SEE1223: Digital Electronics


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1 SEE223: Digital Electronics 3 Combinational Logic Design Zulkifil Md Yusof Dept. of Microelectronics and Computer Engineering The aculty of Electrical Engineering Universiti Teknologi Malaysia
2 Karnaugh Maps (KMap) KMap structure 2,3, and 4variable Karnaugh Maps KMap Grouping and Logic Simplification KMap SOP and POS terms KMap Don t Care Conditions Logic Design using KMaps 2/8/22 A.A.H AbRahman, Z.MdYusof 2
3 Introduction Karnaugh Map (KMap) is a tool for simplifying digital logic with 26 variables KMap, if properly used will produce the simplest SOP and POS expression possible, known as the minimum expression KMap simplifies logic through SOP and POS boolean expressions, and truth table In this class, we ll look at logic simplification of 2, 3, and 4 variables 2/8/22 A.A.H AbRahman, Z.MdYusof 3
4 2variable KMaps KMap is a representation of a truth table, but can be used to obtain Boolean expressions (A,B) truth table A B 2variable KMap B A Cell location of each entry 2/8/22 A.A.H AbRahman, Z.MdYusof 4
5 3variable KMap (A,B,C) truth table A B C 3variable KMap Gray code ordering BC A 2/8/22 A.A.H AbRahman, Z.MdYusof 5
6 4variable KMap (A,B,C,D) truth table 4variable KMap A B C D A B C D CD 2/8/22 A.A.H AbRahman, Z.MdYusof 6
7 KMap Example Given the following standard form of SOP, complete the truth table and Kmap A B C 2/8/22 A.A.H AbRahman, Z.MdYusof 7 A BC
8 KMap Example Given the following SOP expression, complete the KMap BC D D D = when B = and C = ( st minterm) when A = and B = (2 nd minterm) when A =, B =, and C = (3 rd minterm) when A =, B =, C =, and D = (4 th minterm) when A =, B =, C =, and D = (5 th minterm) when A =, B =, C =, and D = (6 th minterm) CD 2/8/22 A.A.H AbRahman, Z.MdYusof 8
9 KMap Grouping After SOP expression has been mapped, minimum expression is obtained by grouping the s and determining the minimum SOP expression from the map When grouping the s, the goal is to maximize the size of the groups, and minimize the number of groups 2/8/22 A.A.H AbRahman, Z.MdYusof 9
10 KMap Grouping (cont.) Rules for grouping of s A group must contain either, 2, 4, 8, or 6 cells. or xvariable Kmap, 2 x cells is maximum Each cell in a group must be adjacent to one or more cells in that same group, but all cells in the group don t have to be adjacent to each other Always include the largest possible number of s in a group Each on the map must be included in at least one group. The s already in a group can be included in another group as long as the overlapping groups include common s 2/8/22 A.A.H AbRahman, Z.MdYusof
11 KMap Minimum Product Term or 3variable KMap cell group yields a 3variable product term 2 cell group yields a 2variable product term 4 cell group yields a variable product term 8 cell group yields a value of for the expression or 4variable KMap cell group yields a 4variable product term 2 cell group yields a 3variable product term 4 cell group yields a 2variable product term 8 cell group yields a variable product term 6 cell group yields a value of for the expression 2/8/22 A.A.H AbRahman, Z.MdYusof
12 KMap Simplification Group the s and find the minimum SOP expression in the KMap below A BC BC BC Expression is minimized when taking large cell possible What is the SOP expression if each cell is taken as a group? 2/8/22 A.A.H AbRahman, Z.MdYusof 2
13 KMap Simplification ind the minimum SOP expression for the logic expression: ( A, B, C) A BC (3,5) C C 2/8/22 A.A.H AbRahman, Z.MdYusof 3
14 KMap Simplification Group the s and find the minimum SOP expression AC CD D AC D 2/8/22 A.A.H AbRahman, Z.MdYusof 4
15 KMap Simplification ind the minimum expression for the logic expression: ( A, B, C, D) BC CD (,2,4,5,6,8,,,2,3,4) D D BC 2/8/22 A.A.H AbRahman, Z.MdYusof 5
16 Don t Care Conditions Don t Care is the condition when the output can either be or, which is denoted by x in the truth table or KMap or both SOP and POS minimum expression, x can be included or ignored 2/8/22 A.A.H AbRahman, Z.MdYusof 6
17 Don t Care Condition (cont.) ind minimum SOP expression for the following KMap CD AD x x AD If the x is replaced by, find the minimum SOP expression AD D 2/8/22 A.A.H AbRahman, Z.MdYusof 7
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