Electrical Machines — DC Machines and Induction Motors

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

GATE Weightage: ~6–10 marks/year (Electrical branch); torque-speed characteristics and starting methods are most frequently tested.

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DC Machine — Generated EMF: E = kΦω = (PΦNZ)/60A (volts)

DC Motor — Torque: T = kΦI_a (Nm)

Speed Regulation: SR = (N_no-load – N_full-load) / N_full-load × 100%

DC Motor Types

Type	Series Field	Shunt Field	Compound
Series	High current winding	None	None
Shunt	Few turns, many turns	High resistance parallel	Both
Compound	Series + Shunt	—	Series + Shunt

• Series motor: High starting torque, no-load speed dangerously high (must never be disconnected)
• Shunt motor: Constant speed, good regulation
• Compound: Starting torque of series + speed stability of shunt

Induction Motor — Synchronous Speed: N_s = 120f/P (rpm) Slip: s = (N_s – N_r)/N_s Rotor EMF frequency: f_r = s·f Induced Torque: T = (3/ω_s) × (V_th² × R_r’/s) / ((R_th + R_r’/s)² + (X_th + X_r’)²)

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

DC Machines — Construction and Working

Basic Principle

A DC machine converts mechanical energy to electrical energy (generator) or vice versa (motor) using electromagnetic induction and commutation.

Generated EMF (Generator)

E = kΦZ N / 60A = (PΦNZ) / 60A

Where:

• P = number of poles
• Φ = flux per pole (Wb)
• N = speed (rpm)
• Z = total number of conductors
• A = number of parallel paths (A = 2 for lap winding, A = P for wave winding)
• k = PZ/60A (machine constant)

No-load terminal voltage (generator): V = E (since I_a ≈ 0)

Loaded terminal voltage: V = E – I_a R_a (armature voltage drop)

Commutation

Commutation reverses the current in coils as they pass through the brushes. Sparking at brushes indicates poor commutation.

Causes of sparking: Mechanical issues, brush misalignment, high reactance of commutation.

DC Motor — Types and Characteristics

Armature Reaction

The armature flux (from armature current) distorts and weakens the main field flux from the field winding:

• Armature reaction reduces total flux → reduces generated EMF
• Demagnetizing effect: Cross-magnetizing component at quadrature axis
• Compensating winding or interpoles reduce armature reaction effects

Solutions:

• Compensating windings embedded in pole faces
• Interpoles (commutating poles) between main poles

Shunt Motor

Circuit: Field winding in parallel with armature

• Field current I_sh = V/R_sh (constant, since V is constant)
• Armature current I_a = I_L – I_sh
• Torque: T ∝ Φ·I_a ∝ I_a (since Φ constant)
• Speed: N = (V – I_aR_a)/kΦ ≈ constant (since Φ constant, small speed drop)

Characteristics:

• Constant speed (good regulation)
• Starting torque limited (I_a limited by armature resistance)
• Suitable for: fans, blowers, conveyors, machine tools

Series Motor

Circuit: Field winding in series with armature → I_a = I_sh = I_L

• Torque: T ∝ Φ·I_a ∝ I_a² (at low saturation) — high starting torque
• Speed: N ∝ V/(kΦ·I_a) — speed inversely proportional to load current
• No-load condition: I_a → 0 → Φ → 0 → N → dangerously high (runaway)

Critical safety point: Series motors should NEVER be belt-driven (load can disconnect → runaway).

Characteristics:

• Very high starting torque
• Variable speed (widely used in traction)
• Suitable for: cranes, elevators, traction, locomotives

Compound Motor

Circuit: Series + Shunt field windings

Cumulative compound: Series and shunt fields aid each other (same direction)

• Torque ∝ I_a(Φ_sh + Φ_se) — higher starting torque than shunt
• Less speed drop than series motor

Differential compound: Fields oppose (rarely used — unstable)

• Torque ∝ I_a(Φ_sh – Φ_se) — can cancel out

Starting Methods for DC Motors

Three-Point Starter

• Series resistance gradually cut out as motor speeds up
• Problem: If shunt field circuit opens → motor runs away
• Used with shunt and compound motors

Four-Point Starter

• Separate overload and no-voltage release coils
• Shunt field current independent of armature current
• Field cannot open accidentally → safer than 3-point

Series Motor Starters

• No-field-current-limiting (series field is armature current)
• Typically just a heavy-duty starting resistor

Soft Starters / Electronic Starters

• SCR-based phase angle control
• Gradually increases voltage to armature
• Modern replacement for resistor-type starters

Induction Motor — Working Principle

Squirrel cage rotor or wound rotor inside a rotating magnetic field from the stator.

Synchronous speed: N_s = 120f/P (rpm)

• f = supply frequency (50 Hz in India)
• P = number of poles

Slip: s = (N_s – N)/N_s

• s = 0 at synchronous (no torque)
• s = 1 at standstill (starting)
• s typically 0.01–0.05 at full load

Rotor quantities (at slip s):

• Rotor frequency: f_r = s·f
• Rotor induced EMF: E₂r = s·E₂
• Rotor impedance: Z₂r = R_r + jsX_r

Induction Motor — Torque-Speed Characteristic

Developed torque: T = (3/ω_s) × (I_r² × R_r/s)

Using Thevenin equivalent from stator side:

T = (3/ω_s) × (V_th² × R_r'/s) / ((R_th + R_r'/s)² + (X_th + X_r')²)

Maximum (pull-out) torque: Occurs when R_r’/s = √((R_th)² + (X_th + X_r’)²)

Condition for T_max: s_max = R_r’/√(R_th² + (X_th + X_r’)²)

Starting torque (s = 1): T_start = (3/ω_s) × (V_th² × R_r’) / ((R_th + R_r’)² + (X_th + X_r’)²)

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Induction Motor — Thevenin Equivalent

The exact equivalent circuit referred to stator:

V₁ → R₁ + jX₁ → → (+) → R_c || jX_m → R₂' + jX₂' → [s/R_r'] → (−)

For analysis, simplify to Thevenin equivalent seen by rotor circuit:

• V_th = V₁ × (jX_m || R_c) / (R₁ + jX₁ + jX_m || R_c)
• Z_th = (R₁ + jX₁) || (jX_m || R_c)
• R_th = real part of Z_th
• X_th = imaginary part of Z_th

Torque equation in terms of Thevenin values: T = (3/ω_s) × (V_th² × (R_r’/s)) / ((R_th + R_r’/s)² + (X_th + X_r’)²)

Effect of Rotor Resistance on Torque

Key Insight

Adding external resistance to rotor circuit (wound rotor):

• T_max unchanged (pull-out torque independent of R_r)
• s_max increases (slip at max torque increases linearly with R_r)
• Starting torque increases (initially low R_r gives low starting torque)

This is the primary advantage of wound rotor motors — adjustable starting torque.

Rotor added resistance used in:

• Soft starting (gradually reduce added R)
• Speed control (maintain torque at different speeds)

Power Flow in Induction Motor

Air-gap power P_ag = Mechanical power developed + Rotor copper loss

P_ag = T_dev × ω_s (synchronous mechanical power)

Rotor copper loss: P_cu2 = s × P_ag

Developed mechanical power: P_dev = (1 – s) × P_ag

Output (shaft) power: P_out = P_dev – friction & windage losses

Stage	Power	Formula
Input	P_in	√3 V_L I_L cosφ
Stator losses	P_cu1 + P_core	Fixed
Air-gap	P_ag	P_in – stator losses
Rotor copper loss	P_cu2 = s·P_ag	Proportional to slip
Developed	P_dev = (1–s)·P_ag	Useful mechanical
Output	P_out = P_dev – P_rot	Shaft power

Efficiency: η = P_out/P_in = (1 – s) × (P_ag/P_in)

Starting Methods for Induction Motors

Direct-On-Line (DOL) Starter

• Full voltage applied → high starting current (6–7× rated)
• Used for small motors (< 5 kW) where supply can handle inrush

Star-Delta Starter

• Motor starts in STAR → reduced voltage → reduced starting current
• After acceleration, switches to DELTA (full voltage)
• Starting current reduced to 1/3 of DOL
• Starting torque reduced to 1/3 of DOL

Auto-Transformer Starter

• Variable tap on auto-transformer reduces voltage
• Adjustable starting current/torque (50%, 65%, 80% taps)
• Less harsh than DOL for large motors

Soft Starter (Electronic)

• SCR phase-angle control gradually increases voltage
• Controlled starting current
• Smooth acceleration profile
• Can also provide soft stopping

Variable Frequency Drive (VFD)

• AC → DC → AC with variable frequency
• Speed control N_s = 120f/P over wide range
• Best efficiency and control
• Can maintain constant V/f for constant torque

Speed Control of Induction Motor

Method	Principle	Range
V/f control	Change frequency	Wide range, constant torque
Pole changing	Change P	Discrete speeds (2:1, 4:1)
Rotor resistance	Change s_max	Limited to wound rotor
Supply voltage	Change torque ∝ V²	Narrow range

V/f method is most common in modern drives because it maintains constant flux (Φ ∝ V/f).

DC Machine — Armature Reaction in Detail

Cross-magnetizing effect: Distorts main field flux Demagnetizing effect: Weakens main flux (at leading pole tips in generator, trailing in motor)

Neutral plane shift: Commutation plane shifts in direction of rotation.

• Generator: shifts in direction of rotation
• Motor: shifts opposite to direction of rotation

Solutions:

• Compensating winding: In pole faces, cancels cross flux
• Interpoles: Small poles between main poles, generated EMF opposes commutation

Example Problem — DC Motor

A 220 V DC shunt motor has R_a = 0.1 Ω, R_sh = 110 Ω. At no-load, current drawn = 5 A, speed = 1200 rpm. Find speed at full load when line current = 50 A.

Solution:

• No-load: I_sh = V/R_sh = 220/110 = 2 A

• I_a0 = I_0 – I_sh = 5 – 2 = 3 A

• E_0 = V – I_a0 R_a = 220 – 3(0.1) = 219.7 V

• Full-load: I_a = I_L – I_sh = 50 – 2 = 48 A

• E = V – I_a R_a = 220 – 48(0.1) = 215.2 V

• Since E = kΦω and Φ is constant (shunt motor):

• ω_0/ω = E_0/E → 1200/ω = 219.7/215.2

• ω = 1175 rpm

Example Problem — Induction Motor

A 4-pole, 50 Hz induction motor has s = 0.04 at full load. Find: (a) N_s, (b) N, (c) rotor frequency.

Solution: (a) N_s = 120×50/4 = 1500 rpm (b) N = N_s(1–s) = 1500(0.96) = 1440 rpm (c) f_r = s·f = 0.04 × 50 = 2 Hz

GATE Exam Strategy — DC Machines and Induction Motors

Expected question types:

1. DC motor: Find speed/armature current for shunt/series motor
2. DC motor: Armature reaction effects
3. Induction motor: Find N_s, N, s, f_r from given data
4. Induction motor: Torque-slip characteristic (T_max, T_start)
5. Starting methods comparison
6. Power flow diagram in induction motor
7. Efficiency calculation

Common GATE mistakes:

• Forgetting that Φ varies with load in series motor (T ∝ I_a² not I_a)
• Confusing s = 0 (no-slip, synchronous) with s = 1 (standstill)
• Using line values instead of phase values in three-phase induction motor
• Confusing N_s and N in torque equation
• Forgetting that V_th in Thevenin equivalent depends on magnetizing branch

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