Unit 3: Fundamentals of electrical machines - Subjective Questions

Mutual Inductance ():

• Mutual inductance is the property by which a changing current in one coil induces an EMF in a neighbouring coil due to the magnetic flux linkage between them.
• It is measured in Henry (H).
• Mathematically, the EMF induced in the second coil is:

where is the mutual inductance and is the rate of change of current in the first coil.

Mutual Coupling in a Transformer:

• A transformer consists of two windings (primary and secondary) wound on a common magnetic core.
• When alternating current flows in the primary winding, it produces an alternating magnetic flux in the core.
• This flux links with the secondary winding and induces an EMF in it — this is the phenomenon of mutual coupling.
• The degree of coupling is expressed by the coefficient of coupling ():

where and are the self-inductances of the two coils.

• For an ideal transformer, (perfect coupling, no leakage flux).

Working Principle:

A transformer works on the principle of Faraday's Law of Electromagnetic Induction and mutual induction.

Construction:

• It has two windings: primary and secondary, wound on a common laminated iron core.

Operation:

• When an AC voltage is applied to the primary winding, an alternating current flows and sets up an alternating magnetic flux in the core.
• This flux links with both windings.
• By Faraday's law, the EMF induced is:

• In the primary:
• In the secondary:

where = frequency, = number of turns, = maximum flux.

Key Points:

• A transformer transfers electrical energy from one circuit to another without any electrical connection and without change in frequency.

• It only works with AC supply, not DC.

  V1 ---[Primary N1]Core[Secondary N2]--- V2
  (Magnetic Flux links both windings)

Turns Ratio ():

The turns ratio of a transformer is the ratio of the number of turns in the primary winding to the number of turns in the secondary winding.

Voltage Relationship:

Since and :

Current Relationship:

For an ideal transformer, input power = output power:

Therefore:

Summary:

Applications of Turns Ratio:

• Step-up transformer (): increases voltage (used in power transmission).
• Step-down transformer (): decreases voltage (used in distribution and household supply).
• Impedance matching in electronic circuits.

Working Principle:

A DC motor works on the principle that when a current-carrying conductor is placed in a magnetic field, it experiences a mechanical force. The direction of this force is given by Fleming's Left Hand Rule.

The force experienced is given by:

where = magnetic flux density, = current, = length of conductor.

Construction (Main Parts):

• Yoke: Outer frame that provides mechanical support and a path for magnetic flux.
• Field Winding (Poles): Produces the magnetic field.
• Armature: Rotating part carrying conductors.
• Commutator: Converts AC induced in the armature to unidirectional torque; reverses current direction.
• Brushes: Carry current from external supply to the commutator.

Operation:

• DC supply is given to the field winding and armature.
• Current-carrying armature conductors in the magnetic field experience a force, producing torque.
• The commutator ensures that the torque acts in one direction, causing continuous rotation.
• A back EMF () is induced which opposes the applied voltage:

DC motors are classified based on how the field winding is connected with respect to the armature winding:

1. Separately Excited DC Motor:

• The field winding is energized by a separate DC source independent of the armature.
• Provides precise speed control.

2. Self-Excited DC Motor:

The field winding is supplied by the motor's own armature. Further classified as:

(a) Shunt Motor:

• Field winding connected in parallel with the armature.
• Nearly constant speed characteristic.
• Applications: lathes, fans, pumps, blowers.

(b) Series Motor:

• Field winding connected in series with the armature.
• Produces high starting torque.
• Applications: electric trains, cranes, hoists.

(c) Compound Motor:

• Has both series and shunt field windings.
• Combines characteristics of both types.
• Types: Cumulative and Differential compound.
• Applications: rolling mills, elevators, heavy machinery.

Voltage equation:

Need for a Starter:

The armature current of a DC motor is given by:

At the moment of starting:

• The motor is at rest, so speed .
• Since back EMF , we have at start.
• Therefore, the starting armature current becomes:

• Since the armature resistance is very small, this current becomes dangerously high (15 to 20 times the full load current).

Consequences of High Starting Current:

• Can damage the armature winding and commutator.
• Produces excessive heat and sparking at brushes.
• Causes a large voltage dip in the supply.

Function of a Starter:

• A starter adds external resistance in series with the armature at the time of starting.
• This limits the starting current to a safe value.
• As the motor gains speed, builds up and the resistance is gradually cut out.
• Common starters: 3-point starter and 4-point starter.

The speed of a DC motor is given by:

From this equation, speed can be controlled by three main methods:

1. Flux Control Method (Field Control):

• A variable resistance (rheostat) is connected in series with the field winding.
• Increasing resistance reduces field current, reducing flux , which increases speed.
• Allows speed above rated speed.
• Advantages: simple, economical, low power loss.

2. Armature Control Method (Rheostatic Control):

• A variable resistance is connected in series with the armature.
• This reduces the voltage across the armature, reducing and thus decreasing speed.
• Allows speed below rated speed.
• Disadvantage: large power loss in the resistance.

3. Voltage Control Method (Ward-Leonard System):

• The voltage applied to the armature is varied using a separate variable DC source.
• Provides smooth speed control over a wide range.
• Expensive but gives excellent control; used in elevators, rolling mills.

Fleming's Left Hand Rule (for Motors):

• Stretch the thumb, forefinger, and middle finger of the left hand mutually perpendicular to each other.
• Forefinger → direction of magnetic Field ()
• Middle finger → direction of Current ()
• Thumb → direction of Force/Motion ()

Application: Determines the direction of force (motion) experienced by a current-carrying conductor in a magnetic field. Used in DC motors.

Fleming's Right Hand Rule (for Generators):

• Stretch the thumb, forefinger, and middle finger of the right hand mutually perpendicular.
• Forefinger → direction of magnetic Field ()
• Thumb → direction of Motion of the conductor
• Middle finger → direction of induced Current/EMF

Application: Determines the direction of induced EMF/current in a conductor moving in a magnetic field. Used in DC/AC generators.

Key Difference:

Working Principle:

A three-phase induction motor works on the principle of electromagnetic induction and the production of a rotating magnetic field (RMF).

Rotating Magnetic Field (RMF):

• When a three-phase supply is given to the three-phase stator winding (phases spaced apart), it produces a magnetic field that rotates in space at synchronous speed:

where = supply frequency, = number of poles.

Operation:

• The rotating magnetic field cuts across the stationary rotor conductors.
• By Faraday's law, an EMF is induced in the rotor conductors.
• Since the rotor circuit is closed (short-circuited), currents flow in the rotor.
• These rotor currents produce their own magnetic field, which interacts with the RMF, producing torque (Lenz's law — the rotor tries to catch up with the field).
• The rotor rotates in the same direction as the RMF but at a speed slightly less than synchronous speed.

Slip ():

The motor is called asynchronous because the rotor never reaches synchronous speed.

Working Principle:

A single-phase induction motor consists of a single-phase stator winding and a squirrel-cage rotor. When single-phase AC supply is given to the stator, it produces an alternating (pulsating) magnetic field, not a rotating one.

Why it is NOT self-starting — Double Field Revolving Theory:

• The pulsating (alternating) magnetic field can be resolved into two equal and opposite rotating fields, each of half the magnitude.
• One field rotates in the forward direction and the other in the backward direction.
• At standstill, both fields produce equal and opposite torques, so the net torque is zero.
• Hence, the motor cannot start on its own.

Making it Self-Starting:

An auxiliary (starting) winding is added to create a phase difference and produce a rotating field. Common methods:

• Split-phase motor

• Capacitor-start motor

• Capacitor-start capacitor-run motor

• Shaded-pole motor

• Once the motor starts rotating in any direction, one field dominates and the motor continues to run in that direction.

Auto-Transformer:

An auto-transformer is a transformer with a single winding that acts as both the primary and secondary. A part of the winding is common to both circuits, so energy is transferred by both conduction and induction.

Working:

• The single winding is tapped at a certain point.
• The portion between the tapping and one end serves as the secondary.
• The voltage ratio depends on the number of turns:

A o----------
| N1 (full winding)
B o---- tap --- V2 output
| N2 (part)
C o----------

Advantages:

• Less copper required → cheaper and lighter.
• Higher efficiency due to lower copper and iron losses.
• Better voltage regulation.
• Smaller in size.

Disadvantages:

• No electrical isolation between primary and secondary (direct connection).
• If the common winding breaks, full primary voltage appears across the secondary — dangerous.
• Not suitable for large voltage ratios.

Applications: Starting of induction motors, voltage regulation (variac/dimmerstat), interconnecting power systems.

Transformer on DC Supply:

A transformer works only on the principle of mutual induction, which requires a changing (alternating) magnetic flux.

What happens with DC:

• When DC is applied to the primary, the current is constant (steady), so:

• Since the flux does not change with time, no EMF is induced in either winding:

• There is no induced voltage in the secondary → the transformer does not work.

Harmful Effects:

• With DC, there is no back EMF (self-induced EMF) in the primary to oppose the applied voltage.
• The primary current is limited only by the small winding resistance:

• This results in a very large current, which can:
  • Overheat and burn the primary winding.
  • Damage the insulation.

Conclusion: A transformer cannot operate on DC and may get damaged if connected to a DC supply. Transformers are strictly AC devices.

Instrument Transformers:

Instrument transformers are special transformers used to measure high voltages and currents safely by stepping them down to measurable, standardized low values for use with meters and relays.

1. Current Transformer (CT):

• Used to measure high currents.
• The primary has very few turns (sometimes a single bar) connected in series with the line.
• The secondary has many turns connected to an ammeter (standard 5 A or 1 A).
• It steps down the current.

• Caution: The secondary should never be left open when primary is energized (dangerous high voltage).

2. Potential Transformer (PT) / Voltage Transformer (VT):

• Used to measure high voltages.
• The primary has many turns connected in parallel (across) the line.
• The secondary has fewer turns connected to a voltmeter (standard 110 V).
• It steps down the voltage.

Advantages:

• Provide isolation between high voltage and measuring instruments → operator safety.
• Allow use of standard low-range instruments.
• Extend the range of measuring instruments.

Difference between Step-up and Step-down Transformer:

Applications of Transformers:

• Power Transmission: Step-up transformers raise voltage to reduce transmission losses ().
• Power Distribution: Step-down transformers reduce voltage for domestic and industrial use.
• Electronic Equipment: Supply low voltage to devices (chargers, adapters).
• Impedance Matching: In audio and communication circuits.
• Isolation: Isolation transformers for safety.
• Measurement: Instrument transformers (CT & PT).

EMF Equation of a Transformer:

Let:

• = number of turns in primary
• = number of turns in secondary
• = maximum value of flux (Wb)
• = supply frequency (Hz)

Derivation:

The flux in the core varies sinusoidally:

The flux increases from to in one quarter cycle, i.e., in time seconds.

Average rate of change of flux:

Average EMF induced per turn = volts.

RMS value (Form factor of sine wave = 1.11):

For the primary winding ( turns):

For the secondary winding ( turns):

Note: Dividing the two equations gives the turns ratio:

Comparison between DC Series Motor and DC Shunt Motor:

Applications:

DC Series Motor:

• Electric trains and traction
• Cranes and hoists
• Lifts and conveyors
• Where high starting torque is required

DC Shunt Motor:

• Lathes and drilling machines
• Fans and blowers
• Centrifugal pumps
• Where constant speed is required

Important: A series motor should never be started on no-load as it may reach dangerously high speeds.

A three-phase induction motor has two main parts: the stator (stationary) and the rotor (rotating). Based on rotor construction, there are two types:

1. Squirrel Cage Rotor:

• Consists of a cylindrical laminated iron core with slots on the surface.
• Heavy copper or aluminium bars are placed in the slots.
• The bars are short-circuited at both ends by end rings, forming a cage-like structure.
• No external connections or slip rings.

Advantages:

• Simple, rugged, and cheap
• Low maintenance (no brushes/slip rings)
• High efficiency

Disadvantages:

• Low starting torque
• External resistance cannot be added

2. Slip-Ring (Wound) Rotor:

• The rotor has a three-phase winding similar to the stator.
• Winding ends are connected to three slip rings mounted on the shaft.
• External resistance can be added through brushes on the slip rings.

Advantages:

• High starting torque
• Starting current can be controlled by external resistance
• Speed control is possible

Disadvantages:

• More expensive and complex
• Requires more maintenance (brushes, slip rings)

Applications: Squirrel cage — pumps, fans, lathes; Slip-ring — cranes, hoists, elevators (high torque applications).

Applications of AC Motors:

A. Three-Phase Induction Motors:

Used in industries where large power and robust operation are required:

• Industrial drives: Pumps, compressors, conveyors
• Machine tools: Lathes, drilling and grinding machines
• Fans and blowers: Ventilation and cooling systems
• Cranes and hoists (slip-ring type for high torque)
• Rolling mills, crushers, and mixers
• Agricultural pumps for irrigation

Advantages: Rugged, reliable, self-starting, high efficiency, low maintenance.

B. Single-Phase Induction Motors:

Used in domestic and small-power applications (where only single-phase supply is available):

• Fans: Ceiling fans, table fans, exhaust fans
• Household appliances: Mixers, grinders, washing machines, refrigerators
• Pumps: Small water pumps
• Air conditioners and coolers
• Blowers and small tools

C. Specific Motor Types:

• Capacitor-start motors: Refrigerators, compressors, pumps (need high starting torque)
• Shaded-pole motors: Small fans, toys, hair dryers (low power)
• Synchronous motors: Power factor correction, constant-speed drives (clocks, timers)

Torque Equation of a DC Motor:

Torque is the turning or twisting force developed in the motor shaft.

Derivation:

The mechanical power developed in the armature is equal to the electrical power converted:

Also, mechanical power in terms of torque and angular speed:

where is speed in rpm.

Equating both:

The back EMF is given by:

where = total conductors, = poles, = parallel paths.

Substituting :

Solving for torque :

Relationship:

Since , , and are constants:

• Torque is directly proportional to the flux () and the armature current ().
• For a shunt motor, is nearly constant, so .
• For a series motor, (before saturation), so — giving very high starting torque.

Back EMF ():

When the armature of a DC motor rotates in the magnetic field, the armature conductors cut the magnetic flux and an EMF is induced in them (according to Faraday's law). By Lenz's law, this induced EMF opposes the applied voltage that causes it. Hence it is called back EMF or counter EMF.

Expression:

where = flux per pole, = conductors, = speed, = poles, = parallel paths.

Since are constants:

Voltage Equation:

Therefore armature current:

Significance of Back EMF:

• Self-regulating action: Back EMF makes the DC motor self-regulating. It automatically adjusts the armature current according to the load.
  • When load increases → speed drops → decreases → increases → more torque developed to meet the load.
  • When load decreases → speed rises → increases → decreases → less torque.
• It limits the armature current during normal running.
• It governs the energy conversion — the power represents the electrical power converted to mechanical power.

Conclusion: Back EMF is essential as it keeps the motor operation stable and ensures the current drawn is proportional to the mechanical load.