Volume 48, Number 1, 2007
43
Analysis of Constant Power Region for
the Induction Motor Control by Field
Weakening Technique
Alexandru MORAR
Abstract: The aim of the study we made in this paper is to establish the output conditions that the inverter
can provide to the motor inputs, from the current and voltage limits point of view. The study cases are
useful for the control by field weakening principle, under some imposed restriction. It is important for a
control system to be able to operate at a constant power, for example in spindle drives applications, in
traction and electrical drive of vehicles.
Keywords: induction motor, constant power region, inverter output conditions.
1. INTRODUCTION
In many papers were studied the control
systems of induction motors by field
weakening technique at high speeds above
the rated base speed [1],…,[6].
We
shall
use
the
following
nomenclature:
mb = motor necessary torque at the base
speed ωb;
mM = motor necessary torque at the
maximum speed ωmax;
Mb = maximum torque of the motor at
the base speed;
MM = maximum torque of the motor at
the maximum speed;
kmb = overload factor at the base speed,
kmb=Mb/mb;
kmM = overload factor at the maximum
speed, kmM=MM/mM;
kv = speeds ratio in the constant power
region, kv=ωmax/ωb.
Considering the case of a classical rotor
flux oriented control system, it has to
establish primarily the maximal torque value.
In the field weakening principle, it is well
known that (if the stator voltage Us reaches
the limit) the stator flux is necessary to be
reduced for obtaining the speed increasing.
This fact result subjecting the stator flux to
the constraint [1]:
Ψs <Ψs max
(1)
2. CONSTANT POWER REGION
ANALISYS
On the basis of the mathematical model
of the rotor flux oriented control, given by the
known equations:
Lm
3
(2)
me = z p
Ψ sq Ψ rd
2 σ Ls Lr
Ψ rd =
Lm
Ψ sq Ψ rd ; ird = 0
Ls
(3a,b)
we maximise the torque expression , provided
that:
Ψ 2sd + Ψ 2sq = Ψ s max
(4)
For a system with a correct adjustment of
its parameters, the torque expression is given
by:
⎛L
⎞
Lm
3
me = z p
Ψ rd Ψ 2s max − ⎜ s Ψ rd ⎟
2 σ Ls Lr
⎝ Lm
⎠
2
(5)
44
ACTA ELECTROTEHNICA
Looking for the maximum torque value
from (5), the optimal rotor flux reference
signal can be obtained now as:
Ψ *rd =
2 Lm
Ψ s max
2 Ls
(6)
The maximal torque value results as
following:
3 1−σ 2
3 1 − σ U s2max
zp
Ψ s max = z p
⋅
4 σ Ls
4 σ Ls ωλ2r
(7)
In the constant power region, the torque
mM and mb, at the maximum and,
respectively, at the base speed, can be related
by:
mω
m
(8)
mM = b b = b
ωmax
kv
M e max =
In the same way, we can establish a
relation between the maximal torque
corresponding to the speeds ωb and ωmax:
2
2
⎛U ⎞
M b U sb2 ωmax
(9)
= 2 ⋅ 2 = ⎜ sb ⎟ kv2
M M ωb U sM ⎝ U sM ⎠
Taking into account the overload factor
and the speed ratio expressions, it is got the
relation between voltages (from eq. (8) and
(9));
2
sM
2
sb
U
U
=
knM
kv
kmb
(10)
The drive, using the field weakening
technique, is shown in Fig. 1. Some proposal
for divers drive solutions can be advanced,
Fig.1. –Field weakening technique drive, the
curves of: power (1), maximal torque (2), stator
voltage (3) and load torque (4).
depending on the overload factor at the base
speed and on the inverter voltage, as well.
For these variants analysis, we need the
relation between electromagnetic torque and
the absorbed current of the induction motor,
in the case of the control by rotor flux
oriented method. Further, we shall present the
relations between the rotor current and the
torque, which issue in a simpler manner.
Because the inverter current is the stator one,
we shall consider, finally, the relation among
the rotor current, the stator current and the
magnetisation current.
Starting from eq. (2), (3 a,b) and (11):
1
(11)
isq =
Ψ
σ Ls sq
the flux components in the torque expression,
depending on the rotor current, will be
eliminated. Finally, the ratio between the
absorbed rotor current at the maximum speed
and, respectively, at the base speed, is
obtained:
2
rM
2
rb
i
i
1− 1−
1
2
knM
⎛ U sM 1 ⎞
=
⎜
⎟
1 U k
1 − 1 − 2 ⎝ sb v ⎠
kmb
2
(12)
The following study cases are significant
in analysis:
Case 1. In the first case, it is fixed: Usb = UsM
= constant; MM = mM, therefore kmM = 1.
From equation (10), we derive the
relation between the speeds ratio, which
decides the constant power region width, and
the overload factor at the base speed:
kv = kmb
(13)
From equation (13) it is obvious the
advantage of using a designed motor with an
overload factor as bigger as possible. From
equations (12) and (13), we emphasise the
relation between the absorbed rotor currents
at maximum and, respectively, at base speed,
considering the factor kmb > 1:
irM > irb
(14)
The equation (14) is equivalent with the
corresponding relation between the stator
currents at the same speed:
Volume 48, Number 1, 2007
irM < isb
(15)
For this reason, the inverter will be
designed taking into account the absorbed
stator current at the base speed.
Case 2. In this second case, it is fixed kmM =
kmb, imposing a constant overload through the
entire field weakening region.
From (10) we can deduce:
U sM = kv U sb
(16)
Namely, in order to have the same
overload factor through the entire constant
power region, the feeding voltage cannot be
maintained constant, but it must be increased
according ti (16).
Referring to the currents, from (12) cit
results:
irm < irb
(17)
and due to the magnetisation current
reduction, one obtains:
isM < isb
(18)
For this reason, the inverter will be
designed according to the absorbed stator
current at the base speed.
Case 3. In this case, a particular speeds ratio
kv is imposed and the overload factor at the
maximum speed is considered to be kmM = 1.
Therefore, the voltage at the maximum
speed can be written as:
U sM =
kv
U sb
kmb
(19)
so, by setting the speeds ratio kv, the feeding
voltage depends on the overload factor at the
base speed kmb. Referring to the currents,
from equation (12) it results:
2
irM
=
irb2
1
(20)
⎛
1 ⎞
kmb kv ⋅ ⎜1 − 1 − 2 ⎟
kmb ⎠
⎝
The rotor current irM, absorbed at the
maximum speed, depending on the overload
factor kmb and the speeds ratio kv, as well.
Thus, if:
kmb
then
k 2 +1
< v
2kv
(21)
45
irM < irb
(22)
and if the equation (21) is not satisfied, then
irM > irb.
3.
CONCLUSIONS
This study deals with some design
criteria useful for spindles induction motors,
fed by current regulated PWM inverters,
controlled by field orientation method,
particularly for operating in the flux
weakening region. Analysing the constant
power region, it is revealed witch heavier
conditions must be verified, from the motor
and the inverter absorbed currents point of
view. There are carried aut useful conclusions
for the increasing of the constant power
region, without oversizing the motor or the
inverter.
REFERENCES
1. Xu X., De Doncker R., Novotny D.W., Stator flux
orientation control of induction machines in the
field weakening region. Rec. IEEE-IAS, Annual
Meeting, Pittsburg, 437-443(1988).
2. Scutaru Gh., Apostoaia C.M., Speed Control in
FluxOriented Induction Motor for Spindle Drives.
Proc. of the 7th International Power Electronics
& Motion Control Conference, Technical
University of Budapest, 3, 410-413 (1996).
3. Xu X., Novotny D.W., Selection of the Flux
Reference for Induction Machine Drives in the
Field Weakening Region. IEEE Transactions on
Industry Applications, 28, No. 6, 1353-1358
(1992).
4. Schäfer U., Feldorienteriente Regelung einer
Asynchronmachine mit Feldschwächung unter
Berűcksichtigung
der
Eisensättigung
und
Erwärmung, Dissertation RWTH Aachen, 1989.
5. Boglietti A., Ferraris P., Lazzari M., Profumo F.,
Evolution of the Basic Induction Motors Project
Criteria based on Sampling Data and Statistical
Evaluations. Proc. of the EMAIM Conference
Record, Torino, 167-173 (1986).
6. Kim S.H., Sul S.K., Maximum Torque Control of
an Induction Machine in the Field Weakening
Region. IEEE Transactions on Industrie
Applications, 31, No. 4, 787-794 (1995).
Ph.D. Assoc. Prof.Eng. Alexandru MORAR
“Petru Maior” University of Tg.-Mures
Faculty of Engineering
1 Nicolae Iorga St., Ro-540088 Targu-Mures
E-mail: morar@upm.ro