Chapter 6: polyphase induction machines

Chapter 6: Polyphase Induction Machines

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

• Study on the behavior of polyphase induction machines:

The analysis begins with the development of single-phase equivalent circuits.

• The general form is suggested by the similarity of an induction machine to a transformer.
• The equivalent circuits can be used to study the electromechanical characteristics of an induction machine as well as the loading presented by the machine on its supply source.

§6.1 Introduction to Polyphase Induction Machines

• An induction machine is one in which alternating current is supplied to the stator directly and to the rotor by induction or transformer action from the stator.

• The stator winding is excited from a balanced polyphase source and produces a magnetic field in the air gap rotating at synchronous speed.
• The rotor winding may one of two types.

• A wound rotor is built with a polyphase winding similar to, and wound with the same number of poles as, the stator. The rotor terminals are available external to the motor.
• A squirrel-cage rotor has a winding consisting of conductor bars embedded in slots in the rotor iron and short-circuited at each end buy conducting end rings. It is the most commonly used type of motor in sizes ranging from fractional horsepower on up.

• The difference between synchronous speed and the rotor speed is commonly referred to as the slip of the rotor. The fractional slip s is

$s=\frac{n_s-n}{n_s}$ (6.1)

• The slip is often expressed in percent.
• n : rotor speed in rpm

$n=(1-s)n_s$ (6.2)

• ${\omega }_m$ : mechanical angular velocity

${\omega }_m=(1-s){\omega }_s$ (6.3)

• $f_r$ : the frequency of induced voltages, the slip frequency

$f_r={\text{sf}}_e$ (6.4)

–A wound-rotor induction machine can be used as a frequency changer.

The rotor currents produce an air-gap flux wave that rotates at synchronous speed and in synchronism with that produced by the stator currents.

• With the rotor revolving in the same direction of rotation as the stator field, the rotor currents produce a rotating flux wave rotating at ${\text{sn}}_s$ with respect to the rotor in the forward direction.
• With respect to the stator, the speed of the flux wave produced by the rotor currents (with frequency ${\text{sf}}_c$ ) equals

${\text{sn}}_s+n={\text{sn}}_s+n_s(1-s)=n_s$ (6.5)

• Because the stator and rotor fields each rotate synchronously, they are stationary with respect to each other and produce a steady torque, thus maintaining rotation of the rotor. Such torque is called an asynchronous torque.
• Equation (4.81) $T=-\frac{\pi }{2}{\frac{\text{poles}}{2}}^2{\Phi }_{\text{sr}}{F}_r\text{sin}{\delta }_r$ can be expressed in the form

$T=-{\text{KI}}_r\text{sin}{\delta }_r$ (6.6)

$I_r$ : the rotor current

${\delta }_r$ : the angle by which the rotor mmf wave leads the resultant air-gap mmf wave

• Fig. 6.1 shows a typical polyphase squirrel-cage induction motor torque-speed curve. The factors influencing the shape of this curve can be appreciated in terms of the torque equation.

Figure 6.1 Typical induction-motor torque-speed

curve for constant-voltage, constant-frequency operation