## 573 Power And Electromagnetic Torque

The power into the three-phase machine needs to be analyzed in terms of dq variables to arrive at the electromagnetic torque expression, which is used in the motor control loop. The power into the induction machine is the product of the phase voltage and phase currents, given as

The input power is embedded in the real part of the product of the voltage space vector and the current space vector conjugate of the stator and rotor variables. The real part can be calculated as follows:

iO-ti>r)Xqr Rf

Idr Vdf

¿-axis equivalent circuit

(/-axis equivalent circuit

FIGURE 5.28 d- and q-axes circuits of the induction machine.

Using the above relation in the power input equation for the induction machine, we get

Now, using Equation 5.37: The input power expression in scalar form is

The multiplying factor 3/2 is due to our choice of the factor 2/3 for the ratio of dq and abc variables.

The input power expression of Equation 5.48 would not have this 3/2 factor, had we chosen as the proportionality constant between the dq and abc variables.

Upon expanding the right-hand side of Equation 5.48 and extracting the output electromechanical power Pe4 we get

RS Llr

¿-axis equivalent circuit

(/-axis equivalent circuit

FIGURE 5.28 d- and q-axes circuits of the induction machine.

where r is the rotor angular velocity of a two-pole machine. The electromagnetic torque for a P-pole machine is, therefore,

Several alternative forms of electromagnetic torque can be derived using the stator and rotor flux linkage expressions. A couple of these torque expressions are as follows :

The torque expressions in terms of dq variables are used in the vector control of induction motor drives. Vector control implementations are accomplished in one of the several available choices of reference frames, such as rotor flux-oriented reference frame, stator flux-oriented reference frame, or air gap flux-oriented reference frame. The abc variables at the input of the controller are converted to dq variables in the chosen reference frame at the input of the controller. The control computations take place in terms of dq variables, and the generated command outputs are again converted back to abc variables. The inverter controller executes the commands to establish the desired currents or voltages in the drive system. Some of these drives will be discussed in Chapter 8.

Torque expression derived in this section is useful for motor controller implementation, while the simplified torque expression ofEquation 5.34 is useful for system level analysis.

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