2 Arithmetic and digital electronics

323 Pages·2008·14.37 MB·English

Series preface
In recent years there have been many changes in the structure of undergraduate
courses in engineering and the process is continuing. With the advent of
modularisation, semesterisation and the move towards student-centred learning
as class contact time is reduced, students and teachers alike are having to adjust to
new methods of learning and teaching.
Essential Electronics is a series of textbooks intended for use by students on
degree and diploma level courses in electrical and electronic engineering and
related courses such as manufacturing, mechanical, civil and general engineering.
Each text is complete in itself and is complementary to other books in the series.
A feature of these books is the acknowledgement of the new culture outlined
above and the fact that students entering higher education are now, through no
fault of their own, less well equipped in mathematics and physics than students of
ten or even five years ago. With numerous worked examples throughout, and
further problems with answers at the end of each chapter, the texts are ideal for
directed and independent learning.
The early books in the series cover topics normally found in the first and second
year curricula and assume virtually no previous knowledge, with mathematics
being kept to a minimum. Later ones are intended for study at final year level.
The authors are highly qualified chartered engineers with wide experience in
higher education and in industry.
R G Powell
Nottingham Trent University
Preface
This book covers the material normally found in first and second year modules on
digital electronics. It is intended for use by degree, diploma and TEC students in
electronic engineering, computer science and physics departments. It is also aimed
at those professionals who are revisiting the subject as a means of updating them-
selves on recent developments.
The philopsophy of this book is to cover the basics in detail and to give a taste
of advanced topics such as asynchronous and synchronous circuit design, and
semi-custom IC design. In other words we have adopted a broad brush approach
to the subject. This will provide the reader with an introduction to these advanced
techniques which are covered in more specialist and comprehensive texts. The
book si sprinkled with numerous practical examples indicating good and
poor design techniques and tips regarding circuit pitfalls. To provide you with
confidence in your understanding each chapter is concluded with both short and
long questions and follows the same format as the other texts in this series.
stnemegdelwonkcA
Thanks are due to David Ross for his patience and understanding when every
deadline was set and missed, and for offering the opportunity for us to be involved
with this series. Finally, and not least, we would like to thank Robert, Emily,
Charlotte, Jessica and Nicki (who also assisted in the proof reading) for their
patience.
Solutions to the self-assessment and other exercises are available from the
publishers to lecturers only. Please e-mail [email protected]
Preface
This book covers the material normally found in first and second year modules on
digital electronics. It is intended for use by degree, diploma and TEC students in
electronic engineering, computer science and physics departments. It is also aimed
at those professionals who are revisiting the subject as a means of updating them-
selves on recent developments.
The philopsophy of this book is to cover the basics in detail and to give a taste
of advanced topics such as asynchronous and synchronous circuit design, and
semi-custom IC design. In other words we have adopted a broad brush approach
to the subject. This will provide the reader with an introduction to these advanced
techniques which are covered in more specialist and comprehensive texts. The
book si sprinkled with numerous practical examples indicating good and
poor design techniques and tips regarding circuit pitfalls. To provide you with
confidence in your understanding each chapter is concluded with both short and
long questions and follows the same format as the other texts in this series.
stnemegdelwonkcA
Thanks are due to David Ross for his patience and understanding when every
deadline was set and missed, and for offering the opportunity for us to be involved
with this series. Finally, and not least, we would like to thank Robert, Emily,
Charlotte, Jessica and Nicki (who also assisted in the proof reading) for their
patience.
Solutions to the self-assessment and other exercises are available from the
publishers to lecturers only. Please e-mail [email protected]
1 Fundamentals
1.1 INTRODUCTION
This chapter introduces the essential information required for the rest of the
book. This includes a description of Boolean algebra, the mathematical language
of digital electronics, and the logic gates used to implement Boolean functions.
Also covered are the 'tools' of digital electronics such as truth tables, timing
diagrams and circuit diagrams. Finally, certain concepts such as duality, positive
and negative assertion level logic and universal gates, that will be used in later
chapters, are introduced.
1.2 BASIC PRINCIPLES
1.2.1 Boolean algebra- an introduction
The algebra of a number system basically describes how to perform arithmetic
using the operators of the system acting upon the system's variables which can
take any of the allowed values within that system. Boolean algebra describes the
arithmetic of a two-state system and is therefore the mathematical language of
digital electronics. The variables in Boolean algebra are represented as symbols
(e.g. A, ,B ,C ,X Y etc.) which indicate the state (e.g. voltage in a circuit). In this
book this state will be either 0 or .1 ~ Boolean algebra has only three operators:
NOT, AND and OR. The symbols representing these operations, their usage and
how they are used verbally are all shown in Table 1.1. Note that whereas the
AND 2 and OR operators operate on two or more variables the NOT operator
works on a single variable.
Table 1.1 Boolean variables and operators
Operator Symbol Usage Spoken sa
NOT - A m not A; or A bar
AND A. B A and B
OR + A+B A or B
nl~ other textbooks, and occasionally later on in this one, you may see these states referred to as HIGH
and LOW or ON and OFF.
-'Sometimes the AND symbol, A- B, si omitted and the variables to be AND'd are just placed together as
AB. This notation will be adopted in later chapters.
2 Fundamentals
Example 1.1
A circuit contains two variables (i.e. signals), X and ,Y which must be OR'd
together. How would this operation be shown using Boolean algebra, and how
would you describe it verbally?
Solution
The operation would be spoken as X or Y and written as X+ .Y
Example 1.2
The output Y of a logic circuit with two inputs, A and B, si given by the Boolean
arithmetic expression, Y= A. .B How would this be described verbally?
Solution
This would be spoken as either Y equals A and B bar, or alternatively Y equals A
and not B.
1.2.2 The three Boolean operators
The basic gates (i.e. circuit elements) available in digital electronics perform the
three Boolean algebraic operations of NOT, AND and OR. The symbols for these
gates are shown in Fig. 1.1. In order to both design and analyse circuits it is neces-
sary to know the output of these gates for any given inputs.
A A
A ~ Y Y
NOT AND OR
Fig. 1.1 The three basic Boolean srotarepo
ehT NOT rotarepo
Since any Boolean variable can only be either 0 or 1 (Boolean algebra is a two- state
system) then if it is 0 its complement is 1 and vice versa. The NOT gate performs
this operation (of producing the complement of a variable) on a logic signal, so if
A is input to the NOT gate then the output is represented by Y= A. Therefore if
A =0 then Y= ,1 or else A = 1 and Y=0 (there are only two possibilities).
The truth table of a logic system (e.g. digital electronic circuit) describes the
output(s) of the system for given input(s). The input(s) and output(s) are used to
label the columns of a truth table, with the rows representing all possible inputs to
the circuit and the corresponding outputs. For the NOT gate there is only one
input (hence one input column, A), which can only have two possible values (0
and ,)1 so there are only two As there is only one output, ,Y there is only
rows. 3
3The number of possible inputs, and hence rows, si given by ;2 (where i is the number of inputs) since each
of the i inputs can only take one of two possible values (0 and 1 .)
Basic principles 3
one output column. The truth table for the NOT gate in Table 1.2 shows that
Y= 1 if A = 0, and Y= 0 if A = .1 So Y= A, the complement of A. The NOT gate is
also sometimes referred to as an inverter due to the fact that it complements
(inverts) its input.
Table 2.1 Truth tables for the three basic logic gates
B Y A B Y
row 1 0 0 0 0 0
A Y row 2 1 0 0 1 1
row 1 0 1 row 3 1 0 0 1 0 1
row 2 1 0 row 4 1 1 1 1 1 1
NOT AND OR
Y=A Y=A .B Y=A +B
The AND operator
The AND operator takes a number of variables as its input and produces one
output whose value is 1 if and only if all of the inputs are .1 That is the output is 1
if input 1 and input 2 and all the other inputs are 1. Hence its name.
Considering a two-input (although it can be any number) AND gate its truth
table will have two input columns, A and B, and one output column, .Y With two
inputs there are 2 2 =4 input combinations (since both A and B can be either 0 or )1
and so four rows. The output of the gate, ,Y will be 0 unless all (i.e. both A and B)
inputs are ,1 so only the last row when A and B are 1 gives an output of .1 The
truth table (see Table 1.2) describes completely the output from an AND gate for
any combination of inputs.
Alternative, but exactly equivalent, descriptions of this operation are given by
use of either the circuit symbol or the Boolean equation, Y= A- B. (This is true of
all combinational logic circuits.)
Example 1.3
Consider a three-input AND gate. How many columns and rows would its truth
table have? What would the Boolean expression describing its operation be? What
would its truth table and circuit symbol be?
Solution
The truth table would have four columns; three for the inputs and one for the
output. Since there are three inputs it would have 23= 8 rows corresponding to all
possible input combinations. Its Boolean algebraic expression would be
Y= A- B (cid:12)9 C, assuming the inputs are A, B and C. Its truth table and circuit symbol
are shown in Fig. 1.2.
4 Fundamentals
A B C A.B.C
0 0 0 0
o o i o
0 1 0 0
o ~ 1 o A m
1 0 0 0
C
1 0 1 _ .O (cid:12)9 _
1 1 0 0
1 1 1 1
Fig. 2.1 Truth table and symbol for a three-input AND gate as discussed in Example 3.1
ehT OR rotarepo
The OR operator takes a number of variables as its input and produces an output
of 1 if any of the inputs are .1 That is the output is i if input I or input 2 or any input
is 1. The layout of the truth table for a two-input OR gate is the same as that for
the two-input AND gate for the same reasons given above (since both have two
inputs and one output). The entries in the output column are all that differ with
Y= 1 whenever any input, either A or B, is 1. 4 Note that this includes an output of
1 if both inputs are 5.1 The Boolean algebraic equation for this gate is Y= A + B.
Example 1.4
Draw the circuit symbol and truth table for a four-input OR gate.
Solution
These are shown in Fig. 1.3.
3.1 BOOLEAN ALGEBRA
Boolean algebra is the mathematical language of digital logic circuits, which are
simply circuits built out of the three gates (operations) introduced above. It
provides techniques for describing, analysing and designing their operation.
Using the above descriptions of the operation of the three basic gates, and their
Boolean descriptions, Y= A, Y= A- B and Y= A + B, the additional rules and laws
of Boolean logic which are needed will now be introduced.
1.3.1 Single-variable theorems
As the heading suggests, this section details those rules which describe the opera-
4Another way oflooking at this si that the output si only 0 if both input A and input B are .0 This (nega-
tive logic) approach si discussed in more detail in Section 1.7.
A5 gate which si similar to the OR gate in all but this aspect, the exclusive-OR (XOR) gate, will be consid-
ered in Section 1.4.
Boolean algebra 5
|
A B C D A+B+C+D
0 0 0 0 i, 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 i 1
0 1 0 l 0 1
0 1 0 i 1 1
0 1 1"0 1
0 1 1 1 ,,, 1
1 ) 0 0 -.. 1
1 ~ 0L 1 i ......
1 3 1 (cid:12)9 0 1
'1 3 1 1 1
1 1 0 0 1
1 1 0 1- 1
1 1 1 0 1
1 1 1 1 1
L
Fig. 1.3 Truth table and symbol for a four-input OR gate as discussed in Example 4.1
tion of logic gates when only one variable is present. Note that these laws, given in
Table 1.3, provide exactly the same information as the truth tables.
Table 3.1 Single-variable Boolean theorems
Idempotent laws: Rows 1 and 4 of the truth tables, demonstrate the effect of a variable oper-
ating upon itself:
A.A=A (1.1)
A+A=A (1.2)
Property of inverse elements: Rows 2 and 3 of the truth tables show the effect of a variable
operating on its complement:
A'A:O (1.3)
A A+ __ = I (1.4)
Involution (NOT) law:
A : A (1.5)
Property of identity elements: The effect of operating on a variable with 0 or :1
A .0:0 (1.6)
A. 1:A (1.7)
A +0: A (1.8)
A + 1 : 1 (1.9)
Idempotent laws
The idempotent ~ laws describe the effect of a variable operating upon itself (i.e.
the same variable goes to all inputs). For the two-input AND gate this gives
Y- A- A which will give 1 if A - 1, and 0 if A -0; hence Y= A. A - A.
nredi" nmans "emas" ni Lalin.
6 Fundamentals
The OR operator gives exactly the same result and these laws give the outputs
in rows 1 and 4 (see Table 1.2) of the truth tables.
Inverse elements
The law of inverse elements describes the effect of operating on a variable, A, with
its complement, A. For the AND gate this gives Y= A. A =0, since A and A must
have complementary values and therefore Y= .0
For the OR gate Y= A + A = ,1 since either A or A must be .1 This law describes
the operations in rows 2 and 3 of the truth tables.
Involution law
This describes the effect of operating on a variable twice with the NOT operator
(i.e. passing a signal through two NOT gates). The effect of this is to return the
variable to its original state. So Y= A = A.
Note that the truth tables could be derived from the above three laws as they give
exactly the same information. It will become apparent that there is always more
than one way of representing the information in a digital circuit, and that you
must be able to choose the most suitable representation for any given situation,
and also convert readily between them.
Properties of identity elements
The above laws give all of the information held in the truth tables. However
another way of expressing this information si as the properties of identity elements.
These just give the output of the AND and OR gates when a variable, A, si oper-
ated on by a constant (an identity element). (So for a two-input gate one of the
inputs si held at either 0 or ).1 Obviously these laws, shown in Table 1.3, can also
be used to completely determine the truth tables.
Note that Equation 6.1 in Table 3.1 states that AND'ing any variable (or
Boolean expression) with 0 gives 0, whilst Equation 9.1 means that OR'ing any
variable (or Boolean expression) with 1 gives .1 However AND'ing with 1
(Equation 1.7) or OR'ing with 0 (Equation 1.8) gives the Boolean value of the
variable or expression used in the operation.
Example 1.5
What is the result of the operations (X. 0) and ((X. Y)+ ?)1
Solution
The output from (X-0) will be 0, since anything AND'd with 0 gives a digital
output of .0 The result of ((X. Y)+ )1 will be ,1 since any expression OR'd with 1
gives .1 Note that in the second example it is the Boolean expression (X- )Y (which
must be either 0 or )1 that si OR'd with .1