Three-Phase Induction Motor – Working, Types, and Torque-Slip

Basic Principle

Rotating Magnetic Field (RMF): Created by 3-phase currents in stator
Synchronous Speed: \(N_s = \dfrac{120f}{P}\) rpm
Slip: \(s = \dfrac{N_s - N_r}{N_s}\) (Always positive for motor action)
Rotor Speed: \(N_r = N_s(1-s)\) rpm
Rotor Frequency: \(f_r = sf\)
Rotor EMF: \(E_r = sE_{r0}\) (At standstill \(E_{r0}\))

GATE Key Point

Operating Modes:
Motor: \(0 < s < 1\) (Normal operation)
Generator: \(s < 0\) (Supersynchronous, \(N_r > N_s\))
Brake: \(s > 1\) (Subsynchronous, \(N_r < 0\))

Types of 3-Phase Induction Motors

Squirrel Cage Type

Rotor bars + end rings (aluminum/copper)
Robust, maintenance-free
Self-starting
Fixed rotor resistance
Lower cost
90% of industrial applications

Wound Rotor Type

3-phase rotor windings
Slip rings + carbon brushes
External resistance control
Higher starting torque
Speed control possible
Higher maintenance cost

GATE Focus

Questions mostly on squirrel cage motors due to widespread use

Per-Phase Equivalent Circuit

Circuit Parameters:

\(R_1, X_1\): Stator resistance and leakage reactance
\(R_c\): Core loss resistance (iron losses)
\(X_m\): Magnetizing reactance
\(R_2', X_2'\): Rotor resistance and reactance referred to stator
\(\dfrac{R_2'}{s}\): Total rotor branch resistance

Referring Rotor to Stator:

\[\begin{aligned} R_2' &= R_2 \left(\dfrac{N_1}{N_2}\right)^2 = R_2 K^2\\ X_2' &= X_2 \left(\dfrac{N_1}{N_2}\right)^2 = X_2 K^2\\ I_2' &= I_2 \left(\dfrac{N_2}{N_1}\right) = \dfrac{I_2}{K} \end{aligned}\]

Power Flow Analysis

Power Flow Sequence:

\[\begin{aligned} P_{input} &\rightarrow P_{stator\ losses} \rightarrow P_{air-gap} \rightarrow P_{rotor\ losses} \rightarrow P_{mechanical} \rightarrow P_{output} \end{aligned}\]

Power Relations:

Input Power: \(P_{in} = \sqrt{3}V_L I_L \cos\phi\)
Stator Losses: \(P_{stator} = 3I_1^2 R_1 + P_{core}\)
Air-gap Power: \(P_{ag} = P_{in} - P_{stator}\)
Rotor Copper Loss: \(P_{rcl} = sP_{ag} = 3I_2'^2 R_2'\)
Mechanical Power: \(P_{mech} = (1-s)P_{ag}\)
Output Power: \(P_{out} = P_{mech} - P_{friction+windage}\)

GATE Key Formula

\(P_{ag} : P_{rcl} : P_{mech} = 1 : s : (1-s)\)

Torque Equations

General Torque Equation:

\[T = \dfrac{3V_1^2 R_2'}{2\pi N_s} \cdot \dfrac{s}{(R_1 + R_2')^2 + s^2(X_1 + X_2')^2}\]

Approximate (when \(R_1 \ll X_1 + X_2'\)):

\[T = \dfrac{3V_1^2 R_2'}{2\pi N_s} \cdot \dfrac{s}{R_1^2 + s^2(X_1 + X_2')^2}\]

Alternative Form:

\[T = \dfrac{P_{ag}}{2\pi N_s/60} = \dfrac{P_{ag} \times 60}{2\pi N_s}\]

From Mechanical Power:

\[T = \dfrac{P_{mech} \times 60}{2\pi N_r} = \dfrac{(1-s)P_{ag} \times 60}{2\pi N_s(1-s)} = \dfrac{P_{ag} \times 60}{2\pi N_s}\]

Maximum Torque Analysis

Maximum Torque:

\[T_{max} = \dfrac{3V_1^2}{4\pi N_s(X_1 + X_2')} \text{ (independent of } R_2' \text{)}\]

Slip at Maximum Torque:

\[s_m = \dfrac{R_2'}{\sqrt{R_1^2 + (X_1 + X_2')^2}} \approx \dfrac{R_2'}{X_1 + X_2'}\]

Starting Torque:

\[T_{st} = \dfrac{3V_1^2 R_2'}{2\pi N_s[(R_1 + R_2')^2 + (X_1 + X_2')^2]}\]

Torque-Slip Relation:

\[\dfrac{T}{T_{max}} = \dfrac{2s/s_m}{s/s_m + s_m/s}\]

GATE Important Points

\(T_{max}\) is independent of rotor resistance
\(s_m\) is proportional to rotor resistance
For low slip: \(T \propto s\) (stable operation)

No-Load Test

Purpose: Determine shunt parameters and losses

Procedure: Motor at rated voltage, no mechanical load

Measurements: \(V_0\), \(I_0\), \(P_0\)

Calculations:

\[\begin{aligned} \cos\phi_0 &= \dfrac{P_0}{\sqrt{3}V_0 I_0}\\ I_c &= I_0 \cos\phi_0 \text{ (core loss component)}\\ I_m &= I_0 \sin\phi_0 \text{ (magnetizing component)}\\ R_c &= \dfrac{V_0}{\sqrt{3}I_c}\\ X_m &= \dfrac{V_0}{\sqrt{3}I_m} \end{aligned}\]

GATE Note

\(P_0 =\) Core losses + Friction & windage losses
(Rotor copper loss \(\approx 0\) since slip is very small)

Blocked Rotor Test

Purpose: Determine series parameters

Procedure: Rotor blocked, reduced voltage applied (15-25% of rated)

Measurements: \(V_{br}\), \(I_{br}\), \(P_{br}\)

Calculations:

\[\begin{aligned} R_{eq} &= R_1 + R_2' = \dfrac{P_{br}}{3I_{br}^2}\\ Z_{eq} &= \dfrac{V_{br}}{\sqrt{3}I_{br}}\\ X_{eq} &= X_1 + X_2' = \sqrt{Z_{eq}^2 - R_{eq}^2} \end{aligned}\]

Typical Parameter Distribution

\(R_1 = R_2' = \dfrac{R_{eq}}{2}\) and \(X_1 = X_2' = \dfrac{X_{eq}}{2}\)
(For wound rotor: \(R_1 = 0.4R_{eq}\), \(R_2' = 0.6R_{eq}\))

Starting Methods - Squirrel Cage

1. Direct-On-Line (DOL)

\(I_{st} = 5-8 \times I_{fl}\)
\(T_{st} = 1.5-2.5 \times T_{fl}\)
Simple and economical
High starting current

2. Star-Delta Starter

\(I_{line} = \dfrac{I_{DOL}}{3}\)
\(T_{st} = \dfrac{T_{DOL}}{3}\)
Motor must be delta-connected
6 terminals required

3. Auto-transformer Starter

\(I_{line} = \dfrac{I_{DOL}}{k^2}\)
\(T_{st} = \dfrac{T_{DOL}}{k^2}\)
Better torque per ampere
Tapping ratio \(k = 0.5, 0.65, 0.8\)

4. Rotor Resistance (Wound Rotor)

External resistance in rotor
High starting torque
Smooth acceleration
Power loss in resistances

GATE Comparison

Auto-transformer starter gives better performance than star-delta

Speed Control Methods

1. Frequency Control (VFD - Most Efficient)

\(N_s = \dfrac{120f}{P}\)
Constant V/f ratio maintained below base speed
Constant flux operation
Wide speed range with high efficiency

2. Voltage Control (Stator Voltage)

\(T \propto V^2\) (at constant slip)
Limited speed range (only speed reduction)
Poor efficiency at reduced speeds
Used for fan/pump applications

3. Pole Changing

Discrete speed steps
Consequent pole method: speeds in ratio 1:2
PAM (pole amplitude modulation) method

4. Rotor Resistance Control (Wound Rotor Only)

Speed below synchronous speed only
High slip operation (inefficient)
Stepless speed control

Efficiency Analysis

Total Losses:

\[\begin{aligned} P_{total\ losses} &= P_{stator\ Cu} + P_{core} + P_{rotor\ Cu} + P_{mechanical}\\ &= 3I_1^2 R_1 + P_{core} + 3I_2'^2 R_2' + P_{friction+windage} \end{aligned}\]

Efficiency:

\[\eta = \dfrac{P_{out}}{P_{in}} = \dfrac{P_{out}}{P_{out} + P_{losses}} = \dfrac{P_{in} - P_{losses}}{P_{in}}\]

Condition for Maximum Efficiency:

\[\text{Variable losses} = \text{Constant losses}\]

\[3I_1^2 R_1 + 3I_2'^2 R_2' = P_{core} + P_{friction+windage}\]

GATE Important

Maximum efficiency occurs when copper losses equal constant losses

Key Formulas for GATE

Basic Relations:

\[\begin{aligned} N_s &= \dfrac{120f}{P} \text{ rpm}\\ s &= \dfrac{N_s - N_r}{N_s}\\ N_r &= N_s(1-s) \text{ rpm}\\ f_r &= sf\\ E_r &= sE_{r0} \end{aligned}\]

Power Relations:

\[\begin{aligned} P_{ag} &= P_{in} - P_{stator\ losses}\\ P_{rcl} &= sP_{ag}\\ P_{mech} &= (1-s)P_{ag}\\ P_{out} &= P_{mech} - P_{mech\ losses} \end{aligned}\]

Torque Relations:

\[\begin{aligned} T &= \dfrac{P_{ag} \times 60}{2\pi N_s}\\ T_{max} &= \dfrac{3V_1^2}{4\pi N_s(X_1 + X_2')}\\ s_m &= \dfrac{R_2'}{X_1 + X_2'}\\ T_{st} &= \dfrac{3V_1^2 R_2'}{2\pi N_s[(R_1 + R_2')^2 + (X_1 + X_2')^2]} \end{aligned}\]

Starting Method Ratios:

\[\begin{aligned} \text{Star-Delta: } &I_{st} = \dfrac{I_{DOL}}{3}, \quad T_{st} = \dfrac{T_{DOL}}{3}\\ \text{Auto-transformer: } &I_{st} = \dfrac{I_{DOL}}{k^2}, \quad T_{st} = \dfrac{T_{DOL}}{k^2} \end{aligned}\]

GATE Problem-Solving Strategy

Power Flow Problems

Always identify: \(P_{in} \rightarrow P_{stator} \rightarrow P_{ag} \rightarrow P_{rcl} \rightarrow P_{mech} \rightarrow P_{out}\)
Use: \(P_{rcl} = sP_{ag}\) and \(P_{mech} = (1-s)P_{ag}\)
Remember: \(P_{ag} : P_{rcl} : P_{mech} = 1 : s : (1-s)\)

Torque Problems

For maximum torque: \(T_{max}\) is independent of \(R_2'\)
For slip at maximum torque: \(s_m \propto R_2'\)
Use \(T = \dfrac{P_{ag} \times 60}{2\pi N_s}\) for quick calculation

Testing Problems

No-load test: Find \(X_m\), \(R_c\), and constant losses
Blocked rotor test: Find \(R_{eq}\), \(X_{eq}\)
Use standard assumptions for parameter distribution

Common GATE Mistakes

Avoid These Errors

Confusing slip \(s\) with rotor speed \(N_r\)
Wrong sequence in power flow analysis
Forgetting to refer rotor parameters to stator side
Using wrong torque formula (mechanical power vs air-gap power)
Mixing up starting current ratios for different methods

Quick Verification Checks

For motor operation: \(0 < s < 1\)
Air-gap power \(>\) Mechanical power \(>\) Output power
Rotor copper loss decreases with increasing load
At no-load: slip is minimum (\(\approx 0.005-0.01\))
At full-load: slip is typically 0.03-0.05

Typical GATE Questions Pattern

High Probability Topics

Power flow analysis and efficiency calculation
Torque-slip characteristics and maximum torque
Starting methods comparison (especially star-delta)
Equivalent circuit parameter determination
Speed control methods and their characteristics

Numerical Problem Types

Given slip, find rotor speed, frequency, and losses
Calculate starting torque and current for different methods
Determine efficiency and losses at given loading
Find equivalent circuit parameters from test data
Compare performance of different speed control methods

Quick Reference

Most Important Relations

\[\begin{aligned} s &= \dfrac{N_s - N_r}{N_s}, \quad N_r = N_s(1-s)\\ P_{ag} : P_{rcl} : P_{mech} &= 1 : s : (1-s)\\ T &= \dfrac{P_{ag} \times 60}{2\pi N_s}\\ T_{max} &= \dfrac{3V_1^2}{4\pi N_s(X_1 + X_2')}\\ s_m &= \dfrac{R_2'}{X_1 + X_2'} \end{aligned}\]

Starting Method Quick Facts

Star-Delta: \(I, T\) both reduce by factor 3
Auto-transformer: \(I, T\) both reduce by factor \(k^2\)
DOL: Highest current, moderate torque
Rotor resistance: High torque, smooth starting

Fundamentals