digital circuits - staff.city.ac.uk

11. Digital Electronics 1

DIGITAL CIRCUITS

• In analogue circuits, signals are represented by levels

of voltage, current or charge which vary continuously

with time

• In digital circuits signals are not measured on a

continuous scale, but are classified as HIGH or LOW

levels to represent only two possible binary values 0

and 1

• Digital circuits are cheap to make, and complex

systems can be made to operate accurately with low

power consumption

• Logic 0 is represented by 0V or low voltage level

• Logic 1 is represented by a higher voltage level

• Most common devices based on TTL standard

• TTL Transistor-Transistor Logic – logic 0

represented by voltage levels 0v to 0,8V (LOW), logic 1

by between 2V and 5V (HIGH)

• So exact values not required for switching between

levels

• Voltages outside ranges are not HIGH or LOW

• Two logic level system leads to the use of binary numbering system


BINARY NUMBERS

• For computers, binary numbers are suitable because:

• They are simple to work with (very fast),

• Use if just two values of voltage, magnetism, or other

signal allows for easy hardware design.

• Most modern computers operate using binary logic

• With only two levels, we can represent exactly two

values

• Values used by convention are 1 and 0

• These two numbers correspond to the only two digits

used in the binary number system

• Computers operate on the principles of the binary

number system

• Binary numbers and arithmetic let you represent any

amount you want using just two digits: 0 and 1


BINARY NUMBERS

REPRESENTATION

• Each digit 1 in a binary number represents a power of

two, in which the power of the digit is the position of the

digit from the right minus 1 – it is a weighted value

• Each 0 represents a zero

• The weighted values for each position is determined as

follows:

01004

00113

00102

00011

BinaryDecimal

1248163264128

2021222324252627


CONVERTING FROM BINARY TO

DECIMAL

• Quite easy and straightforward to convert from a binary

number to a decimal number

• Multiply each digit by its weighted position, then add

each of the weighted values together

• 11001010 represents:

• (1×27)+(1×26)+(0×25)+(0×24)+(1×23)+(0×22)+(1×21)+(0×

20) = 202

• 10011001 = ?

• 11111111 = ?

• 10110110 = ?


CONVERTING FROM DECIMAL TO

BINARY

• Divide the decimal number by 2

• If remainder is 0, write down a 0

• If remainder is 1, write down a 1

• Process is continued by dividing quotient by 2 until the

quotient is zero

• Each remainder representing the binary equivalent is

written from right to left

• Example Convert the decimal number 100 into binary

100100113/2

1100100101/2

00100036/2

01000612/2

10011225/2

0002550/2

0050100/2

Binary

Number

RemainderQuotientDivision


BINARY ADDITION

• Adding two binary numbers together is like adding

decimals, except 1 + 1 = 10 (in binary!), so you have to

carry the 1 to the next column

• 0001 + 0100 = 0101

• 0001 + 0001 = 0010 (1+1 is 10 – carry 1 to next

column)

• 0011 + 0011 = 0110 (1+1 is 10, carry, then 1+1+1 =

11, so carry again)

• 0011 + 0101 = 1000 (carry in every column here)


BINARY NUMBER FORMATS

• Binary numbers are written as a sequence of bits

(binary digits)

• The bigger the number, the more bits needed to

represent it

• The smallest unit of data on a binary computer is a

single bit. Represents true or false (1 or 0)

• A nibble is a collection of 4 bits, and are numbered

from bit 0 (b0) up to three (b3), such that: b3 b2 b1 b0

• The byte is the most important data structure used in

computers

• A byte consists of 8 bits, numbered from bit zero (b0)

up to seven (b7): b7 b6 b5 b4 b3 b2 b1 b0

• b0 is the least significant bit or LSB

• b7 is the most significant bit or MSB

• Byte has 256 different states (28), and represents

decimal values from 0 to 255

• A word is 16 bits: b15 b14 b13 b12 b11 b10 b9 b8 b7

b6 b5 b4 b3 b2 b1 b0

• A word contains 2 bytes: b0 to b7 is the low order byte,

b8 to b15 forms high order bye

• With 16 bits, a word can have 216 = 65,536 values

• 1 kilobyte (1Kb) = 1024 bytes = 210

• 1 megabyte (1Mb) = 1048576 bytes = 220

• A 40 gigabyte (40Gb) hard disk can hold about 40

billion characters


BASIC LOGICAL FUNCTIONS AND

GATES

• There are three fundamental logical operations when

dealing with logical (digital) circuits

• AND, OR and NOT – each has own symbol and clearly

defined behaviour

• The term gate is used to describe the members of a set

of basic electronic components which, when combined

with each other, are able to perform complex logical

and arithmetical operations

• Boolean algebra is the mathematics associated with

binary numbers and logical operations

• Gates are the physical realisation of the simple Boolean

expressions

• A detailed understanding of the electronics within logic

gates is beyond the scope of this course

• Note that logic gates can be designed with a few

electronic components

• There are 5 types of fundamental gates used

• OR, AND, NOT, NAND, NOR, and XOR gates are used

extensively in digital electronics circuits


THE NOT GATE

• Also known as an inverter – always has one input and

one output

• Whatever logical state is applied to the input, the

opposite state will appear at the output

• If the warning light is red = you cannot enter

• If the warning light does not show red = you can enter

• Boolean expression: B = A’ = /A

• A truth table is a table that shows the value of the

output for all possible combinations of outputs

• A NOT gate has only 1 input, so there are 21 = 2 inputs

combinations

• Inputs are in the form of voltages

(1=HIGH=TRUE=+5V) and (0=LOW=FALSE=0V)

• Truth Table for NOT gate:

01

10

B (NOT A, A’)A


THE AND GATE

• AND gate produces a logic 1 output whenever both (or

all) its inputs are also at logic 1 (HIGH, TRUE)

• Verbally: If the switch is ON AND the power lead is

plugged in, the lamp lights up. Both cases have to be

true for the lamp to light up.

• Boolean expression: Z = X.Y

• AND gate can have any number of inputs, but for

practical uses, it is common to use two

• For n inputs, there are a possible 2n combinations, so 2

inputs has 4 combinations, 3 has 8 etc.

• Truth table for a 2-input AND gate:

111

001

010

000

Z = X.YYX


THE OR GATE

• OR gate produces logic 1 (HIGH, TRUE) output

whenever one or more of its inputs are at logic 1 (TRUE

states)

• Verbally: If there is enough light OR the light is on, I can

see

• OR function designated with plus (+) sign

• Boolean expression: Z = X+Y

• OR function and gate can have any number of inputs,

but we will limit our study to two inputs

• Truth table for OR gate:

111

101

110

000

Z = X+YYX


THE NAND GATE

• NAND gate consists of an AND function followed by a

NOT function

• It is an exact inversion of AND function

• Both inputs must have a logic 1 signals applied to them

in order for the output to be a logic 0

• With either (or all) input(s) at logic 0, the output will be a

logic 1

• Boolean expression: Z = (X.Y)’

• Truth Table for NAND gate:

011

101

110

100

Z = (X.Y)’YX


THE NOR GATE

• An OR gate allows the output to be TRUE (logic 1) if

any one or more of its inputs are true

• The NOR gate inverts this and forces the output to logic

1 when any input is HIGH (TRUE, logic 1)

• The output is 1 only when both inputs are 0

• Boolean expression: Z = (X+Y)’

• Truth table for NOR gate:

011

001

010

100

Z = (X+Y)’YX


THE XOR GATE

• The XOR or Exclusive OR function is a variation of the

OR function

• Verbally: If either X or Y are logic 1, but not both, then Z

is a logic 1

• The XOR gate produces a logic 1 output if its two inputs

are different

• If the inputs are the same, the output is a logic 0

• Boolean expression: Z = X ⊕ Y

• Truth table for XOR gate:

011

101

110

000

Z = X ⊕⊕⊕⊕ YYX


GATES AND INTEGRATED

CIRCUITS

• With the exception

of the NOT gate,

all other gates can

have any number

of inputs

• For practical,

commercial

reasons, gates are

manufactured with

2, 3 or 4 inputs

• A standard

Integrated Circuit

(IC) package

contains 14 or 16

pins

• A 14 pin package

can contain four 2-

input gates, three

3-input gates or

two 4-input gates,

and still have

room for two pins

for power supply

connections


BOOLEAN ALGEBRA

• Logic gates described previously can be used in

various combinations to perform tasks of any level of

complexity

• A primary requirement with digital circuits is to find

ways to make them as simple as possible

• This requires that complex logical expressions need to

be reduced to simpler expressions, yet still produce the

same results

• Simpler expression can then be implemented using a

simpler circuit – which saves on power consumption,

space, and cost

• One tool to reduce logical expressions is the

mathematics of logical expressions, introduced by

George Boole in 1854 – Boolean Algebra

• Rules of Boolean Algebra are simple and straight

forward, and can be applied to any logical expression

• Resulting reduced expression can then be readily

tested with a truth table for validation


RULES OF BOOLEAN ALGEBRA

• AND operations (.)

0.0 = 0 A.0 = 0

1.0 = 0 A.1 = A

0.1 = 0 A.A = A

1.1 = 1 A.A’ = 0

• OR operations (+)

0+0 = 0 A+0 = A

1+0 = 1 A+1 = 1

0+1 = 1 A+A = A

1+1 = 1 A+A’ = 1

• NOT operations (‘)

0’ = 1 A’’ = A

1’ = 0

• Associative Law

(A.B).C = A.(B.C) = A.B.C

(A+B)+C = A+(B+C) = A+B+C

• Distributive Law

A.(B+C) = (A.B) + (A.C)

A+(B.C) = (A+B).(A+C)

• Commutative Law

A.B = B.A

A+B = B+A

• Precedence

AB = A.B

A.B+C = (A.B) + C

A+B.C = A + (B.C)

• DeMorgan’s Theorem

NAND:

(A.B)’ = A’ + B’

NOR:

(A+B)’ = A’.B’


TRUTH TABLES

• Truth table shows relationship between inputs and

outputs of a logic circuit

• Each possible combination of input conditions is

considered

• A complete list of all the possible combinations of the

inputs with their corresponding output is summarised in

the truth table

• For the lamp to be

on (1), A must be on

(1) and either B or C

LAMP = (A.C)+(A.B)+(A.B.C)

• Example: Construct a truth table for a logic circuit that

will produce a logic 1 output when two or more of its

three inputs are at logic 1. Hence derive the circuit’s

Boolean equation.

A

B

C

1111

1011

1101

0001

0110

0010

0100

0000

LAMPCBA


READING AND CONSTRUCTION OF

LOGIC GATES

• A Boolean function is an expression formed with binary

variables made up of 0’s and 1’s

• It may be represented as an algebraic expression or in

a truth table

• Operators used: AND, OR, NOT, NAND, NOR, XOR

• For n inputs, there are 2n possible combinations

• Consider the following logic function, with two inputs

(hence 22=4 possible input combinations)

• Q = (A’.B)+(A.B’)

• Construct the corresponding truth table

• The required circuit is shown below


UNIVERSAL GATES

• Combinational logic circuits are more frequently

constructed with a NAND or NOR gates

• NAND and NOR gates are more common from a

hardware point of view

• They are readily available in IC form

• The NAND gate is said to be the universal gate

because any digital system can be implemented with it

• A.B = ((A.B)’)’ – AND gate using NAND gates alone

• A+B = (A’.B’)’ – OR gate using NAND gates alone

• Corresponding logic circuits:

A

B

(A.B)

’AB

A

B

A’

B’

(A’B’)’ = A+B


BOOLEAN SIMPLIFICATION

• Complex digital circuits that implement a Boolean

function is directly related to the complexity of the

algebraic expression from which the function is derived

• It is thus useful t simplify Boolean expressions, which

can be done in two ways

• Algebraically, or by using Karnaugh maps

• Example Using algebraic simplification, simplify the

following:

• Q = (A.B.C)+A.(B’+C’)

• Q = A(B.C+B’+C’)

• Q = A(B.C+(B.C)’)

• Q = A(1) = A

• Example Simplify Q = A’.C.D+A’.B.D+A.C.D+A.B.D

• Q = A’.D(C+B)+A.D(C+D)

• Q = D(C+B)(A’+A)

• Q = D(C+B)(1) = D(C+B)


LOGIC DIAGRAM

• Logic diagram of Q = A’.C.D+A’.B.D+A.C.D+A.B.D and

its simplified version Q = D(C+B)

A

Q

D C B


KARNAUGH MAPS

• Karnaugh Maps (K-map) provides a simple procedure

for minimising Boolean functions

• It is a diagram made of squares

• Each square represents one miniterm of the expression

• By recognising various patterns, a simplified algebraic

expression can be derived for the same function

• The construction of the K-map is such that it consists of

2n where n is the number of variables in the logic

expression

• 0’s and 1’s marked in each square designate the

possible values of the input variables

• Example Q = A.B and Q = A+B can be put into a K-

map as shown:

B

A 0

0

1

1

0 0

0 1

B

A 0

0

1

1

0 1

1 1

Q = A.B Q = A+B


KARNAUGH MAP EXAMPLES

• Q = A’BC + BCD’

• Expression has 4 variables, thus K-map will have 24 =

16 squares (as there are 16 possible input

combinations)

• Although there are 4 1’s in the map, denoting the

expression: A’B

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