# 6.4 Induction Motors

## 6.4 Induction Motors

#### **Introduction to Induction Motors**

The induction motor, also known as the asynchronous motor, is the workhorse of modern industry. It is the most widely used electric motor due to its simple and rugged construction, low cost, minimal maintenance requirements (no brushes or commutator), and excellent reliability. Operating on the principle of a rotating magnetic field, it converts three-phase or single-phase AC electrical energy into mechanical energy. The rotor, which carries no electrical connections to the supply, develops torque through electromagnetic induction—hence the name. This section covers the fundamental principles, construction types, performance characteristics, and control methods of both three-phase and single-phase induction motors.

***

### 1. Principle of Operation (Rotating Magnetic Field)

#### 1.1 Production of a Rotating Magnetic Field (RMF)

1. **Key Concept**: When a balanced three-phase AC supply is fed to a three-phase winding (displaced by 120° in space), it produces a magnetic field of constant magnitude that rotates at a constant speed. This is the foundation of induction motor operation.
2. **Synchronous Speed (**$$N\_s$$**)**: The speed of the rotating magnetic field (RMF) in revolutions per minute (RPM) is given by: $$N\_s = \frac{120 f}{P}$$ where:
   * $$f$$ = Supply frequency (Hz)
   * $$P$$ = Number of poles of the stator winding.
3. **Proof of Rotating Field**: The instantaneous mmf of each phase is sinusoidal and space-displaced. The vector sum of the three mmfs at any instant results in a constant magnitude vector rotating at $$\omega\_s = 2\pi N\_s/60$$ rad/s.

#### 1.2 Torque Production and Slip

1. **Induction Principle**: The rotating stator field cuts the stationary rotor conductors, inducing an electromotive force (EMF) in them according to Faraday's law.
2. **Rotor Current**: Since the rotor circuit is closed (either shorted or connected through resistance), the induced EMF drives a rotor current.
3. **Force and Torque**: The interaction between the rotor current and the stator's rotating magnetic field produces a force (Lorentz force) on the rotor conductors, resulting in a net torque that tends to rotate the rotor in the same direction as the RMF.
4. **Necessity of Slip**: For torque production, there must be a **relative speed** between the RMF and the rotor. If the rotor were to catch up to the RMF (synchronous speed), the relative motion would be zero, no EMF would be induced, and torque would drop to zero.
5. **Slip (s)**: The relative speed difference expressed as a fraction or percentage of synchronous speed. $$s = \frac{N\_s - N\_r}{N\_s}$$
   * $$N\_r$$ = Rotor speed (RPM).
   * At standstill (start): $$N\_r = 0$$, so $$s = 1$$ (or 100%).
   * At synchronous speed (theoretical no-load): $$N\_r = N\_s$$, so $$s = 0$$.
   * Normal operating slip: Typically 2-5% for large motors, 5-10% for small motors.

***

### 2. Construction: Squirrel Cage vs. Wound Rotor

#### 2.1 Stator Construction (Common to Both)

* **Frame**: Outer supporting structure, usually cast iron or aluminum.
* **Stator Core**: Laminated silicon steel to reduce eddy current losses, with slots on the inner periphery.
* **Stator Winding**: Three-phase, balanced winding placed in slots, connected in star (Y) or delta (Δ) configuration.

#### 2.2 Rotor Construction (The Key Difference)

**A. Squirrel Cage Induction Motor (SCIM)**

1. **Rotor Core**: Laminated cylinder mounted on the shaft with slots on the outer periphery.
2. **Rotor Bars**: Uninsulated copper or aluminum bars placed in the rotor slots.
3. **End Rings**: At both ends of the core, the bars are permanently and solidly short-circuited by heavy conductive end rings.
4. **Appearance**: Resembles a "squirrel cage."
5. **Advantages**:
   * Simple, rugged, and cheap to manufacture.
   * Low maintenance (no brushes, slip rings).
   * Explosion-proof (no sliding contacts that could spark).
   * High efficiency.
6. **Disadvantages**:
   * Low starting torque (draws high starting current - 5-8 times full load current).
   * Poor speed control (speed is essentially fixed by supply frequency).
   * Sensitive to supply voltage fluctuations.
7. **Applications**: Fans, pumps, compressors, conveyors, machine tools – any application with constant speed and moderate starting torque requirements.

**B. Wound Rotor Induction Motor (WRIM) or Slip Ring Motor**

1. **Rotor Core**: Similar laminated core, but slots contain a **three-phase, insulated winding**.
2. **Rotor Winding**: Connected in star (Y). The three ends are brought out and connected to three **slip rings** mounted on the shaft.
3. **Brushes**: Stationary carbon brushes ride on the slip rings, providing an external electrical connection to the rotor circuit.
4. **External Resistance**: Through the brushes and slip rings, an external three-phase variable resistance (starter) can be connected in series with the rotor windings.
5. **Advantages**:
   * High starting torque with low starting current (by adding external rotor resistance).
   * Provides some speed control (by varying external rotor resistance).
6. **Disadvantages**:
   * More expensive and complex.
   * Higher maintenance due to brushes and slip rings.
   * Lower efficiency due to additional $$I^2R$$ losses in external resistance during operation.
   * Not suitable for explosive environments.
7. **Applications**: Cranes, hoists, elevators, large compressors – applications requiring high starting torque and limited speed control.

***

### 3. Equivalent Circuit and Concept of Slip

#### 3.1 Per-Phase Equivalent Circuit

The induction motor can be accurately modeled by an equivalent circuit similar to a transformer, with the secondary (rotor) side operating at slip frequency.

1. **Stator Circuit** (Primary):
   * $$R\_1$$: Stator winding resistance per phase.
   * $$X\_1$$: Stator leakage reactance per phase at supply frequency.
2. **Magnetizing Branch**:
   * Shunt branch across the supply representing the core loss and magnetizing current.
   * $$R\_c$$: Core loss resistance.
   * $$X\_m$$: Magnetizing reactance.
3. **Rotor Circuit** (Secondary - referred to stator side):
   * **Key Point**: Rotor parameters are referred to the stator side and to supply frequency.
   * $$R\_2'$$: Rotor winding resistance referred to stator.
   * $$X\_2'$$: Rotor leakage reactance at supply frequency ($$s=1$$), referred to stator.
   * **Slip-dependent Load Resistance**: The most critical element. The electrical power converted to mechanical power is represented by a **variable resistance**: $$R\_L' = R\_2' \left( \frac{1-s}{s} \right)$$
   * **Total Rotor Impedance per phase (referred)**: $$Z\_2' = \frac{R\_2'}{s} + jX\_2'$$ The term $$R\_2'/s$$ can be split as $$R\_2' + R\_2'(\frac{1-s}{s})$$.

#### 3.2 Significance of the Equivalent Circuit

* It allows calculation of all performance parameters: current, power factor, input power, losses, output power, torque, and efficiency.
* The mechanical load is represented electrically by the resistance $$R\_2'(1-s)/s$$. The power dissipated in this resistor represents the **gross mechanical power developed**.
* The circuit clearly shows that at standstill ($$s=1$$), the load resistance is zero, leading to high current (locked rotor current).
* At no-load ($$s \approx 0$$), the load resistance approaches infinity, leading to very low rotor current.

***

### 4. Losses, Efficiency, and Torque-Speed Characteristics

#### 4.1 Losses in an Induction Motor

1. **Constant Losses (Independent of Load)**:
   * **Core (Iron) Losses**: Hysteresis and eddy current losses in the stator and rotor core. Primarily in the stator.
   * **Mechanical Losses**: Friction (bearings) and windage (air resistance).
2. **Variable Losses (Depend on Load)**:
   * **Stator Copper Loss**: $$3 I\_1^2 R\_1$$
   * **Rotor Copper Loss**: $$3 I\_2'^2 R\_2'$$ (where $$I\_2'$$ is referred rotor current).
   * **Stray Load Losses**: Additional losses due to load, difficult to quantify.

#### 4.2 Power Flow and Efficiency

* **Input Power (**$$P\_{in}$$**)**: $$\sqrt{3} V\_L I\_L \cos \phi$$ (for 3-phase).
* **Stator Losses**: Subtracted to get **Air-Gap Power (**$$P\_g$$**)** transferred to the rotor. $$P\_g = P\_{in} - (\text{Stator Cu Loss} + \text{Stator Core Loss})$$
* **Rotor Copper Loss (**$$P\_{cu2}$$**)**: $$P\_{cu2} = s P\_g$$
* **Gross Mechanical Power Developed (**$$P\_m$$**)**: $$P\_m = P\_g - P\_{cu2} = P\_g (1 - s)$$
* **Shaft Output Power (**$$P\_{out}$$**)**: $$P\_{out} = P\_m - \text{Mechanical Losses}$$
* **Efficiency (**$$\eta$$**)**: $$\eta = \frac{P\_{out}}{P\_{in}} \times 100%$$

#### 4.3 Torque-Speed (or Torque-Slip) Characteristics

The torque developed by the motor depends on slip. The relationship is derived from the equivalent circuit.

1. **Torque Equation**:
   * The torque is proportional to the product of rotor current, stator flux, and rotor power factor.
   * **Approximate Torque Equation (Thevenin's equivalent)**: $$T \approx \frac{k s E\_{20}^2 R\_2}{R\_2^2 + (s X\_{20})^2}$$ where $$E\_{20}$$ and $$X\_{20}$$ are rotor-induced EMF and reactance at standstill.
   * **More accurate derived equation**: $$T = \frac{k s V\_1^2 R\_2'}{(R\_{eq} + \frac{R\_2'}{s})^2 + X\_{eq}^2}$$ where $$R\_{eq}$$ and $$X\_{eq}$$ are Thevenin equivalent resistance and reactance of the stator viewed from the rotor.
2. **Shape of the Curve**:
   * At low slip (normal operating region, $$s < s\_{max}$$), torque is approximately proportional to slip: $$T \propto s$$. This is a stable operating region.
   * Torque reaches a maximum value (**Pull-out or Breakdown Torque,** $$T\_{max}$$) at a critical slip $$s\_{max}$$. $$s\_{max} \propto \frac{R\_2'}{X\_{eq}}$$ $$T\_{max} \propto \frac{V\_1^2}{X\_{eq}}$$ (independent of $$R\_2'$$)
   * Beyond $$s\_{max}$$ (high slip), torque decreases with increasing slip. This is an unstable region.
3. **Effect of Changing Parameters**:
   * **Voltage (**$$V$$**)**: Torque is proportional to the square of the voltage ($$T \propto V^2$$). A 10% voltage drop causes a 19% torque reduction.
   * **Rotor Resistance (**$$R\_2$$**)**: Increasing $$R\_2$$ (as in a WRIM) shifts the entire torque-slip curve to the right. $$T\_{max}$$ remains the same but occurs at a higher slip. This is used to **increase starting torque**.

***

### 5. Starting Methods and Speed Control Techniques

#### 5.1 Starting Methods for SCIMs

SCIMs have low starting torque and high starting current (Inrush), which can cause voltage dips. Various methods mitigate this.

1. **Direct-On-Line (DOL) Starting**:
   * Motor is connected directly to full supply voltage.
   * **Disadvantages**: Very high starting current (5-8x Full Load Current, FLC), high mechanical stress.
   * **Application**: Small motors (typically <5-10 HP) where the supply system can handle the inrush.
2. **Star-Delta (Y-Δ) Starting**:
   * **Starting**: Stator windings are connected in **Star (Y)**, reducing phase voltage to $$1/\sqrt{3}$$ of line voltage.
   * **Consequences**: Starting current and torque are reduced to **one-third** of their DOL values.
   * **Running**: After speed builds up, windings are switched to **Delta (Δ)** for full voltage operation.
   * **Application**: Common for motors designed for delta running, medium power ratings.
3. **Auto-transformer (Reduced Voltage) Starting**:
   * A three-phase auto-transformer reduces the voltage applied to the motor during start (e.g., 50%, 65%, 80% taps).
   * **Consequences**: Starting current drawn **from the supply** is reduced by the square of the voltage ratio. Starting torque is also reduced by the square of the voltage ratio.
   * More flexible than star-delta but more expensive.
4. **Soft Starter**:
   * A solid-state device using thyristors or SCRs to gradually ramp up the voltage applied to the motor.
   * Provides smooth acceleration, reduces mechanical stress and inrush current. Does not reduce starting torque as severely as other methods for a given current limit.

#### 5.2 Starting of WRIMs (Slip Ring Motors)

* **Rotor Resistance Starting**: External resistances are connected in series with each rotor phase via slip rings and brushes.
* **Advantage**: High starting torque is achieved with low starting current. The resistance is gradually cut out as the motor speeds up.
* This is the primary reason for using WRIMs.

#### 5.3 Speed Control Techniques

Induction motor speed is given by $$N\_r = N\_s (1-s) = \frac{120f}{P}(1-s)$$. Therefore, speed can be controlled by:

1. **Pole Changing (**$$P$$**)**:
   * Changing the number of stator poles by reconnecting winding coils. Provides discrete speed steps (e.g., 2:1 ratio).
   * Motors built for this are called **multi-speed motors** (e.g., 4/2 pole).
2. **Supply Frequency Control (**$$f$$**) - Variable Frequency Drive (VFD)**:
   * **Most effective and modern method.** Uses a power electronic inverter to vary the frequency and voltage proportionally ($$V/f$$ constant) to maintain constant flux.
   * Allows smooth, wide-range speed control with high efficiency.
   * Also provides soft-start capability.
3. **Supply Voltage Control (**$$V$$**)**:
   * Reducing voltage reduces torque ($$T \propto V^2$$), causing the motor to operate at a higher slip for the same load, thus reducing speed.
   * **Inefficient** and provides only a small speed range. Used for small fans.
4. **Rotor Resistance Control (**$$R\_2$$**) - For WRIMs only**:
   * Inserting external resistance in the rotor circuit increases slip, reducing speed.
   * **Highly inefficient** as the slip power $$sP\_g$$ is dissipated as $$I^2R$$ heat in the external resistors. Used for short-time or intermittent speed control (e.g., in cranes).

***

### 6. Induction Generator Applications

#### 6.1 Principle of Operation

1. An induction machine operates as a generator if it is driven **above synchronous speed** ($$N\_r > N\_s$$) by a prime mover (e.g., wind turbine, micro-hydro turbine).
2. Slip becomes **negative**. The rotor moves faster than the RMF.
3. The direction of induced rotor EMF and current reverses, causing a reversal of torque. The machine now delivers electrical power to the grid, absorbing mechanical power.
4. **Excitation**: It requires an external AC source (the grid) to provide the magnetizing current for the rotating magnetic field. **It cannot generate reactive power; it consumes it.**

#### 6.2 Applications and Limitations

1. **Applications**:
   * **Wind Turbines**: Common in small to medium wind energy conversion systems (Doubly-Fed Induction Generators, DFIG).
   * **Small Hydroelectric Plants**: For standalone or grid-connected systems.
   * **Waste Heat Recovery**: In combined heat and power (CHP) systems.
   * **Braking**: For regenerative braking in some variable speed drives.
2. **Limitations**:
   * Requires an existing AC supply for excitation (not a standalone generator).
   * Poor voltage and frequency regulation unless used with power electronic converters (as in DFIG).

***

### 7. Single-Phase Induction Motors

Used extensively in domestic and light commercial applications (fans, pumps, compressors, washing machines) where only a single-phase supply is available. The key challenge is that a single-phase winding produces a **pulsating** (not rotating) magnetic field, which provides **no starting torque**.

#### 7.1 Need for a Starting Mechanism

A pulsating field can be resolved into two equal and opposite rotating fields. The net starting torque is zero. To make the motor self-starting, a mechanism is needed to create a **phase difference** between two windings, producing a rotating field at start.

#### 7.2 Types of Single-Phase Induction Motors

**A. Split-Phase Motor**

1. **Construction**: Has a **main winding** (thick wire, low resistance, high inductance) and an **auxiliary (starting) winding** (thin wire, high resistance, low inductance). The two windings are displaced by 90° electrical in space.
2. **Operation**: Due to the impedance difference, the current in the auxiliary winding ($$I\_a$$) leads the current in the main winding ($$I\_m$$) by about 30°. This produces a rotating field and starting torque.
3. **Starting Switch**: A centrifugal switch or a relay disconnects the auxiliary winding once the motor reaches 70-80% of synchronous speed. The motor then runs on the main winding only.
4. **Characteristics**: Moderate starting torque, low cost. Used in fans, blowers, washing machine pumps.

**B. Capacitor-Start Motor**

1. **Construction**: Similar to split-phase, but a **capacitor** is connected in series with the auxiliary winding. The capacitor is chosen to create a phase shift closer to 90°.
2. **Operation**: Provides much higher starting torque (2-4 times full load torque) than a split-phase motor because of the better phase split.
3. **Starting Switch**: Uses a centrifugal switch to disconnect both the capacitor and the auxiliary winding after start.
4. **Applications**: Compressors, pumps, conveyors – applications requiring high starting torque.

**C. Permanent Split-Capacitor (PSC) Motor**

1. **Construction**: A capacitor is permanently connected in series with the auxiliary winding. **No centrifugal switch**.
2. **Operation**: Both windings remain energized during running. This improves running power factor, efficiency, and reduces noise. Starting torque is lower than a capacitor-start motor.
3. **Characteristics**: Simple, reliable (no switch), good running performance, low starting torque.
4. **Applications**: Ceiling fans, blowers, HVAC equipment.

**D. Two-Value Capacitor (Capacitor-Start Capacitor-Run) Motor**

1. **Construction**: Uses **two capacitors**. A high-value **starting capacitor** (for high torque) and a lower-value **running capacitor** (for good running performance).
2. **Operation**: At start, both capacitors are in circuit via a centrifugal switch. After start, the switch disconnects the starting capacitor, leaving the running capacitor permanently connected.
3. **Characteristics**: Combines high starting torque of capacitor-start with good running performance of PSC.
4. **Applications**: High-inertia loads like compressors, pumps.

**E. Shaded-Pole Motor**

1. **Construction**: Has salient poles (projecting poles). A portion of each pole is surrounded by a short-circuited copper coil called a **shading ring**.
2. **Operation**: The shading ring causes the flux in the shaded portion to lag the flux in the main portion. This creates a shifting magnetic field from the unshaded to the shaded part, producing a small starting torque.
3. **Characteristics**: Very low starting torque, low efficiency (5-35%), simple, cheap, reliable (no switches or capacitors).
4. **Applications**: Small fans, record players, hair dryers, low-power devices.

**Conclusion**: The induction motor's dominance stems from its elegant simplicity and robustness. From the production of a rotating magnetic field to the intricacies of slip and torque production, a deep understanding of its principles is fundamental. While three-phase motors power industry, the clever adaptations of single-phase motors bring motion to our homes. With the advent of modern power electronics (VFDs), the induction motor's capabilities have been further extended, solidifying its role as the cornerstone of electromechanical energy conversion.
