Computers and Electricity
Gate
A device that performs a basic operation on
electrical signals
Circuits
Gates combined to perform more
complicated tasks
Computers and Electricity
How do we describe the behavior of gates and circuits?
Boolean expressions
Uses Boolean algebra, a mathematical notation for expressing twovalued logic
Logic diagrams
A graphical representation of a circuit; each gate has its
own symbol
Truth tables
A table showing all possible input value and the associated
output values
Gates
Six types of gates
– NOT
– AND
– OR
– XOR
– NAND
– NOR
Typically, logic diagrams are black and white with gates
distinguished only by their shape
We use color for emphasis (and fun)
NOT Gate
A NOT gate accepts one input signal (0 or 1) and returns
the opposite signal as output
AND Gate
An AND gate accepts two input signals
If both are 1, the output is 1; otherwise,
the output is 0
OR Gate
An OR gate accepts two input signals
If both are 0, the output is 0; otherwise,
the output is 1
XOR Gate
An XOR gate accepts two input signals
If both are the same, the output is 0; otherwise,
the output is 1
XOR Gate
Note the difference between the XOR gate
and the OR gate; they differ only in one
input situation
When both input signals are 1, the OR gate
produces a 1 and the XOR produces a 0
XOR is called the exclusive OR
NAND Gate
The NAND gate accepts two input signals
If both are 1, the output is 0; otherwise,
the output is 1
NOR Gate
The NOR gate accepts two input signals
If both are 0, the output is 1; otherwise,
the output is 0
Review of Gate Processing
A NOT gate inverts its single input
An AND gate produces 1 if both input values are 1
An OR gate produces 0 if both input values are 0
An XOR gate produces 0 if input values are the same
A NAND gate produces 0 if both inputs are 1
A NOR gate produces a 1 if both inputs are 0
Gates with More Inputs
Gates can be designed to accept three or more input values
A three-input AND gate, for example, produces an output of 1
only if all input values are 1
Constructing Gates
Transistor
A device that acts either as a wire that conducts
electricity or as a resistor that blocks the flow of
electricity, depending on the voltage level of an input
signal
A transistor has no moving parts, yet acts like
a switch
It is made of a semiconductor material, which is neither a
particularly good conductor of electricity nor a
particularly good insulator
Constructing Gates
A transistor has three terminals
– A source
– A base
– An emitter, typically connected
to a ground wire
If the electrical signal is grounded,
it is allowed to flow through an
alternative route to the ground
(literally) where it can do no
harm
Constructing Gates
The easiest gates to create are the NOT, NAND, and
NOR gates
Circuits
Combinational circuit
The input values explicitly determine the output
Sequential circuit
The output is a function of the input values and the
existing state of the circuit
We describe the circuit operations using
Boolean expressions
Logic diagrams
Truth tables
Are you surprised?
Combinational Circuits
Gates are combined into circuits by using the output of
one gate as the input for another
Combinational Circuits
Three inputs require eight rows to describe all possible input
combinations
This same circuit using a Boolean expression is (AB + AC)
19
Combinational Circuits
Consider the following Boolean expression A(B + C)
Does this truth table look familiar?
Compare it with previous table
Combinational Circuits
Circuit equivalence
Two circuits that produce the same output for identical
input
Boolean algebra allows us to apply provable
mathematical principles to help design circuits
A(B + C) = AB + AC (distributive law) so circuits must
be equivalent
Properties of Boolean Algebra
Adders
At the digital logic level, addition is performed
in binary
Addition operations are carried out
by special circuits called, appropriately,
adders
Adders
The result of adding two
binary digits could
produce a carry value
Recall that 1 + 1 = 10
in base two
Half adder
A circuit that computes the
sum of two bits
and produces the correct
carry bit
Truth table
Adders
Circuit diagram
representing
a half adder
Boolean expressions
sum = A B
carry = AB
Adders
Full adder
A circuit that takes the carry-in value into account
Multiplexers
Multiplexer
A circuit that uses a few input control signals to
determine which of several output data lines is
routed to its output
Multiplexers
The control lines
S0, S1, and S2
determine
which of eight
other input lines
(D0 … D7)
are routed to the
output (F)
Figure 4.11 A block diagram of a multiplexer with three select
control lines
Circuits as Memory
Digital circuits can be used to store information
These circuits form a sequential circuit, because
the output of the circuit is also used as input to
the circuit
Circuits as Memory
An S-R latch stores a
single binary digit
(1 or 0)
There are several ways
an S-R latch circuit
can be designed
using various kinds
of gates
Circuits as Memory
The design of this circuit guarantees
that the two outputs X and Y are
always complements of each
other
The value of X at any point in time
is considered to be the current
state of the circuit
Therefore, if X is 1, the circuit is
storing a 1; if X is 0, the circuit
is storing a 0
Integrated Circuits
Integrated circuit (also called a chip)
A piece of silicon on which multiple gates have
been embedded
Silicon pieces are mounted on a plastic or
ceramic package with pins along the edges
that can be soldered onto circuit boards or
inserted into appropriate sockets
Integrated Circuits
Integrated circuits (IC) are classified by the
number of gates contained in them
Integrated Circuits
CPU Chips
The most important integrated circuit
in any computer is the Central Processing
Unit, or CPU
Each CPU chip has a large number of pins
through which essentially all communication
in a computer system occurs
Karnaugh Maps (K maps)
What are Karnaugh maps?
• Karnaugh maps provide an alternative way of simplifying
logic circuits.
• Instead of using Boolean algebra simplification techniques,
you can transfer logic values from a Boolean statement or a
truth table into a Karnaugh map.
• The arrangement of 0’s and 1’s within the map helps you to
visualise the logic relationships between the variables and
leads directly to a simplified Boolean statement.
Karnaugh maps
• Karnaugh maps, or K–maps, are often used to simplify logic
problems with 2, 3 or 4 variables.
AB
For the case of 2 variables, we form a map consisting of 22=4 cells
as shown in Figure
A
B
0 1
0 1
Cell = 2n ,where n is a number of variables
| 00 | 10 |
| 01 | 11 |
A
B
0 1
0 1
A
B
0 1
0 1
AB
AB AB
A+ B A + B
A + B A + B
Karnaugh maps
• 3 variables Karnaugh map
AB
C 00 01 11 10
0 1
ABC ABC ABC ABC
ABC ABC ABC ABC
Cell = 23=8
Karnaugh maps
• 4 variables Karnaugh map
AB
CD 00 01 11 10
00
01
11
10
• The Karnaugh map is completed by entering a ‘1‘(or
‘0’) in each of the appropriate cells.
• Within the map, adjacent cells containing 1’s (or 0’s)
are grouped together in twos, fours, or eights.
Karnaugh maps
Example
A B
Y
2-variable Karnaugh maps are trivial but can be used to introduce the methods you need
to learn. The map for a 2-input OR gate looks like this:
| A | B | Y |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
A
B
0 1
0 1
| 1 | |
| 1 | 1 |
B
A
A+B
Example
| A | B | C | Y |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
AC
B + AC
AB
C 00 01 11 10
0 1
1 1
| 1 |
1
1
B
Exercise
• Let us use Karnaugh map to simplify the follow
function.
F
1 = m0+m2+m3+m4+m5+m6+m7
F
2 = m0+m1+m2+m5+m7
• Answer
Exercise
| A | B | C | Y |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Given the truth table, find the simplified SOP and POS form.
Exercise
• Design two-level NAND-gate logic circuit from the
follow timing diagram.
A B C D F
Don’t care term
| X | |
| X | 1 |
| X | X |
| X | X |
AB
CD 00 01 11 10
00
01
11
10
AD
Exercise
• Design logic circuit that convert a 4-bits binary code to Excess-3 code
| A | B | C | D | W | X | Y | Z |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 | x | x | x | x |
| 1 | 0 | 1 | 1 | x | x | x | x |
| 1 | 1 | 0 | 0 | x | x | x | x |
| 1 | 1 | 0 | 1 | x | x | x | x |
| 1 1 | 1 1 | 1 1 | 0 1 | x X | X X | X x | X x |
Minimised Expression for each output:
Logic Circuit Diagram
Online Karnaugh map solver with circuit for
up to 6 variables:
http://www.32×8.com/index.html
Thank You M. Darwish