﻿﻿﻿﻿ March 2017 – APSEEE

## Thevenin’s Theorem

Thevenin’s theorem was formulated for resistive networks by French physicist M. Leon. Thevenin, who proposed it in 1883. It may be enunciated as follows

Any two terminal networks consisting of linear impedance and emf sources may be replaced by a single voltage source with a equivalent series resistance. It makes the solution of complicated networks quite quick and simple.

## Explanation of Thevenin Theorem

Consider a simple circuit, to determine the current through load resistance RL, we will proceed as under

Step-1

Remove the resistance RL in which current is to be determined thus creating an open circuit between terminals A and B.

Step-2

Calculate the open circuit voltage VOC (Thevenin voltage Eth) which appears across terminals A and B when they are open (i.e when RL is removed.

Step-3

Replace the source (Battery) by its internal resistance r. When seen from the terminals A and B, the circuit consists of two parallel paths: one containing R2 and the other containing (R1+r). The equivalent resistance Rth of the network, as viewed from these terminals is given as

The equivalent resistance is also called Thevenin resistance.

Step-4

Replace the entire network by a single source (Called Thevenin voltage) source having an emf Eth and internal resistance Rth.  RL (Load Resistance) is now connected back to its terminals A and B from where it was removed.

Determine current flowing through the load resistance RL by applying ohm’s law.

## Applications of Thevenin’s Theorem

• To calculate the current in particular branch in the networks Thevenin Theorem is used.
• Designing of electronic circuits.

The above applications are practical applications of thevenin theorem

## Limitations of Thevenin’s Theorem

• Thevenin’s theorem cannot be applied to a networks which contains non-linear elements. This theorem is applicable only linear circuits or networks.
• Thevenin’s theorem cannot be used for determining the efficiency of the circuit.

## Construction of Synchronous Machines

### Construction of Synchronous Machines

Construction of synchronous motor and synchronous generator is same. In this article, we will discuss about construction of Synchronous machines.

Synchronous machines have following important parts

1. Stator
2. Rotor

### Stator

Stator is outer part of the machine It is the stationary part of a synchronous machine. The stator contains following parts.

• Stator Frame

It is a outer body of a machine. It is made up of cast iron. It protects the inner parts of the machine. The stator core is placed in between the frame. Cast iron is used for the construction of stator frame because it has high mechanical strength.

• Stator Core

The material of a stator core is laminated silicon steel. It is made from number of stamping which are insulated from each other. Laminations are used to reduce eddy current losses and silicon steel is used to reduce hysteresis losses. The function of stator core is to provide an easy path for magnetic flux. The slots are cut on its inner periphery to accommodate the winding.

• Stator Winding

Stator winding is placed in the slots. Enameled copper is used as winding material.

### Rotor

The rotating part of the machine is called Rotor. From construction point of view there are two types of rotors named as

Salient pole type rotor

Non-salient pole type rotor

• Salient Pole Type Rotor

Salient pole type construction is suited for medium and low speeds synchronous generators. In this case, projected poles are provided on the rotor. These rotors are usually employed at hydroelectric plants. The speed of these machine is quite low. These rotors are designed with larger diameter and small axial length. The exciting current is supplied by an exciter in fixed on each alternator shaft.

• Non-Salient Pole Type Rotor

Non-salient pole type construction is suited for the high speeds. In this case rotor is made of silicon steel laminations. These rotors are used for high speed alternators. These rotors have smaller diameter and larger axial length. These types of rotor give noiseless operation and better in dynamic balancing. About 2/3 of the rotor pitch is slotted, leaving 1/3 for the pole centre. The speed may be as high as 3000 r.p.m at 50Hz.

## Universal Motor

### Universal Motor

A motor which can be operated on AC as well as on DC voltage is called universal motor. Single phase AC supply or DC supply is required for the operation of universal motor. The construction as well as working of universal motor is similar to DC series motor. Basically, the applications of universal motor founds in various field of engineering.

### Construction

Basically, Universal motor has two main parts

• Stator
• Rotor

Stator

The stationary part of the machine is called the stator. The stator construction of universal motor is similar to single phase induction motors. It consists of the outer body, pole core, pole shoe and field winding. The pole core and pole shoe are made of laminated silicon steel. Lamination stampings are used to reduce eddy losses and silicon steel reduces hysteresis loss because the core of motor carries alternating flux. In case of DC motors, laminations are not necessary, but in case of universal motor, laminations are necessary because the flux is alternating in nature because the motor is operated on AC supply.

The field winding made of enameled copper is wound around the pole core and produce the required flux.

Rotor

It is rotating part of the machine. It consists of shaft, armature and commutator. The Armature is made up of the stamping of silicon steel since it carries the magnetic flux slots are cut at its outer periphery of the armature and armature winding is placed in these slots.

The Armature is keyed to the shaft is a part of the rotor which transfers mechanical energy to the load.

The Commutator is also keyed to the shaft. The shaft is made of mild steel. Mechanical strength of mild steel is very high.

A compensating winding is used to reduce the reactance, voltage is present in the armature when universal motor operated on AC supply.

Working Principle of Universal Motor

The operation of universal motor based on the principle when a current carrying conductor placed in the magnetic field, a mechanical force is exerted on it and torque develops.

Operation or Working of Universal Motor

In universal motor armature winding and filed winding both are connected in series like DC series motors. This motor develops unidirectional torque whether it is connected on AC supply or DC supply.

When single phase supply given to the motor, during the positive half cycle, current flows through the field winding and armature winding. The magnetic field is set up in both windings. The field of both windings is not aligned with each other. When rotor field tries to align itself with the main field an anti-clockwise torque is developed in the rotor.

When, the universal motor is connected to DC supply it behaves like a DC series motor.

The direction of rotation can be changed by interchanging connections to the field with respect to the armature as in a DC series motor.

### Applications of Universal Motor

This motor is available in small sizes. This is invariably used in following appliances.

• Vacuum cleaners
• Electric hand drills
• Mixer grinders
• Domestic sewing machine etc.
• Mixer and Juicer

## Construction of Three Phase Induction Motor

### Construction of Three Phase Induction Motor

Three phase induction machine can work as an induction generator and induction motor. But for most of the application, its performance as an induction generator is unsatisfactory. In this article we will study about construction of three phase induction motor.

The induction motor has following parts

• Stator
• Rotor

### Stator

Stator is stationary part of the induction motor.

Frame

The function of the frame is to support the inner parts of the motor such as core and winding.

The frame of the motor may be casted or fabricated. For small and medium size machine the frame is made of cast iron or casted. For large machine fabricated steel is used.

The stator frame provided only mechanical support to the inner parts and are not designed to carry the stator flux.

The slots may be open, semi closed or closed.

Stator Core

The stator core is a stack of cylindrical steel laminations (which are usually 0.03 cm to 0.05 cm thick) which are slotted their inner periphery. The three phase winding is housed in these slot. The laminations are used to reduce the eddy current losses.

Stator Winding

The stator core carries a three phase winding. The six terminals of the winding are connected in the terminal box of the machine.

In general, the same material is used for the rotor and the stator, but on e rotor has thicker laminations because the lower frequencies of rotor flux.

The winding may be star connected or delta connected. The winding’s are designed to be delta connected for normal running.

### Rotor

It is a rotating part of the motor. There are two types of rotors, which are employed in three phase induction motors.

### Squirrel Cage Rotor

The motors employing in this type of rotors are known as squirrel cage induction motors. Most of induction motors are employed squirrel cage rotor. It is simple, cheap rugged construction of rotor. It consists of a laminated core with conductors lying. In each slot copper brass, aluminum bar used as rotor conductors. The rotor winding is permanently short circuited and it is not possible to add any external resistance in the rotor circuit.

Phase Wound Rotor or Slip Ring Rotor

Phase wound rotor is also called slip ring rotor. Phase wound rotor carries a three phase wound winding wound for same number of poles as the stator winding. The three finish terminals are connected together forming star point and the three star terminals are connected to three copper slip rings mounted on the shaft. The brush is used to collect the current from the slip rings. These brushes are externally connected to a three phase star connected resistance for the purpose of starting and speed control of motor.

## Transformer

### Transformer

The Transformer is a static device that transfers electrical energy from one electrical circuit to another electrical circuit without change in the frequency but voltage level is usually changed. It may be stepup or step down transformer.

The transformer has two winding one is called primary winding and other is called secondary winding.

• The source is connected to the primary winding.
• The load is connected to the secondary winding.
• The transformer is an electromagnetic energy conversion device, since the energy received by the primary is first converted to magnetic energy and then it is reconverted into electrical energy in the secondary circuit. Thus the primary and secondary winding of a transformer are not connected electrically, but are coupled magnetically.
• The transformers may be stepped up or stepped down. If the secondary winding has more turns than the primary winding, then the secondary voltage transformer is called a stepup transformer. If the primary winding has more turns than the secondary winding, then the secondary voltage is lower than the primary voltage and the transformer is called a step down transformer.

### Necessity of Transformer

The electrical power is generated at 11kv. For economical reasons, ac power is transmitted at very high voltages over long distances. Therefore, a step up transformer is used at generating station to raise the voltage level. Then voltages are stepped down to suitable level by a transformer at various substations. Ultimately for utilization of electrical power, the voltage is stepped down to 400/230 for safely reasons.

### Efficiency

Transformer is a static device, owing to the lack of rotating parts. There are no friction or windage losses, so that the efficiency of a transformer is high. Typical transformer efficiencies at full load lie between 95% and 98%.

### Transformer on DC

A transformer cannot operate on dc supply and never be connected to a dc source. If the primary winding of a transformer is connected to a dc supply mains the flux produced will not vary but remain constant in magnitude and, therefore no emf will be induced in the secondary winding except at the moment of switching ON. There will be no induced emf in the primary winding and therefore, a heavy current will be drawn from the supply mains which may result in the burning out of the transformer winding.

## Induced EMF

### Induced EMF

When flux linking with a conductor changes, an induced emf produced in the conductors. This changes in the flux linkages can be obtained in the following two ways. Dynamically induced emf and statically induced emf.

1. Dynamically Induced emf

In this case, the field is stationary and conductors cut across it or field is moving and kept the conductors stationary.

1. Statically Induced emf

In this, conductors or coil remains stationary and flux linked with it is changed by simply increasing or decreasing the current producing this flux. Statically induced emf is further divided into two types. (i) Self induced emf and (ii) Mutually induced emf.

Dynamically induced emf

By either moving the conductor keeping the magnetic field system stationary or moving the field system keeping the conductor stationary so that flux is cut by the conductor, the emf is induced in this way in the conductor is called dynamically induced emf.

Considering a conductor A having length l placed in the magnetic field of flux density wb/m2.

Consider the condition when conductor A cuts across at right angles to the flux at a velocity V metre/second. Let the conductor be moved through a small distance dx in time dt.

Area swept by the conductor, A = l*dx

Flux cut by the conductor, Φ = B* A

= Bldx

According to Faraday’s laws of electromagnetic induction emf is induced in it.

If the conductor A moves at an angle Φ with the direction of flux or magnetic field at a velocity V metres/second. Then the induced emf is e = BlvsinΦvolts.

Generators work on the principle of production of dynamically induced emf.

### Statically induced e.m.f

Self Induced e.m.f

The emf induced in a coil due to the change of flux produced by it linking with its own turns is called self induced emf.

Consider a coil having N turns and battery is connected when current flows through the coil, it produces flux Φ which also links with its own turns. When the flux linking with the coil, an emf is induced in the coil. This induced emf is called self induced emf.

Mutually Induced emf

The emf induced in a coil due to the change in flux produced by another coil linking with it is called mutually induced emf.

Consider two coil A and B. Coil A is connected source and Galvanometer G is connected to coil B. When current flows coil A flux is produced which links with coil B and induced emf is produced in coil B. Galvanometer G shows deflection.

## Faraday’s Laws of Electromagnetic Induction

### Faraday’s First Law

This Faraday’s laws states that whenever a coil cuts by a magnetic flux or field an emf is induced in the coil.

To explain first law, consider a coil ‘C’ having large number of turns placed in the magnetic field of a permanent magnet to which a galvanometer G is connected.

When permanent magnet NS is moved nearer to coil or away from the coil, there is deflection occurs in the galvanometer needle which indicates that an e.m.f is induced in the conductor. If the conductor is moved parallel with the field, there is no deflection in the needle which indicates that no e.m.f is induced in the conductor.

On the other hand, if the permanent magnet is kept stationary and the coil brought nearer to the magnet or away from the magnet there is again deflection occurs in the galvanometer needle.

However, when the coil and magnet both are kept stationary, no flux cut with coil hence no e.m.f is induced in coil and no deflection occurs.

The magnitude at induced e.m.f depends upon the value of flux linkages with the coil and speed of moving of coil or magnet. Greater speed of magnet or coil greater e.m.f is induced in the coil. The direction of induced e.m.f depends upon the direction of magnetic field and direction of movement of coil.

### Faraday’s Second Law

Second Faraday’s law state that the magnitude of induced e.m.f in the conductor is directly proportional to the product of number of turns and flux linkages with the conductor.

Explanation

Suppose a coil has N turns and flux through it. Changes form an initial value of Φ1 webers to the final value of Φ2 webers in time t.

Initial flux linkages = N Φ1

Final flux linkages = N Φ2

In differential form

Negative sign shows the direction of induced e.m.f. That magnetic effect produced by it opposes very cause producing it.

## Kirchhoff’s Laws

### Kirchhoff’s Laws

Kirchhoff’s Laws are used to solve those networks or circuit where ohms law is may not be readily solved that circuit. Gustav Kirchhoff’s, a German Scientist, summed up his findings in a set of two laws which are called Kirchhoff’s Laws. Resistance of a complicated circuits and for calculating the currents flowing in the various branches of circuits or networks. The two laws are Kirchhoff’s current law and Kirchhoff’s voltage law.

### Kirchhoff’s Current Law (KCL)

This law relates the currents flowing through the circuit that is why this law is called Kirchhoff’s current law. This law is also known as Kirchhoff’s point law.

This law states that, in any electrical network, the algebraic sum of the currents meeting at a junction or node is always zero.

Lets us consider a case in which few conductors meeting at a junction or point A, where some conductors have current entering to the point and some conductors have currents leaving out the point. Assuming currents entering to the point to be positive while the outgoing currents are negative.

We have,

I1-I2-I3+I4+I5=0

I1+I4+I5= I2+I3

Incoming Currents = outgoing Currents

In other words we can say that incoming current is equal to outgoing current.

### Kirchhoff’s Voltage Law (KVL)

This law relates the voltages in a closed circuit of an electrical network. It is also known as Kirchhoff’s mesh law.

Kirchhoff’s voltage law states that the algebraic sum of product of current and resistance in a closed network is equal to the algebraic sum of EMFs in that closed path that is in a closed circuit.

V1+V2 = IR1+IR2

∑V = ∑IR = 0 or ∑IR = ∑V

In other way we can say that, in a closed circuit or mesh, the algebraic sum of all the EMFs plus the algebraic sum of products of currents and resistances is zero.

∑V + ∑IR = 0

## Electric Cells

Electric Cell

An Electric cell is source of electrical energy. The cell gives dc current. The emf developed and current supplied by a cell is very small. The electrical energy can be stored in cell.

Definition of Electric Cell

A cell is source of emf in which chemical energy is converted into electrical energy.

Forming of a Electric Cell

• An electric cell basically consists of two electrodes of different material, so that different potentials are built up when chemical action takes place on them.
• An electrolyte such as an acid, alkali or salt solution so that chemical action takes place between two electrodes. The solution must be capable to react chemically with the two electrodes.

• When two electrodes are immersed in the electrolyte, due to chemical action between electrodes and electrolyte, a potential difference established between two electrodes.

### E.M.F developed in a Cell

The magnitude of emf developed in cell depends upon the following.

• Nature or material of the plates used as electrodes of the cell.
• Type of the electrolyte used in the cell.

### Types of Cells

Electric cells can be divided into two types.

Primary Cells

The cell in which chemical action is not reversible, called primary cells. Voltaic cell, Danial cell, Silver Oxide cell, Dry cell etc are the examples of Primary Cell.

In these types of cells, once the cell is discharge it cannot be recharged because chemical action in this case is not reversible and the cells cannot be recharged. This makes Primary cells expensive source of electrical energy and that is why these cells are rarely used in commercial applications.

Secondary Cells

The cell in which chemical action is reversible, called secondary cell. Lead acid cell, Nickel-iron alkaline cell etc are the examples of secondary cells. These cells are rechargeable.

In these cells chemical action is reversible and cells can be recharged. While recharging, electrical energy is converted in the cell itself.

## Induction Generator

Induction Generator

In this article we will discuss about working of induction generator, various applications of induction generator etc.

We know that when three phase wound motor fed by a three phase supply, the torque develops and rotor rotates. In this case, machine works as a motor and rotor runs less than the speed of synchronous speed of revolving speed at stator.

However, if the induction motor mechanically coupled to a prime mover capable of driving the induction motor at a speed higher than the synchronous speed. When the motor run faster than synchronous speed, an induction motor runs as a generator is known as induction generator. At this stage the induction generator electrical energy to the A.C. mains instead of drawing it. It converts mechanical energy into electrical energy. As the prime movers rotate the rotor runs faster than the synchronous speed, the value of slip becomes negative. The power of the machine under such conditions is also negative.

Show the torque slip curve an induction machine:-

In case of motoring action the slip is positive, whereas in case of generating action the slip is negative.

In above diagram it shows that when the induction motor runs faster than the synchronous power. It start delivering active power to the three-phase line. For creating magnetic field, magnetizing current is required. So it draws reactive power from the line to which it is connected. The reactive power flows always opposite direction to the active power. For magnetizing current, the induction generator must be operated in parallel with another generator which can supply necessary magnetizing current to the induction generator.

The active power is directly proportional to the slop above the synchronous speed. When no source of reactive power connected to the line. Then reactive power required by the machine can supply by a group of capacitors connected across its terminals.

## Difference Between Synchronous Generator and Induction Generator

1. Induction generator does not require direct current excitation.
2. Induction generator will generate power only when it connected to the line.
3. Induction generator delivers active power to the main lines and draw reactive power for magnetizing current from the main lines.