presentation for bde

8/7/2019 Presentation for bde

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ECE-102 BASIC DIGITAL ELECTONICS


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SYLLABUS

Unit- I

Digital system and binary numbers

Characteristics of Digital Systems, Number

system conversions, Boolean algebra, BCDnumbers and their conversions, Binary codes,

Error detecting and correcting codes, Floating

point representation


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Gate-level minimization

Basics Gates, SOP simplification, POS

simplification, K-map method, NAND andNOR implementation, Quine Mc-Clusky

method (Tabular method).


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Unit-II

Logic Circuits

Basics of logic circuits, combinational

circuits, analysis and design Procedure,adder and subtractor, magnitude

comparator, decoders, encoders,

multiplexers.

Introduction to sequential circuits- latches,

flip flops


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TEXT/REFERENCE BOOKS

M. Morris Mano Digital Logic And

Computer Design, PHI publication

M. Morris Mano and M. D. Ciletti, DigitalDesign, 4th Edition, Pearson Education.

Floyd and Jain, Digital Fundamental,

Pearson Education

R.P.Jain , Modern Digital Electronics, 3rd

Edition, TMH Publication.


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Analog vs Digital

The major difference between analog and digital quantities, then, can be simply

stated as follows:

Analog = continuous

Digital = discrete (step by step)


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Advantages and Limitations of Digital Techniques

Advantages

Easier to design. Exact values of voltage or current are not important,

only the range (HIGH or LOW) in which they fall.

Information storage is easy.

Accuracy and precision are greater.

Operation can be programmed. Analog systems can also be

programmed, but the variety and complexity of the available

operations is severely limited.

Digital circuits are less affected by noise. As long as the noise is not

large enough to prevent us from distinguishing a HIGH from a LOW.

More digital circuitry can be fabricated on IC chips.


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Limitations

There is really only one major drawback when using digitaltechniques:

The real world is mainly analog.

Most physical quantities are analog in nature, and it is these

quantities that are often the inputs and outputs that are being

monitored, operated on, and controlled by a system.

To take advantage of digital techniques when dealing with

analog inputs and outputs, three steps must be followed:

1. Convert the real-world analog inputs to digital form. (ADC)

2. Process (operate on) the digital information.

3. Convert the digital outputs back to real-world analog form.

(DAC)


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Section 1.3

Representing Binary Quantities

In digital systems the information that is being

processed is usually presented in binary form.

Binary quantities can be represented by anydevice that has only two operating states or

possible conditions.

Eg. a switch has only open or closed. We

arbitrarily (as we define them) let an open switchrepresent binary 0 and a closed switch represent

binary 1. Thus we can represent any binary

number by using series of switches.


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Binary 1: Any voltage between 2V to 5V

Binary 0: Any voltage between 0V to 0.8V

Not used: Voltage between 0.8V to 2V,this may cause error in a digitalcircuit.

We can see another significant difference between digital and analog systems. In

digital systems, the exact value of a voltage is notimportant; eg, a voltage of 3.6V

means the same as a voltage of 4.3V. In analog systems, the exact value of a

voltage isimportant.

Typical Voltage Assignment


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Digital Number System

Many number systems are in use in digital technology. The

most common are the decimal, binary, octal, and hexadecimal

systems.

The decimal system is clearly the most familiar to us because

it is a tool that we use every day.

Examining some of its characteristics will help us to better

understand the other systems.


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103 102 101 100 10-1 10-2 10-3

=1000 =100 =10 =1 . =0.1 =0.01 =0.001

Most

SignificantDigit

Decimalpoint

Least

Significant Digit

Decimal System

Decimal System: The decimal system is composed of 10

numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5,6, 7, 8, 9; using these symbols as digits of a number, we can

express any quantity. The decimal system, also called the

base-10 system because it has 10 digits.


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Binary System

In the binary system, there are only two symbols orpossible digit values, 0 and 1. This base-2 system can be

used to represent any quantity that can be represented in

decimal or other number system.

23 22 21 20 2-1 2-2 2-3

=8 =4 =2 =1 . =1/2 =1/4 =1/8

Most

Significant

Bit

Binary

point

Least

Significant

Bit


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The Binary counting sequence is shown in the table:

Binary Counting


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and

1 0 1 1 0 1 0 1 2(binary)

2

7

+0+2

5

+2

4

+0+2

2

+0+2

0

= 128+0+32+16+0+4+0+1= 18110 (decimal)

1 1 0 1 1 2 (binary)

24+23+0+21+20 = 16+8+0+2+1

= 2710 (decimal)

Binary-To-Decimal Conversion

Any binary number can be converted to its decimal equivalent simply by

summing together the weights of the various positions in the binary number

which contain a 1

You should noticed the method is find the weights (i.e., powers of 2) for

each bit position that contains a 1, and then to add them up.


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Decimal-To-Binary Conversion

There are 2 methods:

(A) Revese of Binary-To-Digital Method

45 10 = 32 + 0 + 8 + 4 +0 + 1= 25+0+23+22+0+20

= 1 0 1 1 0 12


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(B) Repeat Division

This method uses repeated division by 2. Eg. convert 2510 to binary

25/ 2 = 12+ remainder of 11 (Least Significant

Bit)

12/ 2 = 6 + remaind

er of 0 06 / 2 = 3 + remainder of 0 0

3 / 2 = 1 + remainder of 1 1

1 / 2 = 0 + remainder of 11 (Most Significant

Bit)

Result 2510 = 1 1 0 0 12


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The Flow chart for

repeated-division

method is as follow:


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Octal Number System

The octal number system has a base of eight, meaningthat it has eight possible digits: 0,1,2,3,4,5,6,7.

83 82 81 80 8-1 8-2 8-3

=512 =64 =8 =1 . =1/8 =1/64 =1/512Most

Significant

Digit

Octal

point

Least

Significant

Digit

Octal to Decimal Conversion

eg. 24.68 = 2 x (81) + 4 x (80) + 6 x (8-1) = 20.7510


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Binary-To-Octal / Octal-To-Binary Conversion

Octal

Digit

0 1 2 3 4 5 6 7

BinaryEquiva

lent

000 001 010 011 100 101 110 111

Each Octal digit is represented by three bits of binary digit.

eg. 100 111 0102 = (100) (111) (010)2 = 4 7 28


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Repeat DivisionThis method uses repeated division by 8. Eg. convert 17710 to octal and binary:

177/8 = 22+ remainder of 1 1 (Least Significant Bit)

22/ 8 = 2 + remainder of 6 6

2 / 8 = 0 + remainder of 2 2 (Most Significant Bit)

Result 17710 = 2618

Convert to Binary = 0101100012


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Hexadecimal Number System

The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It

uses the digits 0 through 9 plus the letters A, B, C, D, E, and F as the 16 digitsymbols.

163162 161 160 16-1 16-2 16-3

=4096 =256 =16 =1 . =1/16 =1/256=1/409

6

Most

Signific

ant

Digit

Hexad

ec.point

Least

Signific

ant

Digit


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Hexadecimal to Decimal Conversion

eg. 2AF16 = 2 x (162

) + 10 x (161

) + 15 x (160

) = 68710Repeat Division: Convert decimal to hexadecimalThis method uses repeated division by 16. Eg. convert 37810 to hexadecimal and binary:

378/16= 23+ remainder of 10

A (Least Significant

Bit)

23/ 16 = 1 + remainder of 7 7

1 / 16 = 0 + remainder of 1 1 (Most Significant Bit)

Result 37810 = 17A8

Convert to Binary= 0001 0111 10102= 00000001 0111 1010

(16 bits)


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Binary-To-Hexadecimal /

Hexadecimal-To-Binary Conversion

Hexadecimal

Digit

0 1 2 3 4 5 6 7

Binary

Equivalent0000 0001 0010 0011 0100 0101 0110 0111

Hexadecimal

Digit8 9 A B C D E F

BinaryEquivalent

1000 1001 1010 1011 1100 1101 1110 1111

Each Hexadecimal digit is represented by fourbits of binary digit.

eg. 1011 0010 11112 = (1011) (0010) (1111)2 = B 2 F16


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Octal-To-Hexadecimal /

Hexadecimal-To-Octal Conversion1) Convert Octal (Hexadecimal) to Binary first.

2a) Regroup the binary number in 3 bits a group starts from the LSB if Octal is required.

Go back to Section 2.3 if you are not sure how to group in Octal.

2b) Regroup the binary number in 4 bits a group from the LSB if Hexadecimal is required.

eg. Convert 5A816 to Octal .

5A816= 0101 1010 1000 (Binary)

= 2 6 5 0 (Octal)


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Boolean Variables & Truth Tables

Boolean algebra differs in a major way from ordinary algebra in that 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.

Some common representation of 0 and 1 is shown in the following diagram.

Logic 0Logic 1

False True

Off On

Low High

No Yes

Open Switch Close Switch


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In boolean algebra, there are three basic logic operations:

OR, AND and NOT.

These logic gates are digital circuits constructed from diodes, transistors,

and resistors connected in such a way that the circuit output is the result of

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


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Truth Table

A truth table is a means for describing how a logic circuit's outputdepends on the logic levels present at the circuit's inputs.

In the following two-inputs logic circuit, the table lists all possible

combinations of logic levels present at inputs A and B along with

the corresponding output level X.

When either input A OR B is 1, the output X is 1. Therefore the "?"

in the box is an OR gate. Go to next section to explore more on

the OR gate.


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OR Operation

The expression X = A + B reads as "X equals A OR B".

The + sign stands for the OR operation, not for ordinary addition.

The OR operation produces a result of 1 when any of the input variable is 1.

The OR operation produces a result of 0 only when all the input variables are 0.


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An example of three input OR gate and its truth table is as follows:

With the OR operation, 1 + 1 = 1, 1+ 1 + 1 = 1 and so on.


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AND Operation

The expression X = A * B reads as "X equals A AND B".

The multiplication sign stands for the AND operation, same for ordinarymultiplication of 1s and 0s.

The AND operation produces a result of 1 occurs only for the single case when

all of the input variables are 1.

The output is 0 for any case where one or more inputs are 0


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NOT Operation

The NOT operation is unlike the OR and AND operations in that it can be

performed on a single input variable. For example, if the variable A is subjected to

the NOT operation, the result x can be expressed asx = A'

where the prime (') represents the NOT operation. This expression is read as:

x equals NOT A

x equals the inverse of A

x equals the complement of A

Each of these is in common usage and all indicate that the logic value of x = A' isopposite to the logic value of A.

The truth table of the NOT operation is as follows:

The NOT operation is also referred to as inversion or complementation, and

these terms are used interchangeably.

1' = 0 because NOT 1 is 00' = 1 because NOT 0 is 1


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NOR Operation

NOR and NAND gates are used extensively in digital circuitry. These gates

combine the basic operations AND, OR and NOT, which make it relatively easyto describe then using Boolean Algebra.

NOR is the same as the OR gate symbol exceptthat it has a small circle on the

output. This small circle represents the inversion operation. Therefore the output

expression of the two input NOR gate is:

X = ( A + B )'


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NAND Operation

NAND is the same as the AND gate symbol exceptthat it has a small circle onthe output. This small circle represents the inversion operation. Therefore the

output expression of the two input NAND gate is:

X = ( AB )'


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Section 4.1


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Describing Logic Circuits Algebraically

Any logic circuit, no matter how complex, may be completely describedusing the Boolean operations, because the OR gate, AND gate, and NOT

circuit are the basic building blocks of digital systems.

This is an example of the circuit using Boolean expression:

If an expression contains both AND and OR operations, the AND operations

are performed first (X=AB+C : AB is performed first), unless there are

parentheses in the expression, in which case the operation inside the

parentheses is to be performed first (X=(A+B)+C : A+B is performed first).


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Circuits containing Inverters

Whenever an INVERTER is present in a logic-circuit diagram, its output

expression is simply equal to the input expression with a prime (') overit.


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Section 4.2

Evaluating Logic Circuit Outputs

Once the Boolean expression for a circuit output has been obtained, the

output logic level can be determined for any set of input levels.

This are two examples of the evaluating logic circuit output:

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

X = A'BC (A+D)'

= 0'*1*1* (0+1)'

= 1 *1*1* (1)'

= 1 *1*1* 0

= 0


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

X

= [D+ ((A+B)C)'] * E

= [1 + ((0+0)1 )'] * 1

= [1 + (0*1)'] * 1

= [1+ 0'] *1

= [1+ 1 ] * 1

= 1

In general, the following rules must always be followed when evaluating a Boolean

expression:

1. First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0.

2. Then perform all operations within parentheses.

3. Perform an AND operation before an OR operation unless parentheses indicate

otherwise.

4. If an expression has a bar over it, perform the operations of the expression first

and then invert the result.


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Determining Output Level from a Diagram

The output logic level for given input levels can also be determined

directly from the circuit diagram without using the Boolean expression.


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Section 4.3

Implementing Circuits From Boolean Expression

If the operation of a circuit is defined by a Boolean expression, a logic-

circuit diagram can he implemented directly from that expression.

Suppose that we wanted to construct a circuit whose output is y = AC+BC'

+ A'BC. This Boolean expression contains three terms (AC, BC', A'BC),which are ORed together. This tells us that a three-input OR gate is

required with inputs that are equal to AC, BC', and A'BC, respectively.

Each OR-gate input is an AND product term, which means that an AND

gate with appropriate inputs can be used to generate each of these terms.

Note the use of INVERTERs to produce the A' and C' terms required in

the expression.


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Section 4.4

Boolean Theorems

Investigating the various Boolean theorems (rules) can help us to simplify logic

expressions and logic circuits.


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Multivariable TheoremsThe theorems presented below involve more than one variable:

(9)x + y = y + x (commutative law)

(10) x * y = y * x (commutative law)

(11) x+ (y+z) = (x+y) +z = x+y+z (associative law)

(12) x (yz) = (xy) z = xyz (associative law)

(13a) x (y+z) = xy + xz

(13b) (w+x)(y+z) = wy + xy + wz + xz

(14) x + xy = x [proof see below]

(15) x + x'y = x + y


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Proof of (14)

x + xy= x (1+y)

= x * 1 [using theorem (6)]

= x [using theorem (2)]


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Section 4.5

DeMorgan's Theorem

DeMorgan's theorems are extremely useful in simplifying

expressions in which a product or sum of variables is inverted.

The two theorems are:

(16) (x+y)' = x' * y'(17) (x*y)' = x' + y'

Theorem (16) says that when the OR sum of two variables is

inverted, this is the same as inverting each variable individually

and then ANDing these inverted variables.

Theorem (17) says that when the AND product of two variables

is inverted, this is the same as inverting each variableindividually and then ORing them.


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Example

X= [(A'+C) * (B+D')]'

= (A'+C)' + (B+D')' [by theorem (17)]

= (A''*C') + (B'+D'') [by theorem (16)]

= AC' + B'D


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Three Variables DeMorgan's Theorem

(18) (x+y+z)' = x' * y' * z'

(19) (xyz)' = x' + y' + z'


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Implications of DeMorgan's Theorem

For (16): (x+y)' = x' * y'


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For (17): (x*y)' = x' + y'


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Section 4.6

Universality of NAND & NOR Gates

It is possible to implement anylogic expression using only NAND gates and

no other type of gate. This is because NAND gates, in the proper

combination, can be used to perform each of the Boolean operations OR,

AND, and INVERT.


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In a similar manner, it can be shown that NOR gates can

be arranged to implement any of the Boolean operations.


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Section 4.7

Alternate Logic Gate Representations

The left side of the illustration shows the standard symbol for each logic

gate, and the right side shows the alternate symbol. The alternate

symbol for each gate is obtained from the standard symbol by doing thefollowing:

1. 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.

2. Change the operation symbol from AND to OR, or from OR to AND.

(In the special case of the INVERTER, the operation symbol is notchanged.)


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Several points should be stressed regarding the logic symbol

equivalences:

1. The equivalences are valid for gates with any number of inputs.

2. None of the standard symbols have bubbles on their inputs, and all

the alternate symbols do.

3. The standard and alternate symbols for each gate represent thesame physical circuit: there is no difference in the circuits represented

by the two symbols.

4. NAND and NOR gates are inverting gates, and so both the standard

and alternate symbols for each will have a bubble on either the input or

the output. AND and OR gates are noninverting gates, and so the

alternate symbols for each will have bubbles on both inputs and

output.


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Section 4.8

Logic Symbol Interpretation

Concept of Active Logic Levels:

When an input or output line on a logic circuit symbol has no bubble on it,

that line is said to be active-HIGH. When an input or output line does have

a bubble on it, that line is said to be active-LOW. The presence or

absence of a bubble, then, determines the active-HIGH/active-LOW status

of a circuit's inputs and output, and is used to interpret the circuitoperation.

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