Unit 3: Fundamentals of electrical machines

I. Orientation — Electromagnetic Energy Conversion

Electrical machines convert energy through electromagnetic fields: transformers transfer AC electrical energy between circuits, motors convert electrical energy into mechanical energy, and generators perform the reverse conversion. Their operation rests mainly on Faraday’s law of electromagnetic induction, the Lorentz force principle, and magnetic coupling.

A. Governing Principles and Conventions

The following principles provide the foundation for all machines in this unit.

• Faraday’s law: A changing magnetic flux linkage induces an emf.
  

  TEXT

    e = -N(dΦ/dt)

  
  where (e) is induced emf in volts, (N) is the number of turns, and (\Phi) is magnetic flux in webers.

• Lenz’s law: The negative sign indicates that induced emf opposes the change producing it.

• Force on a conductor: A current-carrying conductor in a magnetic field experiences force.
  

  TEXT

    F = B I l sinθ

  
  where (F) is force in newtons, (B) is flux density in teslas, (I) is current, (l) is active conductor length, and (\theta) is the field-current angle.

• Motor torque: Electromagnetic force acting at a radius produces torque, measured in newton-metres.

• Machine losses: Practical machines have copper (I^2R), iron, mechanical, and stray-load losses; therefore, output power is less than input power.

II. Direction Rules — Motor and Generator Actions

A. Fleming's left-hand rule

Fleming’s left-hand rule determines the direction of force or motion in a motor.

• Finger arrangement: Hold the left thumb, forefinger, and middle finger mutually perpendicular.

  • Forefinger: Magnetic field direction, from north to south.
  • Middle finger: Conventional current direction.
  • Thumb: Force or motion direction.

• Application: It predicts the direction in which an armature conductor moves when it carries current within a magnetic field.

• Reversal: Reversing either current or magnetic field reverses force; reversing both leaves the force direction unchanged.

B. Fleming's right-hand rule

Fleming’s right-hand rule determines the direction of induced current in a generator.

• Finger arrangement: Hold the right thumb, forefinger, and middle finger mutually perpendicular.

  • Forefinger: Magnetic field direction.
  • Thumb: Motion of the conductor.
  • Middle finger: Induced conventional current.

• Application: The rule applies when a conductor cuts magnetic flux and generator emf is produced.

• Contrast: The left hand describes motor action, whereas the right hand describes generator action.

III. Transformers — Static AC Energy Transfer

A. Mutual inductance and mutual coupling phenomena in transformer

Mutual inductance is the property by which changing current in one coil induces emf in another magnetically linked coil.

• Mutual induction: If primary current (i_1) produces flux linking the secondary, the secondary emf is:
  

  TEXT

    e₂ = -M(di₁/dt)

  
  where (M) is mutual inductance in henrys.

• Coupling coefficient: Magnetic linkage quality is represented by:
  

  TEXT

    M = k√(L₁L₂),    0 ≤ k ≤ 1

  
  where (L_1) and (L_2) are self-inductances and (k) is the coupling coefficient.

• Core function: A laminated iron or steel core provides a low-reluctance path, increasing common flux and reducing leakage flux.

B. Transformer working

A transformer transfers AC power between two circuits at the same frequency through mutual induction.

• Primary action: Applied AC voltage causes a magnetizing current and alternating core flux.

• Secondary action: The alternating flux links secondary turns and induces emf according to Faraday’s law.

• Sinusoidal emf equation:
  

  TEXT

    E = 4.44 f N Φm

  
  where (E) is rms induced emf, (f) is frequency, (N) is turns, and (\Phi_m) is maximum flux.

• Loaded operation: Secondary current produces opposing magnetomotive force; the primary draws additional current to keep core flux nearly constant.

C. Concept of turns ratio and applications

The turns ratio determines the relationship between primary and secondary voltage, current, and impedance.

• Ideal relationships:
  

  TEXT

    a = N₁/N₂ = V₁/V₂ = I₂/I₁

  
  where (a) is turns ratio and subscripts 1 and 2 identify primary and secondary quantities.

• Types:

  1. Step-up: (N_2>N_1), so secondary voltage increases and current decreases.
  2. Step-down: (N_2<N_1), so secondary voltage decreases and current increases.

• Impedance transformation:
  

  TEXT

    Z₁ = a²Z₂

  
  This is useful for impedance matching in electronic and communication circuits.

• Applications: Step-up transformers support transmission; step-down units supply domestic and electronic loads; isolation transformers commonly use a 1:1 ratio.

D. Transformer on DC

A normal transformer must not be connected to a DC supply because steady DC produces no continuously changing flux.

• Initial transient: Flux changes briefly at switching, so a momentary secondary emf may occur.

• Steady state: Since (d\Phi/dt=0), secondary induced emf becomes zero and primary counter-emf disappears.

• Danger: Primary current is then limited mainly by low winding resistance:
  

  TEXT

    I ≈ V/R

  
  The resulting current may overheat and burn the winding or saturate the core.

IV. DC Machines — Commutator-Based Energy Conversion

A. Working principles of DC machines

A DC machine can operate as either a generator or motor because electromagnetic energy conversion is reversible.

• Generator operation: Rotating armature conductors cut magnetic flux and induce emf; the commutator mechanically rectifies internal alternating emf into DC at the brushes.

• Motor operation: Armature current interacts with field flux to produce torque:
  

  TEXT

    T ∝ ΦIₐ

  
  where (T) is torque, (\Phi) is flux per pole, and (I_a) is armature current.

• Back emf: Rotation produces an emf opposing the supply:
  

  TEXT

    E_b = V - IₐRₐ

  
  where (V) is supply voltage and (R_a) is armature resistance.

B. Classification of DC motors

DC motors are classified by how their field winding is connected to the armature.

• Separately excited motor: The field winding receives power from an independent DC source, enabling flexible flux control.

• Shunt motor: The field is connected parallel to the armature; nearly constant flux gives approximately constant speed.

• Series motor: Field and armature carry the same current; high starting torque results, but the motor must not run unloaded.

• Compound motor: Both shunt and series fields are used.

  • Cumulative compound: Series flux aids shunt flux.
  • Differential compound: Series flux opposes shunt flux.

C. Starting of DC motors

A DC motor requires a starter because back emf is zero when the rotor is stationary.

• Starting current:
  

  TEXT

    Iₐ,start = V/Rₐ

  
  Since (R_a) is very small, direct starting can produce destructive current.

• Starter resistance: External series resistance limits current and is gradually removed as speed and back emf increase.

• Common starters: Three-point starters serve shunt and compound motors; four-point starters separate the no-volt coil from the field circuit.

• Protection: No-volt release protects against supply failure, while overload release disconnects excessive current.

D. Speed control of DC motors

DC motor speed is controlled by changing armature voltage, armature resistance, or field flux.

• Speed relation:
  

  TEXT

    n ∝ (V - IₐRₐ)/Φ

  
  where (n) is rotational speed.

• Field control: Reducing (\Phi) raises speed; it is efficient and generally used above base speed.

• Armature-voltage control: Varying (V) gives smooth, efficient control below base speed.

• Armature-resistance control: Adding resistance lowers speed but wastes power and produces poor regulation.

E. Applications of DC motors

Each DC motor type is selected according to its starting torque and speed-regulation characteristics.

• Shunt motors: Used for lathes, fans, blowers, and machine tools requiring nearly constant speed.

• Series motors: Used for cranes, hoists, electric traction, and automobile starter motors because of high starting torque.

• Cumulative compound motors: Used for elevators, presses, conveyors, and rolling mills where high starting torque and fairly constant speed are needed.

• Separately excited motors: Used in precision drives, laboratories, and variable-speed industrial systems.

V. AC Motors — Rotating Magnetic-Field Machines

A. Working principle of single-phase induction motors

A single-phase induction motor operates by induction but is not inherently self-starting.

• Pulsating field: A single-phase stator current creates a pulsating magnetic field.

• Double-revolving-field theory: The pulsating field is resolved into two equal fields rotating in opposite directions at synchronous speed.

• Zero starting torque: At standstill, the two fields produce equal opposite torques, giving zero net torque.

• Starting methods: Split-phase, capacitor-start, capacitor-run, and shaded-pole arrangements create phase displacement and an initial rotating field.

• Running action: Once rotation begins, forward torque exceeds backward torque and the motor continues running.

B. Working principle of three-phase induction motors

A three-phase induction motor works because balanced stator currents create a naturally rotating magnetic field.

• Synchronous speed:
  

  TEXT

    Nₛ = 120f/P

  
  where (N_s) is speed in revolutions per minute, (f) is supply frequency, and (P) is number of poles.

• Rotor induction: The rotating field cuts rotor conductors, inducing emf and current; their interaction with the field produces torque.

• Slip:
  

  TEXT

    s = (Nₛ - Nᵣ)/Nₛ

  
  where (N_r) is rotor speed. Slip must remain positive for induction and motor torque.

• Rotor types: Squirrel-cage rotors are rugged and simple; wound rotors permit external resistance for improved starting control.

C. Applications of AC motors

AC motors dominate industrial and domestic drives because they are robust, economical, and compatible with AC supplies.

• Single-phase motors: Used in ceiling fans, refrigerators, washing machines, pumps, mixers, and small machine tools.

• Three-phase induction motors: Used for compressors, conveyors, pumps, crushers, elevators, and factory machinery.

• Synchronous motors: Used for constant-speed drives and power-factor improvement in large installations.

• Selection basis: Required torque, supply phase, speed regulation, starting method, efficiency, and duty cycle determine the motor type.

VI. Special Transformers — Coupling, Economy, and Measurement

A. Mutual inductance and mutual coupling phenomena in transformer

Practical transformer performance depends on how effectively primary flux links the secondary winding.

• Useful mutual flux: Common core flux links both windings and transfers energy through mutual induction.

• Leakage flux: Flux linking only one winding behaves as leakage reactance and causes voltage drop under load.

• Coupling improvement: Closely placed, interleaved, or concentric windings increase (k) and reduce leakage.

• Electrical isolation: Mutual coupling transfers energy without conductive connection, except in an auto-transformer where windings are electrically connected.

B. Auto-transformer

An auto-transformer uses one continuous tapped winding, part of which is common to both input and output.

• Operation: Power is transferred partly by induction and partly by direct electrical conduction.

• Advantages: It requires less copper, has smaller size, higher efficiency, better voltage regulation, and lower cost than a two-winding transformer of similar rating.

• Limitations: It provides no galvanic isolation and can produce dangerous fault currents because of its low impedance.

• Applications: Used for induction-motor starting, voltage regulation, laboratory variable supplies, and interconnection of systems with close voltage ratios.

C. Instrument transformers

Instrument transformers reduce high currents or voltages to safe standardized values for measurement and protection.

• Current transformer (CT): Connected in series; its secondary current is proportional to line current, commonly rated at 5 A or 1 A.

  • Safety rule: Never open-circuit an energized CT secondary because a dangerously high voltage may develop.

• Potential transformer (PT): Connected in parallel; it reduces system voltage to a measurable value, often around 110 V secondary.

• Functions: CTs and PTs isolate meters and protective relays from high-energy power circuits while extending their measurement ranges.

• Errors: Ratio error and phase-angle error arise from magnetizing current, winding impedance, and connected burden.