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Unformatted text preview: ircuits (Fig.4.13) are
used to determine I 2 , in Equation (4.54), Vth , Rth , and X th
should be replaced by V1 , R1 , and X 1 , respectively. The
prediction of performance based on the approximate equivalent
circuit (Fig.4.13) may differ by 5 percent from those based on the
equivalent circuit of Fig.4.8 or 4.16.
For a threephase machine, Equation (4.54) should be multiplied
by three to obtain the total torque developed by the machine. The
torquespeed characteristic is shown in Fig.4.14. At low values of
slip, Rth + ′
R2
R′
′
>> X th + X 2 and 2 >> Rth
s
s Thus Tmech 2
Vth
* *s
≅
′
ω syn R2 1 (4.55)
(4.56) The linear torquespeed relationship is evident in Fig.4.14 near
the synchrouns speed. Note that if the approximate equivalent
circuits (Fig.4.13) are used, in Equation (4.56) Vth should be
replaced by V1 . At larger values of slip, Rth + ′
R2
′
<< X th + X 2
s And Tmech 2
Vth
R′
=
*
*2
ω syn ( X th + X 2 )2 s
′ 1 (4.57)
(4.58) ThreePhase Induction Machine 253 The torque varies almost inversely with slip near s = 1, as seen
from Fig.4.14. Fig.4.14 Torquespeed profile at different voltages.
Equation (4.49) also indicates that at a particular speed (i.e., a
fixed value of s ) the torque varies as the square of the supply
voltage Vth (hence V1 Fig.4.14) shows the Tn profile at various
supply voltages.
An expression for maximum torque can be obtained by setting dT / ds = 0 . Differentiating Equation (4.54) with respect to slip s
and equating the results to zero gives the following condition for
maximum torque: ′
R2
2
′2
= Rth + ( X th + X 2 )
STmax (4.59) 254 Chapter Four
This expression can also be derived from the fact that the
condition for maximum torque corresponds to the condition for
maximum air gar power (Equation (4.51)). This occurs, by the
familiar impedancematching principle in circuit theory, when the ′
impedance of R2 / s equals in magnitude the impedance between
it and the supply voltage V1 (Fig.4.19) as shown in Equation (4.59).
The slip STmax at maximum torque Tmax is: STmax = ′
R2 2
Rth (4.60) ′
+ ( X th + X 2 ) 2 The maximum torque per phase from Equations (4.49) and
(4.60) is: Tmax = 1
2ω syn * 2
Vth Rth + 2
Rth ′
+ ( X th + X 2 ) 2 (4.61) Equation (4.61) shows that the maximum torque developed by
the induction machine is independent of the rotor circuit
resistance. However, from Equation (4.60) it is evident that the
value of the rotor circuit resistance R2 determines the speed at
which this maximum torque will occur. The torque speed
characteristics for various values of R2 are shown in Fig.4.15. In a
wound rotor induction motor, external resistance is added to the
rotor circuit to make the maximum torque occur at standstill so
that high starting torque can be obtained. As the motor speeds up, ThreePhase Induction Machine 255 the external resistance is gradually decreased and finally taken out
completely. Some induction motors are, in fact, designed so that
maximum torque is available at start, that is, at zero speed. Fig.4.15 Torque speed characteristics for varying R2 .
If the stator resistance R1 is small (hence Rth is negligibly
small), from Equations (4.60) and (4.61), sTmax ≅
Tmax ′
R2
′
X th + X 2 2
Vth
*
=
′
2ω syn X th + X 2 1 (4.62) (4.63) 256 Chapter Four
Equation (4.63) indicates that the maximum torque developed
by an induction machine is inversely proportional to the sum of the
leakage reactances.
From Equation (4.54), the ratio of the maximum developed
torque to the torque developed at any speed is: ′
′
(Rth + R2 / s )2 + ( X th + X 2 )2 * s
Tmax
=
2
T
′
′2
Rth + R2 / sTmax + ( X th + X 2 ) sTmax ( ) (4.64) If R1 (hence Rth ) is negligibly small, ′
′
(R2 / s ) + ( X th + X 2 ) * s
Tmax
=
2
T
′
′2
R2 / sTmax + ( X th + X 2 ) sTmax
2 ( 2 ) (4.65) From Equations (4.62) and (4.65) ( ′
′
Tmax (R2 / s ) + R2 / sTmax
=
2
T
′
2 R2 / sTmax
2 ( 2
sTmax + s 2
Tmax
=
T
2 * sTmax * s ) ) 2 * s
sTmax (4.66) (4.67) Equation (4.67) shows the relationship between torque at any
speed and the maximum torque in terms of their slip values. ThreePhase Induction Machine 257 4.11 Efficiency
In order to determine the efficiency of the induction machine as
a power converter, the various losses in the machine are first
identified. These losses are illustrated in the power flow diagram
of Fig.4.16. For a 3 φ machine the power input to the stator is: Pin = 3V1I1 cosθ1 (4.68) The power loss in the stator winding is: P = 3I12 R1
1 (4.69) Where R1 is the AC resistance (including skin effect) of each
phase winding at the operating temperature and frequency.
Power is also lost as hysteresis and eddy current loss in the
magnetic material of the stator core.
The remaining power, Pag , crosses the air gap. Part of it is lost
in the resistance of the rotor circuit.
2
P2 = 3 I 2 R2 (4.70) Where R2 is the ac resistance of the rotor winding. If it is a
woundrotor machine, R2 also includes any external resistance
connected to the rotor circuit through slip rings.
Power is also lost in the rotor core. Because the core losses are
dependent on the freq...
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This document was uploaded on 03/12/2014 for the course ENGINEERIN electrical at University of Manchester.
 Spring '14

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