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Dr. D. J. Jackson Lecture 1-1Electrical & Computer Engineering
ECE380 Digital Logic
Introduction
Dr. D. J. Jackson Lecture 1-2Electrical & Computer Engineering
Digital hardware
• Logic circuits are used to build computer
hardware as well as other products (digital
hardware)
• Late 1960’s and early 1970’s saw a
revolution in digital capability
– Smaller transistors
– Larger chip size
• More transistors/chip gives greater
functionality, but requires more complexity in
the design process
2
Dr. D. J. Jackson Lecture 1-3Electrical & Computer Engineering
Digital hardware
• Integrated circuits are
fabricated on silicon
wafers
• Wafers are cut &
packaged to form
individual chips
• Chips have from tens to
millions of transistors
Dr. D. J. Jackson Lecture 1-4Electrical & Computer Engineering
How complex is a digital design?
• Complexity can, and generally does, surpass
human capability
– 16 million transistors/cm
2
now
– 100 million transistors/cm
2
in 10 years (?)
• Provides motivation for computer-based
design techniques
• Most engineering work is done with CAD
packages
3
Dr. D. J. Jackson Lecture 1-5Electrical & Computer Engineering
Two design approaches
•Traditional
– Relies on mathematical
models
– Analytical approaches
– Provides insight and
understanding of problem
– Useful for small problems
– Inadequate for large
(real) problems
•CAD
– Software relies on
mathematical model and
analytical approach
– Transparent to user
– Many details are
abstracted
– Useful/required for real
problems
Dr. D. J. Jackson Lecture 1-6Electrical & Computer Engineering
Traditional versus CAD design
• CAD tool usage is essential
• Insight and basic understanding offered by
traditional approach is still important
– Initial conceptualization is still traditional
– Effective use of CAD tools requires some
understanding of what the tools are doing
– Use of design options requires insight
4
Dr. D. J. Jackson Lecture 1-7Electrical & Computer Engineering
Types of chips
•Standard chips
– Contain a small amount of circuitry
(<100 transistors)
– Performs simple functions
– 7400 series devices
• Programmable logic devices (PLD)
– Collection of gates with programmable
interconnections
– Function is configurable by designer/user
– Design with PLD is via a CAD tool
Dr. D. J. Jackson Lecture 1-8Electrical & Computer Engineering
Types of chips
•Custom-designed chips
– Optimized for a specific task – better performance
– Larger amount of logic circuitry
– Cost of production is high
– Large volume required to justify cost
5
Dr. D. J. Jackson Lecture 1-9Electrical & Computer Engineering
A field-programmable gate array
Memory block
Group of 8 logic cells
Interconnection
wires
Dr. D. J. Jackson Lecture 1-10Electrical & Computer Engineering
The development process (1)
Design
Correct?
Required Product
Define
specifications
Initial design
Simulation
Redesign
no
6
Dr. D. J. Jackson Lecture 1-11Electrical & Computer Engineering
The development process (2)
Prototype
implementation
Testing
Make
corrections
Meets
specs?
Minor
errors?
Redesign
Finished
product
no
no
yes
yes
Dr. D. J. Jackson Lecture 1-12Electrical & Computer Engineering
What should you expect to gain
from this course?
• Understanding of concepts, models,
algorithms and processes for digitallogic
design
– Relevance of the material to subsequent courses
and to your career
• Problem solving skills
– Formulating and attacking new problems
– Need to struggle with problems – evolve your
problem solving skills
• Communicate solutions in a clear, concise
manner
1
Dr. D. J. Jackson Lecture 2-1Electrical & Computer Engineering
ECE380 Digital Logic
Introduction to Logic Circuits:
Variables, functions, truth tables,
gates and networks
Dr. D. J. Jackson Lecture 2-2Electrical & Computer Engineering
Logic circuits
• Logic circuits perform
operations on digital
signals
– Implemented as
electronic circuits where
signal values are
restricted to a few
discrete values
•Inbinary logic circuits
there are only two
values, 0 and 1
• The general form of a
logic circuit is a
switching network
Switching
Network
X
1
X
2
X
3
X
m
Y
1
Y
2
Y
3
Y
n
discrete values
2
Dr. D. J. Jackson Lecture 2-3Electrical & Computer Engineering
Boolean algebra
• Direct application to switching networks
– Work with 2-state devices Æ 2-valued Boolean
algebra (switching algebra)
– Use a Boolean variable (X, Y, etc.) to represent an
input or output of a switching network
– Variable may take on only two values (0, 1)
–X=0, X=1
– These symbols are not
binary numbers, they
simply represent the 2 states of a Boolean
variable
– They are not
voltage levels, although they
commonly refer to the low or high voltage
input/output of some circuit element
Dr. D. J. Jackson Lecture 2-4Electrical & Computer Engineering
Variables and functions
• The simplest binary element is a switch that
has two states
• If the switch is controlled by x, we say the
switch is open if x =0 and closed if x =1
x 1 = x 0 =
(a) Two states of a switch
S
x
(b) Symbol for a switch
3
Dr. D. J. Jackson Lecture 2-5Electrical & Computer Engineering
Variables and functions
• Assume the switch
controls a lightbulb as
shown
– The output is defined as
the state of the light L
• If the light is on -> L=1
• If the light is off -> L=0
• The state of L, as
function of x is
– L(x)=x
•L(x) is a logic function
• x is an input variable
(a) Simple connection to a battery
S
x
(b) Using a ground connection
as the return path
L
Battery
Light
x
Power
supply
S
L
Dr. D. J. Jackson Lecture 2-6Electrical & Computer Engineering
Variables and functions (AND)
• Consider the possibility of two switches
controlling the state of the light
• Using a series connection, the light will be on
only if both switches are closed
–L(x
1
, x
2
)= x
1
· x
2
– L=1 iff (if and only if) x
1
AND x
2
are 1
The logical AND function (series connection)
S
x
1
L
Power
supply
S
x
2
Light
“·” AND operator
The circuit implements
a logical AND function
x
1
· x
2
=x
1
x
2
4
Dr. D. J. Jackson Lecture 2-7Electrical & Computer Engineering
Variables and functions (OR)
• Using a parallel connection, the light will be
on only if either or both switches are closed
–L(x
1
, x
2
)= x
1
+ x
2
–L=1 if x
1
OR x
2
is 1 (or both)
S
x
1
L
Power
supply
S
x
2
The logical OR function (parallel connection)
Light
“+” OR operator
The circuit implements
a logical OR function
Dr. D. J. Jackson Lecture 2-8Electrical & Computer Engineering
Variables and functions
• Various series-parallel connections would
realize various logic functions
–L(x
1
, x
2
, x
3
)= (x
1
+ x
2
)· x
3
S
x
1
L
Power
supply S
x
2
Light
S
x
3
[...]... Computer Engineering f’= ΠM Dr D J Jackson Lecture 4-16 8 ECE380 Digital Logic Introduction to Logic Circuits: Design Examples Electrical & Computer Engineering Dr D J Jackson Lecture 5-1 Design examples • Logic circuits provide a solution to a problem • Some may be complex and difficult to design • Regardless of the complexity, the same basic design issues must be addressed 1 Specify the desired behavior... f(a,b,c)=(a’+b)c – f(a,b,c)=a’b+c Electrical & Computer Engineering Dr D J Jackson Lecture 3-17 9 ECE380 Digital Logic Introduction to Logic Circuits: Synthesis using AND, OR, and NOT gates Electrical & Computer Engineering Dr D J Jackson Lecture 4-1 Example logic circuit design • Assume we want to design a logic circuit with three inputs x, y, and z • The circuit output should be 1 only when x=1 and either... 1 Dr D J Jackson Lecture 2-14 7 Logic gates and networks • Each basic logic operation (AND, OR, NOT) can be implemented resulting in a circuit element called a logic gate • A logic gate has one or more inputs and one output that is a function of its inputs x1 x2 x1 x2 x1 ⋅ x2 ⋅ … ⋅ xn x1 ⋅ x2 xn AND gates Electrical & Computer Engineering Dr D J Jackson Lecture 2-15 Logic gates and networks x1 x2 x1... Electrical & Computer Engineering Dr D J Jackson Lecture 2-16 8 Logic gates and networks • A larger circuit is implemented by a network of gates – Called a logic network or logic circuit x x x 1 2 f = (x + x ) ⋅ x 1 2 3 3 Electrical & Computer Engineering Dr D J Jackson Lecture 2-17 Logic gates and networks • Draw the truth table and the logic circuit for the following function – F(a,b,c) = ac+bc’ a... function could be written as – f(x,y,z)=xy’z+xyz’+xyz • sum of products form Electrical & Computer Engineering Dr D J Jackson Lecture 4-2 1 Example logic circuit design Electrical & Computer Engineering Dr D J Jackson Lecture 4-3 Example logic circuit design • Implements f correctly, f(x,y,z)=xy’z+xyz’+xyz but is not the simplest such network xy’z+xyz’+xyz xy’z+xy from 14a y z f x 0 0 0 0 x(y’z+y)... Jackson Lecture 2-18 9 Analysis of a logic network • To determine the functional behavior of a logic network, we can apply all possible input signals to it x x 0→ 0→1→ 1 1→ 1→0→ 0 1 A 0→ 1→0→ 1 0→ 0→0→ 1 1→1→ 0→ 1 f B 2 Network that implements = x1 + x1 ⋅ x2 f Electrical & Computer Engineering Dr D J Jackson Lecture 2-19 Analysis of a logic network • The function of a logic network can also be described... manipulation can be used to simplify Boolean expressions – Simpler expression => simpler logic circuit • Not practical to deal with complex expressions in this way • However, the theorems & properties provide the basis for automating the synthesis of logic circuits in CAD tools – To understand the CAD tools the designer should be aware of the fundamental concepts Electrical & Computer Engineering Dr... by a timing diagram (gives dynamic behavior of the network) 1 x 1 0 x 1 2 0 A 1 0 B 1 0 f 1 0 Time Timing diagram Electrical & Computer Engineering Dr D J Jackson Lecture 2-20 10 ECE380 Digital Logic Introduction to Logic Circuits: Boolean algebra Electrical & Computer Engineering Dr D J Jackson Lecture 3-1 Axioms of Boolean algebra • Boolean algebra: based on a set of rules derived from a small number... xy’z+xy from 14a y z f x 0 0 0 0 x(y’z+y) from 12a 0 1 0 0 x(y+z) from 16a 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Electrical & Computer Engineering Dr D J Jackson Lecture 4-4 2 Example logic circuit design • Obviously, the cost (in terms of gates and connections) of this network is much less than the initial network • The process of generating a circuit from a stated desired functional behavior... OR and AND operations are often called the logical sum and product operations • The expression – ABC+A’BD+ACE’ – Is a sum of three product terms • The expression – (A+B+C)(A’+B+D)(A+C+E’) – Is a product of three sum terms Electrical & Computer Engineering Dr D J Jackson Lecture 3-15 Precedence of operations • In the absence of parentheses, operations in a logical expression are performed in the order . Engineering ECE380 Digital Logic Introduction to Logic Circuits: Variables, functions, truth tables, gates and networks Dr. D. J. Jackson Lecture 2-2Electrical & Computer Engineering Logic circuits • Logic. 1-1Electrical & Computer Engineering ECE380 Digital Logic Introduction Dr. D. J. Jackson Lecture 1-2Electrical & Computer Engineering Digital hardware • Logic circuits are used to build computer. & Computer Engineering Logic gates and networks • Each basic logic operation (AND, OR, NOT) can be implemented resulting in a circuit element called a logic gate • A logic gate has one or more