1 EENG 2710 Chapter 3 Simplification of Switching Functions

2 Chapter 3 Homework 3.1b, 3.2b, 3.3b, 3.3c,

3 Simplification By Karnaugh Mapping A Karnaugh map, called a K-map, is a graphical tool used for simplifying Boolean expressions.

4 Construction of a Karnaugh Map Square or rectangle divided into cells. Each cell represents a line in the truth table. Cell contents are the value of the output variable on that line of the truth table.

5 Construction of a Karnaugh Map

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10 Construction of a Karnaugh Map (From Minterm) F(BA) =  m(0, 2)

11 Construction of a Karnaugh Map (From Truth Table) SOP Method  m(1, 3, 6, 7) F(ABC) = A’C + AB

12 Construction of a Karnaugh Map (From Truth Table) POS Method  M(0, 2, 4, 5) F(ABC) = (A + C)(A’ + B)

13 Grouping Cells Cells can be grouped as pairs, quads, and octets. A pair cancels one variable. A quad cancels two variables. An octet cancels three variables.

14 Grouping Cells Keep: A’Discord: A, B, & B’ Y = A’ Y = A’B’ + A’B

15 Grouping Cells Keep: C Discord: A, A’, B, &B’ Y = C Y = A’B’C + A’BC + ABC + AB’C

16 Grouping Cells Keep: B Discard: A, C & D Y = B Y = A’BC’D’ + A’BC’D + A’BCD + A’BCD’ + ABC’D’ + ABC’D + ABCD + ABCD’

17 Grouping Cells along the Outside Edge The cells along an outside edge are adjacent to cells along the opposite edge. In a four-variable map, the four corner cells are adjacent.

18 Grouping Cells along the Outside Edge (SOP Method Shown)

19 Multiple Groups Each group is a term in the maximum SOP expression. A cell may be grouped more than once as long as every group has at least one cell that does not belong to any other group. Otherwise, redundant terms will result.

20 Multiple Groups

21 Maximum Simplification Achieved if the circled group of cells on the K-map are as large as possible. There are as few groups as possible.

22 Maximum Simplification Keep: B’ & D Y = B’ + D

23 Using K-Maps for Partially Simplified Circuits Fill in the K-map from the existing product terms. Each product term that is not a minterm will represent more than one cell. Once completed, regroup the K-map for maximum simplification.

24 Using K-Maps for Partially Simplified Circuits

25 Using K-Maps for Partially Simplified Circuits

26 Don’t Care States The output state of a circuit for a combination of inputs that will never occur. Shown in a K-map as an “x”.

27 Value of Don’t Care States In a K-map, set “x” to a 0 or a 1, depending on which case will yield the maximum simplification.

28 Value of Don’t Care States

29 POS Simplification Using Karnaugh Mapping Group those cells with values of 0. Use the complements of the cell coordinates as the sum term.

30 POS Simplification Using Karnaugh Mapping

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Problem 3.2d Minimize the following function containing don’t-cares the using K-map F(A,B,C,D,E) =  m(3,4,6,9,11,13,15,18,25,26,27,29,31)) 33 f(A,B,C,D,E) = BE + A’B’CE’ + A’C’DE + AC’DE’ A’B’CE’ A’C’DE BE AC’DE’

Problem 3.3 a Minimize the following function containing don’t-cares the using K-map F(A,B,C,D) =  m(2,9,10,12,13) + d(1,5,14) 34 f(A,B,C,D,) = C’D + B’CD’ + ABC’ ABC’ C’D B’CD

Problem 3.3 a (Continued) 35 C’D B’CD ABD’ f(A,B,C,D,) = C’D + B’CD’ + ABD’

36 Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) b.Combine minterm groups with same number of 1’s. 1.Build List 1. a.Start by setting minterms to binary. Minterm = ABCD 2 = 0010  4 = 0100  6 = 0110  8 = 1000  9 = 1001  10 = 1010  12 = 1100  13 = 1101  15 = 1111 

Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) 2. Build List is in 6 2 is in 10 8 is in 9 4 is in 12 - = place where the two differ

38 Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) 3.Build List 3

39 Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) 4. Build Table 1. f(A,B,C,D) = PI 1 + PI 7

40 Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) 5. Build Table 2. f(A,B,C,D) = PI 1 + PI 3 + PI 4 + PI 7

41 Quine-McCluskey Method f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15)

K-Map f(A,B,C,D) =  m(2, 4, 6, 8, 9, 10, 12, 13, 15) 42 AC’ ABD B’CD’ A’B’D’ f(ABCD) = AC’ + B’CD’ + A’BD’ +ABD