Lecture 1 MENG 4779 Mechatronics and Digital Systems -- Spring 2016 (1)

8/20/2019 Lecture 1 MENG 4779 Mechatronics and Digital Systems -- Spring 2016 (1)

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AUC_MENG 4779 Spring 2016

Prof. Dr. Maki K. Habib

AUC_MENG 4779 Spring 2016 Prof. Dr. Maki K. Habib

Mechatronics

and

Digital Systems

Maki K. Habib

Mechanical Engineering Department

School of Sciences and Engineering

The American University in Cairo

[email protected]


The technical term, Mechatronics,

is the concept created in 1969 by

Mr. Tetsuro Mori, CEO / President, SeibuElectric and Machinery Co. Ltd.,

when he worked for Yaskawa Electric

Corporation in Kitakyushu/Japan

He proposed the new technology to produce new

machine tools to unite

• mechanism and electronics supported by• semiconductor power devices and

• CPUs which is necessary to develop ‘intelligent’

products and manufacturing systems.


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The circumstances that lead to the growth

of Mechatronics Products are,

1. Cheap mass produced integrated circuits have enabled the

situation of mechanical functions by electronics,

2. The advent of Microprocessor has made it possible to introduce

“intelligence” in the control functions of mechanical processes,

3. The advent of sensor technology has made it possible to

integrate mechanic and electronic technologies, and

4. The reliability of electronic components and circuits has become high enough to withstand the hostile conditions of

mechanical environment.


1969 1980 1990 2000

The birth of

Mechatronics

System Engineering

2010

Mechatronics as

engineering and

quality of technology

Mechatronics as

interdisciplinary

education and

research identity

Mechatronics as

engineering

science discipline

Mechatronics

as technology

and practices

(team based)

T e c h n o

l o g y, p

r a c t i c e

s, i n t e

l l i g e n c

e, i n t e

r d i s c i p

l i n a r y,

e d u c a t

i o n, i n

n o v a t i o

n, l e a r

n i n g , d

i s c i p l i n e

Servo technology,

Microprocessors

Numerically controlled

systems, Semiconductor

technology boom,

Automotive industry,

Consumer electronics

Technology developed

individually and independently.

Interactions between software,

mechanical and electronics

elements.

The increase in variety and

complexity of product design,and the wide range of growing

industries initiated the demand

for engineers with

Mechatronics thinking

knowledge and actions.

Rapid prototyping, Opto-

electronics, Embedded

systems, Micro-technology

and MEMs, Human

computer interaction,

Electronic and Advanced

manufacturing, Knowledge

based systems, Automation,

Informatics and networking

Uniqueness of Mechatronics as a

significant design trend. Interactive

design process with consideration

on: innovation, human factors, lifecycle factors, quality, reliability,

functionality, smartness, portability,

compactness, low cost, etc.

Mechatronics has gained attention

and its importance was widely

recognized

Nanotechnology, Processor

speeds, High memory capacity

biotechnology, Consumer

electronics, Intelligent systems,

Information and communication

technologies, Biomimetic,

HAFM

Durability, multi-functionality,

flexibility,, recycle and

environmental considerations.

Mechatronics education has

gained international recognition

witnessed by the growing number

of universities offering under and

postgraduate Mechatronics

degree courses due to its role as a

unifying interdisciplinary and

intelligent engineering science

paradigm.

Time line

Fig. 1. The evolution of Mechatronics.


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fuses, permeates (to be diffuse/penetrate through), andcomprehends (understand the nature, perceive) modern

engineering science and technologies.

Mechatronics

regarded as a philosophy that supports

new ways of thinking,

innovations,

design methodologies (synthesis and analysis), and

practices

in the design of new intelligent products and engineeringsystems.

Mechatronics


Mechatronics

is a concurrent, and interdisciplinary engineering science

discipline that concentrates on achieving optimum

functional synergy from the earliest conceptual stages of

the design process.


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The main goals of Mechatronics are to

• bring out novel possibilities of synergizing and fusing

different disciplines and to

• develop

products,

processes, and

systems

that exhibit quality performance in terms of

Reliability,

Precision,

Smartness (thinking and decision making capabilities),

Flexibility,

Adaptability, Robustness,

Compactness, and

Economical features.


New

professional

skills

Continuous

Self learning

Biotechnology

The General Knowledge Space

Medical imaging &

Instrumentation

Precision

Engineering

MEMs, VLSI,

Microsystems

Seeking knowledge

relevant to new

projects and area

of research

Human Adaptive

/Friendly

Mechatronics

New topics

of interest

New topicsIntelligent Control,

Real-time Systems

Mechatronics

Knowledge Space

Team based

experience through

projects

Fig.2. Mechatronics Knowledge Space paradigm.

Automotive

Engineering

Smart

Structures Computational

Intelligence

Mechatronics’Foundational

and Core knowledge, such as

Mathematics, Physics, Electrical, Electronics,

Electrical machines, Mechanics, System dynamics

and modeling, Control, Sensors and Perception,

Algorithms, Mechatronics design-analysis, Machine

design, Fluid power, Smart materials and MEMs,

Computer network,, Programming and IT,

Microcontrollers, Embedded and real-time systems,

Robotics and Automation,AI, Simulation and

interactive virtual modeling, Manufacturing

processes and production systems, Projects,

Engineering management, Professional

practices, and electives.

Biomimetics

Wireless Sensor

Networks and Ambient

Intelligence

New topicsOther topics of

interest

Other new topics

of interestOther new topics

of interest

High Voltage

Power Systems

Note:

The selection/overlapping of specialized topics

(beyond the foundational and core knowledge),

and their details depend on the interest of each

individual and the professional needs of the

relevant carrier.


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Manufacturing

personnel

Finance and

Sales personnel

Design team

Others personnel

as necessary

Technical and

Production personnel

Explore and analyze relevant existing

systems, and patents. Establish technical

feasibility and list functional requirements.

Understand and define the nature of the

problem/target, and identify the needs. Set

goals and constraints.• Define market segments,

• Ident ify lead users ,

• Ident ify competit ive

products,

Synergetic and interactive

development environment

for Mechatronics design

and development process.

Final design for

production with full

documentation

Project coordinator Schedule, milestones,

resources, constrains, work

assignment, documentation

Brainstorming for new ideas and

solutions development. List

potential solutions, scenarios, logic

flow, priorities.

Short list, assess, shapeup

solutions, and select solution. Set

target functional requirements

and details specifications.

Plan, schedule, roles and

responsibilities, design

details, simulate, test, build

prototype, evaluate, optimize.

Design review, evaluation,

enhancement, life cycle

design factors, and human

factors considerations.

Estimate

manufacturing cost, and

assess production

feasibility

Interaction with environment

and other personnel for

information, discussion,

presentation, etc. and as

necessary

Fig. 3. Synergetic and interactive development environment for Mechatronics design and

development process with its interactive stages.

Stage 1


In general Smart Mechatronics Products/processes and systems constitute various

technology

that include

• Wide range of sensors,

• Actuators, and intelligent mechanisms,

• Microcontrollers,

• Decision making (Intelligence)

• Control strategies, artificial perception• Smart materials, Micro- and Nano-technology

• Information and communication technologies,

• etc.


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Examples of Mechatronics applications

There are many Mechatronical applications,

some of them listed as follows,

Robots (all its shapes and purposes),

Automation,

Cars,

Automatic guided vehicles,

Computer controlled machine tools,

Planes and space technology,

Medical equipment,

Cash dispenser,


Type-writer,

Fax machine, Computer Disk Drives

Video Camera,

Video recorder,

CD Rom Players,

Walk-man,

Auto-camera,

Cell phone,

Watches,

Microwave,Washing machine,

Sewing machines,

Air-condition,

etc.


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AUC_MENG 4779 Spring 2016Prof. Dr. Maki K. Habib

Digital Logic Design

Number Systems, Logic Gates and

Design

Maki K. Habib

Mechanical Engineering Department

School of Sciences and EngineeringAmerican University in Cairo

[email protected]


In DIGITAL electronics, current & voltage can assume only discrete values(usually two).

e.g. V

In ANALOG systems, current & voltage levels are continuous & may

assume any value.

Real

World


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Digital Electronics

The advantage of digital electronics are,

• Greater accuracy & reliability

• Versatile, cheaper and low-power,

• Comprehensive theory and algorithms

• Availability of CAD tools

• Optimized device processes

Digital circuits used in:

• Digital Computers Data Processing

• Electronic Calculators Instrumentation

• Control Devices etc.

• Communication Equipment• Telephone Networks, Cell Phones,

• CD Players, Medical Equipment,

• Modern TV sets, Modern Radios, etc.


Disadvantages of digital electronics

1. Signal precision is limited by the number

of bits used to encode each sample,

2. Analogue-to-digital converters and

digital-to-analogue converters are

required to interface a digital,

3. system with real-world analogue signals


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Most physical phenomena of interest are analogue

• Transducers are simple

• Potentially high precision

Analogue Systems

Disadvantages of analogue systems

Behaviour of analogue components is subject to

drift distortion, noise, offsets, etc.,

Errors in analogue signals accumulate during

processing, transmission, and storage,

Only relatively simple signal processing is

practical for most applications.


DigitA digit is a symbol or numeral given to an element of a number

system.

Radix

The radix, or base of a counting system is defined as the number ofunique digits (It is the total number of digits allowed) in a given

number system.

Numbers play an important part in our lives.

Example, for the decimal number system:Radix, r = 10, Digits allowed = 0,1, 2, 3, 4, 5, 6, 7, 8, 9

Number Systems

There are many number systems, such as decimal number

system. Each number constitutes at least one digit.


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Number systems and codes

Decimal(base 10)

Octal(base 8)

Binary(base 2)

Hexadecimal

(base16)

Conversion from decimal to binary

Conversion from binary to decimal


Examples

• Decimal numbers(base 10)

36.210 9810

• Hexadecimal number(base 16)

3F216

• Binary number(base 2)

10112


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Positional system

Each digit carries a certain weight based on its

position.

346.17463.71 Position matters

Weight vs Position


Decimal Positional System

(Base 10 or radix 10)

… 104 103 102 101 100 . 10-1 10-2 …

hundreds positiontens positionones position

tenths position

hundredth position

decimal point


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Binary Positional System

(Base 2 or radix 2)

… 24 23 22 21 20 . 2-1 2-2 …

fours position

twos positionones position

halves position

quarters position

binary point


0.07 0.1 6 40 300

)107()101()106()104()103(17.346 2101210

Decimal Example

Binary Example

10

210123

2

13.25 .25 0 1 0 4 8

21202120212101.1101

Examples


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Binary to Decimal Conversion

What is 1101012 in decimal?

10

012345

2

53

1 0 4 0 16 32

212021202121110101


n 2n

0 20=1

1 21=1

2 22=4

3 23=8

4 24=16

5 25=32

6 26=64

7 27=128

n 2n

8 28=256

9 29=512

10 210=1024

11 211=2048

12 212=4096

20 220=1M

30 230=1G


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Decimal-To-Binary Conversions(method 1)

• The decimal number is simply expressed as a sum of

powers of 2, and then 1s and 0s are written in the

appropriate bit positions.

210

145

10

11001050

212121

21632

183250

210

13468

10

101011010346

2121212121

281664256

101664256

2664256

90256346


(Method 2)Flowchart for Repeated Division


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Example for Repeated Division

Quotient Remainder

50/2 = 25 0 LSB

25/2 = 12 1

12/2 = 6 0

6/2 = 3 0

3/2 = 1 1

1/2 = 0 1 MSB

5010=1100102


Example for Repeated Division

Quotient Remainder

346/2 173 0

173/2 86 1

86/2 43 0

43/2 21 1

21/2 10 1

10/2 5 0

5/2 2 1

2/2 1 0

1/2 0 1

34610=1010110102


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How many different values can be represented with Nbinary digits? Decimal digits? Octal digits? Radix Z

digits?

• Decimal: 1 digit 0-9 10 different values

2 digits 10X10=100 different values

.. 6 digits 106=1,000,000 different values

• Binary: 1 digit 0,1 2 different values=21

2 digits 00,01,10,11 4 different values=22

n digits 2n

different values• Radix Z digits: n digits Zn different values(0 thru. Zn-1)

Examples


Octal-to-Decimal Conversion

Octal-to-Decimal Conversion

1020.75

)1

8(6)0

8(4)1

8(286.24

10250

1287643

)08(2)18(7)28(38

372


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Decimal-to-Octal Conversion

Convert 26610 to Octal

Quotient Remainder

266/8 = 33 2 LSB

33/8 = 4 1

4/8 = 0 4 MSB

26610=4128


Octal-to-Binary Conversion

• Convert 4728 to binary

4 7 2

100 111 010

• Convert 54318 to binary

5 4 3 1

101 100 011 001


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Binary-to-Octal Conversion

• Convert 1001110102 to octal

• Convert 110101102 to octal

82 7 4

010111001

86 2 3

011010110


Octal-to-Hex Conversion

• Convert B2F16 to octal

B2F16 =1011 0010 1111 {convert to binary}

=101 100 101 111

{group into three-bit groupings}

= 5 4 5 78 {Convert to octal}


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BCD Code

• If each digit of a decimal number is represented by

its binary equivalent, the result is a code called

binary-code-decimal (BCD).

8 7 4 (decimal)

1000 0111 0100 (BCD)

9 4 3 (decimal)

1001 0100 0011 (BCD)


• Convert 0110100000111001(BCD) to its decimal

equivalent.

• Convert the BCD number 011111000001 to its decimal

equivalent.

Examples


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Comparison of BCD and Binary

• A straight binary code takes the complete

decimal number and represents it in binary.

• A BCD code converts each decimal digit to

binary individually.

13710=100010012 (binary)

13710=0001 0011 0111 (BCD)

• BCD uses more bits, easier to convert to andfrom decimal.


Review Questions

• Represent the decimal value 178 by its straight

binary equivalent. Then encode the same

decimal number using BCD.

• How many bits are required to represent an

eight-digit decimal number in BCD?

• What is an advantage of encoding a decimal

number in BCD as compared with straight

binary? What is a disadvantage?


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Putting it ALL together


The Nibble and Byte

• A string of 4 bits is called a nibble

• A string of 8 bits is called a byte.

• How many bytes are in a 32-bit string?

• What is the largest decimal value that can be

represented in binary using two bytes?

• How many bytes are needed to represent thedecimal value 846,569 in BCD?


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Review Questions

• How many bytes are needed to represent 23510in binary?

• What is the largest decimal value that can be

represented in BCD using two bytes?


Alphanumeric Codes

• Codes representing letters of the alphabet,

punctuation marks, and other special characters as

well as numbers are called alphanumeric codes.

• The most widely used alphanumeric code is the

American Standard Code for Information

Interchange (ASCII).

The ASCII(pronounced “askee”) code is a seven-

bit code.


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Logic Gates and Boolean

Algebra


Boolean Constants and Variables

• Boolean constants and variables are allowed to have only two

possible values, 0 or 1.

• Boolean 0 and 1 do not represent actual numbers but instead

represent the state of a voltage variable, or what is called its

logic level.

• 0/1 and Low/High are used most of the time.

• Three Logic operations: AND, OR, NOT

• Logic Gates – Digital circuits constructed from diodes, transistors, and

resistors whose output is the result of a basic logic

operation(OR, AND, NOT) performed on the inputs.


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Truth Tables

• How a logic circuit’s output depends on the logic

levels present at the inputs.


Signals, Logic Operators, and Gates

Basic elements of digital logic circuits


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Summary of OR operation

• Produce a result of 1 whenever any input is 1.

Otherwise 0,

• An OR gate is a logic circuit that performs an

OR operation on the circuit's input,

• The expression X = A + B is read as

“X equals A OR B”


Example of the use of an OR gate in an

Alarm system


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Example


Example3


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Review Questions

• What is the only set of input conditions that will produce a LOW output for any OR gate?

• Write the Boolean expression for a six-input ORgate

• If the A input in previous example is permanently kept at the 1 level, what will theresultant output waveform be?


AND Operation with AND Gates

• Truth Table and Gate symbol


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Truth Table and Symbol for a three-

input AND gate


Summary of the AND operation

• The AND operation is performed the same asordinary multiplication of 1s and 0s.

• An AND gate is a logic circuit that performs theAND operation on the circuit’s inputs.

• An AND gate output will be 1 only for the casewhen all inputs are 1; for all other cases theoutput will be 0.

• The expression X = AB is read as

“X equals A AND B.”


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Review Questions

• What is the only input combination that will produce a HIGH at the output of a five-inputAND gate?

• What logic level should be applied to the secondinput of a two-input AND gate if the logic signalat the first input is to be inhibited(prevented) fromreaching the output?

• True or false: An AND gate output will alwaysdiffer from an OR gate output for the same input

conditions.


NOT operation

• Truth table, Symbol, Sample waveform


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Summary of Boolean Operations

• OR

0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1

• AND

0 • 0 = 0 0 • 1 = 0 1 • 0 = 0 1 • 1 = 1

• NOT

1’=0 0’=1 (NOTE THE SYMBOL USED FOR NOT!)


Describing logic circuits algebraically

• Any logic circuit, no matter how complex, can be

completely described using the three basic Boolean

operations: OR, AND, NOT.

• Example: logic circuit with its Boolean expression


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Parentheses

(Often needed to establish precedence;

sometimes used optionally for clarity)

• How to interpret AB+C?

– Is it AB ORed with C ?

– Is it A ANDed with B+C ?

• Order of precedence for Boolean algebra: AND before OR.

Parentheses make the expression clearer, but they are not

needed for the case on the preceding slide.

• Note that parentheses are needed here :


Circuits Contains INVERTERs

• Whenever an INVERTER is present in a logic-circuit

diagram, its output expression is simply equal to the

input expression with a bar over it.


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More Examples


Precedence

• First, perform all inversions of single terms,

• Perform all operations with parentheses,

• Perform an AND operation before an OR,

• operation unless parentheses indicate otherwise,

• If an expression has a bar over it, perform theoperations inside the expression first and then invert

the result


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Determining output level from a

diagram

Determine the output for the

condition where all inputs are LOW.


Implementing Circuits From Boolean

Expressions

• When the operation of a circuit is defined by a

Boolean expression, we can draw a logic-circuit

diagram directly from that expression.


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Example

• Draw the circuit diagram to implement the expression

))(( C B B A x


Review Question

• Draw the circuit diagram that implements the

expression

Using gates having no more than three inputs.

)( D A BC A x


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NOR GATES AND NAND GATES

• NOR Symbol, Equivalent Circuit, Truth Table


Example


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Example

• Determine the Boolean expression for a three-input

NOR gate followed by an INVERTER


NAND Gate

• Symbol, Equivalent circuit, truth table


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Example


Example

• Implement the logic circuit that has the expression

using only NOR and NAND gates

DC AB x


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Example

• Determine the output level in last example for

A=B=C=1 and D=0


Review Questions

• What is the only set of input conditions that will

produce a HIGH output from a three-input NOR

gate?

• Determine the output level in last example for

A=B=1, C=D=0,

• Change the NOR gate at last example to a NANDgate, and change the NAND to a NOR. What is the

new expression for x?


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Boolean Theorems (single-variable)


Multivariable Theorems

x+y = y+x xy = yx commutativity

(x+y) + z = x + (y + z) (xy)z = x(yz) associativity

x(y+z) = xy + xz x + yz = (x+y) (x+z) distributivity

x + xy = x pf: x+xy = x1 + xy = x(1+y) = x1 = x


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Examples

• Simplify the expression

• Simplify

• Simplify

D B A D B A y

B A B A z

BCD A ACD x


Review Questions

• Simplify

• Simplify

• Simplify

C ABC A y

DC B A DC B A y

ABD D A y


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Demorgan’s Theorems

y x y x

y x y x


Example

• Simplify the expression to one having

only single variables inverted.

D BC A z


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Implications of DeMorgan’s Theorems(I)


Implications of DeMorgan’s

Theorems(II)


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Example

• Determine the output expression for the below circuit

and simplify it using DeMorgan’s Theorem


Review Questions

• Using DeMorgan’s Theorems to convert the

expressions to one that has only single-variable

inversions.

• Use only a NOR gate and an INVERTER to

implement a circuit having output expression:

• Use DeMorgan’s theorems to convert below

expression to an expression containg only single-

variable inversions.

C B A z QT S R y C B A z

DC B A y


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Universality of NAND and NOR gates


Universality of NOR gate


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Example


Example


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Alternate Logic-Gate Representations

• Standard and alternate symbols for various logic

gates and inverter.


How to obtain the alternative symbol

from standard ones

• Invert each input and output of the standard symbol,

This is done by adding bubbles(small circles) on

input and output lines that do not have bubbles and by

removing bubbles that are already there.

• Change the operation symbol from AND to OR, or

from OR to AND.(In the special case of theINVERTER, the operation symbol is not changed)


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Points of Consideration

• The equivalences can be extended to gates with any numberof inputs.

• None of the standard symbols have bubbles on their inputs,and all the alternate symbols do.

• The standard and alternate symbols for each gate representthe same physical circuit; there is no difference in thecircuits represented by the two symbols.

• NAND and NOR gates are inverting gates, and so both thestandard and the alternate symbols for each will have a bubble on either the input or the output, AND and OR gates

are non-inverting gates, and so the alternate symbols foreach will have bubbles on both inputs and output.


Logic-symbol interpretation

• Active High/Low

– When an input or output line on a logic circuit

symbol has no bubble on it, that line is said to be

active-High, otherwise it is active-Low.


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Interpretation of the two NAND gate

symbols


Interpretation of the two OR gate

symbols


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Review Questions

• Write the interpretation of the operation performed by

the below gate symbols,

– Standard NOR gate symbol

– Alternate NOR gate symbol

– Alternate AND gate symbol

– Standard AND gate symbol


XOR Gate

The XOR gate is an e xclusive OR gate.

It will output a logic 1 if there is an exclusive logic 1 at input A or B.Exclusive means: Only one input can be high at one time.

Input AOutput X

Input B

XOR

BAX

The Boolean Equationfor XOR :

A B X

0 0 0

0 1 1

1 0 1

1 1 0


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The XNOR gate is an e xclusive OR gate with an NOT gate at the output. Itwill output a logic 0 if there is an exclusive logic 1 at input A or B.

A B X

0 0 1

0 1 0

1 0 0

1 1 1

Input AOutput X

Input B

XNOR

BAX

The Boolean Equation

for XNOR :

XNOR


Truth Tables

• Another way (in addition to logic equations) to define certain

functionality

• Problem: their sizes grow exponentially with number of

inputs.

11111

01011

01101

01001

01110

01010

01100

00000

y2y1x3x2x1

inputs outputs

What are logic equations

corresponding to this table?

Design corresponding circuit.

y1 = x1 + x2 + x3

y2 = x1 * x2 * x3


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Input Minterms Maxterms

A B C Terms Designation Terms Designation

0 0 0 P0 S0

0 0 1 P1 S1

0 1 0 P2 S2

0 1 1 P3 S3

1 0 0 P4 S4

1 0 1 P5 S5

1 1 0 P6 S6

1 1 1 P7 S7

Minterms and Maxterms for Three Binary Variables

What is the signif icance of Minterms and Maxterms?In short, minterms and maxterms may be used to define the two standard forms for logic

expressions, namely the sum of p roducts (SOP), or sum of minterms, and the product of

sums (POS), or product of maxterms. These standard forms of expression aid the logic

circuit designer by simplifying the derivation of the function to be implemented.

3


Row Input Output

Number A B C X

0 0 0 0 0

1 0 0 1 0

2 0 1 0 0

3 0 1 1 1

4 1 0 0 05 1 0 1 1

6 1 1 0 1

7 1 1 1 0

Example

2 X = P3 + P5 + P6

1

SOP

1

2X = S0S1S2S4S7

3

POS


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Logic Equations in Sum of Products Form

• Systematic way to obtain logic equations from a given truth table.

01111

00011

10101

10001

01110

10010

11100

11000

y2y1x3x2x1

inputs outputs

• A product term is included for

each row where yi has value 1

• A product term includes all input

variables.

• At the end, all product terms are

ORed

+ x1*x2*x3y1 = x1*x2*x3 + x1*x2*x3+ x1*x2*x3

+ x1*x2*x3y2 = x1*x2*x3 + x1*x2*x3 + x1*x2*x3+ x1*x2*x3


Minimization Applying Boolean Laws

+ x1*x2*x3y1 = x1*x2*x3 + x1*x2*x3+ x1*x2*x3

• Consider one of previous logic equations:

= x1*x2*(x3 + x3) + x2*x3*(x1 + x1)

= x1*x2 + x2*x3

But if we start grouping in some other way we may not

end up with the minimal equation.


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1. Boolean functions expressed as a sum

of products (SOP) or a product

of sums (POS) are said to be incanonical form.

1. Note the POS is not the complement

of the SOP expression.

Note:


Minimization Using Karnough Maps (1/5)

Provides more formal way to minimization

1. A Karnaugh Map is a grid-like representation of a truth table.

2. It is just another way of presenting a truth table, but the mode of

presentation gives more insight.

3. A Karnaugh map has zero and one entries at different positions. Each

position in a grid corresponds to a truth table entry.

A B C V

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

In the case of the Karnaugh Map the

advantage is that the Karnaugh Map is

designed to present the information in

a way that allows easy grouping of

terms that can be combined.

The

Truth Table


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It includes 3 steps

1. Form Karnough maps from the given truth table.

There is one Karnough map for each output variable.

2. Group all 1s into as few groups as possible with groups as large as

possible.

3. Each group makes one term of a minimal logic equation for the

given output variable.

Forming Karnough maps• The key idea in the forming the map is that

horizontally and vertically adjacent squares correspond to input

variables that differ in one variable only.Thus, a value for the first column (or row) can be arbitrary, but

labeling of adjacent columns (or rows) should be such that those

values differ in the value of only one variable.




Grouping ( This step is critical)When two adjacent squares contain 1s,

They indicate the possibility of an algebraic simplification and they may be

combined in one group of two.

Similarly, two adjacent pairs of 1s may be combined to form a group of four,

Then, two adjacent groups of four can be combined to form a group of eight,

and so on.

In general, the number of squares in any valid group must be equal to 2k.

Note that one 1 can be a member of more than one group and keep in mind

that you should end up with as few as possible groups which are as

large as possible.

The product term that corresponds to a given group is the product of

variables whose values are constant in the group.

If the value of input variable xi is 0 for the group, then xi is entered in the

product, while if xi has value 1 for the group, then xi is entered in the product.

Finding Product Terms


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+ x2*x3y = x1*x2


0111

0001

00 01 11 10

0

1

x1 x2

x3

1

0

0

0

1

0

1

1

111

011

101

001

110

010

100

000

yx3x2x1

Example 1: Given truth table, find minimal circuit



Example 2:

y = x1*x3 + x2

0000

0110

0011

0001

00 01 11 10

00

01

11

10

x1x2

x3x4

1100

1001

00101100

00 01 11 10

00

01

11

10

x1x2

x3x4

Example 3:

Example 4:y = x1*x2*x3 + x1*x2*x4 + x2*x3*x4

y = x1*x4 + x2*x3*x4 + x1*x2*x3*x4

x1 x2

1001

1011

00 01 11 10

0

1

x3


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Gates as Control Elements

An AND gate and a tristate buffer act as controlled switches

or valves.


Wired OR and Bus Connections

Wired OR allows tying together of several

controlled signals.


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Decoders/ Demultiplexers

A decoder allows the selection of one of 2 a options usingan a- bit address as input. A demultiplexer (demux) is a decoder that

only selects an output if its enable signal is asserted.


1 to 4 Line Demux

A 74155 can also be used as demultiplexer. It can function like a rotary

switch to demultiplex a single input to four different output lines.


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Encoders

A 2a -to- a encoder outputs an a-bit binary number

equal to the index of the single 1 among its 2a inputs.


Multiplexers

Multiplexer (mux), or selector, allows one of several inputs to be selected and

routed to output depending on the binary value of a set of selection or address

signals provided to it.


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Half-adder (HA): Truth table and block d iagram

Full-adder (FA): Truth table and block d iagram

x y c c s

----------------------

0 0 0 0 0

0 0 1 0 1

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 1

Inputs Outputs

cout cin

outin x y

s

FA

x y c s

----------------

0 0 0 00 1 0 1

1 0 0 1

1 1 1 0

Inputs Outputs

HA

x y

c

s

Design Example


Half-Adder Implementations

Three implementations of a half-adder.

c

s

(b) NOR-gate half-adder.

x

y

x

y

(c) NAND-gate half-adder with complemented carry.

x

y

c

s

s

x

y

x

y

(a) AND/XOR half-adder.

_

_

_c


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Full-Adder Implementations

HA

HA

xy

cin

cout

(a) Built of half-adders.

s

(b) Built as an AND-OR circuit.

(c) Suitable for CMOS realization.

cout

s

cin

xy

0

1

23

0

1

2

3

xy

cin

cout

s

0

1

Mux

Possible designs for a fu ll-adder in

terms of half-adders, logic gates, and

Multiplexers

Documents

Lecture 1 MENG 4779 Mechatronics and Digital Systems -- Spring 2016 (1)