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component is. Consequently, the saddle in the torque-speed curve will be more
visible in low power machines where the stator resistance is larger in relative
values rs = RsIn/Vn. Moreover the frequency f12 of stator current is close to f1 and
thus visible stator backward current pulsations occur. This may be used to
determine the slip S as f2 f1 = 2Sf1. Low frequency and noise at 2Sf1 may be
encountered in such cases.
7.16 ONE ROTOR PHASE IS OPEN
An extreme case of unbalanced rotor winding occurs when one rotor phase
is open (Figure 7.33). Qualitatively the phenomenon occurs as described for the
general case in paragraph 7.13. After reducing the rotor variables to stator, we
have:
j
Iar 'f = -Iar 'b = - Ibr ' (7.144)
3
1
Vr 'f = Vr 'b = (Var '-Vbr ') (7.145)
3
a
I =0
Ib I
=-
n
Vb V
=
ar
br
cr
i
bc
Figure 7.33 One rotor phase is open
Now, with (7.140) and (7.142), the unknowns are Isf, Isb, Ir2 f, Ir2 b, and, Vr2 f.
This time we may get (from (7.142) with (7.144 and 7.145)):
© 2002 by CRC Press LLC
Author: Ion Boldea, S.A.Nasar& & & & ..& & & ..
- jÉ1(1- 2S)L1m Ir 'b
Isb = (7.146)
Rs + jÉ1(1- 2S)Ls
It is again clear that with Rs = 0, the stator backward current is
-L1m
(Isb) = Ir 'b (7.147)
R =0
s
Ls
Consequently the backward torque component in (7.143) becomes zero.
7.17 WHEN VOLTAGE VARIES AROUND RATED VALUE
It is common practice that, in industry, the local power grid voltage varies
around rated values as influenced by the connection (disconnection) of other
loads and of capacitors used for power factor correction. Higher supply
voltages, even when balanced, notably influence the induction machine under
various load levels.
A ±10% voltage variation around the rated value is, in general, accepted in
many standards. However, the IMs have to be designed to incur such voltage
variations without excessive temperature rise at rated power.
In essence, if the voltage increases above the rated value, the core losses
piron in the machine increase notably as
piron H" (Chf1 + Cef12)V2 (7.148)
In the same time, with the core notably more saturated, L1m decreases
accordingly.
The rated torque is to be obtained at lower slips. The power factor decreases
so the stator current tends to increase while, for lower slip, the rotor current
does not change much. So the efficiency, not only the power factor, decreases
for rated load.
The temperature tends to rise. In the design process, a temperature reserve is
to be allowed for nonrated voltage safe operation.
On the other hand, if low load operation at higher than rated voltage occurs,
the high core losses mean excessive power loss. Motor disconnection instead of
no-load operation is a practical solution in such cases.
If the voltage decreases below rated value, the rated torque is obtained at
higher slip and, consequently, higher stator and rotor currents occur. The
winding losses increase while the core losses decrease, because the voltage (and
magnetizing flux linkage) decreases.
The efficiency should decrease slowly, while the power factor might also
decrease slightly or remain the same as with rated voltage.
A too big voltage reduction leads, however, to excessive losses and lower
efficiency at rated load. With partial load, lower voltage might be beneficial in
terms of both efficiency and power factor, as the core plus winding losses
decrease toward a minimum.
© 2002 by CRC Press LLC
Author: Ion Boldea, S.A.Nasar& & & & ..& & & ..
When IMs are designed for two frequencies 50 or 60Hz care is
exercised to meet the over temperature limitation for the most difficult situation
(lower frequency).
7.18 SUMMARY
" The relative difference between mmf speed n1 = f1/p1 and the rotor speed n
is called slip, S = 1-np1/f1.
" The frequency of the emf induced by the stator mmf field in the rotor is f2 =
Sf1.
" The rated slip Sn, corresponding to rated nn, Sn = 0.08 0.01; larger values
correspond to smaller power motors (under 1 kW).
" At zero slip (S = 0), for short-circuited rotor, the rotor current and the
torque is zero; this is called the ideal no-load mode n = n1 = f1/p1.
" When the rotor windings are fed by balanced three-phase voltages Vr2 of
frequency f2 = Sf1, the zero rotor current and zero torque is obtained at a
slip S0 = Vr2 /E1, where E1 is the stator phase emf. S0 may be positive
(subsynchronous operation) or negative (supersynchronous operation)
depending on the phase angle between Vr2 and E1.
" The active power traveling through an airgap (related to Pointing s vector)
is called the electromagnetic power Pelm.
" The electromagnetic torque, for a short-circuited rotor (or in presence of a
passive, additional, rotor impedance) is
P1
Te = Pelm
É1
" At ideal no-load speed (zero torque), the value of Pelm is zero, so the
machine driven at that speed absorbs power in the stator to cover the
winding and core loss.
" Motor no-load mode is when there is no-load torque at the shaft. The input
active power now covers the core and stator winding losses and the
mechanical losses.
" The induction motor operates as a motor when (0
" For generator mode S 1.
" In all operation modes, the singly fed IM motor absorbs reactive power
for magnetization.
" Autonomous generating mode may be obtained with capacitors at terminals
to produce the reactive power for magnetization.
" At zero speed, the torque is nonzero with stator balanced voltages. The
starting torque Tes = (0.5 2.2)Ten. Ten rated torque; starting current at
rated voltage is Istart = (5 7(8))In; In rated current. Higher values of
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