Electrical Machines
Transformers, DC machines, and AC rotating machines
1. Transformers
Operating Principle
Transformers work on the principle of mutual inductance (Faraday's Law). AC voltage in the primary creates changing magnetic flux that induces voltage in the secondary.
Ideal Transformer Equations
Turns Ratio: a = N₁/N₂ = V₁/V₂ = I₂/I₁
Power In = Power Out: V₁I₁ = V₂I₂
Step-Up Transformer
N₂ > N₁ (more secondary turns)
V₂ > V₁ (increases voltage)
I₂ < I₁ (decreases current)
Step-Down Transformer
N₂ < N₁ (fewer secondary turns)
V₂ < V₁ (decreases voltage)
I₂ > I₁ (increases current)
Transformer Losses
| Loss Type | Cause | Varies With |
|---|---|---|
| Core/Iron Loss (Pi) | Hysteresis + Eddy currents | Constant (voltage dependent) |
| Copper Loss (Pcu) | I²R in windings | Load (current squared) |
Efficiency
η = Output Power / Input Power × 100%
η = (kVA × pf × 100) / (kVA × pf + Pi + k²Pcu)
k = load factor (fraction of full load)
Maximum efficiency occurs when: Pi = Pcu
Voltage Regulation
VR = (VNL - VFL) / VFL × 100%
VNL = no-load voltage, VFL = full-load voltage
Lower VR = better regulation
Transformer Tests
Open-Circuit Test
• Performed at rated voltage on LV side
• HV side open (no load)
• Determines core loss (Pi)
• Finds Rc and Xm
Short-Circuit Test
• Performed at reduced voltage on HV side
• LV side shorted
• Determines copper loss (Pcu)
• Finds Req and Xeq
2. DC Machines
Construction
DC machines have field winding (stator) and armature winding (rotor) with commutator for DC conversion.
EMF Equation
E = (PφZN) / (60A)
P = number of poles, φ = flux per pole (Wb)
Z = total armature conductors, N = speed (rpm)
A = parallel paths (A = P for lap, A = 2 for wave)
DC Motor Types
Shunt Motor
Field winding parallel with armature
V = Eb + IaRa
- • Constant flux (approximately)
- • Good speed regulation
- • Speed control by field rheostat
- • Applications: Lathes, fans, blowers
Series Motor
Field winding in series with armature
V = Eb + Ia(Ra + Rse)
- • High starting torque (T ∝ I²)
- • Poor speed regulation
- • Never run without load (runaway)
- • Applications: Cranes, hoists, traction
Compound Motor
Both shunt and series field windings
- Cumulative: Fields aid - high starting torque
- Differential: Fields oppose - nearly constant speed
- • Applications: Presses, elevators, rolling mills
Torque Equation
T = (PφZIa) / (2πA)
T ∝ φIa
Shunt: T ∝ Ia (constant flux)
Series: T ∝ Ia² (flux ∝ Ia)
Speed Control Methods
Field Control
• Vary field current (flux)
• Above base speed
• Constant power region
N ∝ 1/φ
Armature Control
• Vary armature voltage
• Below base speed
• Constant torque region
N ∝ V
DC Generator
V = E - IaRa
Terminal voltage less than generated EMF due to armature drop
Self-excited: Field current from generator output (residual magnetism required)
Separately-excited: Field current from external source
3. Synchronous Machines
Synchronous Speed
Synchronous machines operate at fixed speed determined by supply frequency and number of poles.
Ns = 120f / P
Ns = synchronous speed (rpm), f = frequency (Hz), P = number of poles
Synchronous Generator (Alternator)
EMF Equation
E = 4.44 fφN Kw
Kw = winding factor = Kd × Kp
Kd: Distribution factor (accounts for distributed windings)
Kp: Pitch factor (accounts for short-pitched coils)
Voltage Regulation
VR = (E0 - V) / V × 100%
E0 = no-load terminal voltage, V = full-load terminal voltage
Unity PF
VR positive (small)
Lagging PF
VR positive (large)
Leading PF
VR negative
Synchronous Motor
- Not self-starting: Requires starting mechanism (induction motor action, damper windings)
- Runs at synchronous speed only: Speed independent of load (within limits)
- Power factor control: Can operate at any power factor
Overexcited
Leading power factor
Supplies reactive power
(Synchronous condenser)
Underexcited
Lagging power factor
Consumes reactive power
Power Angle Equation
P = (VEf/Xs) sin δ
δ = power angle (torque angle)
Maximum power at δ = 90° (stability limit)
4. Induction Motors
Operating Principle
Rotating magnetic field in stator induces current in rotor, producing torque. Self-starting motor.
Slip
s = (Ns - Nr) / Ns
s = (Ns - Nr) / Ns × 100%
Ns = synchronous speed, Nr = rotor speed
Slip Conditions
At Standstill
Nr = 0, s = 1 (100%)
At Synchronous Speed
Nr = Ns, s = 0
Normal Operation
s = 2-5% (typical)
Rotor Parameters
- Rotor frequency: fr = sf
- Rotor EMF: Er = sE2 (E2 = standstill EMF)
- Rotor reactance: Xr = sX2 (X2 = standstill reactance)
- Rotor impedance: Zr = √(R2² + (sX2)²)
Power Flow
Input Power (Pin) = √3 VLIL cos θ
↓ (minus stator losses)
Air Gap Power (Pg) = Pin - Pcu1 - Pcore
↓ (minus rotor copper loss)
Mechanical Power (Pm) = Pg(1-s) = Pg - sPg
↓ (minus friction & windage)
Output Power (Pout)
Important Relationships
Pcu2 = sPg
Rotor copper loss
Pm = (1-s)Pg
Mechanical power developed
Pg : Pcu2 : Pm = 1 : s : (1-s)
Power ratio
η ≈ (1-s) × 100%
Approximate efficiency
Torque
T = Pm / ωr = Pg / ωs
T ∝ sV² / (R2² + (sX2)²)
Starting Torque (s=1):
Tst ∝ V² / (R2² + X2²)
Maximum Torque:
smax = R2/X2
Tmax ∝ V²/X2
Starting Methods
| Method | Voltage | Current | Torque |
|---|---|---|---|
| DOL (Direct On Line) | V | 6-8 × FLC | 100% |
| Star-Delta | V/√3 | 1/3 DOL | 1/3 DOL |
| Autotransformer | xV | x² × DOL | x² × DOL |
| Soft Starter / VFD | Variable | Controlled | Controlled |
5. Speed Control of Induction Motors
Speed Equation
Nr = Ns(1-s) = (120f/P)(1-s)
Speed Control Methods
Pole Changing
• Discrete speed steps
• Ns = 120f/P
• Used: Fans, pumps, crushers
Frequency Control (VFD)
• Variable speed
• V/f ratio constant
• Most efficient method
Rotor Resistance
• Slip ring motors only
• Increases slip
• Energy loss in external resistance
Voltage Control
• Limited range
• T ∝ V² reduces torque
• Used for fans (T ∝ N²)
V/f Control
To maintain constant flux (and thus constant torque capability), the V/f ratio is kept constant.
V/f = constant
Below base frequency: Constant torque region (V/f constant)
Above base frequency: Constant power region (V max, f increases)
6. Single-Phase Motors
Starting Problem
Single-phase motors produce pulsating (not rotating) magnetic field. Need auxiliary starting mechanism.
Types of Single-Phase Induction Motors
Split-Phase Motor
- • Starting winding with higher R/X ratio
- • Centrifugal switch disconnects starting winding
- • Low starting torque (150-175% rated)
- • Applications: Fans, blowers, small pumps
Capacitor-Start Motor
- • Capacitor in series with starting winding
- • Higher starting torque (250-350% rated)
- • Centrifugal switch cuts off capacitor
- • Applications: Compressors, pumps, conveyors
Capacitor-Start Capacitor-Run
- • Two capacitors: starting (large) and running (small)
- • Good starting torque and running efficiency
- • Better power factor
- • Applications: Air conditioners, refrigerators
Shaded-Pole Motor
- • Simplest single-phase motor
- • Copper shading ring on pole face
- • Low efficiency (30-40%)
- • Fixed direction of rotation
- • Applications: Small fans, toys, timers
Universal Motor
Series wound motor that operates on AC or DC
- • High speed (up to 20,000 rpm)
- • High starting torque
- • Speed varies with load
- • Applications: Drills, vacuum cleaners, mixers, blenders
7. Special Machines
Stepper Motor
Moves in discrete steps - precise positioning without feedback
- Step angle: θ = 360°/(Nr × m) where m = number of phases
- Types: Variable reluctance, Permanent magnet, Hybrid
- Applications: Printers, CNC machines, robotics, 3D printers
Servo Motor
Closed-loop control system with feedback for precise position/speed control
- • AC or DC types available
- • High torque-to-inertia ratio
- • Applications: Robotics, CNC, automation
Brushless DC Motor (BLDC)
DC motor with electronic commutation instead of brushes
- • Higher efficiency than brushed DC
- • No brush wear, longer life
- • Requires electronic controller
- • Applications: Computer drives, electric vehicles, drones
Linear Induction Motor
"Unrolled" induction motor producing linear motion
- • Linear synchronous speed: v = 2τf (τ = pole pitch)
- • Applications: Maglev trains, conveyor systems, sliding doors
8. Machine Testing and Maintenance
Induction Motor Tests
No-Load Test
• Motor runs at rated voltage, no load
• Determines: Core loss, friction & windage
• Finds: Rc and Xm
Blocked-Rotor Test
• Rotor locked, reduced voltage applied
• Determines: Full-load copper loss
• Finds: Req and Xeq
Insulation Resistance Testing
Minimum IR = (kV + 1) MΩ
Megger test at 500V or 1000V DC
Polarization Index (PI): 10-min reading / 1-min reading
Good insulation: PI > 2
Dry insulation: PI > 4
Common Faults
| Symptom | Possible Cause |
|---|---|
| Motor won't start | Open circuit, low voltage, seized bearings |
| Motor overheating | Overload, low voltage, poor ventilation |
| Excessive vibration | Unbalanced rotor, worn bearings, misalignment |
| Low insulation resistance | Moisture, contamination, aging |
Key Takeaways for EE Board Exam
Must-Know Formulas
- ✓ Transformer: a = N₁/N₂ = V₁/V₂ = I₂/I₁
- ✓ DC EMF: E = PφZN/(60A)
- ✓ Sync speed: Ns = 120f/P
- ✓ Slip: s = (Ns - Nr)/Ns
- ✓ Power ratio: 1 : s : (1-s)
- ✓ Max efficiency: Pi = Pcu
- ✓ smax = R₂/X₂ (max torque)
Critical Concepts
- ✓ OC test → core loss, SC test → copper loss
- ✓ Series motor: high torque, never no-load
- ✓ Shunt motor: constant speed
- ✓ Overexcited sync motor = leading pf
- ✓ Star-delta: Ist = 1/3 DOL
- ✓ V/f control for speed control
- ✓ Tmax independent of R₂