boss_notes/T-15a_Torque_Starting.md

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← T-14: IM Equivalent Circuit | 🏠 Index | T-15b: Running & Max Torque →

T-15a: Starting Torque & Maximum Starting Torque

Section: B | Priority: 🟠 HIGH | Exam Frequency: 4/7 years Sources: Theraja Ch-34 (Art. 34.18-34.22), VK Mehta Ch-8 (Art. 8.11-8.12), Slides L-04

Why This Topic Matters

Starting torque questions appeared in 4 out of 7 papers. The derivation of starting torque and the condition for maximum starting torque ( $R_{2} = X_{2}$ ) are standard 4-6 mark questions. Every wound-rotor motor problem uses this concept. The starting torque formula is also the entry point for all torque analysis.

📝 Key Definitions

Starting Torque ( $T_{s t}$ ): "The torque developed by the motor at the instant of starting is called starting torque. At starting, the rotor is at standstill ( $N = 0$ , $s = 1$ )." — VK Mehta, Art. 8.11

Condition for Maximum Starting Torque: "The starting torque of an induction motor is maximum when rotor resistance per phase equals standstill rotor reactance per phase ( $R_{2} = X_{2}$ )." — VK Mehta, Art. 8.12

Starting Torque Derivation

At standstill, $s = 1$ . All rotor quantities are at their standstill values:

$E_{2 s} = E_{2}, X_{2 s} = X_{2}, I_{2} = \frac{E _{2}}{R _{2}^{2} + X _{2}^{2}}, cos ϕ_{2} = \frac{R _{2}}{R _{2}^{2} + X _{2}^{2}}$

Torque is proportional to $E_{2} I_{2} cos ϕ_{2}$ :

$T_{s t} \propto E_{2} \cdot \frac{E _{2}}{R _{2}^{2} + X _{2}^{2}} \cdot \frac{R _{2}}{R _{2}^{2} + X _{2}^{2}}$

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

where $k = \frac{3}{2 π N _{s}}$ (with $N_{s}$ in rps).

For squirrel-cage motors: $R_{2}$ is small and fixed. Starting torque is typically 1.5-2 times full-load torque. Starting current is 5-8 times rated.

For slip-ring motors: External resistance $R_{e x t}$ can be added through slip rings to increase $R_{2}$ at starting. This increases starting torque and reduces starting current.

Condition for Maximum Starting Torque

To find the $R_{2}$ that maximizes $T_{s t}$ , differentiate with respect to $R_{2}$ and set to zero:

$\frac{d T _{s t}}{d R _{2}} = k E_{2}^{2} \cdot \frac{( R _{2}^{2} + X _{2}^{2} ) ( 1 ) - R _{2} ( 2 R _{2} )}{( R _{2}^{2} + X _{2}^{2} ) ^{2}} = 0$

Numerator = 0:

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0$

$X_{2}^{2} = R_{2}^{2}$

$R_{2} = X_{2}$

Maximum starting torque (substituting $R_{2} = X_{2}$ ):

$T_{s t, m a x} = \frac{k E _{2}^{2} \cdot X _{2}}{X _{2}^{2} + X _{2}^{2}} = \frac{k E _{2}^{2}}{2 X _{2}}$

Effect of Rotor Resistance on Starting Torque

Condition	Starting Torque	Starting Current	Rotor pf
$R_{2} ≪ X_{2}$	Low	Very high	Low (mostly reactive)
$R_{2} = X_{2}$	Maximum	Moderate	$cos 45° = 0.707$
$R_{2} ≫ X_{2}$	Decreasing	Low	High (mostly resistive)

In wound-rotor motors, external resistance is gradually removed as the motor accelerates. At full speed, slip rings are short-circuited.

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

Appeared: 2019 Q8(a) — (5 marks)

Full Answer:

Pull-out torque: The maximum torque a 3-phase IM can develop while running. Also called breakdown torque or $T_{m a x}$ . If the mechanical load exceeds pull-out torque, the motor stalls.

Maximum starting torque derivation:

Starting torque at $s = 1$ :

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

To find $R_{2}$ for maximum starting torque, differentiate w.r.t. $R_{2}$ and set to zero:

$\frac{d T _{s t}}{d R _{2}} = k E_{2}^{2} \cdot \frac{( R _{2}^{2} + X _{2}^{2} ) - 2 R _{2}^{2}}{( R _{2}^{2} + X _{2}^{2} ) ^{2}} = 0$

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0 ⟹ R_{2} = X_{2}$

Substituting $R_{2} = X_{2}$ :

$T_{s t, m a x} = \frac{k E _{2}^{2} X _{2}}{2 X _{2}^{2}} = \frac{k E _{2}^{2}}{2 X _{2}}$

Note: $T_{s t, m a x} = T_{m a x}$ . When $R_{2} = X_{2}$ , the starting torque equals the maximum running torque.

🎯 Q2: Determine the starting torque of an IM.

Appeared: 2021 Q5(b) — (4 marks)

Full Answer:

Starting torque is the torque developed at standstill ( $s = 1$ ).

From the general torque equation at $s = 1$ :

$T_{s t} = \frac{k \cdot 1 \cdot E _{2}^{2} R _{2}}{R _{2}^{2} + ( 1 ) ^{2} X _{2}^{2}} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

where $k = 3/ (2 π N_{s})$ and $E_{2}$ is standstill rotor EMF per phase.

Effect of rotor resistance: To maximize $T_{s t}$ , set $d T_{s t} / d R_{2} = 0$ :

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0 ⟹ R_{2} = X_{2}$

Maximum starting torque at $R_{2} = X_{2}$ :

$T_{s t, m a x} = \frac{k E _{2}^{2}}{2 X _{2}}$

In wound-rotor motors, external resistance is added to achieve $R_{total} = X_{2}$ for maximum starting torque with reduced starting current.

Exam Variants

Year	Question	Type	Key Result
2019 Q8(a)	Define pull-out torque + max $T_{s t}$ derivation	Theory + derivation	$R_{2} = X_{2}$ for max $T_{s t}$
2021 Q5(b)	Determine starting torque of IM	Derivation	$T_{s t} = k E_{2}^{2} R_{2} / (R_{2}^{2} + X_{2}^{2})$

⚡ Exam Tips & Common Mistakes

Don't confuse $T_{s t}$ and $T_{m a x}$ . $T_{s t}$ is torque at $s = 1$ for any $R_{2}$ . $T_{m a x}$ is the peak of the torque-slip curve.
$R_{2} = X_{2}$ is for maximum STARTING torque. The condition for maximum RUNNING torque is $s = R_{2} / X_{2}$ .
Show the differentiation step. Writing just " $R_{2} = X_{2}$ " without the calculus loses marks.
State $k = 3/ (2 π N_{s})$ . Don't leave $k$ undefined.

T-15b: Running & Max Torque — General torque and breakdown torque
T-19: Starting Methods — How to implement reduced starting current
T-14: IM Equivalent Circuit — Circuit basis for torque equation

← T-14: IM Equivalent Circuit | 🏠 Index | T-15b: Running & Max Torque →

boss_notes/T-15a_Torque_Starting.md

6.1 KB · commit d2f3b6e

← T-14: IM Equivalent Circuit | 🏠 Index | T-15b: Running & Max Torque →

T-15a: Starting Torque & Maximum Starting Torque

Section: B | Priority: 🟠 HIGH | Exam Frequency: 4/7 years Sources: Theraja Ch-34 (Art. 34.18-34.22), VK Mehta Ch-8 (Art. 8.11-8.12), Slides L-04

Why This Topic Matters

📝 Key Definitions

Starting Torque ( $T_{s t}$ ): "The torque developed by the motor at the instant of starting is called starting torque. At starting, the rotor is at standstill ( $N = 0$ , $s = 1$ )." — VK Mehta, Art. 8.11

Condition for Maximum Starting Torque: "The starting torque of an induction motor is maximum when rotor resistance per phase equals standstill rotor reactance per phase ( $R_{2} = X_{2}$ )." — VK Mehta, Art. 8.12

Starting Torque Derivation

At standstill, $s = 1$ . All rotor quantities are at their standstill values:

$E_{2 s} = E_{2}, X_{2 s} = X_{2}, I_{2} = \frac{E _{2}}{R _{2}^{2} + X _{2}^{2}}, cos ϕ_{2} = \frac{R _{2}}{R _{2}^{2} + X _{2}^{2}}$

Torque is proportional to $E_{2} I_{2} cos ϕ_{2}$ :

$T_{s t} \propto E_{2} \cdot \frac{E _{2}}{R _{2}^{2} + X _{2}^{2}} \cdot \frac{R _{2}}{R _{2}^{2} + X _{2}^{2}}$

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

where $k = \frac{3}{2 π N _{s}}$ (with $N_{s}$ in rps).

For squirrel-cage motors: $R_{2}$ is small and fixed. Starting torque is typically 1.5-2 times full-load torque. Starting current is 5-8 times rated.

For slip-ring motors: External resistance $R_{e x t}$ can be added through slip rings to increase $R_{2}$ at starting. This increases starting torque and reduces starting current.

Condition for Maximum Starting Torque

To find the $R_{2}$ that maximizes $T_{s t}$ , differentiate with respect to $R_{2}$ and set to zero:

$\frac{d T _{s t}}{d R _{2}} = k E_{2}^{2} \cdot \frac{( R _{2}^{2} + X _{2}^{2} ) ( 1 ) - R _{2} ( 2 R _{2} )}{( R _{2}^{2} + X _{2}^{2} ) ^{2}} = 0$

Numerator = 0:

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0$

$X_{2}^{2} = R_{2}^{2}$

$R_{2} = X_{2}$

Maximum starting torque (substituting $R_{2} = X_{2}$ ):

$T_{s t, m a x} = \frac{k E _{2}^{2} \cdot X _{2}}{X _{2}^{2} + X _{2}^{2}} = \frac{k E _{2}^{2}}{2 X _{2}}$

Effect of Rotor Resistance on Starting Torque

Condition	Starting Torque	Starting Current	Rotor pf
$R_{2} ≪ X_{2}$	Low	Very high	Low (mostly reactive)
$R_{2} = X_{2}$	Maximum	Moderate	$cos 45° = 0.707$
$R_{2} ≫ X_{2}$	Decreasing	Low	High (mostly resistive)

In wound-rotor motors, external resistance is gradually removed as the motor accelerates. At full speed, slip rings are short-circuited.

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

Appeared: 2019 Q8(a) — (5 marks)

Full Answer:

Pull-out torque: The maximum torque a 3-phase IM can develop while running. Also called breakdown torque or $T_{m a x}$ . If the mechanical load exceeds pull-out torque, the motor stalls.

Maximum starting torque derivation:

Starting torque at $s = 1$ :

$T_{s t} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

To find $R_{2}$ for maximum starting torque, differentiate w.r.t. $R_{2}$ and set to zero:

$\frac{d T _{s t}}{d R _{2}} = k E_{2}^{2} \cdot \frac{( R _{2}^{2} + X _{2}^{2} ) - 2 R _{2}^{2}}{( R _{2}^{2} + X _{2}^{2} ) ^{2}} = 0$

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0 ⟹ R_{2} = X_{2}$

Substituting $R_{2} = X_{2}$ :

$T_{s t, m a x} = \frac{k E _{2}^{2} X _{2}}{2 X _{2}^{2}} = \frac{k E _{2}^{2}}{2 X _{2}}$

Note: $T_{s t, m a x} = T_{m a x}$ . When $R_{2} = X_{2}$ , the starting torque equals the maximum running torque.

🎯 Q2: Determine the starting torque of an IM.

Appeared: 2021 Q5(b) — (4 marks)

Full Answer:

Starting torque is the torque developed at standstill ( $s = 1$ ).

From the general torque equation at $s = 1$ :

$T_{s t} = \frac{k \cdot 1 \cdot E _{2}^{2} R _{2}}{R _{2}^{2} + ( 1 ) ^{2} X _{2}^{2}} = \frac{k E _{2}^{2} R _{2}}{R _{2}^{2} + X _{2}^{2}}$

where $k = 3/ (2 π N_{s})$ and $E_{2}$ is standstill rotor EMF per phase.

Effect of rotor resistance: To maximize $T_{s t}$ , set $d T_{s t} / d R_{2} = 0$ :

$R_{2}^{2} + X_{2}^{2} - 2 R_{2}^{2} = 0 ⟹ R_{2} = X_{2}$

Maximum starting torque at $R_{2} = X_{2}$ :

$T_{s t, m a x} = \frac{k E _{2}^{2}}{2 X _{2}}$

In wound-rotor motors, external resistance is added to achieve $R_{total} = X_{2}$ for maximum starting torque with reduced starting current.

Exam Variants

Year	Question	Type	Key Result
2019 Q8(a)	Define pull-out torque + max $T_{s t}$ derivation	Theory + derivation	$R_{2} = X_{2}$ for max $T_{s t}$
2021 Q5(b)	Determine starting torque of IM	Derivation	$T_{s t} = k E_{2}^{2} R_{2} / (R_{2}^{2} + X_{2}^{2})$

⚡ Exam Tips & Common Mistakes

Don't confuse $T_{s t}$ and $T_{m a x}$ . $T_{s t}$ is torque at $s = 1$ for any $R_{2}$ . $T_{m a x}$ is the peak of the torque-slip curve.
$R_{2} = X_{2}$ is for maximum STARTING torque. The condition for maximum RUNNING torque is $s = R_{2} / X_{2}$ .
Show the differentiation step. Writing just " $R_{2} = X_{2}$ " without the calculus loses marks.
State $k = 3/ (2 π N_{s})$ . Don't leave $k$ undefined.

T-15b: Running & Max Torque — General torque and breakdown torque
T-19: Starting Methods — How to implement reduced starting current
T-14: IM Equivalent Circuit — Circuit basis for torque equation

← T-14: IM Equivalent Circuit | 🏠 Index | T-15b: Running & Max Torque →

azazarfin60/Electrical_Machine_I_Semester_Final

boss_notes/T-15a_Torque_Starting.md

T-15a: Starting Torque & Maximum Starting Torque

Why This Topic Matters

📝 Key Definitions

Starting Torque Derivation

Condition for Maximum Starting Torque

Effect of Rotor Resistance on Starting Torque

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

🎯 Q2: Determine the starting torque of an IM.

Exam Variants

⚡ Exam Tips & Common Mistakes

Report a problem

azazarfin60/Electrical_Machine_I_Semester_Final

boss_notes/T-15a_Torque_Starting.md

T-15a: Starting Torque & Maximum Starting Torque

Why This Topic Matters

📝 Key Definitions

Starting Torque Derivation

Condition for Maximum Starting Torque

Effect of Rotor Resistance on Starting Torque

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

🎯 Q2: Determine the starting torque of an IM.

Exam Variants

⚡ Exam Tips & Common Mistakes

Report a problem

T-15a: Starting Torque & Maximum Starting Torque

Why This Topic Matters

📝 Key Definitions

Starting Torque Derivation

Condition for Maximum Starting Torque

Effect of Rotor Resistance on Starting Torque

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

🎯 Q2: Determine the starting torque of an IM.

Exam Variants

⚡ Exam Tips & Common Mistakes

🔗 Related Topics

T-15a: Starting Torque & Maximum Starting Torque

Why This Topic Matters

📝 Key Definitions

Starting Torque Derivation

Condition for Maximum Starting Torque

Effect of Rotor Resistance on Starting Torque

🏆 Golden Questions (Past Exam Archive)

🎯 Q1: What is pull-out torque? Derive the equation for maximum starting torque of a 3-phase IM.

🎯 Q2: Determine the starting torque of an IM.

Exam Variants

⚡ Exam Tips & Common Mistakes

🔗 Related Topics