Digital Systems - Chapter03.pdf - po ang - Januszek66

CHAPTER 3

DESCRIBING LOGIC

CIRCUITS

■

OUTLINE

3-1

Boolean Constants and

Variables

3-12

Universality of NAND Gates

and NOR Gates

3-2

Truth Tables

3-13

Alternate Logic-Gate

Representations

3-3

OR Operation with OR

Gates

3-14

Which Gate Representation

to Use

3-4

AND Operation with AND

Gates

3-15

IEEE/ANSI Standard Logic

Symbols

3-5

NOT Operation

3-16

Summary of Methods to

Describe Logic Circuits

3-6

Describing Logic Circuits

Algebraically

3-17

Description Languages

Versus Programming

Languages

3-7

Evaluating Logic-Circuit

Outputs

3-8

Implementing Circuits from

Boolean Expressions

3-18

Implementing Logic

Circuits with PLDs

3-9

NOR Gates and NAND

Gates

3-19

HDL Format and Syntax

3-20

Intermediate Signals

3-10

Boolean Theorems

3-11

DeMorgan’s Theorems

■ OBJECTIVES

Upon completion of this chapter, you will be able to:

■

Perform the three basic logic operations.

Describe the operation of and construct the truth tables for the AND,

NAND, OR, and NOR gates, and the NOT (INVERTER) circuit.

■

Draw timing diagrams for the various logic-circuit gates.

■

Write the Boolean expression for the logic gates and combinations of

logic gates.

■

Implement logic circuits using basic AND, OR, and NOT gates.

■

Appreciate the potential of Boolean algebra to simplify complex logic

circuits.

■

Use DeMorgan’s theorems to simplify logic expressions.

■

Use either of the universal gates (NAND or NOR) to implement a

circuit represented by a Boolean expression.

■

Explain the advantages of constructing a logic-circuit diagram using the

alternate gate symbols versus the standard logic-gate symbols.

■

Describe the concept of active-LOW and active-HIGH logic signals.

■

Draw and interpret the IEEE/ANSI standard logic-gate symbols.

■

Use several methods to describe the operation of logic circuits.

■

Interpret simple circuits defined by a hardware description language

(HDL).

■

Explain the difference between an HDL and a computer programming

language.

■

Create an HDL file for a simple logic gate.

■

Create an HDL file for combinational circuits with intermediate

variables.

■

■ INTRODUCTION

Chapters 1 and 2 introduced the concepts of logic levels and logic circuits.

In logic, only two possible conditions exist for any input or output: true and

false. The binary number system uses only two digits, 1 and 0, so it is perfect

for representing logical relationships. Digital logic circuits use predefined

voltage ranges to represent these binary states. Using these concepts, we

can create circuits made of little more than processed beach sand and wire

that make consistent, intelligent, logical decisions. It is vitally important

that we have a method to describe the logical decisions made by these cir-

cuits. In other words, we must describe how they operate. In this chapter,

we will discover many ways to describe their operation. Each description

C HAPTER 3/ D ESCRIBING L OGIC C IRCUITS

method is important because all these methods commonly appear in techni-

cal literature and system documentation and are used in conjunction with

modern design and development tools.

Life is full of examples of circumstances that are in one state or an-

other. For example, a creature is either alive or dead, a light is either on or

off, a door is locked or unlocked, and it is either raining or it is not. In 1854,

a mathematician named George Boole wrote An Investigation of the Laws of

Thought, in which he described the way we make logical decisions based on

true or false circumstances. The methods he described are referred to today

as Boolean logic, and the system of using symbols and operators to describe

these decisions is called Boolean algebra. In the same way we use symbols

such as x and y to represent unknown numerical values in regular algebra,

Boolean algebra uses symbols to represent a logical expression that has one

of two possible values: true or false. The logical expression might be door is

closed, button is pressed, or fuel is low . Writing these expressions is very te-

dious, and so we tend to substitute symbols such as A, B, and C .

The main purpose of these logical expressions is to describe the rela-

tionship between a logic circuit’s output (the decision) and its inputs (the

circumstances). In this chapter, we will study the most basic logic circuits—

logic gates —which are the fundamental building blocks from which all other

logic circuits and digital systems are constructed. We will see how the oper-

ation of the different logic gates and the more complex circuits formed

from combinations of logic gates can be described and analyzed using

Boolean algebra. We will also get a glimpse of how Boolean algebra can be

used to simplify a circuit’s Boolean expression so that the circuit can be re-

built using fewer logic gates and/or fewer connections. Much more will be

done with circuit simplification in Chapter 4.

Boolean algebra is not only used as a tool for analysis and simplifica-

tion of logic systems. It can also be used as a tool to create a logic circuit

that will produce the desired input/output relationship. This process is

often called synthesis of logic circuits as opposed to analysis. Other tech-

niques have been used in the analysis, synthesis, and documentation of

logic systems and circuits including truth tables, schematic symbols, timing

diagrams, and—last but by no means least—language. To categorize these

methods, we could say that Boolean algebra is a mathematic tool, truth ta-

bles are data organizational tools, schematic symbols are drawing tools,

timing diagrams are graphing tools, and language is the universal descrip-

tion tool.

Today, any of these tools can be used to provide input to computers. The

computers can be used to simplify and translate between these various

forms of description and ultimately provide an output in the form neces-

sary to implement a digital system. To take advantage of the powerful bene-

fits of computer software, we must first fully understand the acceptable

ways for describing these systems in terms the computer can understand.

This chapter will lay the groundwork for further study of these vital tools

for synthesis and analysis of digital systems.

Clearly the tools described here are invaluable tools in describing, ana-

lyzing, designing, and implementing digital circuits. The student who ex-

pects to work in the digital field must work hard at understanding and

becoming comfortable with Boolean algebra (believe us, it’s much, much

easier than conventional algebra) and all the other tools. Do all of the ex-

amples, exercises, and problems, even the ones your instructor doesn’t

assign. When those run out, make up your own. The time you spend will be

well worth it because you will see your skills improve and your confidence

grow.

S ECTION 3-2/ T RUTH T ABLES

3-1

BOOLEAN CONSTANTS AND VARIABLES

Boolean algebra differs in a major way from ordinary algebra because

Boolean constants and variables are allowed to have only two possible values,

0 or 1. A Boolean variable is a quantity that may, at different times, be equal

to either 0 or 1. Boolean variables are often used to represent the voltage

level present on a wire or at the input/output terminals of a circuit. For ex-

ample, in a certain digital system, the Boolean value of 0 might be assigned

to any voltage in the range from 0 to 0.8 V, while the Boolean value of 1 might

be assigned to any voltage in the range 2 to 5 V.*

Thus, Boolean 0 and 1 do not represent actual numbers but instead repre-

sent the state of a voltage variable, or what is called its logic level . A voltage

in a digital circuit is said to be at the logic 0 level or the logic 1 level, depend-

ing on its actual numerical value. In digital logic, several other terms are used

synonymously with 0 and 1. Some of the more common ones are shown in

Table 3-1. We will use the 0/1 and LOW/HIGH designations most of the time.

TABLE 3-1

Logic 0

Logic 1

False

True

Off

Low

High

Yes

Open switch

Closed switch

As we said in the introduction, Boolean algebra is a means for expressing

the relationship between a logic circuit’s inputs and outputs. The inputs are

considered logic variables whose logic levels at any time determine the out-

put levels. In all our work to follow, we shall use letter symbols to represent

logic variables. For example, the letter A might represent a certain digital

circuit input or output, and at any time we must have either

if not one, then the other.

Because only two values are possible, Boolean algebra is relatively easy

to work with compared with ordinary algebra. In Boolean algebra, there are

no fractions, decimals, negative numbers, square roots, cube roots, loga-

rithms, imaginary numbers, and so on. In fact, in Boolean algebra there are

only three basic operations: OR, AND, and NOT.

These basic operations are called logic operations. Digital circuits called

logic gates can be constructed from diodes, transistors, and resistors con-

nected so that the circuit output is the result of a basic logic operation (OR,

AND, NOT) performed on the inputs. We will be using Boolean algebra first

to describe and analyze these basic logic gates, then later to analyze and de-

sign combinations of logic gates connected as logic circuits.

3-2

TRUTH TABLES

A truth table is a means for describing how a logic circuit’s output depends

on the logic levels present at the circuit’s inputs. Figure 3-1(a) illustrates a

truth table for one type of two-input logic circuit. The table lists all possible

*Voltages between 0.8 and 2 V are undefined (neither 0 nor 1) and should not occur under normal cir-

cumstances.

C HAPTER 3/ D ESCRIBING L OGIC C IRCUITS

FIGURE 3-1 Example

truth tables for (a) two-

input, (b) three-input, and

Output

Inputs

(b)

(a)

(c)

combinations of logic levels present at inputs A and B, along with the corre-

sponding output level x . The first entry in the table shows that when A and B

are both at the 0 level, the output x is at the 1 level or, equivalently, in the 1

state. The second entry shows that when input B is changed to the 1 state, so

that and the output x becomes a 0. In a similar way, the table

shows what happens to the output state for any set of input conditions.

Figures 3-1(b) and (c) show samples of truth tables for three- and four-

input logic circuits. Again, each table lists all possible combinations of input

logic levels on the left, with the resultant logic level for output x on the right.

Of course, the actual values for x will depend on the type of logic circuit.

Note that there are 4 table entries for the two-input truth table, 8 entries

for a three-input truth table, and 16 entries for the four-input truth table.

The number of input combinations will equal 2 N for an N -input truth table.

Also note that the list of all possible input combinations follows the binary

counting sequence, and so it is an easy matter to write down all of the com-

binations without missing any.

REVIEW QUESTIONS

1. What is the output state of the four-input circuit represented in Figure

3-1(c) when all inputs except B are 1?

2. Repeat question 1 for the following input conditions:

1, B

0, C

3. How many table entries are needed for a five-input circuit?

3-3

OR OPERATION WITH OR GATES

The OR operation is the first of the three basic Boolean operations to be

learned. An example can be found in the kitchen oven. The light inside the

oven should turn on if either the oven light switch is on OR if the door is

opened . The letter A could be used to represent the oven light switch is on and

B could represent door is opened. The letter x could represent the light is on.

The truth table in Figure 3-2(a) shows what happens when two logic inputs,

A and B, are combined using the OR operation to produce the output x .The

table shows that x is a logic 1 for every combination of input levels where one

or more inputs are 1. The only case where x is a 0 is when both inputs are 0.

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