Unit 3: Fundamentals of electrical machines - Subjective Questions

Mutual inductance is the property by which a change of current in one coil induces an electromotive force in a nearby coil.

If a changing current in the first coil produces flux linkage in the second coil, the induced emf is

Similarly,

where is the mutual inductance measured in henry.

The mutual inductance can also be expressed as

where is the number of turns in the second coil and is the flux produced by coil 1 that links coil 2.

The coefficient of coupling is

where .

Factors affecting mutual inductance:

• Number of turns in both coils
• Permeability of the magnetic core
• Cross-sectional area of the core
• Distance between the coils
• Relative orientation of the coils
• Amount of leakage flux
• Magnetic path length

A transformer operates on the principle of mutual induction, with alternating flux produced by the primary winding linking the secondary winding.

A transformer is a static electrical device that transfers AC power from one circuit to another at the same frequency through mutual induction.

Main constructional parts:

• Magnetic core: Made of laminated silicon-steel sheets to provide a low-reluctance flux path and reduce eddy-current loss.
• Primary winding: Connected to the AC supply.
• Secondary winding: Connected to the load.
• Insulation: Electrically isolates windings from each other and from the core.

Working principle:

1. When an AC voltage is applied to the primary winding, an alternating current flows through it.
2. The primary current produces an alternating magnetic flux in the core.
3. This common flux links both primary and secondary windings.
4. According to Faraday's law, emfs are induced in both windings:

5. When a load is connected to the secondary, secondary current flows and electrical power is transferred magnetically from the primary to the secondary.

A transformer changes voltage and current levels but does not change the supply frequency.

Let the core flux be sinusoidal:

The instantaneous induced emf in a winding of turns is

Therefore, the maximum value of induced emf is

The RMS value is

Hence,

For the primary and secondary windings:

Dividing the equations gives

For an ideal transformer, winding voltage drops are neglected, so

Since input power equals output power,

Therefore,

Thus:

• If , it is a step-up transformer.
• If , it is a step-down transformer.
• If , it is an isolation transformer.

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

For an ideal transformer,

Significance:

• It determines whether the transformer increases or decreases voltage.
• It determines the inverse change in current.
• It is used for impedance matching.

The impedance referred from the secondary to the primary is

Applications:

• Step-up transformers: Used at generating stations to raise voltage for transmission.
• Step-down transformers: Used in distribution networks, chargers, and electronic power supplies.
• Isolation transformers: Used for electrical safety and noise isolation.
• Impedance matching transformers: Used in audio, communication, and electronic circuits.
• Instrument transformers: Use a known turns ratio to measure high voltages and currents safely.

A transformer must not be connected to a DC supply because transformer action requires continuously changing magnetic flux.

The induced primary emf is

With a DC supply, after the short switching transient, the flux becomes constant. Therefore,

and no counter emf is induced in the primary. The primary current is then limited only by the small winding resistance:

This current can be extremely large.

Consequences of connecting a transformer to DC:

• Excessive primary current flows.
• The core quickly reaches magnetic saturation.
• Severe copper loss occurs.
• The winding overheats and insulation may fail.
• The protective fuse may operate.
• No continuous secondary voltage is obtained.

A momentary secondary pulse may appear only while the DC is being switched on or off because the flux changes during those instants.

An auto-transformer has a single continuous winding on a laminated magnetic core. A portion of the winding is common to both the primary and secondary circuits, while the remaining portion acts as the series winding.

If the supply is connected across turns and the load across turns, then ideally

Power is transferred in two ways:

• Conductively through the common electrical connection
• Inductively through mutual induction

Comparison with a two-winding transformer:

• An auto-transformer uses one tapped winding, whereas a conventional transformer has separate primary and secondary windings.
• An auto-transformer does not provide electrical isolation.
• It requires less copper and core material for a given rating.
• It generally has lower leakage reactance and better voltage regulation.
• It is more economical when the input and output voltages are close.

Advantages:

• Smaller size and lower cost
• Higher efficiency
• Better voltage regulation
• Higher kVA output for the same quantity of material
• Easy production of variable voltage using a sliding contact

Limitations:

• No galvanic isolation between input and output
• Higher short-circuit current due to low impedance
• Not suitable where the voltage ratio is very large
• A winding fault may expose the low-voltage side to high voltage

Applications:

• Induction motor starting
• Variable AC supplies
• Voltage regulation in distribution feeders
• Interconnection of systems having nearly equal voltages
• Laboratory voltage-control equipment

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

Current Transformer

A current transformer, or CT, has its primary connected in series with the line. Its primary may consist of one or a few turns, while its secondary has many turns and is connected to ammeters, energy meters, or protective relays.

Ideally,

Therefore,

The common rated secondary currents are and .

CT safety: The secondary of an energized CT must never be open-circuited. An open secondary can produce dangerously high voltage, overheating, and insulation failure. It should be short-circuited before disconnecting the instrument.

Potential Transformer

A potential transformer, or PT, is connected in parallel with a high-voltage line. It operates like a step-down transformer and supplies a low voltage to voltmeters, meters, and relays.

Ideally,

A common rated secondary voltage is .

PT safety: The secondary of a PT should not be short-circuited because excessive current may flow. One secondary terminal is generally earthed for safety.

Main purposes:

• Safe measurement of large currents and voltages
• Electrical isolation of instruments from high-voltage systems
• Extension of instrument ranges
• Operation of protective relays

A DC machine consists mainly of a magnetic field system, armature, commutator, brushes, shaft, and bearings. The same construction can operate either as a generator or as a motor.

DC generator operation

A DC generator converts mechanical energy into electrical energy. When armature conductors rotate and cut magnetic flux, an emf is induced according to Faraday's law. Its direction is determined by Fleming's right-hand rule.

The generated emf is

where is the number of poles, is flux per pole, is the total armature conductors, is speed in rpm, and is the number of parallel paths.

The emf generated in each conductor is alternating, but the commutator converts it into a unidirectional output at the brushes.

DC motor operation

A DC motor converts electrical energy into mechanical energy. A current-carrying conductor placed in a magnetic field experiences a force. The direction of force is determined by Fleming's left-hand rule.

The motor torque is proportional to flux and armature current:

As the armature rotates, a back emf is induced:

and

Thus, a DC machine works as a generator when mechanical input is supplied and as a motor when electrical input is supplied.

Fleming's left-hand rule

Stretch the thumb, forefinger, and middle finger of the left hand so that they are mutually perpendicular.

• Forefinger: Direction of magnetic field from north to south
• Middle finger: Direction of conventional current
• Thumb: Direction of force or motion

It is used to determine the direction of force on a current-carrying conductor and is therefore called the motor rule.

Fleming's right-hand rule

Stretch the thumb, forefinger, and middle finger of the right hand so that they are mutually perpendicular.

• Forefinger: Direction of magnetic field
• Thumb: Direction of motion of the conductor
• Middle finger: Direction of induced current

It is used to determine the direction of induced emf or current and is therefore called the generator rule.

Difference: The left-hand rule is applied when electrical energy produces mechanical motion, whereas the right-hand rule is applied when mechanical motion produces electrical energy.

DC motors are classified according to the connection and source of their field windings.

1. Separately excited DC motor

• The field winding is supplied from an independent DC source.
• Flux can be controlled separately from the armature current.
• It provides a wide and precise range of speed control.

2. Shunt motor

• The field winding is connected in parallel with the armature.
• Field flux is approximately constant.
• It has nearly constant speed and moderate starting torque.

3. Series motor

• The field winding is connected in series with the armature.
• Before magnetic saturation, flux is approximately proportional to armature current.
• It develops very high starting torque.
• It must not be operated without load because its speed may rise dangerously.

4. Compound motor

A compound motor has both series and shunt field windings.

• Cumulative compound motor: Series field assists the shunt field. It provides high starting torque with reasonably good speed regulation.
• Differential compound motor: Series field opposes the shunt field. Its operating characteristics may be unstable, so it is rarely used.

Compound motors may also be connected as long-shunt or short-shunt motors.

The voltage equation of a DC motor is

Therefore,

At starting, the armature is stationary, so its back emf is zero:

Hence,

Since armature resistance is very low, an excessive starting current would flow if the motor were connected directly to the supply. A starter inserts external resistance in series with the armature and gradually removes it as speed and back emf increase.

Three-point starter

Its three terminals are:

• L: Line terminal
• A: Armature terminal
• F: Field terminal

Main parts:

• Starting resistance divided into sections
• Movable handle and studs
• No-volt release coil
• Overload release coil
• Return spring

Operation:

1. The handle is moved from the OFF position across the studs.
2. Initially, full starting resistance is connected in series with the armature.
3. As the motor accelerates, the resistance is progressively cut out.
4. At the RUN position, the armature receives the full supply voltage.
5. The no-volt release holds the handle in the RUN position. It releases the handle during supply failure or field-circuit failure.
6. The overload release operates when armature current exceeds a safe limit and disconnects the motor.

A limitation is that weakening the shunt field for speed control may also weaken the no-volt release coil and cause unwanted tripping.

For a DC motor, the back emf is

The generated emf equation is

Therefore,

Hence, the speed relationship is

This equation shows that speed can be controlled by varying flux, armature voltage, or armature-circuit resistance.

1. Field or flux control

A rheostat is connected in series with the shunt field winding. Increasing field resistance reduces field current and flux. Since , the speed increases.

• Used for speeds above the base speed
• Efficient because field current is small
• Limited by commutation and mechanical considerations

2. Armature resistance control

An external resistance is inserted in series with the armature. This increases voltage drop and reduces armature voltage and speed.

• Provides speeds below the base speed
• Simple and inexpensive
• Has poor efficiency and poor speed regulation due to power loss

3. Armature voltage control

The voltage applied to the armature is varied while the flux remains approximately constant.

• Provides smooth speed control below the base speed
• Offers good efficiency and speed regulation
• Implemented using controlled rectifiers, DC choppers, or variable DC supplies

Modern drives combine armature-voltage control below base speed with field weakening above base speed.

The speed of a DC series motor is approximately given by

where is the series-field resistance.

Speed-control methods:

• Armature resistance control: An external resistance is connected in series with the motor. The increased voltage drop reduces speed, but considerable energy is lost.
• Field diverter control: A resistor is connected in parallel with the series field. Part of the current bypasses the field, reducing flux and increasing speed.
• Armature diverter control: A resistor is connected in parallel with the armature. It changes the armature current and is used to obtain lower speeds.
• Tapped-field control: The number of active series-field turns is varied. Fewer turns produce weaker flux and higher speed.
• Variable-voltage control: A controlled rectifier or DC chopper supplies a variable armature voltage for efficient control.
• Series-parallel control: Traditionally used in electric traction with two or more motors.

At no load, the required torque and armature current are very small. Therefore, series-field flux becomes weak. Since , the motor may accelerate to a dangerously high speed. This can cause mechanical damage, so a DC series motor must always be coupled to a load.

DC shunt motor

Characteristics:

• Nearly constant flux
• Approximately constant speed
• Moderate starting torque
• Good speed regulation

Applications:

• Lathes and drilling machines
• Fans and blowers
• Centrifugal pumps
• Machine tools
• Printing machines

DC series motor

Characteristics:

• Very high starting torque
• Large speed variation with load
• Dangerously high speed at no load
• Suitable for heavy starting duties

Applications:

• Electric traction
• Cranes and hoists
• Elevators
• Winches
• Automobile starter motors

Cumulative compound motor

Characteristics:

• Starting torque higher than that of a shunt motor
• Better speed regulation than a series motor
• Series-field flux assists the shunt-field flux
• Suitable for fluctuating and sudden loads

Applications:

• Conveyors
• Rolling mills
• Presses and shearing machines
• Reciprocating pumps
• Heavy planers and compressors

Motor selection is based mainly on starting torque, speed regulation, permissible speed variation, and load characteristics.

When a single-phase AC supply is applied to the stator winding, it produces a pulsating magnetic field. A stationary pulsating field does not create a definite starting direction.

According to the double-revolving-field theory, the pulsating field can be resolved into two rotating magnetic fields of equal magnitude:

• One rotates in the forward direction.
• The other rotates in the backward direction.
• Each has half the amplitude of the original pulsating field.
• Both rotate at synchronous speed in opposite directions.

At standstill, the forward and backward fields produce equal and opposite torques. Therefore, the net starting torque is zero, and the motor is not self-starting.

If an initial rotation is provided, the slip with respect to the forward field becomes smaller, whereas the slip with respect to the backward field becomes greater. The forward torque then exceeds the backward torque, and the motor continues to accelerate in the forward direction.

Common starting methods:

• Split-phase resistance starting
• Capacitor-start method
• Capacitor-start capacitor-run method
• Permanent-split-capacitor method
• Shaded-pole method

These methods create an initial phase difference between stator currents and thereby produce a rotating starting field.

When a balanced three-phase supply is applied to stator windings displaced by electrical, the three currents produce a rotating magnetic field of constant magnitude.

The rotating field completes electrical cycles per second. For a machine having poles, one mechanical revolution corresponds to electrical cycles. Therefore, the synchronous speed in revolutions per second is

Converting to revolutions per minute gives

The rotating stator field cuts the rotor conductors and induces rotor emf and current. The interaction between rotor current and the rotating field produces electromagnetic torque, causing the rotor to rotate in the same direction as the field.

The rotor cannot reach synchronous speed. If , there would be no relative motion between the rotating field and rotor, so no rotor emf, current, or torque would be produced.

The slip is

The percentage slip is

The rotor-current frequency is

At standstill, and . During normal operation, the rotor speed is slightly less than synchronous speed and the slip is small.

Three-phase motors are generally preferred for industrial drives because of their higher efficiency, rugged construction, smooth torque, and self-starting capability.

Major AC motor types and applications:

Single-phase induction motors

• Split-phase motor: Small machine tools, grinders, and blowers
• Capacitor-start motor: Pumps, compressors, and refrigerators requiring high starting torque
• Capacitor-start capacitor-run motor: Air conditioners and applications requiring quiet operation
• Permanent-split-capacitor motor: Fans and blowers
• Shaded-pole motor: Small fans, toys, and low-power appliances

Three-phase induction motors

• Squirrel-cage motor: Pumps, fans, compressors, conveyors, and machine tools
• Slip-ring motor: Cranes, hoists, elevators, crushers, and loads requiring high starting torque

Synchronous motors

• Constant-speed industrial drives
• Large compressors and pumps
• Power-factor improvement when overexcited

Universal motors

• Portable drills
• Mixers and vacuum cleaners
• Sewing machines and other high-speed appliances

Selection factors:

• Available supply and voltage
• Required power rating
• Starting torque
• Speed and speed-control range
• Duty cycle and load variation
• Efficiency and power factor
• Environmental conditions
• Maintenance requirements
• Initial and operating cost

The coefficient of coupling indicates the fraction of flux produced by one winding that links the other winding. It is given by

where is mutual inductance and and are the self-inductances of the two windings.

• If , all the flux produced by one winding links the other winding, representing perfect coupling.
• If , part of the flux does not link both windings.

Leakage flux is the portion of magnetic flux produced by a winding that links only that winding and not the other. Its effect is represented by primary and secondary leakage reactances.

Effects of leakage flux:

• Produces internal reactive voltage drops
• Reduces secondary terminal voltage under load
• Causes imperfect voltage regulation
• Limits short-circuit current
• Affects load sharing and transient performance

Ideal transformer

• Coupling coefficient
• No leakage flux
• No winding resistance
• No core loss
• Infinite core permeability
• Input power equals output power

Practical transformer

• Coupling coefficient is slightly less than unity
• Leakage flux and leakage reactance are present
• Windings have resistance and copper loss
• Hysteresis and eddy-current losses occur in the core
• Magnetizing current is required

Close placement of windings and the use of a high-permeability core improve magnetic coupling.

Given:

1. Secondary voltage

For an ideal transformer,

Therefore,

2. Maximum core flux

Using the transformer emf equation,

Assuming ,

Thus,

3. Ideal primary current

For an ideal transformer,

Therefore,

Conclusion

The secondary voltage is , the maximum core flux is approximately , and the ideal primary current is . Since and , it is a step-down transformer.