## 551 Perphase Equivalent Circuit

Steady state analysis of induction motors is often carried out using the per-phase equivalent circuit. A single-phase equivalent circuit is used for the three-phase induction machine, assuming a balanced set as shown in Figure 5.18. The per-phase equivalent circuit consists of the stator loop and the rotor loop, with the magnetic circuit parameters in the middle. The inductance representing the magnetization current path is in the middle of the circuit, along with an equivalent core loss resistance. For the stator and rotor electrical parameters, the circuit includes the stator winding resistance and leakage reactance and the rotor winding resistance and leakage reactance. A slip-dependent equivalent resistance represents the mechanical power delivered at the shaft due to the energy conversion in the air gap coupled FIGURE 5.18 Steady state per-phase equivalent circuit of an induction motor.

electromagnetic circuit. The electrical input power supplied at the stator terminals converts to magnetic power and crosses the air gap. The air gap power Pag is converted to mechanical power delivered at the shaft after overcoming losses in the rotor circuit.

Although the per-phase equivalent circuit is not enough to develop controllers that demand good dynamic performance like in an EV or HEV, the circuit provides a basic understanding of induction machines. The vast maj ority of applications of induction motors are for adjustable speed drives, where controllers designed for good steady state performance are adequate. The circuit allows the analysis of a number of steady state performance features. The parameters of the circuit model are as follows:

£MS=Stator-induced emf per phase Fs=Stator terminal voltage per phase Is= Stator terminal current Rs=Stator resistance per phase Xls= Stator l eakage reactance Xra=Magnetizing reactance Xlr =Rotor leakage reactance referred to stator Rr =Rotor resi stance referred to stator Ir=Rotor current per phase referred to stator

Note that the voltages and currents described here in relation to the per-phase equivalent circuit are phasors and not space vectors. The power and torque relations are FIGURE 5.19 Steady-state torque-speed characteristics of an induction motor. The steady-state torque-speed characteristics of the machine are as shown in Figure 5.19. The torque produced by the motor depends on the slip and the stator currents, among other variables. The induction motor starting torque, while depending on the design, is lower than the peak torque achievable from the motor. The motor is always operated in the linear region of the torque-speed curve to avoid the higher losses associated with high slip operation. In other words, operating the machine at small slip values maximizes the efficiency.

The value of the rotor circuit resistance determines the speed at which the maximum torque will occur. In general, the starting torque is low, and the maximum torque occurs close to the synchronous speed, when the slip is small. The motor draws a large current during line starting from a fixed AC source, which gradually subsides as the motor reaches the steady state speed. If the load requires a high starting torque, the motor will accelerate slowly. This will make a large current flow for a longer time, thereby creating a heating problem.

Nonlinearity at speeds below the rated condition is due to the effects of leakage reactances. At higher slip values, the frequencies of the rotor variables are large, resulting in dominating impedance effects from rotor leakage inductance. The air gap flux cannot be maintained at the rated level under this condition. Also, large values of rotor current (which flows at high slip values) cause a significant voltage drop across the stator winding leakage impedance (Rs+j Llwhich reduces the induced voltage and, in turn, the stator mmf flux density FIGURE 5.20 Torque-speed characteristics of an induction motor for rated flux condition.
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