Digital Systems - Chapter04.pdf - po ang - Januszek66

CHAPTER 4

COMBINATIONAL

LOGIC CIRCUITS

■

OUTLINE

4-1

Sum-of-Products Form

4-10

Troubleshooting Digital

Systems

4-2

Simplifying Logic Circuits

4-11

Internal Digital IC Faults

4-3

Algebraic Simplification

4-12

External Faults

4-4

Designing Combinational

Logic Circuits

4-13

Troubleshooting Case Study

4-5

Karnaugh Map Method

4-14

Programmable Logic

Devices

4-6

Exclusive-OR and

Exclusive-NOR Circuits

4-15

Representing Data in HDL

4-7

Parity Generator and

Checker

4-16

Truth Tables Using HDL

4-17

Decision Control Structures

in HDL

4-8

Enable/Disable Circuits

4-9

Basic Characteristics of

Digital ICs

■ OBJECTIVES

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

■

Convert a logic expression into a sum-of-products expression.

Perform the necessary steps to reduce a sum-of-products expression to

its simplest form.

■

Use Boolean algebra and the Karnaugh map as tools to simplify and

design logic circuits.

■

Explain the operation of both exclusive-OR and exclusive-NOR circuits.

■

Design simple logic circuits without the help of a truth table.

■

Implement enable circuits.

■

Cite the basic characteristics of TTL and CMOS digital ICs.

■

Use the basic troubleshooting rules of digital systems.

■

Deduce from observed results the faults of malfunctioning

combinational logic circuits.

■

Describe the fundamental idea of programmable logic devices (PLDs).

■

Outline the steps involved in programming a PLD to perform a simple

combinational logic function.

■

Go to the Altera user manuals to acquire the information needed to do

a simple programming experiment in the lab.

■

Describe hierarchical design methods.

■

Identify proper data types for single-bit, bit array, and numeric value

variables.

■

Describe logic circuits using HDL control structures IF/ELSE, IF/ELSIF,

and CASE.

■

Select the appropriate control structure for a given problem.

■

■ INTRODUCTION

In Chapter 3, we studied the operation of all the basic logic gates, and we

used Boolean algebra to describe and analyze circuits that were made up of

combinations of logic gates. These circuits can be classified as

combinational logic circuits because, at any time, the logic level at the out-

put depends on the combination of logic levels present at the inputs. A

combinational circuit has no memory characteristic, so its output depends

only on the current value of its inputs.

In this chapter, we will continue our study of combinational circuits. To

start, we will go further into the simplification of logic circuits. Two methods

will be used: one uses Boolean algebra theorems; the other uses a mapping

technique. In addition, we will study simple techniques for designing

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C HAPTER 4/ C OMBINATIONAL L OGIC C IRCUITS

combinational logic circuits to satisfy a given set of requirements. A com-

plete study of logic-circuit design is not one of our objectives, but the meth-

ods we introduce will provide a good introduction to logic design.

A good portion of the chapter is devoted to the troubleshooting of com-

binational circuits. This first exposure to troubleshooting should begin to

develop the type of analytical skills needed for successful troubleshooting.

To make this material as practical as possible, we will first present some of

the basic characteristics of logic-gate ICs in the TTL and CMOS logic families

along with a description of the most common types of faults encountered in

digital IC circuits.

In the last sections of this chapter, we will extend our knowledge of pro-

grammable logic devices and hardware description languages. The concept

of programmable hardware connections will be reinforced, and we will pro-

vide more details regarding the role of the development system. You will

learn the steps followed in the design and development of digital systems

today. Enough information will be provided to allow you to choose the cor-

rect types of data objects for use in simple projects to be presented later in

this text. Finally, several control structures will be explained, along with

some instruction regarding their appropriate use.

4-1

SUM-OF-PRODUCTS FORM

The methods of logic-circuit simplification and design that we will study

require the logic expression to be in a sum-of-products (SOP) form. Some ex-

amples of this form are:

ABC

A B C

A B

A B C

C D

Each of these sum-of-products expressions consists of two or more AND terms

(products) that are ORed together. Each AND term consists of one or more

variables individually appearing in either complemented or unco mp l e -

mented form. For example, in the sum-of-products expression

the first AND product contains the variables A , B , and C in their uncomple-

mented (not inverted) form. The second AND term contains A and C in their

complemented (inverted) form. Note that in a sum-of-products expression,

one inversio n sig n c an not cover more than one variable in a term (e.g., we

cannot have

ABC

ABC ,

ABC

RST

Product-of-Sums

Another general form for logic expressions is sometimes used in logic-

circuit design. Called the product-of-sums (POS) form, it consists of two or

more OR terms (sums) that are ANDed together. Each OR term contains

one or more variables in complemented or uncomplemented form. Here

are some product-of-sum expressions:

( A

C )( A

C )

( A

B )( C

D ) F

( A

C )( B

D )( B

C )( A

E )

The methods of circuit simplification and design that we will be using

are based on the sum-of-products (SOP) form, so we will not be doing much

121

S ECTION 4-3/ A LGEBRAIC S IMPLIFICATION

with the product-of-sums (POS) form. It will, however, occur from time to

time in some logic circuits that have a particular structure.

REVIEW QUESTIONS

1. Which of the following expressions is in SOP form?

(a) AB

(b) AB ( C

D )

B )( C

F )

(d)

2. Repeat question 1 for the POS form.

4-2

SIMPLIFYING LOGIC CIRCUITS

Once the expression for a logic circuit has been obtained, we may be able to re-

duce it to a simpler form containing fewer terms or fewer variables in one or

more terms. The new expression can then be used to implement a circuit that is

equivalent to the original circuit but that contains fewer gates and connections.

To illustrate, the circuit of Figure 4-1(a) can be simplified to produce the

circuit of Figure 4-1(b). Both circuits perform the same logic, so it should be ob-

vious that the simpler circuit is more desirable because it contains fewer gates

and will therefore be smaller and cheaper than the original. Furthermore, the

circuit reliability will improve because there are fewer interconnections that

can be potential circuit faults.

FIGURE 4-1 It is often

possible to simplify a logic

circuit such as that in part

(a) to produce a more

efficient implementation,

shown in (b).

A + BC

x = A B(A + BC)

(a)

x = A B C

(b)

In subsequent sections, we will study two methods for simplifying logic

circuits. One method will utilize the Boolean algebra theorems and, as we

shall see, is greatly dependent on inspiration and experience. The other

method (Karnaugh mapping) is a systematic, step-by-step approach. Some

instructors may wish to skip over this latter method because it is somewhat

mechanical and probably does not contribute to a better understanding of

Boolean algebra. This can be done without affecting the continuity or clarity

of the rest of the text.

4-3

ALGEBRAIC SIMPLIFICATION

We can use the Boolean algebra theorems that we studied in Chapter 3 to

help us simplify the expression for a logic circuit. Unfortunately, it is not al-

ways obvious which theorems should be applied to produce the simplest

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C HAPTER 4/ C OMBINATIONAL L OGIC C IRCUITS

result. Furthermore, there is no easy way to tell whether the simplified

expression is in its simplest form or whether it could have been simplified

further. Thus, algebraic simplification often becomes a process of trial and

error. With experience, however, one can become adept at obtaining reason-

ably good results.

The examples that follow will illustrate many of the ways in which the

Boolean theorems can be applied in trying to simplify an expression. You

should notice that these examples contain two essential steps:

1. The original expression is put into SOP form by repeated application of

DeMorgan’s theorems and multiplication of terms.

2. Once the original expression is in SOP form, the product terms are

checked for common factors, and factoring is performed wherever possi-

ble. The factoring should result in the elimination of one or more terms.

EXAMPLE 4-1

Simplify the logic circuit shown in Figure 4-2(a).

A C

A B(A C)

z = ABC + AB(AC)

A B C

(a)

B + C

z = A(B + C)

(b)

FIGURE 4-2

Example 4-1.

Solution

The first step is to determine the expression for the output using the method

presented in Section 3-6. The result is

AB # ( A C )

ABC

Once the expression is determined, it is usually a good idea to break down

all large inverter signs using DeMorgan’s theorems and then multiply out

all terms.

ABC

A B ( A

C )

[theorem (17)]

ABC

A B ( A

C )

[cancel double inversions]

ABC

A B A

A BC

[multiply out]

ABC

[ A A

A ]

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