5.1 INTRODUCTION

Induction machines are the most common form of electromechanical energy-conversion devices for industrial, commercial, and domestic appliances that operate at constant speed. The need for energy processing and conversion from electrical to mechanical energy to do mechanical work abounds in every sector of human endeavor. About one-third of the world's electricity consumption is used for running induction motors that drive pumps, fans, compressors, elevators, and machinery of various types to meet industrial, commercial, and domestic needs.

The AC induction motor is a common form of asynchronous motor whose operation depends on three electromagnetic phenomena:

1. Motor action. When an iron rod (or other magnetic material) is suspended in a magnetic field so that it is free to rotate, it will align itself with the field. If the magnetic field is moving or rotating, the iron rod will move with the moving field so as to maintain alignment.
2. Rotating field. A rotating magnetic field can be created from fixed-stator poles by driving each pole pair from a different phase of the AC supply.
3. Transformer action. The current in the rotor windings is induced from the current in the stator windings, avoiding the need for a direct connection from the power source to the rotating windings. Consequently, the induction motor can be considered an AC transformer with a rotating secondary winding.

5.2 CONSTRUCTION AND TYPES OF INDUCTION MOTORS

The induction motor stator resembles the stator of a revolving-field, three-phase, synchronous generator. The stator or the stationary part consists of three-phase winding held in place in the slots of a laminated steel core enclosed and supported by a cast-iron steel frame, as seen in Figure 5.1 [5].

The phase windings are placed 120° electrical apart and may be externally connected in Y or delta for which six leads are brought out to a terminal box mounted on the frame of the motor. When the stator is energized from a three-phase voltage, it produces a rotating magnetic field in the stator core.

Three-phase induction motors are of two kinds:

• cage-rotor or squirrel-cage induction motors
• wound-rotor or slip-ring induction motors

Cage Rotors

Cage-rotor machines (also called squirrel-cage machines) are the most common type of induction motors. In a cage-rotor design, solid conductors are in slots on the rotor. The ends of the conductors are short circuited at each end of the rotor using an “end-ring.” For small to medium-sized machines (up to a few hundred horsepower) the rotor conductors are cast using aluminum. This construction makes the rotor relatively cheap to produce. In larger machines, rotors are usually made by manually hammering solid copper bars into the rotor slots, then manually brazing an end-ring into place. Fabricated rotor cages are significantly more expensive that cast rotor cages. Figure 5.2 [5] shows a cage-rotor induction motor.

Wound Rotors

Wound-rotor induction machines have a three-phase winding, similar to the stator winding, on the rotor as seen in Figure 5.3 [5]. The rotor is usually Y connected with the terminals of the three rotor phases connected to slip rings. In normal operation, the windings at the slip rings are short circuited to allow currents to flow. An advantage of wound-rotor machines is that external circuits can be connected to the rotor, allowing external control of the machine. While all induction machines can be controlled to operate at different torques and speeds, wound-rotor control is particularly attractive in some applications. However, wound-rotor induction machines are usually significantly more expensive than cage rotor machines. Possible applications for wound-rotor machines include:

• speed control of very large machines (multi-milliwatt)
• reduced cost control of large machines
• doubly fed induction generation (used in some wind turbines)

5.3 OPERATING PRINCIPLE

The basic idea behind the operation of an induction machine is quite simple. The three-phase stator winding is connected to a three-phase supply. Currents flow in the stator winding, producing an mmf and flux density that rotates at synchronous speed:

(5.1)

The magnetic field passes conductors on the rotor and induces a voltage in those conductors. Since the conductors are short circuited, current flows in the rotor conductors, producing a second rotor magnetic field, which acts to oppose the stator magnetic field and rotates at synchronous speed. With two magnetic fields rotating at constant speed, a torque is produced.

5.4 BASIC INDUCTION-MOTOR CONCEPTS

The following induction-motor concepts are important to understanding the operation and application of these machines.

Leakage Reactance

The leakage reactance is for magnetizing the armature of the cylindrical-rotor machines. It is given by

(5.2)

The mutual air-gap flux between the rotor and the stator is φ (weber, or Wb) per pole and rotates at an angular speed of ω radians per second. The flux linkage of N-turns of conductors will be at maximum when ωt = 0 and also at ωt = 90°. Assuming distributed winding, the flux linkage λ_a will vary as the cosine of the angle ωt. Thus, the flux linkage with coil a is

(5.3)

The induced voltage

(5.4)

(5.5)

where

The rms value of the generated voltage is then

(5.6)

(5.7)

In actual machines, the armature conductors are distributed in slots such that the phasor sum of the induced emf is less than the numerical sum. Hence, a factor to account for the difference is introduced, which we shall call the winding factor K_w (note that K_w varies between 0.85 and 0.95 for three-phase windings). Consequently, for distributed-phase windings, the rms value of the generated voltage is:

(5.8)

Let stator emf be E₁ and rotor emf E_r. Hence, we may write

(5.9)

(5.10)

Since E_r was induced by transformer action, the induction motor behaves like a transformer with a rotating secondary winding. Hence, we may write an expression for the transformation ratio in terms of the stator and rotor parameters as:

(5.11)

where f is frequency in Hz, K_w1 and K_w2 are stator and rotor winding factors, N_ph1 and N_ph2 are the number of turns in the stator and rotor windings, and a₁and a₂ are the corresponding number of current paths.

5.5 INDUCTION-MOTOR SLIP

Suppose that the rotor circuit is open and is rotated at a speed n rpm by some external means in the duration of the rotating flux φ_m. If n_syn is given in rpm, the rotational speed of φ_m is defined as the slip s

(5.12)

When motor rotor is at slip s

(5.13)

the rotor frequency is given by

(5.14)

This frequency also affects the resulting voltage E′_s:

(5.15)

5.6 ROTOR CURRENT AND LEAKAGE REACTANCE

If r₂₂ is the resistance of the rotor winding in ohms/φ, with the corresponding leakage inductance L₂₂, the wound-rotor leakage inductance in henrys/φ leads to leakage reactance of the rotor, given by:

(5.16)

and with slip s accounted for,

(5.17)

Therefore, rotor current

(5.18)

This mimics the transformer-equivalent circuit from the secondary side comprising of in series with the reactance of the secondary side given as the equivalent circuit, as shown in Figure 5.4.

The primary side is equivalent to the stator-winding operating at stator frequency regardless of the slip, i.e.,

(5.19)

This follows from the fact that the polyphase rotor currents at slip s have a synchronous frequency and therefore produce a rotor mmf that rotates at sn_sync rpm relative to the rotor in the same direction as the stator flux while the rotor is rotating at a speed of (1 − s)n_syn rpm in the same direction.

The resultant speed of the rotor mmf relative to the stator is the sum of those two speeds:

(5.20)

Synchronous speed n_syn = sn of speed of stator at stator frequency f (Hz).

The ideal transformer may be used to model the equivalent circuit of an induction motor as shown in Figure 5.5.

Equivalent rotor resistance is shown in two parts, namely:

1. Rotor resistance r₂ is dynamic resistance.
2. Mechanical load on the machine is modeled as .

Example 1

A four-pole, three-phase induction motor operates from a supply whose frequency is 50 Hz. Calculate:

1. speed at which the magnetic field of the stator is rotating
2. speed of the rotor when the slip is 0.04
3. frequency of the rotor current when the slip is 0.03
4. frequency of the rotor currents at standstill

Solution:

Stator field revolves at synchronous speed, given by:

rotor speed

frequency of rotor current

at standstill

Example 2

A 110 V, 50 Hz, delta-connected induction motor has a star-connected, slip-ring rotor with phase transformation ration of 3.8. The rotor resistance and standstill leakage reactance are 0.012Ω and 0.25Ω per phase, respectively. Neglecting stator impedance and magnetizing current, determine:

1. rotor current at start with slip rings shorted
2. rotor power factor at start with slip rings shorted
3. rotor current at 4 percent slip with slip rings shorted
4. rotor power factor at 4 percent slip with slip rings shorted
5. external rotor resistance per phase required to obtain starting current of 100A in the stator supply lines

It should be noted that in a Δ/Y connection, primary-phase voltage is the same as line voltage. The rotor-phase voltage can be found by using the phase transformation ratio of 3.8, i.e., K = 1/3.8.

Solution:

Rotor-phase voltage at standstill = 1,100 × 1/3.8 = 289.5 V

1. rotor impedance/phase
2. rotor phase current at start = 289.5/0.2503 = 1, 157A
3. power factor = R₂/Z₂ = 0.012/0.2503 = 0.048 lagging
4. at 4 percent slip
   
5. external resistance required/phase = 0.7196 – 0.012 = 0.707Ω

5.7 ROTOR COPPER LOSS

As shown in Figure 5.4, for an induction-machine equivalent circuit, we can compute

(5.21)

(5.22)

(5.23)

from which we compute

(5.24)

Given the value of P_in and P_losses for mechanical power

(5.25)

If we account for friction and stray losses (P_fw − P_stray), we have the net mechanical power as follows:

(5.26)

where P_fw is power lost as friction and windage.

The induction motor can become a generator when I₂₂ reverses its direction and

(5.27)

This is only possible when Q is supplied to the motor at the bus where it is connected.

5.8 DEVELOPING THE EQUIVALENT CIRCUIT OF POLYPHASE, WOUND-ROTOR INDUCTION MOTORS

Assuming the machine is three-phase connected and accounting for its losses in steady-state conditions, the currents and voltages align with neutral values at the stator side. The machine's rotating air-gap flux is defined by balanced, three-phase emf of the stator. This gives a terminal voltage, which differs from the stator emf by the voltage drop in the stator leakage impedance given as

(5.28)

In terms of rotor emf

where V₁ is stator terminal voltage, E₁ is counter emf generated by resultant field flux, I₁ is the stator current, R₁ is the stator effective resistance,  X₁ is the stator leakage reactance.

As an analog to the transformer model, we have the stator equivalent circuit as shown in Figure 5.6.

• I₂  produces mmf linking mmf of the rotor current.
• I_m is the stator current required to create the resultant air-gap flux as a function of E₁.
• I_c is in phase with E₁ and magnetizing component I₂₂ lagging E₁ by 90°.

These quantities are included in the circuit diagram in Figure 5.6.

As before, we can refer to the cage-wound rotor as a function of slip and then refer to the secondary side as seen in Figure 5.7.

(5.29)

(5.30)

(5.31)

Thus, referring to the stator is essentially the same as referring the secondary quantities to the primary side of the static transformer.

Earlier, we defined as part of slip

(5.32)

where

• Z_2s is the slip-frequency rotor-leakage impedance per phase referred to the stator, R₂ is the effective resistance, and sX₂ is the leakage reactance at slip frequency.

Thus

(5.33)

(5.34)

The redrawn equivalent circuits for polyphase induction motors are shown in Figure 5.8.

The equivalent circuit is used to analyze the induction motor in general since V, I, losses, and torque are based on the model parameters.

We derive them simply as follows:

P_g transferred across the air gap from the stator is

(5.35)

The total rotor loss I²R can be written as a₁I²₂R₂.

Net power is

(5.36)

Motor mechanical power is

(5.37)

Total power delivered to the rotor is a fraction of P_g(mech).

(1 − s) is converted into mechanical power because of P_g1 (electrical power). The conversion (1 − s) mechanical is dissipated as rotor circuit I²R loss.

The internal mechanical power per stator phase is equal to the power absorbed by the resistance

(5.38)

Example

A 115 V, 60 Hz, three-phase, Y-connected, six-pole induction motor has an equivalent circuit consisting of stator impedance (0.07 + j0.3)Ω and an equivalent rotor impedance at standstill of (0.08 + j0.3)Ω. The magnetizing branch has G_c = 0.022 mho and B_m = 0.158 mho. Find:

1. secondary current
2. primary current
3. primary power factor
4. gross power output
5. gross torque
6. input power
7. gross efficiency using equivalent circuit—assume a slip of 29 percent

Solution:

1. Using the equivalent circuit shown in Figure 4.8,
   
2. 
   
3. Primary power factor = cos 36.5° = 0.804.
4. P_g = 3I₂²R_L = 3 × 16.17² × 3.92 = 3, 075W.
5. Synchronous speed N_s = 120 × 60/6 = 1,200 rpm.
   Actual rotor speed N = (1 − s)N_s = (1 − 0.02) × 1, 200 = 1, 176 rpm
   Therefore, .
6. Primary power input = 
   
7. 

5.9 COMPUTING CORRESPONDING TORQUE OF INDUCTION MOTORS

(5.39)

(5.40)

where

Note that torque and power do not include stray, frictional, windage, and load losses.

Example

A three-phase induction motor has a four-pole, star-connected stator winding and runs on a 220 V, 50 Hz supply. The rotor resistance per phase is 0.1Ω and reactance 0.9Ω. The ratio of stator to rotor turns is 1.75. The full load slip is 5 percent. Calculate for this load:

1. load torque in Nm
2. speed at maximum torque
3. rotor emf at maximum torque

Solution:

1. K = rotor turns/stator turns = 1/1.75

   Stator voltage/phase,

   

   Standstill rotor emf/phase,

   
   
   

   Rotor copper loss

   
2. For maximum torque,
   
   
3. Rotor emf/phase at maximum torque = 1/9 × 72.6
   

5.10 APPROXIMATION MODEL FOR INDUCTION MACHINES

To approximate the model further, the exciting branch is usually neglected to allow for 30 to 50 percent full-load current and because the leakage is also higher. This is drawn as either of the two diagrams in Figure 5.9.

Example

A three-phase, Y-connected, 220 V, _L-L, 10Hp, 60 Hz, six-pole, induction motor has the following constants in ohms:

The total friction, windage, and core loss are assumed constant at 403 W, load slip of 2.0 percent.

Determine torque, power, I_stator, pf, and η of motor at rated frequency.

Solution:

The impedance  Z_f is represented in Figure 5.10 as

Sum of

Applied voltage

Deducting losses of 403W

Efficiency : Total stator I²R loss = 3(18.8)² × 0.294 = 312 W

Friction + windage + core losses = 403W

Total = 403 + 369.25 + 312 = 1,084.25W

Output = 17,690

5.11 SPEED CONTROL OF INDUCTION MOTORS

To allow for multi-application of induction motors, engineers use many options to adjust the speed of induction motors, including:

1. Pole changing motors. By simply changing coil connections, the number of poles can be changed in the ratio 2:1, typically for cage-rotor-induction motors.
2. Line frequency control. Varying the line frequency can change the speed of the motor by using a wound-rotor induction motor as a frequency changer. The most effective and economic source of adjustable frequency must be determined for this method.
3. Line-voltage control. This can be achieved by reduction of speed from n₁ to n₂ rpm by changing the control torque of the induction motor, which is proportional to the square of voltage at the primary side terminals.
4. Rotor resistance control. This is done by adjusting the rotor-circuit resistance at the primary winding.
5. Control of slip by auxiliary devices. By varying the slip s through changes in frequency, the speed of an induction motor can be controlled.

5.12 APPLICATION OF INDUCTION MOTORS

The following are some applications of induction motors.

1. Squirrel-cage induction motors are more widely used in industrial applications than slip-ring induction motors since they are cheaper in cost, rugged in construction, and low maintenance. Squirrel-cage induction motors are suitable for applications where the drive requires constant speed, low starting torque, and no speed-control drives.
2. Wound-rotor induction motors are suitable for loads requiring high starting torque and low starting current, and loads that have high inertia, which results in higher energy losses. They also are applicable for loads that require a gradual buildup of torque and speed control. Specific applications include in conveyors, cranes, pumps, elevators, and compressors.

5.13 INDUCTION-GENERATOR PRINCIPLES

The conditions under which polyphase induction machines behave as induction generators include the following.

1. When the slip of a polyphase induction machine becomes negative, the rotor current and rotor emf attain negative values. The prime-mover torque becomes opposite to the electric torque. This condition can be achieved by coupling a prime mover whose speed can be controlled by the machine's rotor. If the speed of the prime mover is increased such that the slip becomes negative (i.e., the speed of the prime mover becomes greater than the synchronous speed) then the induction machine behaves like a generator.
2. If the speed of the prime mover is further increased, such that it exceeds the negative maximum value of the torque produced, then the generating effect of the generator vanishes. Clearly the speed of the induction generator during the whole operation is not synchronous; therefore, the induction generation is also called asynchronous generation.

An induction generator is not a self-excited machine. In order to develop the rotating magnetic field it requires magnetizing current and reactive power. The induction generator obtains its magnetizing current and reactive power from the supply mains or from a synchronous generator connected in parallel with it. Thus, normally, the induction generator cannot work in isolation because it continuously requires reactive power from some external supply system.

The only way to make an induction generator self-excited in isolated mode is to use capacitor banks for reactive power supply instead of an AC-supply system.

5.13.1 Isolated Induction Generators

Isolated induction generators are also known as self-excited generators. A bank of capacitors is connected across the generator's stator terminals as shown in Figure 5.10.

The function of the capacitor bank is to provide the lagging reactive power to the induction generator as well as the load. So

(5.41)

where Q is the total lagging reactive power supplied by the capacitor bank, Q₁ is the reactive power drawn by the induction generator, and Q₂ is the reactive power fed to the load.

5.14 CHAPTER SUMMARY

Principles of induction machines have been presented in this chapter. This class of electromechanical machines is important as it serves dual purposes when in an electric power system depending on its mode of excitation. We have found that, as an induction motor, it is a nonlinear load, which is of interest because of the power-quality problem of harmonic distortion it creates in an electric power system. As an induction generator, its application in wind energy systems is of importance. Consequently, irrespective of the configuration of an induction machine, its place as a load or an energy source in any power system is of utmost interest, especially in the design, operation, and management of a microgrid. Considerable resources have therefore been included in this chapter for the benefit of the student to facilitate overall knowledge of this machine in preparation for its applications in a microgrid system. Several exercises to illustrate computational modelling, performance, and speed regulation of induction machines have been presented in this chapter.