CT_Questions/CT_02.md

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Department of Electrical & Computer Engineering (ECE), RUET

2nd Year Even Semester (Session 2023-24)

Class Test (CT) Questions — CT-02

Course Code: ECE 2207
Course Title: Electrical Machines-I
Date: 03/08/2026
Time: 20 minutes
Total Marks: 20 (10 marks per question)

Question Paper Scan

CT-02 Question Paper Scan

Question Paper Transcript

Sl.	Question	COs	POs	Marks
01.	Justify the following: "The value of rotor resistance does not alter the value of the maximum torque but only the value of slip at which it occurs."	CO1	PO1	10
02.	Explain the blocked rotor test of an induction motor. Also enlist the necessities of performing this test.	CO1	PO1	10

Comprehensive Solutions & Analysis

Question 01: Maximum Torque and Rotor Resistance Independence

Statement to Justify: "The value of rotor resistance does not alter the value of the maximum torque but only the value of slip at which it occurs." [Marks: 10, CO: 1, PO: 1]

1. Mathematical Derivation of Rotor Torque

Let:

$E_{2}$ = Rotor induced EMF per phase at standstill
$X_{2}$ = Standstill rotor leakage reactance per phase
$R_{2}$ = Rotor circuit resistance per phase
$s$ = Operating slip of the motor

At any running slip $s$ :

Induced rotor EMF per phase: $E_{2 s} = s E_{2}$
Rotor leakage reactance per phase: $X_{2 s} = s X_{2}$
Rotor impedance per phase: $Z_{2 s} = R_{2}^{2} + (s X_{2})^{2}$

The rotor current per phase is: $I_{2} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

The rotor power factor is: $cos θ_{2} = \frac{R _{2}}{Z _{2 s}} = \frac{R _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

The developed electromagnetic torque is proportional to the rotor power input: $T = k_{t} \cdot E_{2} \cdot I_{2} \cdot cos θ_{2}$ $T = \frac{k \cdot s E _{2}^{2} R _{2}}{R _{2}^{2} + s ^{2} X _{2}^{2}}$

Here $k = \frac{3}{2 π N _{s}}$ is a machine constant.

2. Slip for Maximum Torque ( $s_{m T}$ )

To find the slip at which torque reaches its peak, differentiate $T$ with respect to slip $s$ and set it to zero: $\frac{d T}{d s} = 0$

Using the quotient rule: $\frac{d}{d s} [\frac{s R _{2}}{R _{2}^{2} + s ^{2} X _{2}^{2}}] = \frac{( R _{2}^{2} + s ^{2} X _{2}^{2} ) ( R _{2} ) - ( s R _{2} ) ( 2 s X _{2}^{2} )}{( R _{2}^{2} + s ^{2} X _{2}^{2} ) ^{2}} = 0$

Set the numerator to zero: $R_{2} (R_{2}^{2} + s^{2} X_{2}^{2}) - 2 s^{2} R_{2} X_{2}^{2} = 0$ $R_{2}^{2} - s^{2} X_{2}^{2} = 0$ $s^{2} = \frac{R _{2}^{2}}{X _{2}^{2}}$ $s_{m T} = \frac{R _{2}}{X _{2}}$

This shows that the slip at maximum torque $s_{m T}$ is directly proportional to rotor resistance $R_{2}$ : $s_{m T} \propto R_{2}$

If rotor resistance increases, maximum torque occurs at a higher slip (a lower rotor speed).

3. Value of Maximum Torque ( $T_{max}$ )

Substitute the condition $s = s_{m T} = \frac{R _{2}}{X _{2}}$ into the general torque equation:

$T_{max} = \frac{k ( \frac{R _{2}}{X _{2}} ) E _{2}^{2} R _{2}}{R _{2}^{2} + ( \frac{R _{2}}{X _{2}} ) ^{2} X _{2}^{2}}$ $T_{max} = \frac{k ( \frac{R _{2}^{2}}{X _{2}} ) E _{2}^{2}}{R _{2}^{2} + R _{2}^{2}} = \frac{k ( \frac{R _{2}^{2}}{X _{2}} ) E _{2}^{2}}{2 R _{2}^{2}}$ $T_{max} = \frac{k E _{2}^{2}}{2 X _{2}}$

4. Physical Justification & Conclusion

$T_{max}$ is independent of $R_{2}$ : The term $R_{2}$ cancels out completely from the final expression for $T_{max}$ . Maximum torque depends only on supply voltage ( $E_{2} \propto V$ ) and standstill rotor leakage reactance $X_{2}$ .
Peak position shifts with $R_{2}$ : The slip at which the peak occurs is $s_{m T} = \frac{R _{2}}{X _{2}}$ . Adding external resistance to the rotor circuit shifts the torque peak toward higher slip (lower speed).
Starting torque improves: If we insert external rotor resistance such that $R_{2} = X_{2}$ , then $s_{m T} = 1$ . The motor develops its maximum possible torque right at start ( $N = 0$ ).
Summary: Varying rotor resistance changes the location of the maximum torque on the speed axis. But it never alters the magnitude of the peak itself. (Justified)

Torque-slip characteristics of 3-phase induction motor with varying rotor resistance

Question 02: Blocked Rotor Test of an Induction Motor

Question: Explain the blocked rotor test of an induction motor. Also enlist the necessities of performing this test. [Marks: 10, CO: 1, PO: 1]

1. Test Description & Operating Setup

The blocked rotor test is analogous to the short-circuit test of a transformer. The rotor is held stationary so that it cannot rotate ( $N = 0$ , slip $s = 1$ ).

3-Phase AC    3-Phase        Wattmeters      Stator Terminals
Supply -----> Variac -------> [ W1, W2 ] ------> [ A1, B1, C1 ]
                              Ammeters [A]
                              Voltmeter [V]        Rotor Shaft
                                                    [ LOCKED / BLOCKED ]

Procedure:

Lock the rotor shaft mechanically using a clamp or brake band so it cannot rotate.
Short-circuit the slip rings if using a wound rotor motor.
Connect a three-phase autotransformer (variac) to the stator through a voltmeter, ammeters, and two wattmeters.
Start with zero applied voltage.
Gradually increase the voltage until rated full-load stator current flows through the ammeters.
Record the applied line voltage $V_{b r}$ , line current $I_{b r}$ , and total input power $P_{b r} = W_{1} + W_{2}$ .
Keep the test brief to avoid overheating the rotor and stator windings.

Why Core Loss is Neglected:

The applied voltage $V_{b r}$ during this test is small. It is typically only 10% to 15% of rated line voltage. Core loss is proportional to voltage squared ( $V^{2}$ ). Because the test voltage is very low, the iron losses in the stator and rotor are negligible. The measured input power $P_{b r}$ is taken entirely as full-load copper loss of both windings.

2. Parameter Calculations

Assuming a star-connected stator:

Per-phase values: $V_{phase} = \frac{V _{b r}}{3}, I_{phase} = I_{b r}$
Total equivalent impedance referred to stator: $Z_{01} = \frac{V _{b r} / 3}{I _{b r}}$
Total equivalent resistance referred to stator: $P_{b r} = 3 \cdot I_{b r}^{2} \cdot R_{01} ⟹ R_{01} = \frac{P _{b r}}{3 I _{b r}^{2}}$
Total equivalent leakage reactance referred to stator: $X_{01} = Z_{01}^{2} - R_{01}^{2}$
Separating stator and rotor parameters: Measure DC resistance of stator winding $R_{1, dc}$ and multiply by 1.2 to 1.3 for AC skin effect to get $R_{1}$ . The rotor resistance referred to stator is: $R_{2}^{'} = R_{01} - R_{1}$ The leakage reactance is usually split equally: $X_{1} = X_{2}^{'} = 0.5 X_{01}$

3. Necessities of Performing the Blocked Rotor Test

The blocked rotor test is necessary for the following engineering reasons:

Measurement of Full-Load Copper Losses: It gives the total $I^{2} R$ loss of the stator and rotor windings at rated current. This is required to determine motor efficiency.
Extraction of Equivalent Circuit Series Parameters: It supplies $R_{01}$ , $X_{01}$ , and $Z_{01}$ . These values form the series branch of the exact and approximate equivalent circuits.
Calculation of Short-Circuit Current at Rated Voltage: Because impedances are linear, the starting current with full rated voltage applied across the terminals is: $I_{sc} = I_{b r} \times (\frac{V _{rated}}{V _{b r}})$
Estimation of Starting Torque: Starting torque depends directly on short-circuit current and rotor resistance ( $T_{s t} \propto I_{sc}^{2} R_{2}$ ). The test values help determine starting torque without running the motor at full voltage.
Construction of Circle Diagram: The short-circuit power factor $cos ϕ_{sc} = \frac{P _{b r}}{3 V _{b r} I _{b r}}$ and short-circuit current $I_{sc}$ are needed to draw the short-circuit line and complete the circle diagram of the induction motor.

Cross-References & Study Vault Links

Classroom Board Notes on Torque Derivation: ClassNoteByRaidah/Class_07.md
Classroom Board Notes on Maximum Torque: ClassNoteByRaidah/Class_08.md
Classroom Notes on Motor Testing: ClassNoteByRaidah/Class_09.md
Faculty Lecture Slides: SlidesByMaam/L-03_ECE-2107.md, SlidesByMaam/L-04_ECE-2107.md
Textbook Chapter: Books/B.L._Theraja/Ch-34_Induction_Motor.md, Books/B.L._Theraja/Ch-35_Computation_and_Tests_on_Induction_Motors.md
Semester Final Repeats: PrevYearQuestions/2023.md, PrevYearQuestions/2020.md

← CT-01 | 🏠 Index | CT-03 →

CT_Questions/CT_02.md

9.0 KB · commit d2f3b6e

← CT-01 | 🏠 Index | CT-03 →

Department of Electrical & Computer Engineering (ECE), RUET

2nd Year Even Semester (Session 2023-24)

Class Test (CT) Questions — CT-02

Course Code: ECE 2207
Course Title: Electrical Machines-I
Date: 03/08/2026
Time: 20 minutes
Total Marks: 20 (10 marks per question)

Question Paper Scan

CT-02 Question Paper Scan

Question Paper Transcript

Sl.	Question	COs	POs	Marks
01.	Justify the following: "The value of rotor resistance does not alter the value of the maximum torque but only the value of slip at which it occurs."	CO1	PO1	10
02.	Explain the blocked rotor test of an induction motor. Also enlist the necessities of performing this test.	CO1	PO1	10

Comprehensive Solutions & Analysis

Question 01: Maximum Torque and Rotor Resistance Independence

Statement to Justify: "The value of rotor resistance does not alter the value of the maximum torque but only the value of slip at which it occurs." [Marks: 10, CO: 1, PO: 1]

1. Mathematical Derivation of Rotor Torque

Let:

$E_{2}$ = Rotor induced EMF per phase at standstill
$X_{2}$ = Standstill rotor leakage reactance per phase
$R_{2}$ = Rotor circuit resistance per phase
$s$ = Operating slip of the motor

At any running slip $s$ :

Induced rotor EMF per phase: $E_{2 s} = s E_{2}$
Rotor leakage reactance per phase: $X_{2 s} = s X_{2}$
Rotor impedance per phase: $Z_{2 s} = R_{2}^{2} + (s X_{2})^{2}$

The rotor current per phase is: $I_{2} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

The rotor power factor is: $cos θ_{2} = \frac{R _{2}}{Z _{2 s}} = \frac{R _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}}$

Here $k = \frac{3}{2 π N _{s}}$ is a machine constant.

2. Slip for Maximum Torque ( $s_{m T}$ )

To find the slip at which torque reaches its peak, differentiate $T$ with respect to slip $s$ and set it to zero: $\frac{d T}{d s} = 0$

This shows that the slip at maximum torque $s_{m T}$ is directly proportional to rotor resistance $R_{2}$ : $s_{m T} \propto R_{2}$

If rotor resistance increases, maximum torque occurs at a higher slip (a lower rotor speed).

3. Value of Maximum Torque ( $T_{max}$ )

Substitute the condition $s = s_{m T} = \frac{R _{2}}{X _{2}}$ into the general torque equation:

4. Physical Justification & Conclusion

$T_{max}$ is independent of $R_{2}$ : The term $R_{2}$ cancels out completely from the final expression for $T_{max}$ . Maximum torque depends only on supply voltage ( $E_{2} \propto V$ ) and standstill rotor leakage reactance $X_{2}$ .
Peak position shifts with $R_{2}$ : The slip at which the peak occurs is $s_{m T} = \frac{R _{2}}{X _{2}}$ . Adding external resistance to the rotor circuit shifts the torque peak toward higher slip (lower speed).
Starting torque improves: If we insert external rotor resistance such that $R_{2} = X_{2}$ , then $s_{m T} = 1$ . The motor develops its maximum possible torque right at start ( $N = 0$ ).
Summary: Varying rotor resistance changes the location of the maximum torque on the speed axis. But it never alters the magnitude of the peak itself. (Justified)

Torque-slip characteristics of 3-phase induction motor with varying rotor resistance

Question 02: Blocked Rotor Test of an Induction Motor

Question: Explain the blocked rotor test of an induction motor. Also enlist the necessities of performing this test. [Marks: 10, CO: 1, PO: 1]

1. Test Description & Operating Setup

The blocked rotor test is analogous to the short-circuit test of a transformer. The rotor is held stationary so that it cannot rotate ( $N = 0$ , slip $s = 1$ ).

3-Phase AC    3-Phase        Wattmeters      Stator Terminals
Supply -----> Variac -------> [ W1, W2 ] ------> [ A1, B1, C1 ]
                              Ammeters [A]
                              Voltmeter [V]        Rotor Shaft
                                                    [ LOCKED / BLOCKED ]

Procedure:

Lock the rotor shaft mechanically using a clamp or brake band so it cannot rotate.
Short-circuit the slip rings if using a wound rotor motor.
Connect a three-phase autotransformer (variac) to the stator through a voltmeter, ammeters, and two wattmeters.
Start with zero applied voltage.
Gradually increase the voltage until rated full-load stator current flows through the ammeters.
Record the applied line voltage $V_{b r}$ , line current $I_{b r}$ , and total input power $P_{b r} = W_{1} + W_{2}$ .
Keep the test brief to avoid overheating the rotor and stator windings.

Why Core Loss is Neglected:

2. Parameter Calculations

Assuming a star-connected stator:

Per-phase values: $V_{phase} = \frac{V _{b r}}{3}, I_{phase} = I_{b r}$
Total equivalent impedance referred to stator: $Z_{01} = \frac{V _{b r} / 3}{I _{b r}}$
Total equivalent resistance referred to stator: $P_{b r} = 3 \cdot I_{b r}^{2} \cdot R_{01} ⟹ R_{01} = \frac{P _{b r}}{3 I _{b r}^{2}}$
Total equivalent leakage reactance referred to stator: $X_{01} = Z_{01}^{2} - R_{01}^{2}$
Separating stator and rotor parameters: Measure DC resistance of stator winding $R_{1, dc}$ and multiply by 1.2 to 1.3 for AC skin effect to get $R_{1}$ . The rotor resistance referred to stator is: $R_{2}^{'} = R_{01} - R_{1}$ The leakage reactance is usually split equally: $X_{1} = X_{2}^{'} = 0.5 X_{01}$

3. Necessities of Performing the Blocked Rotor Test

The blocked rotor test is necessary for the following engineering reasons:

Measurement of Full-Load Copper Losses: It gives the total $I^{2} R$ loss of the stator and rotor windings at rated current. This is required to determine motor efficiency.
Extraction of Equivalent Circuit Series Parameters: It supplies $R_{01}$ , $X_{01}$ , and $Z_{01}$ . These values form the series branch of the exact and approximate equivalent circuits.
Calculation of Short-Circuit Current at Rated Voltage: Because impedances are linear, the starting current with full rated voltage applied across the terminals is: $I_{sc} = I_{b r} \times (\frac{V _{rated}}{V _{b r}})$
Estimation of Starting Torque: Starting torque depends directly on short-circuit current and rotor resistance ( $T_{s t} \propto I_{sc}^{2} R_{2}$ ). The test values help determine starting torque without running the motor at full voltage.
Construction of Circle Diagram: The short-circuit power factor $cos ϕ_{sc} = \frac{P _{b r}}{3 V _{b r} I _{b r}}$ and short-circuit current $I_{sc}$ are needed to draw the short-circuit line and complete the circle diagram of the induction motor.

Cross-References & Study Vault Links

Classroom Board Notes on Torque Derivation: ClassNoteByRaidah/Class_07.md
Classroom Board Notes on Maximum Torque: ClassNoteByRaidah/Class_08.md
Classroom Notes on Motor Testing: ClassNoteByRaidah/Class_09.md
Faculty Lecture Slides: SlidesByMaam/L-03_ECE-2107.md, SlidesByMaam/L-04_ECE-2107.md
Textbook Chapter: Books/B.L._Theraja/Ch-34_Induction_Motor.md, Books/B.L._Theraja/Ch-35_Computation_and_Tests_on_Induction_Motors.md
Semester Final Repeats: PrevYearQuestions/2023.md, PrevYearQuestions/2020.md

← CT-01 | 🏠 Index | CT-03 →

Department of Electrical & Computer Engineering (ECE), RUET

2nd Year Even Semester (Session 2023-24)

Class Test (CT) Questions — CT-02

Question Paper Scan

Question Paper Transcript

Comprehensive Solutions & Analysis

Question 01: Maximum Torque and Rotor Resistance Independence

1. Mathematical Derivation of Rotor Torque

2. Slip for Maximum Torque (smT​)

3. Value of Maximum Torque (Tmax​)

4. Physical Justification & Conclusion

Question 02: Blocked Rotor Test of an Induction Motor

1. Test Description & Operating Setup

Procedure:

Why Core Loss is Neglected:

2. Parameter Calculations

3. Necessities of Performing the Blocked Rotor Test

Cross-References & Study Vault Links

Department of Electrical & Computer Engineering (ECE), RUET

2nd Year Even Semester (Session 2023-24)

Class Test (CT) Questions — CT-02

Question Paper Scan

Question Paper Transcript

Comprehensive Solutions & Analysis

Question 01: Maximum Torque and Rotor Resistance Independence

1. Mathematical Derivation of Rotor Torque

2. Slip for Maximum Torque (smT​)

3. Value of Maximum Torque (Tmax​)

4. Physical Justification & Conclusion

Question 02: Blocked Rotor Test of an Induction Motor

1. Test Description & Operating Setup

Procedure:

Why Core Loss is Neglected:

2. Parameter Calculations

3. Necessities of Performing the Blocked Rotor Test

Cross-References & Study Vault Links

2. Slip for Maximum Torque ( $s_{m T}$ )

3. Value of Maximum Torque ( $T_{max}$ )

2. Slip for Maximum Torque ( $s_{m T}$ )

3. Value of Maximum Torque ( $T_{max}$ )