555 Timer IC

The 555 timer is a very popular IC used in electronic industry which is reliable, easy to use and cheaper in cost. It was invented in 1971 by American company signets. It is an integrated circuit (chip) used in variety of timer, pulse generation, and oscillator applications. The 555 can be used to provide time delays, as an oscillator, as a flip-flop element, temperature measurement,monostable and astable multivibrator.It operates basically in one of the two modes either as a monostable multivibrator or as an astable multivibrator.The name 555 timer has come from the three resistor of 5k each within the timer circuits.

                                                                                           Figure 1: 555 Timer

Pin configuration

Pin1: Grounded terminal

All voltage is measured with respect to this terminal.

Pin 2: Trigger terminal

It is an inverting input to a comparator that is responsible of flip flop from set to reset. The output of the timer depends on the amplitude of the external trigger pulse applied to this pin.

Pin 3: output terminal

There are two ways in which load can be connected to the output terminal either between pin 3 and ground or between Pin 3 and Pin 8 and is usually called a load and normally off load respectively.

Pin 4: Reset terminal

To disable or reset the timer a negative pulse is applied to this pin due to which it is referred to as reset terminal. When this pin is not to be used for reset purpose, it should be connected to Vcc to avoid any possibility of false triggering.

Pin5: Control voltage terminal

Its function is to control the threshold and trigger level. Thus either the external voltage or a pot connected to this pin determines the pulse width of the output waveform. The external voltage applied to this pin can also be used to modulate the output waveform. When this pin is not used it should be connected to ground through a 0.01uF capacitor to avoid any noise pollution.

Pin 6: Threshold terminal

This is non inverting input terminal of comparator , which compares the voltage applied to this terminal with a reference voltage of +2/3 v. the amplitude of voltage applied to this terminal is responsible for the set state of flip flop.

Pin 7: Discharge terminal

This pin is connected internally to the collector of transistor and mostly a capacitor is connected between this terminal and ground. It is called discharge terminal because when transistor is at saturation capacitor discharge through the transistor. When the transistor is at cut off, the capacitor changes at a rate determined by external resistor and capacitor.

Circuit Operation

·        The voltage divider consisting of 5k resistors provides a trigger level of 1/3 VCC and a threshold level of 2/3 VCC.

·        The control voltage input (pin 5) can be used to externally adjust the trigger and threshold levels to other values of necessary.

·        When the trigger input (at +5v) momentarily goes below 1/3 VCC the output of comparator B switches from 0 to 5v and sets the S-R latch, causing the output pin3 to go +5v and turning the discharge transistor Q1 OFF.

·        The output will stay at +5V until the threshold input (at 0V) goes above 2/3 VCC and causes the output of comparator A to switch from 0 to +5 V.

·        This resets the latch causing the output to go back to 0Vand turning the discharge transistor ON  the external reset input can be used to reset the latch independent of the threshold input the trigger threshold inputs (pin 2 and 6) are controlled by external components connected to produce either monostable or astable action.

Monostable operation of 555 timer

A monostable vibrator is often called a one shot multivibrator.In such vibrator one state of output is stable while the other is unstable.It makes use of external resistor(R) and a capacitor(C).The pulse width of the output is given by

                         t_p = 1.1 R₁ C₁      

Before a trigger pulse is applied, the output is 0V.When a negative (0V) pulse is applied to the trigger input (pin2), the internal comparator detects the input and sets the state of flip-flop.At 2/3 VCC, the comparators output goes high and resets the flip-flop.Monostable 555 Timer circuit triggers on a negative going pulse applied to pin 2. Once triggered, the 555 monostable will remain in this HIGH unstable output state until the time period set up by the R, XC.

Figure 2: Monostable Multivibrator

Astable Operation

                                                          Figure 3: Astable Multivibrator

It is often called as a free-running multivibrator which is used to generate rectangular wave. It does not require external trigger to change the state of the output. It can be produced by adding two resistors and a capacitor to the basic timer IC.The output is either high or low and is determined by the two resistors and capacitors.

Pin 1 is grounded, Pins 4 and 8 are shorted and then tied to supply +vcc, Output is taken from pin3, Pin 2 and 6 are shorted and then connected to ground through a capacitor (C).Pin 7 is connected to supply +Vcc through a resistor RA and between pins 6 and 7 are resistor RB is connected.At pin 5 either a bypass capacitor of 0.01 uF s connected or Modulation input is applied.

Circuit operation

·        When the power is turned ON, the capacitor is uncharged and thus the trigger voltage (pin2) is at 0v.

·        The output of comparator b is at +5v and the output of comparator A to be at 0v forcing the output of the latch and the base of Q1 to be at 0v.As a result the transistor Q1 is OFF, c1 is charging through R1 and R2.

·        When the capacitor voltage reaches 1/3 Vcc, comparator B switches to its 0 output state and when the capacitor reaches 2/3Vcc, comparator A switches to its +5V state.

·        This resets the latch causing the base of Q1 to go +5V. As a result Q1 turns ON. This sequence creates a discharge path for the capacitor through R2.

·        The capacitor now begins to discharge causing comparator A to go to 5V at the point where the capacitor discharges down to 1/3 Vcc, comparator B switches to +0V level.

·        This sets the latch, making the base of Q1 to go down to 0V.

·         As a result Q1 turns OFF. Another charging cycle begins and the entire process repeats.

·        This produce a rectangular wave output whose duty cycle depends on the values of R1 and R2.

·        The frequency of oscillator of a 555 timer in astable mode is given by the expression,

F=1.44/(R1 + 2R2) C1

·        By selecting R1 and R2, the duty cycle of the output can be adjusted since C1 charges through R1+R2 and discharges only through R2, a duty cycle approaching 50% can be achieved of R2>>R1 so that charging and discharging times are approximately equal.

RAM and ROM

Read-only memory, or ROM, is a form of data storage in computers and other electronic devices that can not be easily altered or reprogrammed. RAM is referred to as volatile memory and is lost when the power is turned off whereas ROM in non-volatile and the contents are retained even after the power is switched off.

Random-access memory, or RAM, is a form of data storage that can be accessed randomly at any time, in any order and from any physical location in contrast to other storage devices, such as hard drives, where the physical location of the data determines the time taken to retrieve it. RAM is measured in megabytes and the speed is measured in nanoseconds and RAM chips can read data faster than ROM.

Comparison chart

RAM versus ROM comparison chart

	RAM	ROM
Definition	Random Access Memory or RAM is a form of data storage that can be accessed randomly at any time, in any order and from any physical location., allowing quick access and manipulation.	Read-only memory or ROM is also a form of data storage that can not be easily altered or reprogrammed.Stores instuctions that are not nescesary for re-booting up to make the computer operate when it is switched off.They are hardwired.
Stands for	Random Access Memory	Read-only memory
Use	RAM allows the computer to read data quickly to run applications. It allows reading and writing.	ROM stores the program required to initially boot the computer. It only allows reading.
Volatility	RAM is volatile i.e. its contents are lost when the device is powered off.	It is non-volatile i.e. its contents are retained even when the device is powered off.
Types	The two main types of RAM are static RAM and dynamic RAM.	The types of ROM include PROM, EPROM and EEPROM.

SOURCE; Google

Managerial Skills

Managerial Skills

Primary Skills
• Conceptual Skill
• Technical Skill
• Human Skill

Secondary Skills
• Design Skill
• Communication Skills
• Leadership Skill

Human or interpersonal skills:
• How should I understand myself to
understand others and achieved the
common goals and objectives of the
organization? Also focus on how
something is done?
• This includes the ability to
understand other people and
interact effectively with them. The
human skills are also important in
creation of an environment in which
people feel secure and free to
express their opinions.
• In short it is the ability to work with
other people in teams

Design skill:
These skills enable a manager to
handle and solve any kind of
unforeseen or actual problems, that
may crop up in the organization. Such
problems could arise due to internal
factors or external factors and/or
both.
In short it is the problem solving skill

Conceptual skill:
What should be the right thing
and focus on why something is
done?
• This refers to the ability to
think and conceptualize
abstract situations. These
abilities are required for
making complex decisions.
• In short it is the mental
capacity to develop plans,
strategies and vision

Technical Skills
How can we do better and focus
on what is done?
These skills include the
knowledge, abilities of and
proficiency in activities involving
methods, processes and
procedures in the relevant fields as
accounting, engineering,
manufacturing etc.
In short it is the ability to use the
knowledge or techniques of a
particular discipline to attain ends.

Communicational skill
The abilities of exchanging
ideas and information
effectively.
To understand others and let
others understand
comprehensively.

Leadership Skills
The abilities to influence
other people to achieve the
common goal. �

Kuensel retrieved

Library developed and functional

July 1, 2016 News 1 Comment 992 Views

Council: The government has developed an e-Library portal and e-Library studio.

This was revealed in its responses to the National Council (NC).

The government also stated that the installation and commissioning of the entire network will be completed by August end.

“The e-Library would be a collection of digital objects that will include text, visuals, video, audio among others stored as standardised and customised electronic media along with means for organising, storing, and retrieving the contents from the e-library repository,” the government states in its response which is available on the NC’s website.

The response also stated that the Centre for Development of Advanced Computing (CDAC) assigned with the project has developed e-library portal and is functional now. It can be viewed by visiting: http://14.139.61.149/elib/.

The syllabus for all classes from preprimary to higher secondary along with textbooks is available.

As of June 2016, the CDAC project engineer stationed in Thimphu visited 49 schools and identified 40 schools for site readiness to implement the project.

The project engineer also discussed with the Royal University of Bhutan and education ministry on 12 schools and three colleges where the project could be launched.The education ministry is currently processing custom clearances for the equipment to be used for the e-Library.

Once the e-Library is established, training on its use and management will be conducted in September. Content creation for Bhutanese would be initiated from October this year.

Each of the 49 schools will be provisioned with three computers connected through local servers so local content can be accessed and stored.

A central repository is proposed to be located at a data centre in the country, preferably the Thimphu Tech Park, to host the e-Library portal.

In places with poor Internet connectivity, the contents of the local server will be periodically updated with the use of a movable media such as hard drives. The contents will also be accessible by users on their own devices outside the e-Library section using Wi-Fi facility.

The e-Libraries will also be facilitated with televisions so that content can be broadcasted to students. While each schools would be given three computer, each of the 12 tertiary institutes will be provided with five computers.

The e-Library will also have a studio to create digital content for digitalising books, documents and audio-video, which will also be connected to central repository to upload contents to the library’s portal either with use of internet or movable data.

Meanwhile, the purchase order for supply, installation and commissioning of IT hardware for the central repository, colleges, schools and studio has been placed. The vendor is processing to dispatch hardware from India and Singapore. Delivery and distribution of IT hardware to all locations in Bhutan would be done between June-July this year.

Tempa Wangdi

Digital game

1. Educational Game

Educational games are games explicitly designed with educational purposes, or which have incidental or secondary educational value. All types of games may be used in an educational environment. Educational games are games that are designed to help people to learn about certain subjects, expand concepts, reinforce development, understand a historical event or culture, or assist them in learning a skill as they play. Game types include board, card, and video games. An educational game is a game designed to teach humans about a specific subject and to teach them a skill. As educators, governments, and parents realize the psychological need and benefits of gaming have on learning, this educational tool has become mainstream. Games are interactive play that teach us goals, rules, adaptation, problem solving, interaction, all represented as a story. They satisfy our fundamental need to learn by providing enjoyment, passionate involvement, structure, motivation, ego gratification, adrenaline, creativity, social interaction and emotion in the game itself while the learning takes place.

2. Action Games

The action game is a video game genre that emphasizes physical challenges, including hand–eye coordination and reaction-time. The genre includes diverse subgenres such as fighting games, shooter games and platform games which are widely considered the most important action games, though multiplayer online battle arena and some real-time strategy games are also considered to be action games.

In an action game, the player typically controls the protagonist or avatar. The avatar must navigate a level, collecting objects, avoiding obstacles, and battling enemies with various attacks. At the end of a level or group of levels, the player must often defeat a boss enemy that is more challenging and often larger than other enemies. Enemy attacks and obstacles deplete the avatar's health and lives, and the player receives a Game over when they run out of lives. Alternatively, the player wins the game by finishing a sequence of levels. But some action games, usually arcade games, are unbeatable and have an indefinite number of levels; and the player's only goal is to maximize their score by collecting objects and defeating enemies.

3. Excergaming

Fitness ,^[1] exergaming or exer-gaming (a portmanteau of "exercise" and "gaming"), or gamercising^[2][3] is a term used for video games that are also a form of exercise.^[4] Exergaming relies on technology that tracks body movement or reaction. The genre has been credited with upending the stereotype of gaming as a sedentary activity, and promoting an active lifestyle.^[5][6] Research indicates that exergames can produce real improvements in fitness.^[7] Exergames are seen as evolving from technology changes aimed at making video games more fun.^[8]

4. Racing Game

The racing video game genre is the genre of video games, either in the first-person or third-person perspective, in which the player partakes in a racing competition with any type of land, water, air or space vehicles. They may be based on anything from real-world racing leagues to entirely fantastical settings. In general, they can be distributed along a spectrum anywhere between hardcore simulations, and simpler arcade racing games. Racing games may also fall under the category of sports games.

Digital game

1. Educational Game
Educational games are games explicitly designed with educational purposes, or which have incidental or secondary educational value. All types of games may be used in an educational environment. Educational games are games that are designed to help people to learn about certain subjects, expand concepts, reinforce development, understand a historical event or culture, or assist them in learning a skill as they play. Game types include board, card, and video games. An educational game is a game designed to teach humans about a specific subject and to teach them a skill. As educators, governments, and parents realize the psychological need and benefits of gaming have on learning, this educational tool has become mainstream. Games are interactive play that teach us goals, rules, adaptation, problem solving, interaction, all represented as a story. They satisfy our fundamental need to learn by providing enjoyment, passionate involvement, structure, motivation, ego gratification, adrenaline, creativity, social interaction and emotion in the game itself while the learning takes place.

2. Action Games
The action game is a video game genre that emphasizes physical challenges, including hand–eye coordination and reaction-time. The genre includes diverse subgenres such as fighting games, shooter games and platform games which are widely considered the most important action games, though multiplayer online battle arena and some real-time strategy games are also considered to be action games.

In an action game, the player typically controls the protagonist or avatar. The avatar must navigate a level, collecting objects, avoiding obstacles, and battling enemies with various attacks. At the end of a level or group of levels, the player must often defeat a boss enemy that is more challenging and often larger than other enemies. Enemy attacks and obstacles deplete the avatar's health and lives, and the player receives a Game over when they run out of lives. Alternatively, the player wins the game by finishing a sequence of levels. But some action games, usually arcade games, are unbeatable and have an indefinite number of levels; and the player's only goal is to maximize their score by collecting objects and defeating enemies.

3. Excergaming
Fitness ,^[1] exergaming or exer-gaming (a portmanteau of "exercise" and "gaming"), or gamercising^[2][3] is a term used for video games that are also a form of exercise.^[4] Exergaming relies on technology that tracks body movement or reaction. The genre has been credited with upending the stereotype of gaming as a sedentary activity, and promoting an active lifestyle.^[5][6] Research indicates that exergames can produce real improvements in fitness.^[7] Exergames are seen as evolving from technology changes aimed at making video games more fun.^[8]

4. Racing Game

The racing video game genre is the genre of video games, either in the first-person or third-person perspective, in which the player partakes in a racing competition with any type of land, water, air or space vehicles. They may be based on anything from real-world racing leagues to entirely fantastical settings. In general, they can be distributed along a spectrum anywhere between hardcore simulations, and simpler arcade racing games. Racing games may also fall under the category of sports games.

Flip flop

j flop is an important basic memory element for digital circuit. Flip-flop is designed by assembling different logic gates. Single logic gate does not have any information storing capacity but by combining different such gates one can make such a digital circuit which can store digital information. Flip flop is such circuit. There are different types of flip flop with different characteristics for different application. Flip-flop is the basic building blocks of most sequential circuits. Flip-flop (FF), is also known as a bistable multivibrator, because it has two stable states. It y6zaætwgdxtcan remain in d of the states indefinitely. Its state can be changed ny applying the proper triggering signal. Flip flop is one-bit memory element. There are two outputs in a flip-flop generally marked as Qh and . Either of Q and  can be used as output but normal practice is to take Q as output port and  as inverted output port.

It is to be noted here that a flip flop has always one out complement of other output. The state of a flip-flop is normally determined by the condition of output Q. If Q = 1 the flip-flop is said to be in HIGH state or logic 1 state or SET state. When, Q = 0 the state of flip flop is said to be in LOW state or logic 0 state or RESET state or CLEAR state.The figure below shows a block diagram of a flip flop. It shows a flip-flop may have one or more inputs but only two outputs. The combination of inputs which alter the outputs or state of flip flop is referred as excitation. The excitation is used to switch the flip flop from one state to other. But the typical feature of flip flop is that once the state of flip flop is changed by applying, excitation, it remains unaltered even the excitation is removed from input ports. Hence, momentary application of excitation is enough to change the state a flip-flop. This is how flip flop behaves as memory element. When, Q = 1 it stores a 1 and when, Q = 0, it stores a 0. Flip-flops are the basic components of shift registers and counters. Flip flop is a sequential circuit hence it can be either synchronous or asynchronous. When inputs are controlled by clock pulse it is normally referred as flip flop. Here the inputs are applied but not acted until clock pulse appears and enable the inputs. When the same circuit is made asynchronous that is its inputs is not controlled by clock pulse, it is called latch. It is SET or RESET instantaneously on receiving the input signal. Latch can act independently of clock signal.

Most simple type of flip flop is S R Flip Flop. It has two inputs S and R and two outputs Q and . The state of this latch is determined by condition of Q. If Q is 1 the latch is said to be SET and if Q is 0 the latch is said to be RESET. This S R Latch or Flip flop can be designed either by two cross-coupled NAND gates or two-cross coupled NOR gates. When we design this latch by using NOR gates, it will be an active high S-R latch. That means it is SET when S = 1. When we design this latch by using NAND gates, it will be an active low S-R latch. That means it is SET when S = 0. S R Flip Flop is also called SET RESET Flip Flop. Figure below shows the logic circuit of S R latch.

In the above logic circuit if S = 1 and R = 0, Q becomes 1. Let us explain how.

NOR gate always gives output 0 when at least one of the inputs is 1.So when S is applied as 1 the output of gate G2 i.e.  is 0 irrespective of the condition of second input Q to the gate.Now  is input of gate G1 so both the inputs of G1 become 0 as R is already 0. So, output of G1 is now  or 1.So whatever may be the previous condition of Q, it always becomes Q = 1 and  = 0 when, S = 1 and R = 0. This is called SET condition of the latch.In the above logic circuit if S = 0 and R = 1, Q becomes 0. Let us explain how.As we already said, a NOR gate always gives output 0 when at least one of the inputs is 1.So when R is applied as 1, the output of gate G1 i.e. Q is 0 irrespective of the condition of second input  to the gate.So, whatever may be the previous condition of Q, it always becomes 0 this 0 is then fed back to input of gate G2. As here S is already 0, both inputs of G2 are 0. Hence output of G2 i.e.  will be 1. So, Q = 0 and  = 1 when, S = 0 and R = 1. This is called RESET condition of the latch.In the above logic circuit if S = 0 and also R = 0, Q remains same as it was. Let us explain how.First suppose Q is previously 1.Now the inputs of G2 are 0 and 1 as S=0 and Q=1. So output of G2 i.e.  is  or 0.Now both inputs of G1 are 0 as R=0 and =0. So output of G1 i.e. Q is  or 1.Now suppose Q is previously 0.Now both inputs of G2 are 0 and 1 as S=0 and Q=0. So output of G2 i.e.  is  or 1.Now the inputs of G1 are 0 and 1 as R=0 and =1. So output of G1 i.e. Q is  or 0.So it is proved that Q remains same as it is when S = 0 and also R = 0 in S R latch or flip flop.

In the above logic circuit if S = 1 and also R = 1, the condition of Q is totally unpredictable. Let us explain how.

First suppose Q is previously 1.Now both inputs of G2 are 1 as S=1 and Q=1. So output of G2 i.e.  is  or 0.Now the inputs of G1 are 1 and 0 as R=1 and =0. So output of G1 i.e. Q is  or 0. That means Q is changed.Now Q is 0. So inputs of G2 are 1 and 0 as S = 1 and Q = 0. So output of G2 i.e.  is  or 0. That means  is unchanged.Now the inputs of G1 are 1 and 0 as R=1 and =0. So output of G1 i.e. Q is  or 0. That means Q is unchanged.So, when both S and R are 1, it becomes unpredictable whether the value of output Q will be changed or unchanged. This condition of S R latch normally avoided. As the latch is SET when S = 1(HIGH), the latch is called Active High S R Latch. There is other type of latch which is SET when, S = 0 (LOW), and this latch is known as Active Low S R Latch.

Department of Electrical Engineering (2016)

1. 05160160 APSANA SHARMA

2. 05160161 BIR MAYA RAI

3. 05160162 BIRKHA BDR. SUBBA

4. 05160163 CHEKI ZANGPO

5. 05160164 CHENCHO LHAMO

6. 05160165 CHETEN DORJI

7. 05160166 DORJI SINGYE

8. 05160167 JAMBAY CHOEDA

9. 05160168 JIGME DORJI

10. 05160169 JIMBA CHOPHEL

11. 05160170 KARMA CHODEN

12. 05160171 KARMA GAYLEG

13. 05160172 KARMA THINLEY

14. 05160173 KINLEY ZANGMO

15. 05160174 KUENLEY PENJOR
GYELTEN

16. 05160176 MOHAN PRASAD DHAKAL

17. 05160177 MON BDR GURUNG

18. 05160178 NGAWANG NAMGAY

19. 05160179 NIMA TSHERING BAL

20. 05160180 PEMA TSHERING

21. 05160181 PEMA YANGCHEN

22. 05160182 PHUNTSHO CHODEN

23. 05160183 RATU

24. 05160185 RINCHEN LHAMO

25. 05160186 SANGAY JAMTSHO

26. 05160187 SANGAY TSHERING

27. 05160188 SANJEP RAI

28. 05160189 SARDA CHUWAN

29. 05160190 SHEKHAR CHANDRA
KHATIWARA

30. 05160191 SOM BDR. RAI

31. 05160192 SONAM CHEDUP

32. 05160193 SONAM CHOKI

33. 05160194 SONAM DEKAR

34. 05160195 TASHI

35. 05160196 TASHI TENZIN

36. 05160197 TEMPA GYELTSHEN

37. 05160198 TENZIN CHENJUR

38. 05160200 TIL BDR

39. 05160201 TSHERING DORJI

40. 05160202 TSHERING WANGMO

41. 05160203 TSHEWANG DORJI

42. 05160204 TSHEWANG LHAMO

43. 05160205 UGYEN WANGCHUK

44.  05160206 VIJAY RAI

45. 05160207 YOUNTEN JAMTSHO

46. 05150173 CHOEKI DORJI

47.  05150194 NAKUL GHALLEY

Cast system

This is a system which we follow from the time when the
practice of Hindu religion started in this world. we practice it
because it is our culture that we have to follow it. Brahmin are
to give religious teaching to the people . They are to give
teaching because from the ancient time they are restricted by
religion to take intoxicated substances such as alcohol. Now
world has become so rich with people of high intelligent they
they ask reason for everything. They are so advanced with modern technologies of proving everything in practical that
they even want religious beliefs to be shown in practical field.
This system had been as far as possible by the pace of the
modernization but this system will not be eradicated
completely because if this is eradicated then Hindu religion
won't exist.

It is found that most people talks about the
practice of cast system in Hindu. And this talks
comes from only those people who don't do any
religious practice whether they be Hindu, Buddhist
or in any religion. Those who practice religious
activities (such as Rinpochoes, Pundit, lamas, and
other devotee of God) never says to eradicate this
system because they know it is not like that of
scientific theorem that is accepted only after
experimentally proven. You can't change it all of
sudden by giving drawbacks on it because it is
also one form of culture.

Now this system is changed greatly compared to
the century time ago. Nowadays people are leaving
Hinduism and Buddhism and moving towards
Christianity because it is simpler one. It Christianity
you need not to perform any ritual even when
someone dies in relative.
It doesn't matter if someone doesn't take the thing
you give thinking that one is different but if
someone doesn't give you your share thinking you
are different you have right to raise a quires.
Shekhar Khatiwara

K-Map representation

Digital Teaching AidKarnaugh Mapping - Lesson 4Lesson Plan(introduction...)IntroductionKarnaugh mapTruth table to Karnaugh mapPairs, Quads, and OctetsOverlapping and Rolling

Pairs, Quads, and Octets

Pairs

Fig. 4-4: Four variable simplification

As you see in Fig. 4-4, only one variable goes from uncomplement to complement. Whenever this happens, you can eliminate the variable that changes form.

Proof:

X = A B C

Ex:

Fig. 4-5: Pairs

Whenever you see a pair first encircle it and then simplify to get the simplified Boolean expression:

Quad

Fig. 4-6: Quad

Quad: A group of 4 one's that are horizontally or vertically adjacent. End to end or in form of a square.

A quad eliminates two variables and their complements.

Proof:

(two pairs)

X = A B (C + C)

X = A B

Encircle the quad and step through the different one's in the quad and determine which two variables go from complement to uncomplement (or vs), these are the variables that drop out.

Ex:

Fig. 4-7: Quad

The variables B and D can be eliminated. So we get the following equation:

X = A C

Octet

Fig. 4-8: Octet

An octet eliminates three variables and their complements.

Proof:

(two quads)

X = A (C + C)

X = A

Karnaugh Simplifications

Process:

1. Draw the Karnaugh map
2. Look for octets and encircle them.
3. Look for quads and encircle them.
4. Look for pairs and encircle them.
5. Simplify and write down the equation.

Ex:

Fig. 4-9: Karnaugh map

Overlapping and Rolling

Overlapping groups

Ex:

Fig. 4-10: Karnaugh map

Groups can overlap to get a simpler equation:

Rolling the map

Ex:

Fig. 4-11: Karnaugh map

Instead of encircling two pairs:

We can roll the map and encircle a quad:

HO: Simplify the following map.

Solution:

HO: Simplify the following map.

Solution:

Simplification Procedure for Karnaugh maps

Pair Reduction Rule : Remove the variable which changes its state from complemented to uncomplemented or vice versa.Pair removes one variable only.

Quad Reduction Rule : Remove the two variables which change their states.A quad removes two variables.

Octet Reduction Rule : Remove the three variables which changes their state.Octet removes three variables.

Map Rolling : Map rolling means roll the map considering the map as if its left edges are touching the right edges and top edges are touching bottom edges.While marking the pairs quads and octet, map must be rolled.

Overlapping Groups : Overlapping means same 1 can be encircled more than once. Overlapping always leads to simpler expressions.

Redundant Group : It is a group whose all 1's are overlapped by other groups. Redundant groups must be removed. Removal of redundant group leads to much simpler expression.

 

Ex. 1 : Represent the following boolean expression in a K-map and simplify.

F = x'yz + x'yz' + xy'z' + xy'z

Solution :

The K-map is as follows :

Hence the simplified expression is

F = x'y + xy'

Ex. 2 :Simplify the following boolean expression using K-map.

F = a'bc + ab'c' + abc + abc'

Solution :

The K-map is as follows :

Hence the simplified expression is

F = bc + ac'