nm and 0.1 kgm
2
, respectively.
It is required to determine all the values needed for realisation of indirect rotor flux oriented
control. These are the rated rotor flux, the rated stator d-axis and q-axis currents (all in terms of rms
and peak values) and the value of the slip speed for rated torque operation. Next, since the scheme under
consideration contains only the speed controller, it is necessary to determine the parameters of the speed

PI controller. This is done using the so-called symmetrical optimum method. The time delay due to the
inverter and signal processing may be approximated with a first order delay block whose time constant
is 0.05 ms (the reasons and the need for this will be explained later).
The controller under consideration is the one shown in Fig. 3.10 for the general case of variable
rotor flux reference. Since in this example rotor flux reference is kept constant at all times, the overall
drive configuration is as shown in Fig. 3.11. The complete indirect vector controller (for variable rotor
flux reference case) and its implementation as a part of the indirect feed-forward rotor flux orientation
scheme (for the case of constant rotor flux reference) are shown for convenience once more in the upper
and the lower part of Fig. 3.22. Constants in the implementation in the lower part of Fig. 3.22 follow
directly from (3.30)-(3.31), taking into account that, due to constant rotor flux reference,
ψ
rm
d
s
Li
**
= :
iKT KiT
P
L
LP
L
Li
Ki K i
L
TTi
qs e qs e
r
mr
r

mds
sl qs sl qs
m
rr
r
d
s
** **
**
** **
**
= == =
=
===
11
2
22
2
3
12
3
1
1
ψ
ωω
ψ
(3.32)
On the basis of Fig. 3.22 and (3.32) one concludes that the required values for an indirect vector
controller are the rated rotor flux value and constants
KK

1
2
, . The inductances of the machine and rotor
time constant are needed in (3.32). Note that the stator parameters are not required (although they are
given, for the sake of completeness). From given reactances one easily finds the inductances and the
rotor time constant:
()
()
LX
LX
LLL
TLR
mm
rr
rmr
rrr
===
===
=+=
== =
2 50 132 100 0 42
2 50 12.6 100 0 04
046
046 63 73ms
π
π
ππ
γγ
γ
/.

/.
.
./.
H
H
H
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64
ψ
r
* i
ds
*
T
r
s
1/
L
m
T
e
*
2
L
r
i
qs
*

3
PL
m
ω
sl
*
ω
r
φ
r
e
L
r
1/
s
ω
ψ
r
*
i
ds
*
i
α
s
*
i
a
*
i

a
1/
L
m
2C
R
j
φ
r
e
i
b
*
P
i
b
e
W
ω
* T
e
* i
qs
*
M
PI K
1
3
_
i

β
s
*
i
c
*
i
c
K
2
φ
r
e
ω
sl
*
ω
r
I
1/
s
M
ωω
mech
P
Fig. 3.22 - Full indirect vector controller and its implementation for the case of constant rotor flux
reference operation (base speed region only).
When the machine operates in steady-state under rated operating conditions (index n), then it follows
from (3.9) that
TP

L
L
i
en
m
r
rn qsn
=
3
2
ψ
()
ψ
ωψ
rn m dsn
sl n m qsn r rn
Li
Li T
=
=
(3.33)
since
ψψ ωω
rrnL
e
nsl sln
TT
*
,,===. Furthermore, the stator current has to be equal to its rated value as
well. Note however that the given rated current is rms value, while the one obtained from d-q axis

currents is the peak value. Hence
iii
iI
sn dsn qsn
sn sn
=+
== =
22
222.12.97xA
(3.34)
As the next step, it is necessary to calculate the rated stator d-q axis current components. Torque
equation of (3.33) and stator current equation (3.34) are used for this purpose, since the rated torque
value and the rated stator current are known quantities:
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TP
L
L
iP
L
L
ii
ii
ii i i
en
m
r
rn qsn

m
r
dsn qsn
dsn qsn
dsn qsn qsn dsn
==
=
=
=
3
2
3
2
507 3
042
046
507 115 4.407
2
2
ψ
.
.
.

Substitution into stator current equation (3.34) yields
iii
iii
ii
ii
ii

xi
xx
xi
sn dsn qsn
sn dsn qsn
dsn dsn
dsn dsn
dsn dsn
dsn
dsn
=+=
==+
=+
=+
−+=
=
−+=
=
=
22 2
222
22
24
42
2
2
12 12
2.97
88209
88209 19.42

8821 19.42
8821 19.42 0
8821 19.42 0
4.59
4.23
2.1426 A
2.057 A
/
.
.
.
.
.
,,
The correlation between d-q axis currents yields values of stator q-axis current:
ii
i
qsn dsn
qsn
=
=
4.407
2.0568
12,
A
2.1424 A
One notes that the same two values appear in inverse order as solutions for rated stator d-q axis
currents. Normally the correlation
ii
dsn qsn

< holds true, so that finally
ii
dsn qsn
==2.057 2.1424AA
Once when the stator rated d-q axis currents are known, it becomes possible to determine the rated rotor
flux and the constants of (3.32):
K
P
L
Li
K
Ti
Li
r
mds
rds
rn m dsn
1
22
2
2
3
1
1
3
0
46
042
1
2.057

0 4226
11
0 073x2.057
533rad
0 42 2.057 0864
== =
== =
== =
*
*
.
.
.
.
.

A/Nm
/As
xWb
ψ
The calculated values of the stator d-q axis current components and the rotor flux are the peak values.
Corresponding rms values are
Ψ
rn
dsn
qsn
dsn qsn sn
I
I
II I

==
==
==
+= + = ≡
0 864 2 0 61
2.057 2 14545 A
2.1424 2 1515 A
14545 1515 2.1
22 2 2

.
.

Wb
A
It is now possible to determine the rated angular slip frequency as well. Note however that the slip
frequency is a variable, determined by the instantaneous value of the stator q-axis current command.
Hence the rated value of the angular slip frequency will exist only when the stator q-axis current
command is exactly equal to its rated value. From (3.33)
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()
ωψ
ωω
sl n m qsn r rn qsn r dsn
sl n mech sl n
Li T i Ti
P

rad / s
rad / s
=== =
== =
2.1424 0 073x2.057 14.267
14.267 2 7.1337
/( . )
//
()
Since rated synchronous (mechanical) speed is known (1500 rpm for 50 Hz for a four-pole machine),
one may calculate the rated speed of rotation of the machine as well,
n
n sl n mech
=− =− =1500
60
2
1500 30 7.1337 14319
π
ωπ
xrpm
()
/.
This completes the necessary calculations related to the machine parameters. What now remains to be
done is to design a PI speed controller. In order to perform this task, it is necessary to somehow
represent the whole control system and the machine with a simple transfer function diagram. Consider
the structure of Fig. 3.22. In this scheme the induction machine can be represented with the block
diagram of Fig. 3.2. The resulting complete block diagram is shown in Fig. 3.23.
ψ
r
* i

ds
* i
α
s
* i
a
* i
a
1/
L
m
2C
R
j
φ
r
e
i
b
*
P
i
b
e
W
ω
* T
e
* i
qs

*
M
PI K
1
3
_
i
β
s
* i
c
* i
c
K
2
φ
r
e
ω
sl
*
ω
r
1/
s
ω
ω
i
a
i

αs
i
ds
1
ψ
r
L
m
i
b
3
−
j
φ
r
1+sT
r
e
i
c
2i
βs
i
qs
L
m
ω
sl
T
r

φ
r
3P L
m
T
e
P
ω
2L
r
Js
T
L
ω
r
1
s
Fig. 3.23 - Block diagram of the indirect vector controller and the induction motor.
According to (3.1), reference and actual phase currents can be regarded as being mutually equal. Hence
the current controlled PWM inverter can be omitted from the scheme in Fig. 3.23. Furthermore, in the
absence of detuning, the estimated and the actual rotor flux position angles are the same (i.e. axes d*
and d of Fig. 3.20 coincide). Hence the two co-ordinate transformation blocks exp(jφ
r
) and exp(-jφ
r
)
cancel each other. The same applies to transformation blocks 2/3 and 3/2 in Fig. 3.23. Since stator d-
axis current reference is constant, rotor flux in the machine is, after initial transient (initial excitation)
constant as well. This enables Fig. 3.23 to be simplified to the form shown in Fig. 3.24.
Note that the flux channel does not contain a controller. Furthermore, after initial transient (initial

excitation) rotor flux is constant and equal to its reference. Hence the upper channel of Fig. 3.24 can be
omitted from further consideration. Only the lower channel, in which the speed PI controller is, has to be
considered further on. Note that in this channel the constant K
1
and the block 32PL L
mr
cancel each
other. Thus the transfer function block diagram, required for the design of the speed controller, reduces
to the one shown in Fig. 3.25.
Figure 3.25 is the final structure of the transfer function block diagram. Only one minor modification is
still necessary. Speed PI controller is designed for the response to the input speed command - hence
disturbance term (load torque) can be ignored. The resulting structure is the one of Fig. 3.26.
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Controller Machine
ψ
r
* i
ds
* i
ds
= i
ds
*
ψ
r
=
ψ

r
*
1/L
m
L
m
ω
* T
e
* i
qs
* i
qs
= i
qs
*3PL
m
PI K
1
_2L
r
T
e
−
T
L
ω
mech
ω
P 1/(Js)

Fig. 3.24 - Simplified block diagram of an indirect rotor flux oriented induction machine.
T
L
ω*
T
e
*=
T
e
−
PI
−
ω
P
Js
Fig. 3.25 - Speed control loop.
ω* T
e
*=T
e
P
PI
− Js
ω
Fig. 3.26 - Speed control loop with omitted disturbance term.
Recall that the inverter was taken as ideal in the very beginning. However, some delay always occurs in
the system due to signal processing and delay in inverter response to the command. These delays are
conveniently represented in a somewhat superficial way, by inserting a first order delay block in
between the torque command and the actual torque. Hence the transfer function block diagram that is
used for the speed controller design takes the final form of Fig. 3.27. Time constant of the first order

delay block, T
delay
, is in this example equal to 50 micro-seconds.
Transfer function of a PI controller is given with
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()
Gs K
sT
KsT
sT
p
i
pi
i
()=+=
+
1
1
1
(3.35)
ω* T
e
*1 T
e
P
PI
− 1+sT

delay
Js
ω
Fig. 3.27 - Final structure of the block diagram for the speed controller design.
Parameters that need to be determined are the proportional gain and the integral time constant, K
p
and
T
i
. The transfer function of the process is essentially represented with a first order delay block
(representing all the small delays in the system) and a pure integrator (representing mechanical
subsystem of the motor). Note that, without the first order delay block (that was artificially added at the
end of the transfer function block diagram derivation) the complete indirect rotor flux oriented induction
machine is represented with a single, pure integrator block (Fig. 3.26). This means that the complete
electro-magnetic part of the motor behaves as an ideal plant that instantaneously responds to the given
command (recall that, by definition, torque response is instantaneous in a rotor flux oriented induction
machine and is achieved by an appropriate, again instantaneous, change in the stator q-axis current;
instantaneous change in the current is possible since ideal current feeding was assumed).
Since the complete drive transfer function consists of a first order delay block with very small time
constant and a pure integrator block, so-called symmetrical optimum method is used to calculate the
parameters of the PI controller (this method is in general used when there is a pure integrator in the
plant transfer function; when there is no pure integrator and the plant is represented with first order
delay blocks only, another method called modulus optimum is used).
Calculation of the parameters of the PI controller depends on the ratio of time constants in the plant
transfer function. Let the sum of all the small delays be denoted with σ (in this case sum of all the small
delays equals T
delay
) and let the dominant time constant in the plant transfer function be T
dom
.Dominant

time constant is here the time constant of the integrator block: T
dom
= J/P = 0.1/2 = 0.05 sec. One then
looks at the ratio of the dominant time constant to four times the sum of small delays in the system. For
the data of this drive, shown in Fig. 3.28, one finds that the ratio of dominant time constant to four
times the sum of small time delays is
ω* K
p
(1+sT
i
) T
e
*1 T
e
1
− sT
i
1+0.00005s 0.05s
ω
PI First order delay
Integrator
block block
Fig. 3.28 - Transfer block diagram for the speed PI controller design, using data of the drive under
consideration.
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()
T

dom
4
005
450 10
250 1
6
σ
==>>
−
.
x
When this ratio is much larger than one (this is in general always the case in vector controlled drives),
the proportional gain and the integral time constant of the speed PI controller are determined with
KT
T
pdom
i
=
=
2
4
σ
σ
(3.36)
Therefore, for the numerical values of this drive, one gets
()
()
K
T
T

p
dom
i
== =
== = =
−
−
2
005
250 10
500
4 4 50 10 0 0002 0 2
6
6
σ
σ
.

x
xsms
When the speed controller is designed using symmetrical optimum, an overshoot of 43% will results in
the speed response to application of a step speed command. This in general cannot be tolerated and the
overshoot is reduced by inserting a smoothing element in the channel of the speed command. The
smoothing block is a first order delay block, whose time constant equals four times the sum of small
delays in the system. Hence the transfer function of the smoothing element is
Gs
s
smooth
()
()

=
+
1
14
σ
(3.37)
and its time constant equals the PI controller integrator time constant, 0.2 ms here. The overshoot in the
speed response is reduced, by inserting the smoothing element, to 8.1%. The final outlay of the speed
controller part of the drive is shown in Fig. 3.29, both in general form and with the calculated values for
the drive under consideration.
Limiter
ω
*1
K
p
(1+
sT
i
)
T
e
*
1+(4σ)s −
sT
i
ω
Reference PI
smoothing controller
ω
Limiter

ω* 1 500(1+0.0002
s
)
T
e
*
1+0.0002s − 0.0002
s
ω
Reference PI
smoothing controller
ω
Fig. 3.29 - Speed control loop designed using symmetrical optimum: general outlay and specific
values for the drive under consideration.
Fig. 3.29 includes, apart from the reference smoothing element and the speed PI controller, one specific
control block that has not been discussed so far in any considerable depth. The block is called ‘limiter’
and it serves the purpose of limiting the output of the speed controller to the maximum permissible
value. For the purpose of explaining its role, suppose that the step speed reference, equal to rated
(electrical) angular speed of the machine under consideration, is applied from standstill at time instant
zero. Rated speed has already been calculated and it corresponds to 299.9 rad/s ≈ 300 rad/s. The
smoothing element has very small time constant (meaning that its output becomes equal to the input in
very short time interval of approximately five times 4σ, that is 1 ms) and therefore its existence can be
neglected for the sake of this discussion. Since the reference speed at the input of the summator equals
300 rad/s, while the actual speed is zero, the speed error equals 300 rad/s. Passing of this speed error
through the speed controller whose proportional gain is 500 means that the torque reference at the
output of the speed controller initially equals 300 x 500 = 150,000 Nm! Recall that the rated torque is
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5.07 Nm. This means that stator q-axis current would initially be required to be equal to approximately
30,000 times its rated value. This obviously must not be allowed since the inverter would be destroyed
immediately. The output of the speed controlled must therefore be limited to the maximum permitted
torque value, that corresponds to the maximum permitted stator q-axis current value. The maximum
permitted stator q-axis current value is determined by the rating of the inverter. Typically in high
performance drives, continuous current rating of the inverter would be 100% to 150% of the machine’s
current rating, while the inverter short term (transient rating) may be up to 300% (and rarely higher) of
the continuous current rating. Short term means that such a high current can persist during a transient
only. Hence the torque would be typically limited to 200% to 300% of the machine’s rated torque.
When the output of the speed controller is smaller than the limiter adjustment, the limiter passes through
the torque command (i.e. speed controller output) without affecting its value. If the speed controller
output exceeds the limiter adjustment, the limiter gives at the output the adjusted maximum value.
Hence the limiter is often represented with the block illustrated in Fig. 3.30.
T
e(out)
T
e(lim)
T
e(in)
*T
e(in)
T
e(out)
*
−T
e(lim)
Fig. 3.30 - Limiter.
Adjustment of the limiter for negative and positive inputs is always the same in electric drives. The
action of the limiter can be described with
If T

e(in)
* < T
e(lim)
then T
e(out)
* = T
e(in)
* (3.38a)
If T
e(in)
* >T
e(lim)
then T
e(out)
* = T
e(lim)
(3.38b)
where T
e(lim)
is the limiter adjustment.
3.7. PROBLEMS
1. An induction machine has the following parameters (all rotor quantities are referred to the stator):
RR LL L
sr sr m
== == =10 63 004 04HHΩΩ
γγ
The machine has four poles and rated power and rated speed of 0.75 kW and 1400 rpm, respectively,
for the rated 380 V, 50 Hz three-phase supply with a star connected stator winding.
(a) Give the complete time domain model of the machine in an arbitrary rotating reference frame in
terms of d-q axis variables.

(b) Derive the space vector model of the induction machine in an arbitrary reference frame, using the
model given in (a) as the starting point. Sketch the dynamic space vector equivalent circuit of an
induction machine in the arbitrary reference frame.
(c) Derive and calculate the stator phase voltage d-q axis components and the stator phase voltage
space vector, if the machine is fed with the sinusoidal phase voltages
vVtvVt vVt
ab
c
==−=−2223243cos cos( / ) cos( / )
ωωπ ωπ
of rated rms value and frequency, and the speed of the reference frame is selected as synchronous (d-
axis of the reference frame aligned with the stator phase voltage space vector).
(d) Calculate the stator current space vector and the stator current d-q axis components in the
synchronously rotating frame if the machine operates with rated load torque and is fed with the
voltages given in (c).