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INDUCTION MOTOR # Torque and Power by use of Thevenin’s Theorem (or Torque-Slip Characteristics): When torque and power relations are to be emphasized, considerable simplification results from application of Thevenin's network theorem to the induction-motor equivalent circuit. In its general form, Thevenin's theorem permits the replacement of any network of linear circuit elements and complex voltage sources, such as viewed from two terminals a and b (Fig. 1.15a), by a single complex voltage source Veq in series with a single impedance Zeq (Fig. 1.15b). The Thevenin-equivalent voltage Veq is that appearing across terminals a and b of the original network when these terminals are open-circuited; the Thevenin-equivalent impedance Zeq is that viewed from the same terminals when all voltage sources within the network are set equal to zero. Figure 1.15 (a) General linear network and (b)its equivalent at terminals ab by Thevenin's theorem. For application to the induction-motor equivalent circuit, points a and b are taken as those so designated in Fig. 1.13 a and b. The equivalent circuit then assumes the forms given in Fig. 1.15 c and d where Thevenin's theorem has been used to transform the network to the left of points a and b into an equivalent voltage source VTH in series with an equivalent impedance ZTH RTH jX TH . R1 X1 Ie I1 V1 a Zf X2 R1 I2 jX b Fig. 1.15 (c) X1 E2 V1 X2 Ie I1 R2 s a Zf R2 I2 jX E2 1 s R2 ( ) s b Fig. 1.15 (d) For most Induction Motors (IM), negligible error results from neglecting the stator resistance. The Thevenin’s-equivalent stator impedance ZTH is the impedance between terminals a and b of Fig. 1.15 c and d, viewed toward the source with the source voltage set equal to zero (or equivalently replaced by a short circuit) and therefore is, ZTH ( R1 jX 1 ) || ( jX ) RTH jX TH and VTH V1 jX R1 j ( X 1 X ) Poly-Phase Induction Motor RTH jXTH jX2 a R2 I2 1 s R2 ( ) s E2 VTH b Fig. 1.15(e). Thevenin equivalent of induction motor circuit model. The circuit then reduces to Fig 1.15(e) in which it is convenient to take VTH as the reference voltage VTH From Fig. 1.15(e). I 2 ( RTH R2 s ) j ( X TH X 2 ) Also electromagnetic torque is, Te For ‘m’ phase, Te s {( RTH VTH2 . So, I 22 ( RTH R2 s . ) 2 ( X TH X 2 )2 2 1 I 22 R2 VTH ( R2 s ) . s s s {( RTH R2 s )2 ( X TH X 2 )2 } mVTH2 ( R2 s ) mVTH2 ( R2 s ) . R2 2 R2 2 2 2 ) ( X TH X 2 ) } (2 N s 60){( RTH ) ( X TH X 2 ) } s s Figure 1.15(f). IM torque-slip curve showing braking, motor and generator regions. The variation of torque with slip or speed of an IM can be plotted from the above equation for different values of slip ‘s’ and with the motor connected to constant-frequency voltage source. A general shape of the torque-speed or torque slip curve is shown in figure 1.15 f. depending upon the value of slip an IM can have the following operating regions or modes: ____________________________________________________________________________________ mkdas@imu.ac.in Page 2 of 7 Poly-Phase Induction Motor Motoring Mode: 0 s 1 Under normal operation, rotor revolves in the direction of rotating field produced by the stator currents. As such, the slip varies from 1 at standstill to zero at synchronous speed, i.e. 1 s 0 . The corresponding speed values are zero ( s 1 ) and synchronous speed ( s 0 ). Generating Mode: s 0 For this operating mode, slip is negative, i.e. s 0 . An IM will operate in this region only when its stator terminals are connected to constant-frequency voltage source and its rotor is driven above synchronous speed by a prime mover. The connection of stator terminals to voltage source is essential in order to establish the rotating air-gap field at synchronous speed. In case stator is disconnected from voltage source and rotor is driven above synchronous speed by the prime mover, no generating action would take place. Braking Mode: s 1 For this mode, slip is greater than 1. A slip more than one can be obtained by driving the rotor, with a prime mover, opposite to the direction of rotating field. But such a use in practice is rare. A practical utility of slip more than 1 is obtained by bringing the rotor to a quick stop by braking action, called ‘Plugging’. For obtaining s 0 , or for obtaining plugging, any two stator leads are interchanged. With this the phase sequence is reversed and therefore, the direction of rotating magnetic field becomes suddenly opposite to that of the rotor rotation. The electromagnetic torque Te, now acting opposite to rotor rotation, produces the braking action. Thus the motor can be quickly brought to rest by plugging, but the stator must be disconnected from the supply before the rotor can start in the other direction. All the three regions of operation (braking region, s = 2 to s = 1; motor region, s = 1 to s = 0 and generator region, s = 0 to s = -1) are illustrated in figure 1.15 f. The torque-speed characteristics of an IM in the operating region resembles that of a shunt motor because, even at full load the slip does not exceed 5%, that is, speed varies from 95% to very close to synchronous speed for different loads. Therefore, the IM torque-speed characteristics are stated to have shunt characteristics. # Maximum Torque and Maximum Power: The maximum torque is also referred to as stalling torque or pull-out torque or breakdown torque Tem. The condition for maximum internal torque can be obtained by using the maximum power transfer theorem of circuit theory. The electromechanical torque is a maximum when the power delivered to R2/s in Fig. 1.13a is a maximum. The torque equation of an IM, as given by, Te 1 I 22 R2 mVTH2 ( R2 s ) mVTH2 RL 2 2 s s s {( RTH R2 s ) 2 ( X TH X 2 ) 2 } s {( R RL ) X } ____________________________________________________________________________________ mkdas@imu.ac.in Page 3 of 7 Poly-Phase Induction Motor Where, RL = R2/s, RTH = R and X = XTH + X2. Again the torque can be considered power in the variable resistance R2/s = RL of 1.15f. For maximum torque or maximum power transfer to RL, we have to equate dTe 0 . So, dRL dTe ( R RL ) 2 X 2 2 RL ( R RL ) mVTH2 0 dRL {( R RL )2 X 2 }2 2 ( X TH X 22 ) 2 Or, RL2 R 2 X 2 ; RL R 2 X 2 RTH The above equation is the well-known relationship for maximum power transfer which states that the load resistance is to be equal to the magnitude of internal or the remaining impedance in the series circuit. Let, smt be the slip at maximum torque, then RL So, S mt Or, Tmax R2 2 RTH ( X TH X 2 )2 1 mVTH2 s 2( R R 2 X 2 ) R2 2 RTH ( X TH X 22 ) 2 . smt and maximum torque, Tmax 0.5 2 mVTH mVTH2 R2 X 2 s 2( R R 2 X 2 R 2 X 2 ) 1 2 s RTH RTH ( X TH X 2 )2 . So, the slip at which maximum torque occurs is directly proportional to the rotor resistance R2, but maximum torque Tmax is independent of R2. This means that if R2 is increased by inserting external resistance in the rotor circuit of a WRIM, the magnitude of maximum internal torque is unaffected but the slip at which it occurs is affected proportionally. From maximum torque equation it shows that, Tmax is directly proportional to the square of the stator voltage. Tmax is reduced by an increase in stator resistance R1 (i.e RTH) and Tmax is reduced by an increase in stator leakage reactance X1 and rotor reactance X2. For obtaining higher value of maximum torque, the air gap is kept as small as is possible. A small air gap allows more flux to be mutual between stator and rotor windings. As a consequence, leakage fluxes and therefore leakage reactance are reduced and the magnitude of maximum torque becomes more. Typical torque-slip curves for an IM with variable rotor-circuit resistances are shown in figure 1.16 and The speed of motor can be controlled by varying the rotor circuit resistance but maximum torque remains unaffected. The starting torque can be varied by changing the rotor circuit resistance. The line current taken by the motor shall also vary with rotor circuit resistance. The power factor at starting is also affected by rotor circuit resistance. ____________________________________________________________________________________ mkdas@imu.ac.in Page 4 of 7 Poly-Phase Induction Motor In order to get better performance of the IM, starting torque is increased by inserting a suitable external resistance in the rotor circuit at starting. As the motor accelerates, external resistance is cut out in steps so as to maintain maximum torque during the accelerating period. Finally, external resistance is reduced to zero and the rotor attains normal speed. Figure 1.16 Induction-motor torque-slip curves showing effect of changing rotor-circuit resistance. The internal mechanical power developed is given by, Pm mI 22 R2 R (1 s )VTH2 (1 s ) m 2 . s s ( R R2 ) 2 ( X X ) 2 TH TH 2 s The power at the slip, which gives the maximum torque, Pm t Tmaxs (1 sm t ) = 0.5 Or, Pmt 0.5mVTH2 mVTH2 1 s RTH R ( X TH X 2 ) 2 TH 1 2 RTH RTH ( X TH X 2 )2 (1- 2 s (1- R2 2 TH R R2 2 RTH ( X TH X 2 )2 ( X TH X 2 )2 ) ). However, this is not the maximum power. The maximum power occurs at a different slip, smp, Pm equation can be rewritten as ( 2 Pm mVTH ( RTH R2 R2 R2 ) s s )2 ( X TH X 2 )2 ( 2 mVTH ( RTH R2 R2 R2 R2 ) s s R2 ) 2 ( X TH X 2 ) 2 mVTH2 Rl ( R0 Rl ) 2 X 2 Where, Rl = R2/s – R2, R0 = RTH + R2 and X = XTH + X2. For maximum power, using the maximum power transfer theorem, ____________________________________________________________________________________ mkdas@imu.ac.in Page 5 of 7 Poly-Phase Induction Motor Rl R02 X 2 ( RTH R2 )2 ( X TH X 2 ) 2 which is the internal impedance. Putting the values of Rl, R0 and X. R2 R2 ( RTH R2 )2 ( X TH X 2 ) 2 = Zsc . Where Zsc is short circuit impedance. s Or, R2 R2 R2 Z sc ; smp where smp is the slip at maximum power. smp R2 Z sc As is evident, the equation for maximum power is same as derived for maximum torque. That is’ Pmax 2 mVTH 2( R0 R02 X 2 ) 2 mVTH mVTH2 . 2( RTH R2 Z sc ) 2( Rsc Z sc ) Where Rsc = RTH + R2 = short circuit resistance. Starting Torque: At starting, N r 0 , Hence s 1.0 . Thus, the starting torque Tstart is given by, Tstart s {( RTH 2 mVTH R2 2 R2 ) ( X TH X 2 )2 } The above equation shows that the starting torque can be controlled by controlling the rotor resistance 2 ( X TH X 2 ) 2 . R2 . The maximum torque at starting will occur when smst 1 i.e., when R2 RTH The starting torque increases by adding resistance in the rotor circuit. At the same time the starting current will reduce. The advantage of the slip-ring induction motor in which a high starting torque is obtained at low starting current. # Motor Torque in terms of Maximum Torque: The torque expression of an IM can also be expressed in terms of maximum torque Tmax and the dimensionless ratio s smt . In order to get a simple and approximate expression, if slip‘s’ is very small then the stator resistance R1 or the stator equivalent resistance RTH, is neglected compared to R2/s. The torque equation of an IM, as given by, Te s {( RTH mVTH2 ( R2 s ) mVTH2 ( R2 s ) Where, X = XTH + X2. R2 2 R2 2 2 2 ) ( X TH X 2 ) } s {( RTH ) X } s s The maximum torque equation of an IM, as given by, Tmax mVTH2 1 2 2s RTH RTH X2 2 2 2( RTH RTH X 2 ) R2 Te mVTH2 ( R2 s) mVTH 1 Now, ( ) /( ) 2 2 R Tmax s {( RTH R2 s ) 2 X 2 } 2s RTH RTH X ( RTH 2 ) 2 X 2 s s ____________________________________________________________________________________ mkdas@imu.ac.in Page 6 of 7 Poly-Phase Induction Motor Since R1 or RTH is neglected, We know, S mt So, Te R2 2X . R Tmax ( 2 ) 2 X 2 s s R2 2 RTH ( X TH X 2 ) 2 R2 2 RTH X2 . So, from this equation R2 smt X . ( RTH 0 ). Te smt X 2X . ( Substitution of the value of R2 ). s X 2 2 Tmax ( mt s ) X s Te 2Tmax 2 Or, Te . smt s Tmax smt s s smt s smt ____________________________________________________________________________________ mkdas@imu.ac.in Page 7 of 7