Number System and Logic Families
1. Introduction to digital Electronics & Boolean algebra:
Logic Gate Truth Tables
As well as a standard Boolean Expression, the input and output information of
any Logic Gate or circuit can be plotted into a standard table to give a visual
representation of the switching function of the system. The table used to represent the
boolean expression of a logic gate function is commonly called a Truth Table. A logic
gate truth table shows each possible input combination to the gate or circuit with the
resultant output depending upon the combination of these input(s).
For example, consider a single 2-input logic circuit with input variables labelled
as A and B. There are “four” possible input combinations or 22 of “OFF” and “ON”
for the two inputs. However, when dealing with Boolean expressions and especially
logic gate truth tables, we do not general use “ON” or “OFF” but instead give them bit
values which represent a logic level “1” or a logic level “0” respectively.
Then the four possible combinations of A and B for a 2-input logic gate is given as:

Input Combination 1. – “OFF” – “OFF” or ( 0, 0 )

Input Combination 2. – “OFF” – “ON” or ( 0, 1 )

Input Combination 3. – “ON” – “OFF” or ( 1, 0 )

Input Combination 4. – “ON” – “ON” or ( 1, 1 )
Therefore, a 3-input logic circuit would have 8 possible input combinations or 2 3 and
a 4-input logic circuit would have 16 or 24, and so on as the number of inputs
increases. Then a logic circuit with“n” number of inputs would have 2n possible input
combinations of both “OFF” and “ON”.
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Number System and Logic Families
So in order to keep things simple to understand, in this tutorial we will only deal with
standard 2-input type logic gates, but the principals are still the same for gates with
more than two inputs.
Then the Truth tables for a 2-input AND Gate, a 2-input OR Gate and a single
input NOT Gate are given as:
2-input AND Gate
For a 2-input AND gate, the output Q is true if BOTH input A “AND” input B are
both true, giving the Boolean Expression of: ( Q = A and B ).
Symbol
Boolean Expression Q = A.B
Truth Table
A
B
Q
0
0
0
0
1
0
1
0
0
1
1
1
Read as A AND B gives Q
Note that the Boolean Expression for a two input AND gate can be written as: A.B or
just simply ABwithout the decimal point.
2-input OR (Inclusive OR) Gate
For a 2-input OR gate, the output Q is true if EITHER input A “OR” input B is true,
giving the Boolean Expression of: ( Q = A or B ).
2
Number System and Logic Families
Symbol
Boolean Expression Q = A+B
Truth Table
A
B
Q
0
0
0
0
1
1
1
0
1
1
1
1
Read as A OR B gives Q
NOT Gate
For a single input NOT gate, the output Q is ONLY true when the input is “NOT”
true, the output is the inverse or complement of the input giving the Boolean
Expression of: ( Q = NOT A ).
Symbol
Truth Table
A
Q
0
1
1
0
Boolean Expression Q = NOT A or A Read as inversion of A gives Q
The NAND and the NOR Gates are a combination of the AND and OR Gates with
that of a NOT Gate or inverter.
2-input NAND (Not AND) Gate
3
Number System and Logic Families
For a 2-input NAND gate, the output Q is true if BOTH input A and input B are NOT
true, giving the Boolean Expression of: ( Q = not(A and B) ).
Symbol
Boolean Expression Q = A .B
Truth Table
A
B
Q
0
0
1
0
1
1
1
0
1
1
1
0
Read as A AND B gives NOT-Q
2-input NOR (Not OR) Gate
For a 2-input NOR gate, the output Q is true if BOTH input A and input B are NOT
true, giving the Boolean Expression of: ( Q = not(A or B) ).
Symbol
Boolean Expression Q = A+B
Truth Table
A
B
Q
0
0
1
0
1
0
1
0
0
1
1
0
Read as A OR B gives NOT-Q
4
Number System and Logic Families
As well as the standard logic gates there are also two special types of logic gate
function called anExclusive-OR Gate and an Exclusive-NOR Gate. The actions of
both of these types of gates can be made using the above standard gates however, as
they are widely used functions, they are now available in standard IC form and have
been included here as reference.
2-input EX-OR (Exclusive OR) Gate
For a 2-input Ex-OR gate, the output Q is true if EITHER input A or if input B is true,
but NOT both giving the Boolean Expression of: ( Q = (A and NOT B) or (NOT A
and B) ).
Symbol
Boolean Expression Q = A
Truth Table
A
B
Q
0
0
0
0
1
1
1
0
1
1
1
0
B
2-input EX-NOR (Exclusive NOR) Gate
For a 2-input Ex-NOR gate, the output Q is true if BOTH input A and input B are the
same, either true or false, giving the Boolean Expression of: ( Q = (A and B) or (NOT
A and NOT B) ).
Symbol
Truth Table
A
B
5
Q
Number System and Logic Families
Boolean Expression Q = A
0
0
1
0
1
0
1
0
0
1
1
1
B
Summary of 2-input Logic Gates
The following Truth Table compares the logical functions of the 2-input logic gates
above.
Inputs
Truth Table Outputs For Each Gate
A
B
AND
NAND OR
NOR
EX-OR EX-NOR
0
0
0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
1
1
0
1
0
0
1
The following table gives a list of the common logic functions and their equivalent
Boolean notation.
Logic Function
Boolean Notation
AND
A.B
OR
A+B
NOT
A
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Number System and Logic Families
NAND
A .B
NOR
A+B
EX-OR
(A.B) + (A.B) or A
EX-NOR
(A.B) + or A
B
B
2-input logic gate truth tables are given here as examples of the operation of each
logic function, but there are many more logic gates with 3, 4 even 8 individual inputs.
The multiple input gates are no different to the simple 2-input gates above, So a 4input AND gate would still require ALL 4-inputs to be present to produce the required
output at Q and its larger truth table would reflect that.
The Laws of Boolean
As well as the logic symbols “0” and “1” being used to represent a digital input or
output, we can also use them as constants for a permanently “Open” or “Closed”
circuit or contact respectively. A set of rules or Laws of Boolean Algebra expressions
have been invented to help reduce the number of logic gates needed to perform a
particular logic operation resulting in a list of functions or theorems known commonly
as the Laws of Boolean Algebra.
Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We
can use these “Laws of Boolean” to both reduce and simplify a complex Boolean
expression in an attempt to reduce the number of logic gates required.Boolean
Algebra is therefore a system of mathematics based on logic that has its own set of
rules or laws which are used to define and reduce Boolean expressions.
The variables used in Boolean Algebra only have one of two possible values, a
logic“0” and a logic “1” but an expression can have an infinite number of variables all
labelled individually to represent inputs to the expression, For example,
7
Number System and Logic Families
variablesA, B, C etc, giving us a logical expression of A + B = C, but each variable
can ONLY be a 0 or a 1.
Examples of these individual laws of Boolean, rules and theorems for Boolean
Algebra are given in the following table.
Truth Tables for the Laws of Boolean
Boolean
Expression
A+1=1
A+0=A
A.1=A
A.0=0
A+A=A
A.A=A
Equivalent
Description
A
in
Switching Circuit Law or Rule
parallel
with
closed = "CLOSED"
A
in
parallel
with
series
with
series
with
open = "A"
A
in
closed = "A"
A
in
open = "OPEN"
A
in
parallel
with
series
with
A = "A"
A
in
Boolean
A = "A"
8
Annulment
Identity
Identity
Annulment
Idempotent
Idempotent
Algebra
Number System and Logic Families
NOT A = A
A+A=1
A.A=0
A+B = B+A
A.B = B.A
A+B = A.B
A.B = A+B
NOT
NOT
A
(double negative) = "A"
A
in
parallel
with
NOT A = "CLOSED"
A
in
series
with
NOT A = "OPEN"
A
in
parallel
with
B=
with
B=
B in parallel with A
A
in
series
B in series with A
Double Negation
Complement
Complement
Commutative
Commutative
invert and replace OR with
de
AND
Theorem
invert and replace AND with
de
OR
Theorem
Morgan’s
Morgan’s
The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a
change in position for addition and multiplication, the Associative Law allowing the
removal of brackets for addition and multiplication, as well as the Distributive
Law allowing the factoring of an expression, are the same as in ordinary algebra.
Each of the Boolean Laws above are given with just a single or two variables, but the
number of variables defined by a single law is not limited to this as there can be an
infinite number of variables as inputs too the expression. These Boolean laws detailed
9
Number System and Logic Families
above can be used to prove any given Boolean expression as well as for simplifying
complicated digital circuits.
A
brief
description
of
the
various Laws
of
Boolean are
given
below
with Arepresenting a variable input.
Description of the Laws of Boolean Algebra

Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will
equal 1.

o
A . 0 = 0 A variable AND’ed with 0 is always equal to 0.
o
A + 1 = 1 A variable OR’ed with 1 is always equal to 1.
Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal
that term.

o
A + 0 = A A variable OR’ed with 0 is always equal to the variable.
o
A . 1 = A A variable AND’ed with 1 is always equal to the variable.
Idempotent Law – An input that is AND´ed or OR´ed with itself is equal to that
input.
o
A + A = A A variable OR’ed with itself is always equal to the variable.
o
A . A = A A variable AND’ed with itself is always equal to the variable.
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Number System and Logic Families

Complement Law – A term AND´ed with its complement equals “0” and a
term OR´ed with its complement equals “1”.

o
A . A = 0 A variable AND’ed with its complement is always equal to 0.
o
A + A = 1 A variable OR’ed with its complement is always equal to 1.
Commutative Law – The order of application of two separate terms is not
important.
o
A.B=B.A
The order in which two variables are AND’ed makes no
difference.
o
A+B=B+A
The order in which two variables are OR’ed makes no
difference.

Double Negation Law – A term that is inverted twice is equal to the original term.
o
A = A
A double complement of a variable is always equal to the
variable.

de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems,

(1) Two separate terms NOR´ed together is the same as the two terms inverted
(Complement) and AND´ed for example, A+B = A. B.
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Number System and Logic Families

(2) Two separate terms NAND´ed together is the same as the two terms inverted
(Complement) and OR´ed for example, A.B = A +B.
Other algebraic Laws of Boolean not detailed above include:

Distributive Law – This law permits the multiplying or factoring out of an
expression.

o
A(B + C) = A.B + A.C (OR Distributive Law)
o
A + (B.C) = (A + B).(A + B)
(AND Distributive Law)
Absorptive Law – This law enables a reduction in a complicated expression to a
simpler one by absorbing like terms.

o
A + (A.B) = A (OR Absorption Law)
o
A(A + B) = A (AND Absorption Law)
Associative Law – This law allows the removal of brackets from an expression
and regrouping of the variables.
o
A + (B + C) = (A + B) + C = A + B + C (OR Associate Law)
o
A(B.C) = (A.B)C = A . B . C (AND Associate Law)
12
Number System and Logic Families
Boolean Algebra Functions
Using the information above, simple 2-input AND, OR and NOT Gates can be
represented by 16 possible functions as shown in the following table.
Function Description
Expression
1.
NULL
0
2.
IDENTITY
1
3.
Input A
A
4.
Input B
B
5.
NOT A
A
6.
NOT B
B
7.
A AND B (AND)
A.B
8.
A AND NOT B
A.B
9.
NOT A AND B
A.B
10.
NOT A AND NOT B (NAND) A . B
11.
A OR B (OR)
A+B
12.
A OR NOT B
A+B
13
Number System and Logic Families
13.
NOT A OR B
A+B
14.
NOT OR (NOR)
A+B
15.
Exclusive-OR
A.B + A.B
16.
Exclusive-NOR
A.B + A.B
Laws of Boolean Algebra Example No1
Using the above laws, simplify the following expression: (A + B)(A + C)
Q = (A + B).(A + C)
A.A + A.C + A.B + B.C– Distributive law
A + A.C + A.B + B.C – Idempotent AND law (A.A = A)
A(1 + C) + A.B + B.C – Distributive law
A.1 + A.B + B.C
– Identity OR law (1 + C = 1)
A(1 + B) + B.C
– Distributive law
A.1 + B.C
– Identity OR law (1 + B = 1)
Q = A + (B.C)
– Identity AND law (A.1 = A)
Then the expression: (A + B)(A + C) can be simplified to A + (B.C) as in the
Distributive law.
Number System :
14
Number System and Logic Families
A set of values used to represent different quantities is known as Number
System". For example, a number system can be used to represent the number of
students in a class or number of viewers watching a certain TV program etc. The
digital computer represents all kinds of data and information in binary numbers. It
includes audio, graphics, video, text and numbers. The total number of digits used in a
number system is called its base or radix. The base is written after the number as
subscript such as 51210.
Some important number systems are as follows.

Decimal number system

Binary number system

Octal number system

Hexadecimal number system
The decimal number system is used in general. However, the computers use binary
number system. The octal and hexadecimal number systems are used in the computer.
Decimal number System
See Also: Convert decimal numbers to binary numbers
The Decimal Number System consists of ten digits from 0 to 9. These digits can be
used to represent any numeric value. The base of decimal number system is 10. It is
the most widely used number system. The value represented by individual digit
depends on weight and position of the digit.
Each number in this system consists of digits which are located at different positions.
The position of first digit towards left side of the decimal point is 0. The position of
second digit towards left side of the decimal point is 1. Similarly, the position of first
digit towards right side of decimal point is -1. The position of second digit towards
right side of decimal point is -2 and so on.
15
Number System and Logic Families
The value of the number is determined by multiplying the digits with the weight of
their position and adding the results. This method is known as expansion method. The
rightmost digit of number has the lowest weight. This digit is called Least Significant
Digit (LSD). The leftmost digit of a number has the highest weight. This digit is
called Most Significant Digit (MSD). The digit 7 in the number 724 is most
significant digit and 4 is the least significant digit.
See Also: Number Bases
Example:
The weights and positions of each digit of the number 453 are as follows:
Position
2
1
0
Weights
102
101
100
Face value
4
5
3
The above table indicates that:
The value of digit 4
=
4x102 =
400
The value of digit 4
=
5x10 =
50
The value of digit 3
=
3x10 =
3
The actual number can be found by adding the values obtained by the digits as
follows:
400 + 50 + 3 =45310
Example:
The weights and positions of each digit of the number 139.78 are as follows.
Position
2
1
0
Weights
102
101
100
16
.
-1
-2
10-1
10-2
Number System and Logic Families
Face Value
1
3
9
7
8
The above table indicates that:
The value of digit 1
=
1x102 =
100
The value of digit 3
=
3x101 =
30
The value of digit 9
=
9x100 =
9
The value of digit 7
=
7x10-1 =
0.7
The value of digit 8
=
8x10-2 =
0.08
The actual number can be found by adding the values obtained by the digits as
follows:
100 + 30 + 9 + 0.7 + 0.8
=
139.78
Binary Number System
Digital computer represents all kinds of data and information in the binary system.
Binary Number System consists of two digits 0 and 1. Its base is 2. Each digit or bit in
binary number system can be 0 or 1. A combination of binary numbers may be used to
represent different quantities like 1001. The positional value of each digit in binary
number is twice the place value or face value of the digit of its right side. The weight
of each position is a power of 2.
The place value of the digits according to position and weight is as follows:
Position
3
2
1
0
Weights
23
22
21
20
Example: Convert 101112 decimal number
Position
2
1
0
17
-1
-2
Number System and Logic Families
Weights
102
101
100
10-1
10-2
Face Value
1
3
9
7
8
101112
=
1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20
=
1 x 16 + 0 + 1 x 4 + 1 x 2 + 1 x 1
=
16 + 0 + 4 2 + 1
=
2310
Example: Convert 101.1012
Position
2
1
0
Face Value
1
0
1
Weight
24
21
20
101.1012
=
.
-1
-2
-3
1
0
1
2-1
2-2
2-3
1 x 22 + 0x21 + 1 x 20 + 1x 2-1 + 0 x 2-2 + 1 x 2-3
=
1 x 4 + 0 + 1 x 1 + ½ + 0 + 1/8
=
4 + 0 + 1 + 0.5 + 0.125
=
5.62510
Octal Number System
Octal Number System consists of eight digits from 0 to 7. The base of octal system is
8. Each digit position in this system represents a power of 8. Any digit in this system
is always less than 8. Octal number system is used as a shorthand representation of
long binary numbers. The number 6418 is not valid in this number system as 8 is not a
valid digit.
The place value of each digit according to position and weight is as follows.
Position
4
3
2
18
1
0
Number System and Logic Families
Weight
84
83
82
81
80
Example: convert 458 to decimal number
458
=
4 x 81 + 5 x 80
=
4x8+5x1
=
32 + 5
=
3710
Hexadecimal number system
The Hexadecimal Number System consists of 16 digits from 0 to 9 and A to F. The
alphabets A to F represent decimal numbers from 10 to 15. The base of this number
system is 16. Each digit position in hexadecimal system represents a power of 16. The
number 76416 is valid hexadecimal number. It is different from 76410 which is seven
hundred and sixty four. This number system provides shortcut method to represent
long binary numbers.
The place value of each digit according to position and weight is as follows:
Position
4
3
2
1
0
Weights
164
163
162
161
160
Example: Convert 3A16 to decimal number
3A16
=
3 x 161 + A x 160
=
3 x 16 + 10 x 1
=
48 + 10
=
5810
19
Number System and Logic Families
Conversion Between Different Number Systems
Positional number systems
Our decimal number system is known as a positional number system, because the
value of the number depends on the position of the digits. For example, the
number 123 has a very different value than the number 321, although the same digits
are used in both numbers.
(Although we are accustomed to our decimal number system, which is positional,
other ancient number systems, such as the Egyptian number system were not
positional, but rather used many additional symbols to represent larger values.)
In a positional number system, the value of each digit is determined by which place it
appears in the full number. The lowest place value is the rightmost position, and each
successive position to the left has a higher place value.
In our decimal number system, the rightmost position represents the "ones" column,
the next position represents the "tens" column, the next position represents
"hundreds",
etc.
Therefore,
the
number 123 represents 1 hundred
and 2 tens
and 3 ones, whereas the number 321 represents 3 hundreds and 2 tens and 1 one.
The values of each position correspond to powers of the base of the number system.
So for our decimal number system, which uses base 10, the place values correspond to
powers of 10:
... 1000 100
... 10^3
10
1
10^2 10^1 10^0
20
Number System and Logic Families
Converting from other number bases to decimal
Other number systems use different bases. The binary number system uses base 2, so
the place values of the digits of a binary number correspond to powers of 2. For
example, the value of the binary number 10011 is determined by computing the place
value of each of the digits of the number:
1
0
0
1
1
the binary number
2^4 2^3 2^2 2^1 2^0 place values
So the binary number 10011 represents the value
(1 * 2^4) +
(0 * 2^3) +
(0 * 2^2) +
(1 * 2^1) +
(1 * 2^0)
=
16
0
+
0
2
1
=
19
The
same
to
any
number
2
number in base 5
+
principle
applies
+
base.
number 2132 base 5 corresponds to
2
1
3
5^3 5^2 5^1 5^0 place values
21
+
For
example,
the
Number System and Logic Families
So the value of the number is
(2 * 5^3)
+ (1 * 5^2) + (3 * 5^1) + (2 * 5^0)
= (2 * 125) + (1 * 25)
+ (3 * 5)
+ (2 * 1)
= 250
+ 15
+ 2
+ 25
= 292
Converting from decimal to other number bases
In order to convert a decimal number into its representation in a different number
base, we have to be able to express the number in terms of powers of the other base.
For example, if we wish to convert the decimal number 100 to base 4, we must figure
out how to express 100 as the sum of powers of 4.
100
=
(1 * 64)
+
(2 * 16)
+
(1 * 4)
+
(0 * 1)
=
(1 * 4^3)
+
(2 * 4^2)
+
(1 * 4^1)
+
(0 * 4^0)
Then we use the coefficients of the powers of 4 to form the number as represented in
base 4:
100
=
1210
base 4
22
Number System and Logic Families
One way to do this is to repeatedly divide the decimal number by the base in which it
is to be converted, until the quotient becomes zero. As the number is divided, the
remainders - in reverse order - form the digits of the number in the other base.
Example: Convert the decimal number 82 to base 6:
82/6 =
13 remainder 4
13/6 =
2
remainder 1
2/6
0
remainder 2
=
The answer is formed by taking the remainders in reverse order: 2 1 4 base
There are many methods or techniques which can be used to convert numbers from one
base to another. We'll demonstrate here the following

Decimal to Other Base System

Other Base System to Decimal

Other Base System to Non-Decimal

Shortcut method - Binary to Octal

Shortcut method - Octal to Binary

Shortcut method - Binary to Hexadecimal

Shortcut method - Hexadecimal to Binary
Decimal to Other Base System
23
Number System and Logic Families
Steps

Step 1 - Divide the decimal number to be converted by the value of the new base.

Step 2 - Get the remainder from Step 1 as the rightmost digit (least significant digit)
of new base number.

Step 3 - Divide the quotient of the previous divide by the new base.

Step 4 - Record the remainder from Step 3 as the next digit (to the left) of the new
base number.
Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes
zero in Step 3.
The last remainder thus obtained will be the most significant digit (MSD) of the new
base number.
EXAMPLE
Decimal Number: 2910
Calculating Binary Equivalent:
Step
Operation
Result
Remainder
Step 1
29 / 2
14
1
Step 2
14 / 2
7
0
Step 3
7/2
3
1
Step 4
3/2
1
1
24
Number System and Logic Families
Step 5
1/2
0
1
As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse
order so that the first remainder becomes the least significant digit (LSD) and the last
remainder becomes the most significant digit (MSD).
Decimal Number: 2910 = Binary Number: 111012.
Other base system to Decimal System
Steps

Step 1 - Determine the column (positional) value of each digit (this depends on the
position of the digit and the base of the number system).

Step 2 - Multiply the obtained column values (in Step 1) by the digits in the
corresponding columns.

Step 3 - Sum the products calculated in Step 2. The total is the equivalent value in
decimal.
EXAMPLE
Binary Number: 111012
Calculating Decimal Equivalent:
Step
Step 1
Binary
Number
111012
Decimal Number
((1 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (1
x 20))10
25
Number System and Logic Families
Step 2
111012
(16 + 8 + 4 + 0 + 1)10
Step 3
111012
2910
Binary Number: 111012 = Decimal Number: 2910
Other Base System to Non-Decimal System
Steps

Step 1 - Convert the original number to a decimal number (base 10).

Step 2 - Convert the decimal number so obtained to the new base number.
EXAMPLE
Octal Number: 258
Calculating Binary Equivalent:
STEP 1: CONVERT TO DECIMAL
Step
Octal Number
Decimal Number
Step 1
258
((2 x 81) + (5 x 80))10
Step 2
258
(16 + 5 )10
Step 3
258
2110
Octal Number: 258 = Decimal Number: 2110
STEP 2: CONVERT DECIMAL TO BINARY
Step
Operation
Result
26
Remainder
Number System and Logic Families
Step 1
21 / 2
10
1
Step 2
10 / 2
5
0
Step 3
5/2
2
1
Step 4
2/2
1
0
Step 5
1/2
0
1
Decimal Number: 2110 = Binary Number: 101012
Octal Number: 258 = Binary Number: 101012
Shortcut method - Binary to Octal
Steps

Step 1 - Divide the binary digits into groups of three (starting from the right).

Step 2 - Convert each group of three binary digits to one octal digit.
EXAMPLE
Binary Number: 101012
Calculating Octal Equivalent:
Step
Binary Number
Octal Number
Step 1
101012
010 101
Step 2
101012
28 58
27
Number System and Logic Families
Step 3
101012
258
Binary Number: 101012 = Octal Number: 258
Shortcut method - Octal to Binary
Steps

Step 1 - Convert each octal digit to a 3 digit binary number (the octal digits may be
treated as decimal for this conversion).

Step 2 - Combine all the resulting binary groups (of 3 digits each) into a single binary
number.
EXAMPLE
Octal Number: 258
Calculating Binary Equivalent:
Step
Octal Number
Binary Number
Step 1
258
210 510
Step 2
258
0102 1012
Step 3
258
0101012
Octal Number: 258 = Binary Number: 101012
Shortcut method - Binary to Hexadecimal
Steps

Step 1 - Divide the binary digits into groups of four (starting from the right).

Step 2 - Convert each group of four binary digits to one hexadecimal symbol.
28
Number System and Logic Families
EXAMPLE
Binary Number: 101012
Calculating hexadecimal Equivalent:
Step
Binary Number
Hexadecimal Number
Step 1
101012
0001 0101
Step 2
101012
110 510
Step 3
101012
1516
Binary Number: 101012 = Hexadecimal Number: 1516
Shortcut method - Hexadecimal to Binary
Steps

Step 1 - Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal
digits may be treated as decimal for this conversion).

Step 2 - Combine all the resulting binary groups (of 4 digits each) into a single binary
number.
EXAMPLE
Hexadecimal Number: 1516
Calculating Binary Equivalent:
Step
Hexadecimal Number
Binary Number
29
Number System and Logic Families
Step 1
1516
110 510
Step 2
1516
00012 01012
Step 3
1516
000101012
Hexadecimal Number: 1516 = Binary Number: 101012
Binary Arithmetic
VLSI design components on most microchips perform binary arithmetic. These
components are easier to design than ones performing decimal arithmetic.
Addition:
0
+
0
=
0
0
+
1
=
1
1
+
0
=
1
=
0
1 + 1 = 0 and carry 1 to the next column
Subtraction:
0
-
0
0 - 1 = 1 and borrow 1 from next column (equivalent to subtracting
one from that next column, alternative to binary subtraction is to use
2's
1
complement,
-
see
0
1-1=0
Multiplication
30
later)
=
1
Number System and Logic Families
0
*
0
=
0
0
*
1
=
0
1
*
0
=
0
1*1=1
Examples:
1 1 0 1
_____________
1101
*1011
1 1 1 1
1
1 1 1 1
1101
11101
10000
+1011
- 10011
-
1101
11
1 1 0 1
1 0 1 1 // 1 0 0 1 0 0 0 1
1011
1 1 1 0
1011
0 0 0 0
1101
11000
1010
1101
1 1 0 1
1011
10001111
10
(note on multiplication example: each partial product is either zero or the multiplicand
shifted over an appropriate number of places).
31
Number System and Logic Families
Representation of Negative Numbers
There are several ways of representing negative binary numbers: sign and magnitude
system, 1's complement, 2's complement, R's complement, etc.
1. Sign and Magnitude:
Consider n-bit word: first bit is sign bit (0 for plus, 1 for negative), other
n-1 bits are magnitude. One can represent any one of 2 n-1 positive integers or
2n-1 negative integers.
However with this representation the design of
arithmetic components to perform arithmetic is awkward.
2. 1's complement :
A negative number, -N, is represented by its 1's complement defined as
taking the positive number N and complementing bit by bit to produce the
negative number -N. Alternatively -N = ( 2n - 1 ) - N. As an example consider
n=4, -N is represented by 15 - N , so 0 0 1 0 = N, N=2, -N is 1 1 0 1 = 13 ( or
=15 - N = 15 - 2) . Note that in this n=4 number system , -0 and -8 have no
representation.
3. 2's complement system:
a positive number N, is represented by a 0 followed by the magnitude of
N. However a negative number -N is represented by it's 2's complement, N*.
For n-bit system, N* = 2n - N , or complement N bit by bit and then Add 1. As
an example consider 0010 (N=2), -N is formed by taking 1101 + 1 = 1110 (or
= N*= 2n - N = 16 - 2 = 14) .
4. R's complement system:
For base R system, with word length = n bits, for a positive number N
then R's complement is N* = Rn - N and (R-1)'s complement is (Rn-1)-N. As an
example, R=8 (octal), n=4, then the 8's complement of 03458 is 100008 32
Number System and Logic Families
03458 = 74338 . and the 7's complement is (100008 - 1) -03458 = 7777 - 0345
= 74328 . Note that just as the 1's complement can be formed on a bit by bit
approach, the (R-1)'s complement can be formed on a digit by digit approach,
so for octal numbers the 7's complement can be formed by subtracting each
digit from 7 .
Addition of 2's complement numbers
Perform the addition in the same manner you would do for two positive numbers.
(ignore any carries from the sign position). This way will always be correct, except
when an overflow occurs. For an n bit word an overflow exists if the correct
representation of the sum (including sign) requires more than n bits. If two positive nbit numbers sum to form a negative n-bit number or if two negative n-bit numbers
sum
to
form
a
positive
n-1
sum < 2n-1
0011
0100
sum >= 2
0101
n-bit
-N
>
number,
+N
0110
0111 Correct
1101
occured.
sum > 2n-1
1011
1100
1010
1010
(1)
1011
has
-M sum <= 2n-1
1011
+M
0101
0110
>
then overflow
Wrong
Overflow!
1111
Correct
0001 (1)1001
(1)
Correct, ignore Correct,
carry from sign ignore
bit
Addition of 1's complement numbers
33
carry.
last
0101
Wrong,
Overflow!
Number System and Logic Families
Similar to 2's complement arithmetic, except carry from the sign bit is added to the n
bit sum in the right most position, this is called the "end-around carry".
1010
0110
1100
1010
1011
1001
0101
1001
(1)0000
(1)0111
1
"end-around
carry"
1 "end-around
(1)0011
1
carry"
1110
Correct
0001
correct
overflow)
(no
1000
Correct
(no
overflow)
0100
Wrong
,
Overflow!
Note again, one can extend arithmetic to any base. For example base R, R's
complement representation for negative numbers, add numbers and discard the carry
from
ie.
the
4
digit
octal
last
digit
numbers,
Consider
-0345
position.
+
0672
:
- 0345 in 8's complement is 7433 -345 in 7's complement is 7432
+
7433
7432
0672
+0672
=====
=====
(1)0325 (1)0324
34
Number System and Logic Families
discard carry , 0325
1
"end-around
carry"
=====
0325
Codes
Code is a symbolic representation of discrete information. Codes are of different
types. In digital electronics, codes are used to communicate the information between
computers. These codes represent the information symbolically as a string of bits 0
and 1 and rules defined by the code decide the arrangement of these bits.
Codes can be broadly categorized into three categories
1) Weighted Codes
2) Unweighted Codes
3) Sequential Codes
1)Weighted Codes
These codes are positionally weighted and each position with in the binary equivalent
of the number is assigned a fixed value. Thus, they obey positional weighting
principal.
Binary Coded Decimal(BCD) is an example of weighted codes. In this code, the
binary equivalent of a number will always remain same.
To understand how to compute bcd equivalent of a number, you may refer below link:

Binary Coded Decimal
2)Unweighted Codes
35
Number System and Logic Families
These codes are not positionally weighted i.e. each position with in the binary
equivalent of the number is not assigned a fixed value. Thus, unweighted codes do not
obey positional weighting principal.
Grey code is an example of unweighted codes.In grey code, computation not only
depends on the bit under consideration but also on the neighboring bits. Due to which,
unlike weighted codes, the binary equivalent of a number may return different values
on each computation Excess-3 code is also an example of unweighted code.
To understand how to compute grey code or excess-3 equivalent of a number, you
may refer below link:

Grey Code

Excess-3 Code
3)Sequential Codes
A code is said to be sequential when each succeeding code is one binary number
greater than the preceeding code. BCD and Excess-3 codes are examples of sequential
codes in digital electronics.
Thus, this post summarizes various codes used in digital electronics to communication
information.
Feel free to leave your footprints in the comments section below for any queries,
suggestions or feedback.
Binary Codes
36
Number System and Logic Families
Decimal numbers can be coded in terms of binary signals. The simplest coding is
binary coded decimal or BCD, where each decimal digit is replaced by it's 4 bit binary
equivalent.
Example: 937.25 is 100100110111.00100101
9
3
7.
2
5
1001
0011
0111.
0010
0101
Note that the result is quite different from that obtained by converting the number as a
whole
Below
into
is
a
table
of
binary
binary.
codes
for
decimal
digits:
Decimal
8421
6311
excess-3
2out-of-5
Gray
Digit
(BCD)
Code
Code
Code
Code
0
0000
0000
0011
00011
0000
1
0001
0001
0100
00101
0001
2
0010
0011
0101
00110
0011
3
0011
0100
0110
01001
0010
4
0100
0101
0111
01010
0110
5
0101
0111
1000
01100
1110
6
0110
1000
1001
10001
1010
7
0111
1001
1010
10010
1011
8
1000
1011
1011
10100
1001
9
1001
1100
1100
11000
1000
37
Number System and Logic Families

the 8421 and 6311 are called weighted codes, where if weights are
w3,w2,w1,w0 then code a3,a2,a1,a0 represents decimal number N, N=w3 a3 +
w2 a2 + w1 a1 + w0 a0

excess-3 code is obtained by adding 3 (0011) to each of the BCD codes

2 out of 5 code, has exactly 2 out of 5 bits that are '1', which is useful for error
checking

gray code: codes for successive decimal digits differ in one bit only. Useful for
low power applications, where reduction in switching is important. (ie
difference between 6 and 7 is in bit 4 only)

in general the decimal value of a coded digit cannot be computed by a single
formula when a non-weighted code is used.

other codes, such as ASCII, EBCDIC involve letters, symbols, punctuation
marks, etc...
Code Conversion:
1) Binary to gray code conversion

Binary to gray code conversion is a very simple process. There are several
steps to do this types of conversions. Steps given below elaborate on the idea
on this type of conversion.

(1) The M.S.B. of the gray code will be exactly equal to the first bit of the
given binary number.

(2) Now the second bit of the code will be exclusive-or of the first and second
bit of the given binary number, i.e if both the bits are same the result will be 0
and if they are different the result will be 1.

(3)The third bit of gray code will be equal to the exclusive-or of the second and
third bit of the given binary number. Thus the Binary to gray code conversion
38
Number System and Logic Families
goes on. One example given below can make your idea clear on this type of
conversion.
Let (01001) be the given binary number
(01001)2 = ( ? )Grey
Exor
Exor
Exor
Exor
0
1
0
0
1
0
1
1
0
1
=(01101)grey
Gray code to binary conversion
Gray code to binary conversion is again very simple and easy process. Following
steps can make your idea clear on this type of conversions.
(1) The M.S.B of the binary number will be equal to the M.S.B of the given gray
code.
(2) Now if the second gray bit is 0 the second binary bit will be same as the previous
or the first bit. If the gray bit is 1 the second binary bit will alter. If it was 1 it will be 0
and if it was 0 it will be 1.
(3) This step is continued for all the bits to do Gray code to binary conversion.
39
Number System and Logic Families
(01001)Grey =(?)2
0
1
Ex-Or Ex-Or
0
1
0
0
1
Ex-Or Ex-Or
1
1
0
Answer= (01110)2
BCD (Binary-Coded Decimal) code :
 Four-bit code that represents one of the ten decimal digits from 0 to 9.
 Example - (37)10 is represented as 0011 0111 using BCD code, rather than
(100101)2 in straight binary code.
 Thus BCD code requires more bits than straight binary code.
 Still it is suitable for input and output operations in digital systems.
 Note: 1010, 1011, 1100, 1101, 1110, and 1111 are INVALID CODE in BCD
code.
Excess-3 code:
 4-bit code is obtained by adding binary 0011 to the natural BCD code of the
digit.
 Example - decimal 2 is coded as 0010 + 0011 = 0101 as Excess-3 code.
 It not weighted code.
 Its self-complimenting code, means 1's complement of the coded number yields
9's complement of the number itself.
 Used in digital system for performing substraction operations.
40
Number System and Logic Families
ASCII (American Standard Code Information Interchange) code :
 It is 7-bit or 8-bit alphanumeric code.
 7-bit code is standard ASCII supports 127 characters.
 Standard ASCII series starts from 00h to 7Fh, where 00h-1Fh are used as
control characters and 20h-7Fh as graphics symbols.
 8-bit code is extended ASCII supports 256 symbols where special graphics and
math's symbols are added.
 Extended ASCII series starts from 80h to FFh.
EBCDIC (Extended Binary Coded Decimal Interchange Code) code
 8-bit alphanumeric code developed by IBM, supports 256 symbols.
 It was mainly used in IBM mainframe computers.
Switching Characteristics of BJT & FET:
BJTs are characterized by linear current transfer function between the collector
current and the base current. They have much larger transconductance and they can
achieve much higher input signal gain thanks to their current control. In addition, they
have higher speeds and higher maximum operating frequency. Consequently, they are
preferred in the amplifier circuits and in the linear integrated circuits as well as high
frequency and high power applications. When BJTs are operated as switches, they
consume appreciable power and therefore they are less suitable in VLSI integrated
circuits. They are used in very high speed logic circuits such as TTL and ECL. They
consume
more
area
on
the
chip
than
the
mos
transistors.
The FETs are characterized by high input impedance and some types of FETS operate
41
Number System and Logic Families
as a relay such as the enhancement MOSFETS making them superior as switches.
Their nonlinear transfer characteristics between the drain current and the gate to
source voltage and their smaller trans conductance makes them less suitable in
amplifier circuits. Therefore, we see that the dominating logic family for
implementing memories, CPUs and DSPs are made of MOS transistors especially the
complementary CMOS transistors which have good logic performance parameters.
The
static
power
consumption
of
the
CMOS
is
negligible.
In summary, the applications of a device and its dominance in some applications stem
from its characteristics , availability, cost and familiarity.
1. Transistor–transistor logic (TTL) is a class of digital circuits built from bipolar
junction transistors (BJT) and resistors. It is called transistor–transistor logic because
both the logic gating function (e.g., AND) and the amplifying function are performed
by transistors (contrast with RTL and DTL).
2. A particular characteristic of TTL signals is that the inputs to a gate “float high” —
i.e. rise to a logical ‘1’ — if left unconnected. This means that the main requirement
for driving a TTL input is to “pull down” the level to near 0V. This typically takes a
few milliamps per input. This is usually described by saying that a TTL signal source
has to be able to “sink” a relatively large current. Typically, TTL gates take around
10-20 nanoseconds to switch level. Hence we can ‘clock’ TTL and pass bits through
the gates at rates up to around 50MHz provided the circuits are designed carefully.
With care, speeds approaching 100MHz are possible, but for high speed operation
other
forms
of
logic
Lots of TTL gates are available
42
may
work
better.
Number System and Logic Families
Transistor Transistor Logic or TTL
Under Digital ElectronicsThe full form of TTL is Transistor Transistor Logic. This is a
logic family which is mainly build up of NPN transistors, PN junction diodes and
diffused resistors. The basic building block of this logic family is NAND gate and
there are various subfamilies of this logic gate those are standard TTL, advanced
Schottky TTL, schottky TTL, low power TTL, high power TTL, fast TTL etc. now to
know about this family in a more descriptive way we will discuss the internal
structure and characteristic parameters of some of its subfamilies.
Standard TTL
The above figure shows the internal structure and characteristics of a standard TTL
NAND gate. The NAND gate of it is a quad two input type. And it has four circuits of
5400/740. In plain ways the circuit of this type of TTL operates as follows. The Q1
showed in the figure is a two emitter NPN transistor. This type is NAND gate is
analogous to two transistors whose base and emitter terminals are joined together. The
43
Number System and Logic Families
diodes named as D2 and D3 are used to limit the input voltages which are negative in
nature.
Low Power TTL
This is a subfamily under the main family. This is named so because lower power
consumption and dissipation is achieved. Though the speed at which the operation is
done is somewhat reduced. The above figure is of a low power TTL which is made
using NAND gates. The NAND gate used in this is of 74L00 or 54L00 type and is of
quad two input type. The construction of this type of TTL is almost similar to that of
standard TTL except the resistance which is of a higher value. For this increased value
of the resistance the power dissipation of the circuit is lowered.
High Power TTL
44
Number System and Logic Families
Unlike the
low power TTL the High power TTL is the high speed edition of the standard TTL.
The speed of operation of this type of TTL is more than the previously discussed. The
power dissipation for this higher than other previously discussed TTLs. The above
diagram is of a high power TTL NAND gate. The NAND gate is a quad two input of
type 74H00 or 54H00. The above drawn figure is very similar to that of a standard
TTL except Q3 transistor and D1 diode combination, which has been replaced by an
arrangement of Q3, Q5 and R5. The speed of operation is higher and the power
dissipation is also higher for this type of TTLs.
Schottky TTL
45
Number System and Logic Families
Another TTL subfamily is Schottky TTL. This design was used to speed up the time
of operation. The speed offered by this type of TTL is twice the speed that is offered
by the high power TTL. The power dissipation for both the TTLs are same and there
is no extra power consumption. The figure above represents the basic NAND based
diagram of Schottky TTL. The circuit diagram is pretty much similar to that of a high
power TTL, here the Q transistor of high power TTL is missing. The Schottky
transistor which is used for this type TTL is nothing but a bipolar transistor which has
its base and collector connected by a schottky diode. This Schottky TTL is further
devided in many parts like low power Schottky, Advanced low power schottky and
advanced Schottky whose discussion has been avoided due to complexities.
Basic Properties of some TTL Families.
74 family
Supply Voltage
‘1’ Level Output Current
74LS family
54 family
+5V (+/- 0.5V) +5V (+/- 0.5V) +5V (+/- 0.25V)
0.4mA
0.4mA
46
0.4mA
Number System and Logic Families
‘0’ Level Ouput Current
16mA
8mA
16mA
‘1’ Level Input Voltage (min)
2V
2V
2V
‘0’ Level Input Voltage (max)
0.8V
0.8V
0.8V
‘1’ Level Input Current
0.04mA
0.05mA
0.04mA
‘0’ Level Input Current
1.6mA
0.4mA
1.6mA
Important characteristics of digital ICs
1. Fan out
2. Power dissipation
3. Propagation Delay
4. Noise Margin
5. Fan In
6. Operating temperature
7. Power supply requirements
1. Fan-out
Fan out specifies the number of standard loads that the output of the gate can drive
without impairment of its normal operation
2. Power dissipation
Power dissipation is measure of power consumed by the gate when fully driven by all
its inputs.
3. Propagation delay
Propagation delay is the average transition delay time for the signal to propagate from
input to output when the signals change in value. It is expressed in ns.
47
Number System and Logic Families
4. Noise margin
It is the maximum noise voltage added to an input signal of a digital circuit that does
not cause an undesirable change in the circuit output. It is expressed in volts.
5. Fan in
Fan in is the number of inputs connected to the gate without any degradation in the
voltage level.
6. Operating temperature
All the gates or semiconductor devices are temperature sensitive in nature. The
temperature in which the performance of the IC is effective is called as operating
temperature. Operating temperature of the IC vary from 00 C to 700 c.
TTL NAND Gate
Basic TTL NAND Gate The circuit diagram for a 2-input LS-TTL NAND gate, part
number 74LS00, is shown in Figure TTL-1. The NAND function is obtained by
combining a diode AND gate with an inverting buffer amplifier. The circuit’s
operation is best understood by dividing it into the three parts that are shown in the
figure and discussed in the next three paragraphs: • Diode AND gate and input
protection. • Phase splitter. • Output stage. Q5 VCC = +5
48
Number System and Logic Families
Diodes D1X and D1Y and resistor R1 in Figure TTL-1 form a diode AND gate, as in
Section Diode.2. Clamp diodes D2X and D2Y do nothing in normal operation, but
limit undesirable negative excursions on the inputs to a single diode-drop. Such
49
Number System and Logic Families
negative excursions may occur on HIGH-to-LOW input transitions as a result of
transmission-line effects, discussed in Section Zo. Transistor Q2 and the surrounding
resistors form a phase splitter that controls the output stage. Depending on whether the
diode AND gate produces a “low” or a “high” voltage at VA, Q2 is either cut off or
turned on. The output stage has two transistors, Q4 and Q5, only one of which is on at
any time. The TTL output stage is sometimes called a totem-pole or push-pull output.
Similar to the p-channel and n-channel transistors in CMOS, Q4 and Q5 provide
active pull-up and pull-down to the HIGH and LOW states, respectively. The
functional operation of the TTL NAND gate is summarized in Figure TTL-2(a). The
gate does indeed perform the NAND function, with the truth table and logic symbol
shown in (b) and (c). TTL NAND gates can be designed with any desired number of
inputs simply by changing the number of diodes in the diode AND gate in the figure.
Commercially available TTL NAND gates have as many as 13 inputs. A TTL inverter
is designed as a 1-input NAND gate, omitting diodes D1Y and D2Y in Figure TTL-1.
IILmax The maximum current that an input requires to pull it LOW
IIHmax The maximum current that an input requires to pull it HIGH
IOLmax The maximum current an output can sink in the LOW state while
maintaining an output voltage no more than VOLmax. Since current flows into the
output, IOLmax has a positive value, 8 mA for most LS-TTL outputs
IOHmax The maximum current an output can source in the HIGH state while
maintaining an output voltage no less than VOHmin. Since current flows out of the
output, IOHmax has a negative value, −400 µA for most LS-TTL outputs.
Fanout: fanout is a measure of the number of gate inputs that are connected to (and
driven by) a single gate output.
TTL Subfamilies:
50
Number System and Logic Families
TTL NAND Gate with Totem Pole Outputs
A two input standard TTL NAND gate is a multiple emitter transistor for the inputs A
and B. the output transistors Q3 and Q4 form a totem-pole output arrangement.
Operation:
If A or B is low, the base-emitter junction of Q1 is forward biased and its basecollector junction is reverse biased. Then there is a current from Vcc through R1 ti the
base emitter junction of Q1 and into the LOW input, which provides a path to the
ground for the current. Hence there is no current into the base of Q2 and making it
into cur-off. The collector of Q2 is HIGH and turns Q3 into saturation. Since Q3 acts
as a emitter follower, bu providing a low impedance path from Vcc to the output,
making the output into HIGH. At the same time, the emitter of Q2 is at ground
potential, keping Q4 OFF.
When A and B are high, the two input base emitter junctions of Q1 are reverse niased
and its base collector junction is forward biased. This permits current through R1 and
the base collector junction of Q1 into the base of Q2, thus driving Q2 into saturation.
As a result Q4 is turned ON by Q2, and producing LOW ourput which is near ground
potential. At the same time, the collector of Q2 is sufficiently at LOW voltage level to
keep Q3 OFF.
TTD NAND Gate with open-collector outputs
A two input TTL NAND gate with open collector output is the collector of the
transistor Q3 with nothing connected to it, hence it is known as ‘open collector
output’. But the circuit is not completed and in order to get the proper HIGH and
LOW logic levels at the output of the circuit; an external pull-up resistor must be
connected between Vcc and the collector of Q3.
51
Number System and Logic Families
Operation:
A low on either input A or input B forward biases the base emitter junction of Q1 and
reverse biases it’s base collector junction. There is current through R1 and the base
emitter junction of Q1 out to the low input. Then there is ni current into the base of
Q2 and hence Q2 is in off.
When both inouts are high, the base emitter junction of Q1 is reverse biased and it’s
base collector junction is forward biased. This oermits current through R1 and the
base collector junction of Q1 into the base of Q2, thus driving Q2 into saturation. As a
result on by Q2, thus driving Q2 into saturation. As result, Q3 is turned on bu Q2.
When Q3 is on, the output is connected to near ground through the saturated
transistor; resulting the output as LOW.
TTL – CMOS Introduction:
Theory: A logic family of monolithic digital integrated circuit devices is a group of
electronic logic gates constructed using one of several different designs, usually with
compatible logic levels and power supply characteristics within a family. Many logic
families were produced as individual components, each containing one or a few
related basic logical functions, which could be used as "building-blocks" to create
systems or as so-called "glue" to interconnect more complex integrated circuits. A
"logic family" may also refer to a set of techniques used to implement logic within
VLSI integrated circuits such as central processors, memories, or other complex
functions. Some such logic families use static techniques to minimize design
complexity. Other such logic families, such as domino logic, use clocked dynamic
techniques to minimize size, power consumption, and delay. The first transistor–
transistor logic family of integrated circuits was introduced by Sylvania as Sylvania
Universal High–Level Logic (SUHL) in 1963. Texas Instruments introduced 5400
Series TTL family in 1964. Transistor–transistor logic uses bipolar transistors to form
its integrated circuits. TTL has changed significantly over the years, with newer
versions replacing the older types. Since the transistors of a standard TTL gate are
saturated switches, minority carrier storage time in each junction limits the switching
speed of the device. Variations on the basic TTL design are intended to reduce these
effects and improve speed, power consumption, or both.
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Number System and Logic Families
The German physicist Walter H. Schottky formulated a theory predicting the Schottky
effect, which led to the Schottky diode and later Schottky transistors. Schottky
transistors have a much higher switching speed than conventional transistors because
the Schottky junction does not promote charge storage, leading to faster switching
gates. Gates built with Schottky transistors use more power than normal TTL and
switch faster. With Low-power Schottky (LS), internal resistance values were
increased to reduce power consumption and increase switching speed over the original
version. The introduction of Advanced Low-power Schottky (ALS) further increased
speed and reduced power consumption. A faster logic family called Fast (Schottky)
(F) was also introduced that was faster than normal Schottky TTL.
2 Input TTL NAND Gate
Interfacing considerations
Like DTL, TTL is a current-sinking logic since a current must be drawn from inputs
to bring them to a logic 0 level. At low input voltage, the TTL input sources current
which must be absorbed by the previous stage. The maximum value of this current is
about 1.6 mA for a standard TTL gate.[17] The input source has to be low-resistive
enough (<500 Ω) so that the flowing current creates only a negligible voltage drop
(<0.8 V) across it, for the input to be considered as a logical "0". TTL inputs are
sometimes simply left floating to provide a logical "1", though this usage is not
recommended.
Standard TTL circuits operate with a 5-volt power supply. A TTL input signal is
defined as "low" when between 0 V and 0.8 V with respect to the ground terminal,
and "high" when between 2.2 V and VCC (5 V) and if a voltage signal ranges
between 0.8 V and 2.0 V were to be sent into the input of a TTL gate, there would be
no certain response from the gate and therefore it is considered "uncertain" (precise
logic levels vary slightly between sub-types and by temperature). TTL outputs are
53
Number System and Logic Families
typically restricted to narrower limits of between 0.0 V and 0.4 V for a "low" and
between 2.6 V and VCC for a "high", providing at least 0.4 V of noise immunity.
Standardization of the TTL levels was so ubiquitous that complex circuit boards often
contained TTL chips made by many different manufacturers selected for availability
and cost, compatibility being assured; two circuit board units off the same assembly
line on different successive days or weeks might have a different mix of brands of
chips in the same positions on the board; repair was possible with chips manufactured
years (sometimes over a decade) later than original components. Within usefully
broad limits, logic gates could be treated as ideal Boolean devices without concern for
electrical limitations.
In some cases (e.g., when the output of a TTL logic gate needs to be used for driving
the input of a CMOS gate), the voltage level of the "totem-pole" output stage at output
logical "1" can be increased up to VCC by connecting an external resistor between the
V3 collector and the positive rail. It pulls up the V5 cathode and cuts-off the diode.
However, this technique actually converts the sophisticated "totem-pole" output into a
simple output stage having significant output resistance when driving a high level
(determined by the external resistor).
Sub-types
Successive generations of technology produced compatible parts with improved
power consumption or switching speed, or both. Although vendors uniformly
marketed these various product lines as TTL with Schottky diodes, some of the
underlying circuits, such as used in the LS family, could rather be considered
DTL.[21]
Variations of and successors to the basic TTL family, which has a typical gate
propagation delay of 10ns and a power dissipation of 10 mW per gate, for a power–
delay product (PDP) or switching energy of about 100 pJ, include:




Low-power TTL (L), which traded switching speed (33ns) for a reduction in
power consumption (1 mW) (now essentially replaced by CMOS logic)
High-speed TTL (H), with faster switching than standard TTL (6ns) but
significantly higher power dissipation (22 mW)
Schottky TTL (S), introduced in 1969, which used Schottky diode clamps at
gate inputs to prevent charge storage and improve switching time. These gates
operated more quickly (3ns) but had higher power dissipation (19 mW)
Low-power Schottky TTL (LS) – used the higher resistance values of lowpower TTL and the Schottky diodes to provide a good combination of speed
(9.5ns) and reduced power consumption (2 mW), and PDP of about 20 pJ.
54
Number System and Logic Families


Probably the most common type of TTL, these were used as glue logic in
microcomputers, essentially replacing the former H, L, and S sub-families.
Fast (F) and Advanced-Schottky (AS) variants of LS from Fairchild and TI,
respectively, circa 1985, with "Miller-killer" circuits to speed up the low-tohigh transition. These families achieved PDPs of 10 pJ and 4 pJ, respectively,
the lowest of all the TTL families.
Low-voltage TTL (LVTTL) for 3.3-volt power supplies and memory
interfacing.
Most manufacturers offer commercial and extended temperature ranges: for example
Texas Instruments 7400 series parts are rated from 0 to 70 °C and 5400 series devices
over the military-specification temperature range of −55 to +125 °C.
Special quality levels and high-reliability parts are available for military and
aerospace applications.
Radiation-hardened devices are offered for space applications.
TTL CHARACTERISTICS
Each logic family is characterized by several important parameters. These properties,
and how they relate to the TTL logic families in particular, are explained below:
Fan-in is the maximum number of inputs to a gate. Although physical considerations
limit fan - in, more pragmatic factors, such as limitations on the number of pins
possible on IC packages
and their standardization predominate. TTL NAND gates typically provide 1, 2, 4, or
8 inputs.
If more than eight inputs are required, then a network of NAND gates must be
employed
Fan -out specifies the number of standard loads that the output of a gate can drive
without impairing its normal operation. A standard load is defined to be the amount of
current required to drive an input of another gate in the same logic family.
A voltage transfer curve is a graph of the input voltage to a gate versus its output
voltage; when the input voltage is 0 V, the output is HIGH at 3.3 V. As the input
voltage is increased from 0 to 0.7 V, the output remains relatively constant (Region I).
Beyond 0.7 V to about 1.2 V, the output decreases more gradually with increasing
input voltage (Region II). The threshold voltage, the voltage on the transfer curve at
which Vout= Vin and occurs in Region III, is found at the intersection of the transfer
curve and the line Vout= Vin. Finally, in Region IV, the output remains constant at
0.2 V as the input voltage is increased.
Noise immunity is a measure of the ability of a digital circuit to avert logic level
changes on signal lines when noise causes voltage level changes.
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Number System and Logic Families
The propagation delay time for a gate is the time required for the output to respond to
a change in an input. In all practical gates, a time lag exists between an input change
and the corresponding output response. The time interval between the instants when
the input and output change states are not a satisfactory measure of the delay time of a
logical device for two reasons. First, the input signals to gates and the output signals
produced by gates are not the idealized pulses studied in
theory.
TTL gates are available in three different types of output configurations.
Totem - pole output gates are used in most logic; in this case, the gate by itself drives
its output HIGH or LOW depending on the gate’s inputs. Connecting the outputs of
two or more totem -pole gates together produces undefined output values, may
damage the device and should never be done.
Open - collector output gates can only drive their output LOW; for input combinations
where the
output should be HIGH, an external pullup resistor connected the supply voltage is
needed to
produce the HIGH. The outputs of open - collector gates to be wired together; the
result is to effectively AND all the output signals together.
Three - state (or tri – state) output gates provide, as their name implies, three output
states. Like totem -pole output gates, tri - state gates can drive their output either
HIGH or LOW, as determined by the input combination, but they also have a control
input that overrides the effect of the other inputs and places the gate output in a ‘third
state’. In the third state the internal transistors of the gate are effectively disconnected
from the gate output and the output is in an open circuit or high - impedance state.
This allows a direct wire connection of many tri - state gate outputs to a common line;
however, when this is done, all but at most one of the tri - state output gates connected
together must be in the high - impedance state at any given time.
CMOS
CMOS logic is exemplified by its extremely low power consumption and high noise
immunity. Hence, it is prevalently used in devices demanding low power dissipation,
such as digital wristwatches and other battery powered devices, or in devices operated
in noisy environments, such as industrial plants. A wide variety of CMOS logic
devices in the 4000 series are available. Unlike TTL logic, CMOS logic requires two
supply voltages, VDD and VSS. In typical logical designs, VDD ranges from +3 V to
+16V. The other supply, VSS, is normally grounded. Also, the physical representation
of the binary states in CMOS logic is not entirely compatible with TTL logic. As a
consequence of CMOS's extremely high input impedance, the logic levin CMOS
systems are essentially VDD and ground. If, for example, a 5 volt power supply is
used, LOW typically ranges from 0 to 0.01 V and HIGH from 4.99 to 5.0 V for
56
Number System and Logic Families
CMOS outputs. Input voltages ranging from 3.5 to 5 V are recognized as HIGH and
voltages from 0 to 1.5 V as LOW.
It may appear that CMOS output logic levels, using a 5 V power supply, completely
conform to the TTL logic level ranges of 0 to 0.8 V for LOW and 2.0 to 5.5 V for
HIGH. However, the voltage level ranges representing HIGH and LOW are not the
only factors that determine whether two logic family are compatible or not. The
amount of current that can be supplied by outputs and that can be assimilated by
inputs of gates within each logic family is another consideration. Specifically, when
CMOS drives TTL logic, the crucial question is whether the CMOS output, in the
LOW state, can sink enough of the current originating at the TTL input to ensure that
the voltage at the TTL input does not exceed its maximum LOW level input voltage
of 0.8 V. Typical CMOS gates can sink about 0.4 mA in the LOW state while
maintaining an output voltage of 0.4 V or less. This is sufficient to drive two lowpower TTL inputs, but generally insufficient to drive even one standard TTL input. In
any case, loss of dc noise immunity is an inevitable result. It is better to use a special
buffer such as a 74C901 to drive standard TTL from CMOS. The HIGH state poses no
problems. Similarly, improper circuit operation may result from connecting TTL
outputs to CMOS inputs. In the LOW state, a TTL output can drive CMOS directly.
However, the guaranteed TTL HIGH output level of 2.4 volts is not a valid input level
for CMOS. If the TTL output drives only CMOS inputs, then essentially no current is
drawn and the HIGH output may be 3.5 V or higher. (A pull-up resistor to +5 can be
connected to the gate output to assure that the output is above 3.5V.)
CMOS CHARACTERISTICS
The voltage transfer curve - curves in the transition region are almost vertical. This
narrow transition region is the reason for CMOS logic's high noise immunity. Not
much voltage range is covered in the transition from one state to the other. In contrast
to TTL devices, the threshold voltage depends on the supply voltage and is
approximately half the supply voltage. As with TTL logic, current spiking occurs
during switching. Hence, bypass capacitors are used in CMOS logic design as well.
However, they are not as critical as in TTL logic design because of CMOS's high
noise immunity. Whereas the typical quiescent (static) power dissipation (power
dissipation of a device that is not changing logic states) of TTL IC's was about 40
mW, the power dissipation of CMOS IC's are typically 25 nW. However, as the
frequency of switching increases,
The input impedance, in either state, of CMOS gates is typically 1012 Ω. The input
capacitance
is 5 pF. The output impedance depends on the particular device, and is on the order of
1 kΩ, for
either state.
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Number System and Logic Families
The propagation delay times for CMOS devices are relatively long due to their high
output impedance. Typical delay times are 60 nsec for VDD= 5 V, and 25 nsec for
VDD= 10 V.
Doubling the supply voltage more than doubles the speed of a CMOS gate. The rise
and fall transition times are typically 70 nsec. For VDD= 5V. Thus CMOS devices
operate significantly slower than TTL devices Floating inputs in CMOS logic
guarantee neither LOW nor HIGH outputs and cause increased susceptibility to noise,
as well as excessive power dissipation. Hence, all unused inputs should be connected
to VDD or VSS, as appropriate
The fan-out of CMOS devices is usually greater than 50 because the input current
requirement of CMOS logic is nil (∼pA). However, current is required to charge and
discharge the capacitance of CMOS inputs during logic transitions. Hence, the greater
the fan-out the longer the propagation delays.
58