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chapter outcomes, 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. Describe how to 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 combina. tional logic circuits, Describe the fundamental idea of programmable logic devices PLDs.

Describe the steps involved in programming a PLD to perform a simple. combinational logic function,Describe hierarchical design methods. Identify proper data types for single bit bit array and numeric value. Describe logic circuits using HDL control structures IF ELSE IF. ELSIF and CASE, Select the appropriate HDL 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 combi. national logic circuits because at any time the logic level at the output. depends on the combination of logic levels present at the inputs A combi. national 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 design. ing combinational logic circuits to satisfy a given set of requirements A. complete study of logic circuit design is not one of our objectives but the. methods we introduce will provide a good introduction to logic design. M04 WIDM0130 12 SE C04 indd 137 1 8 16 8 38 PM,138 Chapter 4 Combinational Logic Circuits.

A good portion of this chapter is devoted to the topic of troubleshooting. This term has been adopted as a general description of the process of isolat. ing a problem or fault in any system and identifying a way of fixing it The. analytical skills and efficient methods of troubleshooting are equally appli. cable to any system whether it is a plumbing problem a problem with your. car a health issue or a digital circuit Digital systems implemented using. TTL integrated circuits have for decades provided an exceptional vehicle for. the study of efficient systematic troubleshooting methods As with any sys. tem the practical characteristics of the pieces that make up the system must. be understood in order to effectively analyze its normal operation locate the. trouble and propose a remedy We will present some basic characteristics. and typical failure modes of logic ICs in the TTL and CMOS families that are. still commonly used for laboratory instruction in introductory digital courses. and take advantage of this technology to teach some fundamental trouble. shooting principles, 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. Upon completion of this section you will be able to. Identify the form of a sum of products SOP expression. Identify the form of a product of sums POS expression. 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. examples of this form are,2 AB ABC C D D,3 AB CD EF GK HL. 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 uncomple. mented form For example in the sum of products expression ABC ABC. 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 inversion sign cannot cover more than one variable in a term e g we. cannot have ABC or 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. M04 WIDM0130 12 SE C04 indd 138 1 8 16 8 38 PM,Section 4 2 Simplifying Logic Circuits 139.

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,1 1A B C21A C2,2 1A B21C D2F. 3 1A C21B D21B C21A D E2, The methods of circuit simplification and design that we will be using. are based on the sum of products form so we will not be doing much with. the product of sums form It will however occur from time to time in some. logic circuits that have a particular structure, Outcome 1 Which of the following expressions is in SOP form. Assessment,Questions a AB CD E,c 1A B21C D F2,2 Repeat question 1 for the POS form. 4 2 Simplifying Logic Circuits, Upon completion of this section you will be able to.

Justify the use of simplification, Name two simplification techniques for digital circuits. Once the expression for a logic circuit has been obtained we may be able to. reduce 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. obvious that the simpler circuit is more desirable because it contains fewer. Figure 4 1 It is often,possible to simplify a logic. B BC x 5 A B A 1 BC,circuit such as that in part,a to produce a more C. efficient implementation,shown in b,M04 WIDM0130 12 SE C04 indd 139 1 8 16 8 38 PM.

140 Chapter 4 Combinational Logic Circuits, 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, Another strategic advantage of simplifying logic circuits involves the. operational speed of circuits Recall from previous discussions that logic. gates are subject to propagation delay If practical logic circuits are config. ured such that logical changes in the inputs must propagate through many. layers of gates in order to determine the output they cannot possibly oper. ate as fast as circuits with fewer layers of gates For example compare the. circuits of Figure 4 1 a and b In Figure 4 1 a the longest path a signal. must travel involves three gates In Figure 4 1 b the longest signal path. C only involves two gates Working toward a common form such as SOP. or POS assures similar propagation delay for all signals in the system and. helps determine the maximum operating speed of the system. 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,Outcome 1 List two advantages of simplification. Assessment,Questions 2 List two methods of simplification. 4 3 Algebraic Simplification, Upon completion of this section you will be able to.

Apply Boolean algebra theorems and properties to reduce Boolean. expressions,Manipulate expressions into POS or SOP form. 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. always obvious which theorems should be applied to produce the simplest. 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,M04 WIDM0130 12 SE C04 indd 140 1 8 16 8 38 PM. Section 4 3 Algebraic Simplification 141, 2 Once the original expression is in SOP form the product terms are checked. for common factors and factoring is performed wherever possible 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. z 5 ABC 1 AB AC,C z 5 A B 1 C,Figure 4 2 Example 4 1.

The first step is to determine the expression for the output using the method. presented in Section 3 6 The result is,z ABC AB 1A C2. 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. z ABC AB1A C2 theorem 17,ABC AB1A C2 cancel double inversions. ABC ABA ABC multiply out,ABC AB ABC A A A, With the expression now in SOP form we should look for common variables. among the various terms with the intention of factoring The first and third. terms above have AC in common which can be factored out. z AC1B B2 AB,Since B B 1 then,z AC112 AB,M04 WIDM0130 12 SE C04 indd 141 1 8 16 8 38 PM. 142 Chapter 4 Combinational Logic Circuits,We can now factor out A which results in.

This result can be simplified no further Its circuit implementation is shown. in Figure 4 2 b It is obvious that the circuit in Figure 4 2 b is a great deal. simpler than the original circuit in Figure 4 2 a, Example 4 2 Simplify the expression z AB C ABC ABC. The expression is already in SOP form, Method 1 The first two terms in the expression have the product AB in. common Thus,z AB1C C2 ABC,We can factor the variable A from both terms. Invoking theorem 15b, Method 2 The original expression is z AB C ABC ABC The first two. terms have AB in common The last two terms have AC in common How do. we know whether to factor AB from the first two terms or AC from the last. two terms Actually we can do both by using the ABC term twice In other. words we can rewrite the expression as,z AB C ABC ABC ABC.

where we have added an extra term ABC This is valid and will not change. the value of the expression because ABC ABC ABC theorem 7. Now we can factor AB from the first two terms and AC from the last two. z AB1C C2 AC1B B2,AB AC A1B C2, Of course this is the same result obtained with method 1 This trick of using. the same term twice can always be used In fact the same term can be used. more than twice if necessary,M04 WIDM0130 12 SE C04 indd 142 1 8 16 8 38 PM. Section 4 3 Algebraic Simplification 143,Example 4 3. Simplify z AC1ABD2 ABC D ABC,First use DeMorgan s theorem on the first term. z AC1A B D2 ABC D ABC step 1,Multiplying out yields.

z ACA ACB ACD ABC D ABC 2,Because A A 0 the first term is eliminated. z A BC ACD ABC D ABC 3, This is the desired SOP form Now we must look for common factors among. the various product terms The idea is to check for the largest common fac. tor between any two or more product terms For example the first and last. terms have the common factor BC and the second and third terms share the. common factor A D We can factor these out as follows. z BC1A A2 A D1C BC2 4, Now because A A 1 and C BC C B theorem 15a we have. z BC A D1B C2 5, This same result could have been reached with other choices for the. factoring For example we could have factored C from the first second and. fourth product terms in step 3 to obtain,z C1A B A D AB2 ABC D.

The expression inside the parentheses can be factored further. z C1B A A A D2 ABC D,Because A A 1 this becomes,z C1B A D2 ABC D. Multiplying out yields,z BC AC D ABC D, Now we can factor A D from the second and third terms to get. z BC A D1C BC2, If we use theorem 15a the expression in parentheses becomes B C. In addition we will study simple techniques for design ing combinational logic circuits to satisfy a given set of requirements A complete study of logic circuit design is not one of our objectives but the methods we introduce will provide a good introduction to logic design M04 WIDM0130 12 SE C04 indd 137 1 8 16 8 38 PM

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