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