Necessity of Connecting Alternator in Parallel

For high efficiency of operation, reliability, convenience and economy in maintenance and repair and possibility of additions to plant with the growth in load on the power station the alternators are put in parallel.

Circuit diagram for connecting alternators in parallel
Photo Credit - www.allaboutcircuits.com

Continuity of supply:-

The continuity of supply is one of the important requirement of any electrical apparatus. If one alternator fails, the continuity of supply can be maintained through other units. This will ensure uninterrupted supply to consumer.

Efficiency:-

The load on the power system varies during whole day, being minimum during the late night hours. Single alternator operate most efficiently only when delivering full load, units can be added or put off depending upon the load requirement. This permit the efficient operation of the power system.

Maintenance and repair:-

It is often desirable to carry out routine maintenance and repair of one or more units. For this purpose, the desired unit can be shut down and the continuity of supply can be maintained through other units.

Load Growth:-

The load demand is increasing due to the increasing use of electrical energy. The load growth can be met by adding more units without disturbing the original installation.

Star Delta Starter

If normal supply voltage is applied to the stationary motor then a very large initial current is taken by the stator for short while. This will lead to excess copper losses in the winding which will overheat the motor. This will produce line voltage drop, that in turn, will affect the operation of other electrical equipments connected to the same line and heavy starting current may damage the motor winding. In order to avoid these problem, a starter is used to start the induction motor safely.

Star-Delta Starter

The motor is designed to operate as delta connected motor under normal conditions. The motor starts as a star connected motor and the voltage per phase is ^V_L/_√3. The starting torque is reduced as it is directly proportional to square of stator voltage and there is jerk while switching from star to delta.

Start Mode

The 3 pole 6 way switch is kept in start mode. This will connect terminal R', Y' B' of stator winding to each other. This act as star point. The supply is connected to R,Y,B terminal of the stator winding. Thus in start mode the stator winding is connected to form the star.

Run Mode

The 3 pole 6 way switch is thrown in run position once the motor accelerate. This will connect the terminals in following manner:

R→B', Y→R', B→Y'

Capacitor Start Capacitor Run Induction Motor

Capacitor Run Induction Motor-Representational Image 

We know that single phase induction motor is not self starting due to the absence of rotating magnetic field at starting. In order to produce rotating field there must be some phase difference. In capacitor start capacitor run induction motor we are using two winding, one main winding and other auxiliary winding. For producing the phase difference two capacitors are used since we know that in capacitor the current leads the applied voltage by some angle. The capacitor serves to shift the phase on one of the windings so that the voltage across the winding is at 90° from the other winding, thus making the capacitor run motor a truly two-phase machine at its rated load.

Capacitor start capacitor run induction motor

Instead of one if two capacitors are used in the auxiliary winding of the motor, one of them for starting and other for running, optimum starting and running performances can be obtained. Figure shows the schematic  diagram of the motor. The capacitor Cs is cut out when the motor reaches about 75% of the synchronous speed by centrifugal device. The capacitor C and auxiliary winding A retained in the circuit for running condition. A motor starting capacitor may be a double-anode non-polar electrolytic capacitor which could be two + to + (or - to -) series connected polarized electrolytic capacitors. Such AC rated electrolytic capacitors have such high losses that they can only be used for intermittent duty (1 second on, 60 seconds off) like motor starting. A capacitor for motor running must not be of electrolytic construction, but a lower loss polymer type.

Torque- Speed Characteristics of Capacitor Start Capacitor Run Induction Motor

Capacitor-start/capacitor-run motors have moderate-to-high starting torque compared to other types of single-phase motors. Starting torque generally ranges from 200%-350% of normal full-load torque.

Torque- Speed Characteristic of Capacitor Start Capacitor Run Induction Motor

Their main advantage over the capacitor-start motor is their lower starting-torque that results in lower starting current.
Because of the lower starting current, they are generally used for most single-phase applications between 3-and-10 horsepower  due to their lower starting current. Common applications include: larger single-phase compressors, pumps, grinders, conveyors, and larger single-phase air-conditioning compressors.

Divergence Theorem

It gives the facility to convert surface integral into its equivalent volume integral.

Representational Image
Photo Credit - www.wikipedia.com

Statement:- It states that the surface integral of a normal component of any vector function on a closed surface is equal to the volume integral of the divergence of vector function.
Proof:-  The total charge enclosed by volume △V is given by

........(1)

The total outward flux coming out from the volume is △V equal to the charge enclosed.

△Ψ = △Q = charge density x volume

△Ψ = △Q =ρ_v△V = ρ_v△x△y△z.......(2)

From equation (1) and (2)

........(3)

The divergence of D is given by 

........(4)

From (3) and (4) 

According to Gauss's Law

ψ = Q

Universal Motors

A universal motor is defined as a motor which may be operated on either d.c. or single phase a.c. supply at very high speed in the range of 3000 to 7000 rpm or so. It has starting torque and variable speed characteristics. Universal motor are designed for weak field to minimize commutation difficulties. The stator core is laminated and high resistance brushes are used.

Universal Motor
Photo Credit www.woodgears.ca

Construction

The construction of universal motor is similar to that of DC Machine. The field poles are mounted on stator winding. Field winding is wound on the field poles. Both the stator field poles and armature is laminated to minimize the Eddy current losses while working on a.c. supply. The armature has straight or skewed slots and the commutator and brush arrangement is resting on it. 

Universal Motor - Construction

Working

Such motors develop unidirectional torque regardless of whether they operate on A.C. or D.C. supply. The motor runs on the same principle as d.c. motor i.e., force between the main pole flux and the current carrying armature conductor. The universal motor produces the electric torque proportional to the quadrate of the supply current. Since the same current flows through the field winding and the armature, it follows that ac reversals from positive to negative, or from negative to positive, will simultaneously affect both the field flux polarity and the current direction through the armature. This means that the direction of the developed torque will remain positive, and rotation will continue in the same direction. Thus, a universal motor can run both on dc and ac. So the electric torque has the same torque direction at any current polarity and in addition at AC power. The starting torque of a universal motor is determined by the current that flows through the armature and field windings. Due to the inductive reactance of these windings the AC starting current will always be less than the DC starting current.Consequently, the starting torque on AC power will be lower than the starting torque on DC power. The characteristics of universal motor are very much similar to those of D.C. series motors, but the series motor develops less torque when operating from an A.C. supply than when working from an equivalent D.C. supply.

Torque-Speed Characteristics

Torque speed characteristics of Universal Motor

Application

1. Universal motors are used in most hand held power tools such as jigsaw, routers, drills and sanders.
2. They also find their use in household appliances like vacuum cleaner, drink and food mixers and sewing machine. 

Double Cage Induction Motor

Double Cage Induction motor Stator Construction

Construction

An induction motor with two cage rotor is used for high starting torque. The slotting arrangement for double cage induction motor is as shown in above figure. As the name indicate the double cage induction motor has two winding in rotor. The outer bars consists of rotor bars having low reactance and high resistance. On the other hand, the inner cage consists of rotor bars having high reactance and low resistance.

Working

At start the rotor frequency is high, the outer cage carries most of the current despite its high resistance. The inner cage has low reactance and is mostly ineffective. This gives high starting torque and low starting current. As the motor picks up the speed, the rotor frequency reduces and the inner cage carries most of the current. Under normal running condition, the outer cage and inner cage are in parallel giving low combined resistance and both the cages are active.

When the speed is normal frequency reduces and it is so small that the reactance of both the cages are practically negligible. Hence it has been made possible to construct a single machine which has high starting torque with reasonable starting current which maintains speed regulation and high efficiency.

Torque-Speed characteristics

Torque Speed Characteristic of Double Cage Induction Motor

Advantages

1. It has high efficiency and good speed regulation.
2. It gives higher starting torque.
3. Lower starting current and are cheaper in cost.
4. They are more robust and are explosion proof since the risk of sparking is eliminated by the absence of slip ring and brushes.

Construction and Working of Induction Motor

Construction:-

An induction motor consists of two main parts

1. Stator
2. Rotor

The stationary frame is called stator and the rotating armature is called rotor. The stator of induction motor is similar to that of the synchronous motor or generator. It is made of large number of stampings. These stampings are slotted in order to receive the stator windings. Figure below shows the construction of Induction motor.

Induction Motor

The functions of various parts is as follows:-

• Frame:- Its function is to provide mechanical support to the entire construction. The frame also contains the stator winding of Induction motor.
• Air gap:- Air gap provides the space for the rotating magnetic field between the stator and rotor.
• Fan:- The fan rotates with the rotor. Its function is to cool down the motor.
• Slip rings:- The rotor winding terminals are permanently connected to the slip rings(in slip ring type induction motor). The slip rings are continuously in contact with the brushes which are pressed against slip rings. External connections from the brushes are brought out,

Figure below shows the rotor and stator of induction motor.

Stator and Rotor construction of Induction motor

Rotor drum is provided with slots. The stator is stationary which can be star connected or delta connected to the 3Φ AC supply through a switch as shown in figure below. 

Stator winding connections

The function of stator winding is to produce a rotating magnetic field in the air gap between the stator and rotor. It should be known that the rotor is connected to any external supply. The current flows through the rotor due to principle of induction.

Working:-

The 3Φ induction motor has 3Φ winding which is supplied by 3Φ alternating voltage and 3Φ balanced current flow in the winding. The current produce magnetic flux which is constant in magnitude and rotating at synchronous speed. The rotating magnetic filed swift pass the rotor conductors which as yet are stationary. The relative velocity between the magnetic flux and stationary rotor conductors induces emf in the rotor conductors according to Faraday's Law of Electromagnetic Induction. The induced emf is directly proportional to the relative velocity between the magnetic flux and stationary rotor conductors. The frequency of induced emf is same as the supply frequency and the direction of induced emf is given by Fleming's Right Hand Rule.

Since the rotor conductor forms a closed circuit, rotor current produces which opposes the very cause producing it according to Lenz's Law. In this case, the cause which is producing the rotor current is relative velocity between the magnetic flux and stationary rotor conductors. Hence the rotor starts running in the same direction as rotating magnetic field and always try to catch up the the speed of rotating magnetic field.

Note:-For better understanding of the working of Induction motor, refer following videos.

Commutation in DC Machine

One can be aware of the fact that emf induced into the rotating conductor or winding is always sinusoidal or alternating in nature. If the machine is of DC nature that is DC Generator then it is bounding for someone to convert it (sinusoidal emf) into a unidirectional emf. This process of converting AC voltage into a DC is commonly called as commutation. Hence the commutator is a device which normally acts like a rectifier and perform the process of rectification. Coil(s) current is constant and unidirectional so long as the coil is under the influence of given pole pair(s), while it reverses when the coil passes onto the next pole pair as the armature rotates. Commutation take place when the coil is passing through the interpolar region and during this period the coil is shorted via the commutator segments by the brushes located (electrically) in the interpolar region. Commutation takes place simultaneously for P coils in a lap wound machines and two coil sets of  ^P/₂ coils each in a wave wound machine.
Various symbols used in the figure are:

I_c = coil current 

             I_b = 2I_c = Brush current 

                                       W_c = Width of one commutator segment 

                                                       W_m = Width of mica insulation between segments 

    W_b = Brush width 

                                  V_c = peripheral speed of commutator

At the instant the commutation of the coil begins, the leading tip of the brush makes full contact with the segment x and is just going to make contact with the segment y. At this instant all coils to the right of segment x carry current I_c flowing from left to right and those on the left current I_c in the opposite direction.

During this period of commutation as the coil passes from right to the left of the brush, the coil current must reverse and the brush should short circuit the coil via segments x and y. The contact width x_c between brush and segment x reduces linearly while the contact width y_c between brush and segments y increase. The coil current i_c(t) during this period is changing. If at the end of the commutation period, when the trailing tip of the brush is going to break contact with segment x. At the end of commutation period, the coil current has not reversed and acquired full value I_c but as I_c' < I_c, the breaking of current (I_c-I_c') at the trailing brush tip takes place causing sparking.

Hence sparking at the brushes which results in poor commutation is due to the inability of the current in the short circuited coil to reverse completely by the end of the short circuited period.

Two methods are available for improving commutation:-

Resistance Commutation

In this method the low resistance copper brushes are replaced by comparatively high resistance carbon brushes.

E.m.f. Commutation

In this method an arrangement is made to neutralise the reactance voltage by producing a reversing e.m.f. in the short-circuited coil under commutation.

For better understanding of what is commutation refer the following videos:-

Basic Electrical Engineering - MCQ

DC Generator - MCQ

Faraday's Law of Electromagnetic Induction

Michael Faraday (B. 22 September 1791 D. 25 August 1867)
Photo Credit - geeknewsnetwork.net

On April 21, 1820 Danish Physicist Hans Christian Oersted discovered the magnetic field produced by an electric field. After this discovery scientists began to search for the converse phenomena. that is production of electric current from magnetic field about 1821 onward. The problem they put to themselves was conversion of magnetism into electricity. Michael Faraday was one of those scientist. Michael Faraday had a habit of keeping magnets in his pocket while walking to remind himself of the problem. After 9 years of continuous experiment and research he succeeded to obtain the electricity by converting magnetic field. He postulated the two laws of electromagnetism which is the basis of Electrical Engineering and serves as the laws upon which the working of most of the electrical equipments like motor, generator, transformer, etc are based.

Laws of Electromagnetic Induction  

Any change in the magnetic field of a coil of wire will cause a voltage(emf) to be induced in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic  field strength by moving a magnet toward or away from the coil, rotating the coil relative to the magnet. The expression for emf induced is as follows,

where e = generated emf

                        N = number of turns of coil

                                                                        Φ = flux

Above equation can be mathematically proved as below.

Suppose a coil has N number of turns and the flux through it changes from an initial value of Φ₁ Webers to final value Φ₂ Webers in time "t" seconds.

Initial flux linkages = NΦ₁

Final flux linkages = NΦ₂

Therefore,

Negative sign signify the fact that the induced emf sets up current in such a direction that the magnetic effect produced by it opposes the very cause of producing it. The induced emf in the coil is equal to the negative rate of change of magnetic flux times the number of turns in the coil. It involves interaction of charge with magnetic field. 

First Law:- Whenever a conductor is placed in a varying magnetic field and emf gets induced across the conductor and if the conductor is closed circuit then induced current flows through it.

Second Law:- It states that, "the emf induced is equal to the rate of change of flux linkages with the coil. 

Note:- 

1. The flux linkages is the product of number of turns and the flux associated with the coil.
2. The direction of induced emf is given by Fleming's Right Hand and Left Hand rule,

Cogging And Crawling of Induction Motor

Induction Motor
Photo Credit - www.wikipedia.com

Crawling

 It has been found that induction motor practically this squirrel cage type ,sometimes exhibit,a tendency to run stable at speeds as low as ¹/₇th of their synchronous speed .This is known as crawling of induction motor. This action is due to ,the fact that the AC winding of the stator produces a flux wave which is not pure sine wave .It is a complex wave consisting of a fundamental wave which revolves synchronously and odd harmonics like 3^rd,5^th,7^th etc. which rotate either in the forward or backward direction at ^Ns/₃, ^Ns/₅, ^Ns/₇ speed respectively. As a result in addition to the fundamental torque ,harmonics torques also developed whose synchronous speed for fundamental torque. For Example Ns/n, where N is the order of harmonics torque. Since the 3^rd harmonics current are absent in a balanced three phase system ,they produce  no torque .Hence total motor torque has there components.

1. The fundamental torque rotating with synchronous speed 
2. Fifth harmonics torque rotating at ^Ns/₅ 
3. Seventh harmonics torque having a speed of ^Ns/₇

If we neglect all the higher  harmonics, the resultant torque can be taken as equal to the sum of the fundamental torque and the seventh harmonics torque. When this happen the motor will not accelerate  up-to its normal speed  but will remain running at a speed which is nearly equal to ¹/₇th of its full speed. This is referred as a crawling and  motor starts running with unwanted sound .

Cogging

The rotor of particularly squirrel cage  induction motor sometimes refuse to start at all particularly a  when the voltage is low. This happens of stator teeth is equal to the  number of  rotor teeth ,and therefore  due to the magnetic locking or cogging .It is found that the reluctance of magnetic paths  is minimum  when the stator and rotor teeth comes in front of each other, it is in such position of maximum reluctance that the stator tends to remain fixed thus causes serious trouble during starting.

This can be easily overcome by making number of rotor slots more than the number of stator slots and by giving slightly skew to the rotor slots (skewed).What is meant is to arrange the stack of rotor laminations so that the rotor slots are "skewed" or angled with respect to the axis of rotation. Constructing the rotor with skewed slots and providing more (or fewer) rotor slots than stator slots is the remedy for both cogging and crawling.

Analog Circuits

465 Tektronics Oscilloscope

 

 

A system can be analog or digital. Electronic system/circuits comprises of digital and analog circuits.
So what is this analog circuits?
Analog circuits are those in which voltage and current vary continuously through the given range of time. Analog circuit uses continued value analog signals (continuous valued signals). They can take on infinite values within the specified range. This circuits are usually complex combinations of Op-Amps, resistors, capacitors and other basic electronic components (electronic circuit elements). This circuits can be designed to amplify, attenuate, isolate, distort or modify the applied signal in some way. Analog circuits can be very difficult or complex to design due to the involvement of various connection of circuit elements or they can be simple like combining two resistor to make a voltage divider. Analog circuits are much more sensitive to noise (undesired voltage, harmonics). But sometimes this sensitivity of analog circuit to noise is used as an advantage(This point will be explained in more details where an example of this, is encountered).

Class B Push Pull Amplifeir

The devices which uses analog circuits to process the signals are called analog devices. Examples of analog devices include signal generators, radio frequency transmitter/receiver, electric motors and speed controllers.

Basic Circuit Element - Inductor

Chip Inductor
Image Courtesy - www.directindustry.com  

Inductors are also called as storage element because they store energy in their magnetic field. In electric circuits the inductor shown is an idealization of the physical inductor. The practical inductance or inductor or a two terminal element will be called an inductor if at any time t its flux φ(t) and its current i(t) satisfy the relation defined by a curve in the iφ plane. This curve is called the characteristic of the inductor at time t.
An inductor is represented symbolically as shown in figure below,

Inductor Symbol

In circuit theory, the fundamental characterization of a two terminal element is in terms of voltage and current. The voltage across the inductor is given by Faraday's Law as ,

v = ^dφ/_dt    ..............(1)

where v is in Volts and φ is in Webers (Wb).
But we know that

φ(t)= Li(t) ................(2)

where L is constant and is called inductance. The SI unit of Inductor is "Henry" and it is shown by capital letter H.

Therefore equation (1) will become,

v = L^d/_dt i(t)...............(3)

Now integrating both sides,

v ^i(t)/_L = ^d/_dti(t)

where i(0) is called inductor current at t=0.

Let us verify this relation between voltage and current that above all equation agrees with Lenz's law; which states that "the electromotive force(emf) induced by a rate of change of flux will have a polarity such that it will oppose the cause of such rate of change of flux."

From equation (1), (2) and (3) we can conclude that

1. When the current i(t) increases; that is  ^d/_dt i(t) > 0, the flux φ also increases; that is ^dφ/_dt  >0.
2. Inductors opposes the instantaneous change in current through it.
3. Inductors have memory that it shows the property of causality.

Torque Speed Characteristics Of Double Cage Induction Motor

Torque speed characteristics of Double cage induction motor

The main disadvantage of squirrel cage motor is poor starting torque  because of low rotor resistance .The starting torque could be increased by having a cage of  high resistance but then the motor will have poor efficiency under normal running conditions.The difficulty with squirrel cage induction  motor is that its cage is permanently short circuited ,so no external resistance can be introduced . Many efforts have been made to build a squirrel motor with high starting torque without scarifying its electrical efficiency under normal running conditions ,which cannot be possible easily .Two  independent cages on the rotor was tried and a motor called as double cage motor is invented .The outer cage consist of bare of high resistance where as the inner cage has low resistance .Hence outer cage has high resistance and low ratio reactance to reactance where as the inner cage has low resistance but being situated at deep in the rotor .has large ratio of reactance to reactance .Hence the outer cage develops maximum torque at starting while the inner cage about 15%  slip .At starting and at large slip values ,frequency of induced emf in the rotor is high ,so the reactance of inner cage and therefore its impedance are both high .Hence very little current flows in it . Most of starting current is confused to outer cage ,despite its high resistance .Hence the motor develops high starting torque due to high resistance outer cage .
                                               As the speed increase the frequency of rotor emf decreases ,so that the reactance and hence the impedance of inner cage decreases and becomes very small under normal running conditions . Most of the current then flows through it and hence it develops the grater part of the motor torque .When speed is normal ,frequency of the rotor emf is so small that the reactance of both cages is practically negligible .The current is carried by two cages in parallel giving a low combined resistance .Hence it has been made possible to construct a  single machine which has a good starting torque with reasonable starting current and which maintain high efficiency with good speed regulation under operating conditions.      

Basic Circuit Element-Capacitor

Different types of Capacitor
Photo Credit - www.wikipedia.org

A capacitor stores electrical energy by electrostatic stress in the dielectric.it consist of two conducting surfaces separated by a layer of an insulating medium which is called dielectric.the conducting surface may be in form of either circular plates or be of spherical or cylindrical shape.
Capacitance - Capacitance is the property of capacitor to store electricity.the capacitance of a capacitor is defined as the amount of charge required to create a unit potential difference between its plates.due its property of storing electricity (charge),capacitor is called a storing element.If a charge of Q coulomb is given to one of the two plate of capacitor and if a potential difference of V volts is established between the two ,then its capacitance is

C = ^Q/_V = ^Charge/_{Potential difference}

Capacitance is the charge required per unit potential difference. The unit of capacitance is coulomb/volt which is also called farad.

1 farad =1 coulomb/volt 

One farad is defined as the capacitance of a capacitor which required a charge of one coulomb to establish a potential difference of one volt between its poles.
The characteristic of nearly all physical capacitors is monotonically increasing that, as v increases Q increases.

Characteristic of physical capacitor

In the circuit diagrams a capacitor is represented symbolically as shown in figure below,

Symbol of capacitor

When i(t) is positive, positive charges are brought (at time t) to the top plate whose charge is Q. The rate of change of Q is also positive. Thus we have,

i(t) = ^dQ/_dt

Current - voltage relationship in capacitor

The charge on a capacitor is given by the expression 

Q = Cv

Putting this value in the above equation of i(t) we get,

From the above equation following important parameters are found:

1. Since Q = Cv, the voltage across a capacitor is directly proportional to charge Q where C is capacitance/proportionality constant.
2. A capacitor has the property of causality.
3. Current in the capacitor is present only when voltage on it changes with time that is, AC Voltage. When its voltage is constant or DC voltage, i(t) = 0. Hence the capacitor behaves like open circuit.
4. Rate of change in voltage of a capacitor is inversely proportional to its capacitance.
   
5. Rate of change in capacitor voltage is slower when the value of capacitance C is high. Also the capacitor does not accommodate the change in current instantaneously.

In this way the introduction of capacitor is given briefly. We will also update more details of capacitor including its charging, discharging, types etc in upcoming posts.

Block Diagram Algebra

The transfer function for the input output behaviour of a linear system or element of a linear system is given by

G(s) = ^C(s)/_R(s)

where R(S )-Laplace transform  of the input variable  and 

C(S)-Laplace transform of  the  output variable 

A control system may have many number of components  connected to each  other  and each components have its own transfer function .To show this different transfer function of each component a block diagram is generally referred .
block diagram of  a system is pictorial representation or convenient graphical representation of the input -output behaviour i.e. transfer function of each component and of the flow signals . The block diagram depicts the interrelationship  between different system components .The block diagram  representation is more advantageous than mathematical representation as it identical the flow of signal of the actual system more realistically .
The block itself shows transfer function and it is a system of mathematical  operation on the input signal to the block that produces output signal .The flow  of information is unidirectional from input to the output with the output being equal to the result of product of input and output transfer function. The blocks are connected by lines with arrows indicating the unidirectional flow of information. The arrowhead which points toward the block designates the input and the arrowhead coming out of the block that is leading away from the block represents the output. Such arrows are referred to as signal.

Block Diagram Algebra - Figure

The above figure shows that the signal into the block represents the input R(s) and the signal leading away from the block represents the output C(s), while the block itself represents the transfer function.

Summing Point

Summing Point :- The figure (b) shows the summing operation. The plus indicated that the signal a and b are to be added. The quantities being added have the same dimension and units. Figure (a) shows the differencing operation. The plus and minus sign indicates that the two signals a and b are to be subtracted. If the circle with cross symbol is not shown with any sign convention, by default the two signals are added.
Branch Point :- It is a point from which the signal from block goes concurrently to other block.

Representation of Signals

There are various methods to represent signals. Few of them are listed below :

1. Graphical representation
2. Tabular Form
3. Sequential Form
4. Functional Form

Graphical Representation

In case of graphical representation x[n] value will be plotted for each value of n(-∞ < n < ∞). It is easy to represent x[n] graphically if x[n] is of finite length.

Graphical Representation of a Discrete time signal x[n]

Tabular Representation

In this type of representation, a table is prepared for all values of x[n]. If the signal length increases it is tedious to prepare table for all values of n and x[n].

n	-2	-5	0	1	-3	4	2	3
X[n]	5	3	4	5	4	1	3	2

Sequential Representation

In this type of representation, a finite duration sequence with time origin (n=0). It is indicated by the symbol ↑ is represented as 

The arrow head represent that at 0 instant the amplitude of the signal is 3 while at -1 instant the signal has an amplitude of 2.

An infinite duration sequence can be represented as

Functional Representation

In case of functional representation, sequence x[n] is expressed in mathematical form such as

Basic Circuit Elements- Resistance

Different types of Resistance
Photo Credit - physics.tutorvista.com

It is defined as the property of a substance due to which it opposes the flow of current (i.e. electrons) through it resistance is denoted by capital letter  R  and its symbol is

Resistance Symbol

Resistance is one of the basic element of an electric circuit . The  practical resistance is called a resistor and its unit is "ohm" () .A conductor is said to have a resistance of one ohm if it permits one ampere current to flow through it when one volt is applied across its terminals.

Metals , acids and salts are good conductor of electricity due to presence of a large number of free or loosely  attached -electrons in their atoms. Hence they have a very large resistance. Resistance is a passive element and hence dissipates power in the form of heat. The power dissipated by a resistance can be given as
P = vi

                          =ⁱ/_Ri       OR   ^v/_Rv

Laws of Resistance :-

The resistance R offered by a conductor depends on the following factors :

1. It varies directly as  length l of conductor.
2. It varies inversely as the cross sectional area A of the conductor. 
3. On the nature of material.
4. On the temperature of the conductor.

R ∝^l/_A

R = Ρ^l/_A

where P is proportionality constant or specific resistance or resistivity.