Thevenin’s Theorem

Thevenins theorem states theat “Any linear, bilateral network having a no of voltage sources and resistances can replaced by a simple equivalent circuit consisting of single voltage source in series with a resistance, where the value of the voltage source is equal to the open circuit voltage across the two terminals of the network and the resistance is equal to the equivalent resistance measured between the terminals with all the energy sources replaced by their internal resistances”.
In many pratical applications it is always not necessary to analyze the complete circuit. It requires that the voltage or current or power in only one resistance of a circuit be found. The use of this theorem provides a simple, equivalent circuit, which can be substituted for the original network.

Superposition Principle

In any linear bilateral network, the current flowing in any element/branch when two or more sources are present is equal to the algebraic sum of the currents flowing in individual elements/branches when individual sources are acting alone, while the other sources are non-operative, that is while considering the effect of individual sources other ideal voltage sources and ideal current sources in the network are replaced by ‘short circuit’ and ‘open circuit’ across their terminals respectively.

Norton’s Theorem

It states that “Any two terminal linear network with current sources, voltage sources and resistances can be replaced by an equivalent circuit consisting of a current source in parallel with a resistance”.
The value of the current source is the short circuit current between the two terminals of the network and the resistance of the equivalent resistance measured between the terminals of the network with all the voltage sources replaced by their internal resistances and current sources by open circuit.

Maximum Power Transfer Theorem

This theorem is used to find the value of load resistance for which there would be maximum amount of power transfer from source to load.
Maximum power transfer theorem states that “In any linear bilateral network the maximum power will be delivered by the load, when the load resistance is equal to the source resistance.
i.e RL = RS.

IL = Vth /(RL+Rth) [RS = Rth]
Pmax = Vth
2/4Rth

D.C. Motor Characteristics

The performance of a d.c. motor can be judged from its characteristic curves known as motor characteristics, following are the three important characteristics of a d.c. motor:
(i) Torque and Armature current characteristic (Ta/Ia) It is the curve between armature torque Ta and armature current Ia of a d.c. motor. It is also known as electrical characteristic of the motor.
(ii) Speed and armature current characteristic (N/ia) It is the curve between speed N and armature current Ia of a d.c. motor. It is very important characteristic as it is often the deciding factor in the selection of the motor for a particular application.
(iii) Speed and torque characteristic (N/Ta) It is the curve between speed N and armature torque Ta of a d.c. motor. It is also known as mechanical characteristic.

Speed of a D.C. Motor

Therefore, in a d.c. motor, speed is directly proportional to back e.m.f. Eb and
inversely proportional to flux per pole .

Shaft Torque (Tsh)

The torque which is available at the motor shaft for doing useful work is known as shaft torque. It is represented by Tsh.

The total or gross torque Ta developed in the armature of a motor is not available at the shaft because a part of it is lost in overcoming the iron and frictional losses in the motor. Therefore, shaft torque Tsh is somewhat less than the armature torque Ta. The difference Ta - Tsh is called lost torque.
Clearly,

As stated above, it is the shaft torque Tsh that produces the useful output. If the speed of the motor is N r.p.m., then,

Armature Torque of D.C. Motor

Torque is the turning moment of a force about an axis and is measured by the product of force (F) and radius (r) at right angle to which the force acts.
T = F x r
In a d.c. motor, each conductor is acted upon by a circumferential force F at a distance r, the radius of the armature. Therefore, each conductor exerts a torque, tending to rotate the armature. The sum of the torques due to all armature conductors is known as gross or armature torque (Ta).
Let in a d.c. motor
r = average radius of armature in m

l = effective length of each conductor in m
Z = total number of armature conductors
A = number of parallel paths
i = current in each conductor = Ia/A
B = average flux density inWb/m2
f = flux per pole inWb
P = number of poles
Force on each conductor, F = B i l newtons
Torque due to one conductor = F x r (newton- metre)
Total armature torque, Ta = Z F r (newton-metre)
= Z B i l r
Now i = Ia/A, B = /a where a is the x-sectional area of flux path per pole at
radius r. Clearly,
   a = 2  r l /P.

Since Z, P and A are fixed for a given machine,

 

Hence torque in a d.c. motor is directly proportional to flux per pole and armature
current.
(i) For a shunt motor, flux f is practically constant.

(ii) For a series motor, flux f is directly proportional to armature current Ia
provided magnetic saturation does not take place.

Types of D.C. Motors

•  Shunt Motor
•  Series Motor
•  Compound Motor

Shunt DC Motor

Series DC Motor

Compound DC Motor

Back or Counter E.M.F.

When the armature of a d.c. motor rotates under the influence of the driving torque, the armature conductors move through the magnetic field and hence e.m.f. is induced in them as in a generator.
The induced e.m.f. acts in opposite direction to the applied voltage V(Lenz’s law)and in known as back or counter e.m.f. Eb.
The back e.m.f. Eb = P ZN/60A) is always less than the applied voltage V, although this difference is small when the motor is running under normal conditions.
The electric work done in overcoming and causing the current to flow against Eb is converted into mechanical energy developed in the armature.
It follows, therefore, that energy conversion in a d.c. motor is only possible due to the production of back e.m.f. Eb.

Net voltage across armature circuit = V - Eb

If Ra is the armature circuit resistance, then,

Since V and Ra are usually fixed, the value of Eb will determine the current drawn by the motor. If the speed of the motor is high, then back e.m.f.
Eb = P ZN/60A is large and hence the motor will draw less armature current and vice versa.

Significance of Back E.M.F.

The presence of back e.m.f. makes the d.c. motor a self-regulating machine i.e., it makes the motor to draw as much armature current as is just sufficient to develop the torque required by the load.
Armature current,
(i) When the motor is running on no load, small torque is required to overcome the friction and windage losses. Therefore, the armature current Ia is small and the back e.m.f. is nearly equal to the applied voltage.
(ii) If the motor is suddenly loaded, the first effect is to cause the armature to slow down. Therefore, the speed at which the armature conductors move through the field is reduced and hence the back e.m.f. Eb falls. The decreased back e.m.f. allows a larger current to flow through the armature and larger current means increased driving torque. Thus, the driving torque increases as the motor slows down. The motor will stop slowing down when the armature current is just sufficient to produce the increased torque required by the load.
(iii) If the load on the motor is decreased, the driving torque is momentarily in excess of the requirement so that armature is accelerated. As the armature speed increases, the back e.m.f. Eb also increases and causes the armature current Ia to decrease. The motor will stop accelerating when the armature current is just sufficient to produce the reduced torque required by the load. It follows, therefore, that back e.m.f. in a d.c. motor regulates the flow of armature current

i.e., it automatically changes the armature current to meet the load requirement.

Construction of D.C. Generator

    The d.c. generators and d.c. motors have the same general construction. In fact, when the machine is being assembled, the workmen usually do not know whether it is a d.c. generator or motor. Any d.c. generator can be run as a d.c.motor and vice-versa. All d.c. machines have five principal components viz.,

(i) Field System
(ii) Armature Core
(iii)Armature Winding
(iv) Commutator
(v) Brushes

(i) Field system

The function of the field system is to produce uniform magnetic field within which the armature rotates. It consists of a number of salient poles (of course, even number) bolted to the inside of circular frame (generally called yoke). The yoke is usually made of solid cast steel whereas the pole pieces are composed of stacked laminations. Field coils are mounted on the poles and carry the d.c. exciting current. The field coils are connected in such a way that adjacent poles have opposite polarity.

The m.m.f. developed by the field coils produces a magnetic flux that passes through the pole pieces, the air gap, the armature and the frame. Practical d.c. machines have air gaps ranging from 0.5 mm to 1.5 mm. Since armature and field systems are composed of materials that have high permeability, most of the m.m.f. of field coils is required to set up flux in the air gap. By reducing the length of air gap, we can reduce the size of field coils (i.e. number of turns).

(ii) Armature core

The armature core is keyed to the machine shaft and rotates between the field poles. It consists of slotted soft-iron laminations (about 0.4 to 0.6 mm thick) that are stacked to form a cylindrical core. The laminations are individually coated with a thin insulating film so that they do not come in electrical contact with each other. The purpose of laminating the core is to reduce the eddy current loss. The laminations are slotted to accommodate and provide mechanical security to the armature winding and to give shorter air gap for the flux to cross between the pole face and the armature “teeth”.

(iii) Armature winding

The slots of the armature core hold insulated conductors that are connected in a suitable manner. This is known as armature winding. This is the winding in which “working” e.m.f. is induced. The armature conductors are connected in series-parallel; the conductors being connected in series so as to increase the voltage and in parallel paths so as to increase the current. The armature winding of a d.c. machine is a closed-circuit winding; the conductors being connected in a symmetrical manner forming a closed loop or series of closed loops.

Armature

• The armature windings start and finish at a point on the armature called the commutator.

(iv) Commutator

A commutator is a mechanical rectifier which converts the alternating voltage generated in the armature winding into direct voltage across the brushes. The commutator is made of copper segments insulated from each other by mica sheets and mounted on the shaft of the machine. The armature conductors are soldered to the commutator segments in a suitable manner to give rise to the armature winding. Depending upon the manner in which the armature conductors are connected to the commutator segments, there are two types of armature winding in a d.c. machine viz., (a) lap winding (b) wave winding. Great care is taken in building the commutator because any eccentricity will cause the brushes to bounce, producing unacceptable sparking. The sparks may bum the brushes and overheat and carbonise the commutator.

(v) Brushes

The purpose of brushes is to ensure electrical connections between the rotating commutator and stationary external load circuit. The brushes are made of carbon and rest on the commutator. The brush pressure is adjusted by means of adjustable springs. If the brush pressure is very large, the friction produces heating of the commutator and the brushes. On the other hand, if it is too weak, the imperfect contact with the commutator may produce sparking. Multipole machines have as many brushes as they have poles. For example, a 4- pole machine has 4 brushes. As we go round the commutator, the successive brushes have positive and negative polarities. Brushes having the same polarity are connected together so that we have two terminals viz., the +ve terminal and the -ve terminal.

Action of Commutator

• Since, the connection of the coil side to the external load is reversed at the same instant the current in the coil side reverses, the current through the load will be direct current. This is what a commutator does.
• A commutator having two segments C1 and C2. It consists of a cylindrical metal ring cut into two halves or segments C1 and C2 respectively separated by a thin sheet of mica.
• The commutator is mounted on but insulated from the rotor shaft.
• The ends of coil sides AB and CD are connected to the segments C1 and C2 respectively .
• Two stationary carbon brushes rest on the commutator and lead current to the external load. With this arrangement, the commutator at all times connects the coil side under S-pole to the +ve brush and that under N-pole to the -ve brush.

Loop Generator

       Consider a single turn loop ABCD rotating clockwise in a uniform magnetic field with a constant speed .As the loop rotates, the flux linking the coil sides AB and CD changescontinuously.

      Hence the e.m.f. induced in these coil side also changes but the e.m.f. induced in one coil side adds to that induced in the other.

(i) When the loop is in position no. 1, the generated e.m.f. is zero because the coil sides (AB and CD) are cutting no flux but are moving parallel to it

When the loop is in position no. 2, the coil sides are moving at an angle to the flux
and, therefore, a low e.m.f. is generated as indicated by point 2.
(iii) When the loop is in position no. 3, the coil sides (AB and CD) are at right angle to
the flux and are, therefore, cutting the flux at a maximum rate. Hence at this instant,
the generated e.m.f. is maximum as indicated by point 3.
(iv) At position 4, the generated e.m.f. is less because the coil sides are cutting the flux
at an angle.
(v) At position 5, no magnetic lines are cut and hence induced e.m.f. is zero as
indicated by point 5
(vi) At position 6, the coil sides move under a pole of opposite polarity and hence the
direction of generated e.m.f. is reversed. The maximum e.m.f. in this direction (i.e.,
reverse direction, will be when the loop is at position 7 and zero when at position 1.
This cycle repeats with
each revolution of the coil.
   Note that e.m.f. generated in the loop is alternating one. It is because any coil side, say
AB has e.m.f. in one direction when under the influence of N-pole and in the other
direction when under the influence of S-pole. If a load is connected across the ends of
the loop, then alternating current will flow through the load.
The alternating voltage generated in the loop can be converted into direct voltage by
a device called commutator. We then have the d.c. generator. In fact, a commutator is
a mechanical rectifier.

DC GENERATOR

Introduction

The greater percentage of the electrical machines in service are a.c. machines, the d.c.
machines are of considerable industrial importance.
The principal advantage of the d.c. machine, particularly the d.c. motor, is that it
provides a fine control of speed. Such an advantage is not claimed by any a.c. motor.
However, d.c. generators are not as common as they used to be, because direct current,
when required, is mainly obtained from an a.c. supply by the use of rectifiers.
Nevertheless, an understanding of d.c. generator is important because it represents a
logical introduction to the behaviour of d.c. motors.
Indeed many d.c. motors in industry actually operate as d.c. generators for a brief
period.
DC Generator
An electrical generator is a machine which converts mechanical energy (or power) into
electrical energy (or power).
The energy conversion is based on the principle of the production of dynamically
(or motionally) induced e.m.f.
whenever a conductor cuts magnetic flux, dynamically induced e.m.f. is produced in it
according to Faraday’s Laws of Electromagnetic Induction. This e.m.f. causes a current to
flow if the conductor circuit is closed.
The direction of induced e.m.f. (and hence current) is given by Fleming’s right hand
rule. Therefore, the essential components of a generator are:
(a) a magnetic field
(b) conductor or a group of conductors

(c) motion of conductor w.r.t. magnetic field

DC MOTOR

What is a DC Motor?

         DC stands for "direct current". A DC motor is an electric motor that uses electricity and a magnetic field to produce torque, which turns the DC motor.

         A DC motor consists of two magnets of opposite polarity and an electric coil. When a power supply is added to the coil electric current flows through in a circuit and generates a small magnetic field. The repellent and attractive electromagnetic forces of the magnets provide the torque that causes the armature to turn.

DC MOTOR THEORY

         A DC motor works by converting electric power into mechanical work. This is accomplished by forcing current through a coil and producing a magnetic field that spins the motor. The simplest DC motor is a single coil apparatus, used here to discuss the DC motor theory 

How it Works ???

               Magnets are polarized, with a positive and a negative side.

               A DC motor uses the attraction between opposite poles and the repulsion of like poles to convert         electric energy into kinetic energy.

               As the magnets within the DC motor attract and repel one another, the motor turns. The magnetic         force on the armature works perpendicular to both wire and magnetic field.

               If an electric current goes through the coil, the motor will act like a generator and produce an               electric motive force (EMF). When the motor spins it produces a voltage called the back   EMF           because it opposes  the applied voltage on the motor. Therefore, the voltage drop across the               motor consists of the voltage drop from the back EMF and the voltage drop from the  internal             resistance of the rotation of the armature

               The current through the motor is given by:

               I = (Vapplied Vbackemf) / Rarmature  

    Main component of DC motor

               A DC Motor usually consists of: An armature core, an air gap, poles, and a yoke which form the         magnetic circuit; an armature winding, a field winding, brushes and a commutator   which form the         electric circuit; and a frame, end bells, bearings, brush supports and a shaft which provide the               mechanical support

      

      The basic testing equipment you will need to trouble- shoot DC motors in the field includes..

1.           Megohmmeter
2.           AC voltmeter
3.           DC clamp-on ammeter
4.           Ohmmeter
5.           DC voltmeter
6.           Tachometer

Moving-iron Instruments

In moving–iron instruments the movable system consists of one or more pieces of specially-shaped soft iron,   which are so pivoted as to be acted upon by the magnetic field produced by the current in coil. There are two   general types of moving-iron instruments namely (i) Repulsion (or double iron) type (ii) Attraction (or single-iron) type. The brief description of different components of a moving-iron instrument is given below.

·         Moving element:  a small piece of soft iron in the form of a vane or rod

·         Coil:  to produce the magnetic field due to current flowing through it and also to magnetize the iron pieces.

·         In repulsion type, a fixed vane or rod is also used and magnetized with the same polarity.

·         Control torque is provided by spring or weight (gravity).

·         Damping torque is normally pneumatic, the damping device consisting of an air chamber and a moving vane attached to the instrument spindle.

Deflecting torque produces a movement on an aluminium pointer over a graduated scale. The deflecting torque in any moving-iron instrument is due to forces on a small piece of magnetically ‘soft’   iron that is magnetized by a coil carrying the operating current.

Construction and working of Moving-iron Instruments:

Repulsion Type:

Construction:

Repulsion type moving–iron instrument consists of two cylindrical soft iron vanes mounted within a fixed current-carrying coil. One iron vane is held fixed to the coil frame and other is free to rotate, carrying with it the pointer shaft. The two iron vanes lie in the magnetic field produced by the coil that consists of only few turns if the instrument is an ammeter or of many turns if the instrument is a voltmeter.

Working:

Current in the coil induces both vanes to become magnetized and repulsion between the similarly magnetized vanes produces a proportional rotation. The deflecting torque is proportional to the square of the current in the coil, making the instrument reading is a true ‘RMS’ quantity. Rotation is opposed by a hairspring that produces the restoring torque. Only the fixed coil carries load current, and it is constructed so as to withstand high transient current. Moving iron instruments having scales that are nonlinear and somewhat crowded in the lower range of calibration.

Attraction Type:

The moving system of the instrument is attached to a soft iron piece. The operating current is passing through a coil placed near it. When current flows in the coil, the soft-iron piece is attracted towards the coil and the movement causes pointer to move across the scale.

The pointer will come to rest at a position where deflecting torque is equal to the controlling torque.

If current in the coil is reversed, the direction of magnetic field also reverses and so does the magnetism produce in the soft iron piece. Hence, the direction of the deflecting torque remains unchanged. For this reason, such instruments can be used for both d.c. and a.c. measurements.

Advantages:

•          The instruments are suitable for use in a.c and d.c circuits.
•          The instruments are robust, owing to the simple construction of the moving parts.
•          The stationary parts of the instruments are also simple.
•          Instrument is low cost compared to moving coil instrument.
•      Torque/weight ratio is high, thus less frictional error.

·      

Disadvantages:

• Have non-linear scale.
• Cannot be calibrated with high degree of precision for d.c. on account of the affect of hysteresis in the iron vanes.
• This instrument will always have to be put in the vertical position if it uses gravity control.

Errors:

•           Errors due to temperature variation.
•           Errors due to friction is quite small as torque-weight ratio is high in moving-iron instruments.
•            Stray fields cause relatively low values of magnetizing force produced by the coil. Efficient magnetic screening is essential to reduce this effect.
•                   Error due to variation of frequency causes change of reactance of the coil and also changes the eddy currents induced in neighboring metal.
•            Deflecting torque is not exactly proportional to the square of the current due to non-linear  characteristics of iron material.

Comparison between PMMC and MI instruments:

	PMMC	MI
Suitable for measuring	Direct current and voltage	Direct and alternating currents and voltages
Method of control	Hairsprings	Hairsprings
Method of damping	Eddy current	Air
Frequency limits		20-200 Hz
Scale	Uniform(Linear)	Cramped or Non-linear

DC Voltmeter

1. A dc voltmeter is constructed by a connecting a resistor in series with a PMMC instrument.
2. A voltmeter should have a very high resistance R_se and it is normally connected in parallel with the circuit where the voltage is to be measured.
3. To minimize voltmeter loading, the voltmeter operating current should be very small i.e., the resistance connected in series with the coil should be high. 

The addition of a series resistance or a multiplier limits the current through the movement so as not to exceed the full-scale deflection current. The value of multiplier required to extend the voltage range

      The multiplier is usually mounted inside the case of the voltmeter for moderate ranges upto 500 V. For higher voltages, the multiplier may be mounted separately outside the case on a pair of binding posts to avoid excessive heating inside the case.

Multirange Voltmeter

1. Any one of the several multiplier resistors is selected by means of a rotary switch.              
2. The rotary switch used with the voltmeter should be a break-before make type i.e. moving contact should disconnect from one terminal before connecting to the next terminal.

3. This system has the advantage that all multipliers except the first have standard resistance values and can be commercially obtained in precision tolerances. The low range multiplier R₄, is the only special resistor that must be manufactured to meet the specific circuit requirements.

Precautions

• Observe the correct polarity. Wrong polarity causes the meter to deflect against the mechanical stop and this may damage the pointer.
• Place the voltmeter across the circuit or component whose voltage is to be measured.
• When using a multirange voltmeter, always use the highest voltage range and then decrease the range until a good upscale reading is obtained.
• Always be aware of the loading effect. The effect can be minimised by using as high a voltage range as possible. The precision of the instrument decreases if the indication is at the low end of the scale.

Multirange Ammeter

1. A multi-range ammeter can be constructed simple by employing several values of shunt resistances, with a rotary switch to select the desired range.
2. When an instrument is used in this fashion, care must be taken to ensure shunt does not become open-circuited, even for a very short instant.
3. If the shunt resistance remains open-circuited for a fraction of time, a very large current may   flow through the ammeter and damage the instrument. When the switch is moved from position ‘B’ to ‘C ’ or moved to any positions, the shunt resistance will remain open-circuited for a   fraction of time, resulting a very large current may flow through the ammeter and damage the   instrument.  
4. To avoid such situation, one may use the make-before-break switch. The   wide-ended   moving   contact connected to the next terminal to which it is being moved before it loses contact with   the previous terminals. Thus, during the switching time there are two resistances are parallel   with the instrument and finally the required shunt only will come in parallel to the instrument.

Multirange Ammeter Circuit

Make-before-break switch

Precautions

• Never connect an ammeter across a source of emf. Because of its low resistance it would draw damaging high currents and destroy the delicate movement. Always connect an ammeter in series with a load capable of limiting the current.
• Observe the correct polarity. Reverse polarity causes the meter to deflect against the mechanical stop and this may damage the pointer.
• When using a multirange meter, first use the highest current range; then decrease the current range until substantial deflection is obtained. To increase accuracy of the observation, use a range that will give a reading as near to full-scale as possible.

DC Ammeter

1. An ammeter is required to measure the current in a circuit and it is therefore connected in series with the components carrying the current.
2. For higher range ammeters a low resistance made up of manganin (low temperature coefficient of resistance) is connected in parallel to the   moving coil (see and instrument may be calibrated to read directly to the total current.
3. Both the magnetic field strength and spring tension decrease with an increase in temperature. On other side, the coil resistance increases with an increase in temperature. These changes lead to make the pointer read low for a given current with respect to magnetic field strength and coil resistance.
4. Use of Manganin resistance (known as swamping resistance which has a temperature coefficient practically zero) in series with the coil resistance can reduce the error due to the variation of resistance of the moving coil. The total resistance of the coil and swamping resistor increases slightly with a rise of temperature, but only just enough to counteract the change of springs and magnet, so that the overall temperature effect is zero.
5. The swamping resistance is usually three times that of coil thereby reducing a possible error of, say, 4% to 1%. One disadvantage of use of swamping resistors is reduction in the full-scale sensitivity of the movement, because a higher applied voltage is necessary to sustain full-scale current.

The range can be extended with the addition of shunt. The resistance of the shunt can be calculated by applying conventional circuit analysis where

  

Since shunt resistance is in parallel with the meter movement, the voltage drops across the shunt and the movement must be the same and we can write

   

For each required value of full-scale meter current we can then solve for the value of shunt resistance required. External shunts of this type are normally used for measuring very large currents.