# DIGITAL CIRCUIT LOGIC UNIT 5: KARNAUGH MAPS (K-MAPS)

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1 DIGITAL CIRCUIT LOGIC UNIT 5: KARNAUGH MAPS (K-MAPS) 1

2 Learning Objectives 1. Given a function (completely or incompletely specified) of three to five variables, plot it on a Karnaugh map. The function may be given in minterm, maxterm, or algebraic form. 2. Determine the essential prime implicants of a function from a map. 3. Obtain the minimum SOPs or minimum PoSs form of a function from the map. 4. Determine all of the prime implicants of a function from a map. 5. Understand the relation between operations performed using the map and the corresponding algebraic operations. 2

3 Minimum Form of Switching Functions Minimum Sum-of-Products: A minimum sum-of-products (SOP) expression for a function is defined as a sum of product terms which (a) has a minimum number of terms and (b) of all those expressions which have the same minimum number of terms, has a minimum number of literals. It corresponds directly to a minimum two-level gate circuit which has a minimum number of gates and gate inputs. 3

4 Minimum Form of Switching Functions How to Find a Minimum Sum-of-Products Given a Minterm Expansion: 1. Combine terms by using the uniting theorem XY + XY = X. Do this repeatedly to eliminate as many literals as possible. A given term may be used more than once because X + X = X. 2. Eliminate redundant terms by using the consensus theorem or other theorems. 4

5 Minimum Form of Switching Functions Example 1: 5

6 Minimum Form of Switching Functions Minimum Product-of-Sums: A minimum product-of-sums (PoS) expression for a function is defined as a product of sum terms which (a) has a minimum number of terms and (b) of all those expressions which have the same minimum number of terms, has a minimum number of literals. Given a maxterm expansion, the minimum product of sums can often be obtained by a procedure similar to that used in the minimum sum-of-products case, except that the uniting theorem (X + Y)(X + Y ) = X is used to combine terms. 6

7 Minimum Form of Switching Functions Example 2: 7

8 Two- and Three-Variable Karnaugh Maps Karnaugh Maps: A Karnaugh map is a systematic way of simplifying switching functions and lead directly to minimum cost two-level circuits composed of AND and OR gates. It specifies the value of the function for every combination of values of the independent variables. Two variable K-map 8

9 Two- and Three-Variable Karnaugh Maps Two Variable Karnaugh Maps: Note that the value of F for A = B = 0 is plotted in the upper left square, and the other map entries are plotted in a similar way in the figure below (Figure 5-1 in book). Each 1 on the map corresponds to a minterm of F. For example, a 1 in square 01 indicates that AB is a minterm. Minterms in adjacent squares of the map can be combined since they differ in only one variable. 9

10 Two- and Three-Variable Karnaugh Maps Three-Variable Karnaugh Maps: A three-variable Karnaugh map can be plotted in a similar way to the two-variable map. The value of one variable, A, is listed on the top and the vales of the other two, B and C, are listed on the side. 10

11 Two- and Three-Variable Karnaugh Maps Example 11

12 Two- and Three-Variable Karnaugh Maps Locations of Minterms on a Karnaugh Map: Minterms in adjacent squares of the map differ in only one variable and therefore can be combined using the uniting theorem XY + XY = X. 12

13 Two- and Three-Variable Karnaugh Maps Mapping Minterm and Maxterm Expressions on Karnaugh Maps: Given the minterm or maxterm expansion of a function, it can be mapped on a Karnaugh map as follows: 13

14 iclicker Question 12 Which one is correct? (1 minute) A) K-Maps can show Minterms only. B) K-Maps can show Maxterms only. C) K-Maps can be used to derive and simplify f. D) Two adjacent cells in a K-Map are different in two bits. E) K-Maps cannot be used to prove if two functions are equal. 14

15 iclicker Question 12 Which one is correct? (1 minute) A) K-Maps can show Minterms only. B) K-Maps can show Maxterms only. C) K-Maps can be used to derive and simplify f. D) Two adjacent cells in a K-Map are different in two bits. E) K-Maps cannot be used to prove if two functions are equal. 15

16 Two- and Three-Variable Karnaugh Maps Plotting Product Terms: To plot the term b, 1 s are entered in the four squares of the map where b = 1 as shown below: 16

17 Two- and Three-Variable Karnaugh Maps Plotting A Karnaugh Map using an Expression in Algebraic form: Given f(a,b,c)=abc +b c+a, we would plot the map: 17

18 Two- and Three-Variable Karnaugh Maps Simplifying Expressions: 18

19 Two- and Three-Variable Karnaugh Maps Consensus Theorem in Karnaugh Maps: 19

20 Two- and Three-Variable Karnaugh Maps If a function has two or more minimum sum-ofproducts forms, all of these forms can be determined from a map. Figure 5-9 shows the two minimum solutions for F = Σ m(0, 1, 2, 5, 6, 7). 20

21 Four-Variable Karnaugh Maps Location of terms on a Four-Variable K-map: 21

22 Four-Variable Karnaugh Maps Plotting functions on a Four-Variable Karnaugh Map: This is accomplished in the same way as for two- or threevariable Karnaugh maps. 1 s are plotted for whichever values of the variables would result in the expression yielding 1. 22

23 Four-Variable Karnaugh Maps Simplifying Expressions in Four-Variable Karnaugh Maps: 23

24 Four-Variable Karnaugh Maps Expressions with don t cares : Don t cares are noted as X s in Karnaugh maps. See below: 24

25 Four-Variable Karnaugh Maps From SOP to POS form using Karnaugh Maps: For the function specified below as f, the process of finding the product-of-sums from the sum-ofproducts is shown. 25

26 Determination of Minimum Expressions Using Essential Prime Implicants Prime Implicants: A product term implicant is called a prime implicant if it cannot be combined with another term to eliminate a variable. a b c, a cd, and ac are prime implicants a b c d, abc, and ab c are not prime implicants A sum-of-products expression containing a term which is not a prime implicant cannot be minimum. 26

27 Determination of Minimum Expressions Using Essential Prime Implicants Determination of All Prime Implicants: The minimum solution may not include all prime implicants, as shown below: 27

28 Determination of Minimum Expressions Using Essential Prime Implicants Essential Prime Implicants: If a minterm is covered by only one prime implicant, that prime implicant is said to be essential, and it must be included in the minimum sum of products. In order to find a minimum sum of products from a map, we should first loop all of the essential prime implicants. Simple word: Minimum form considers the essentials! 28

29 Determination of Minimum Expressions Using Essential Prime Implicants Finding Essential Prime Implicants: Sometimes essential prime implicants can be found by inspection. Other times, we must look at all squares adjacent to that minterm. If the given minterm and all of the 1 s adjacent to it are covered by a single term, then that term is an essential prime implicant. If all of the 1 s adjacent to a given minterm are not covered by a single term, then we cannot say whether these prime implicants are essential or not without checking the other minterms. See figure on next page. 29

30 Determination of Minimum Expressions Using Essential Prime Implicants 30

31 Determination of Minimum Expressions Using Essential Prime Implicants Procedure to Obtain a Minimum Sum of Products from a Karnaugh Map: 1. Choose a minterm (a 1) which has not yet been covered. 2. Find all 1 s and X s adjacent to that minterm. (Check the n adjacent squares on an n-variable map.) 3. If a single term covers the minterm and all of the adjacent 1 s and X s, then that term is an essential prime implicant, so select that term. (Note that don t-care terms are treated like 1 s in steps 2 and 3 but not in step 1.) 31

32 Determination of Minimum Expressions Using Essential Prime Implicants Procedure (continued): 4. Repeat steps 1, 2, and 3 until all essential prime implicants have been chosen. 5. Find a minimum set of prime implicants which cover the remaining 1 s on the map. If there is more than one such set, choose a set with a minimum number of literals. See figure 5-19 in book for a flowchart of this procedure (pg 148). 32

33 Five-Variable Karnaugh Maps Five-Variable Karnaugh Maps: A five-variable map can be constructed in three dimensions by placing one four-variable map on top of a second one. Terms in the bottom layer are numbered 0 through 15 and corresponding terms in the top layer are numbered 16 through 31, so that terms in the bottom layer contain A and those in the top layer contain A. To represent the map in two dimensions, we will divide each square in a four-variable map by a diagonal line and place terms in the bottom layer below the line and terms in the top layer above the line (Figure 5-21). 33

34 Five-Variable Karnaugh Maps 34

35 Five-Variable Karnaugh Maps Example of Five-Variable Karnaugh Map: 35

36 Five-Variable Karnaugh Maps Simpler representation (a.k.a. Veitch map) A BC DE ABC DE Every cell in the left-hand side map is logically adjacent with the mirror image cell in the right-hand side map. 36

37 Five-Variable Karnaugh Maps Simpler representation (a.k.a. Veitch map) A BC DE ABC DE Every cell in the left-hand side map is logically adjacent with the mirror image cell in the right-hand side map. 37

38 Five-Variable Karnaugh Maps Simpler representation (a.k.a. Veitch map) A BC DE ABC DE Every cell in the left-hand side map is logically adjacent with the mirror image cell in the right-hand side map. 38

39 Five-Variable Karnaugh Maps Simpler representation (a.k.a. Veitch map) A BC DE ABC DE Every cell in the left-hand side map is logically adjacent with the mirror image cell in the right-hand side map. 39

40 Example Five-Variable Karnaugh Maps A BC DE ABC DE Every cell in the left-hand side map is logically adjacent with the mirror image cell in the right-hand side map. 40

41 Example Five-Variable Karnaugh Maps A BC DE ABC DE Many examples: 41

42 Other Uses of Karnaugh Maps Other Uses: We can prove that two functions are equal by plotting them on maps and showing that they have the same Karnaugh map. We can perform the AND operation (or the OR operation) on two functions by ANDing (or ORing) the 1 s and 0 s which appear in corresponding positions on their maps. A Karnaugh map can facilitate factoring an expression. When simplifying a function algebraically, the Karnaugh map can be used as a guide in determining what steps to take (see pages ). 42

43 Other Forms of Karnaugh Maps Veitch Diagrams: Instead of labeling the sides of a Karnaugh map with 0 s and 1 s, some people prefer to use the labeling shown in Figure For the half of the map labeled A, A=1; and for the other half, A=0. 43

44 Other Forms of Karnaugh Maps Other forms of Five-Variable Karnaugh Maps: One form simply consists of two four-variable maps side-byside as in Figure 5-28(a). Figure 5-28(b) shows mirror image map, in which the first and eighth columns are adjacent as are second and seventh columns, third and sixth columns, and fourth and fifth columns. 44

45 Finance in Engineering What is the most profitable company in the world? Apple (2016 profit was \$18 Billion on \$75.9 Billion in revenue) How do they make this profit? Competitor(s)? Fox Business: Between fiscal 2008 and 2016, Apple's iphone shipments surged from 11.6 million to million. During that period, Apple's annual revenue rose from \$32.5 billion to \$215.6 billion.

46 Finance in Engineering What is the most profitable company in the world? Apple (2016 profit was \$18 Billion on \$75.9 Billion in revenue) How do they make this profit? Competitor(s)? Samsung fabricates some parts of iphones: Application Processor! What makes Apple this successful? 1- Engineering reason: perfect customized design for their products 2- Marketing reason:

47 Finance in Engineering What is the most profitable company in the world? Apple (2016 profit was \$18 Billion on \$75.9 Billion in revenue) How do they make this profit? Competitor(s)? Samsung fabricates some parts of iphones: Application Processor! What makes Apple this successful? 1- Engineering reason: perfect customized design for their products 2- Marketing reason: Apple Stores!

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