54 vth rth and x th should be replaced by v1 r1 and

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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 three-phase machine, Equation (4.54) should be multiplied by three to obtain the total torque developed by the machine. The torque-speed 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 torque-speed 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) Three-Phase Induction Machine 253 The torque varies almost inversely with slip near s = 1, as seen from Fig.4.14. Fig.4.14 Torque-speed 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 T-n 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 impedance-matching 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, Three-Phase 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. Three-Phase 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 wound-rotor 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.

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