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# Pe 4030 digital logic chapter 7 (weeks 11 12)

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digital logic automation mechatronics Boolean Algebra, Karnaugh Mapping, Application of logic gates,Number system

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### Pe 4030 digital logic chapter 7 (weeks 11 12)

1. 1. PE-4030 Weeks 11 & 12 Chapter 8: Digital Logic Professor Charlton S. Inao Professor , Mechatronics System Design Defence University College of Engineering Bishoftu, Ethiopia
2. 2. Instructional Objectives To explain and understand the following concept: 1. Digital Logic 2. Number Systems 3. Logic Gates 4. Boolean Algebra 5. Karnaugh Maps
3. 3. Digital Logic • Digital circuitry is the basis of digital computers and microprocessor controlled systems. • Digital logic operates with digital signals where there are only two possible signal levels. • This circuitry evolved from the transistor circuits being able to output at one of two voltage levels depending on the levels at its inputs. The two levels usually 5V and 0V are the high and low signals and represented by 1 and 0.
4. 4. Number System In Mechatronics Decimal Binary 0 0000 1 BCD Octal Hexadecimal 00000000 0 0 0001 0000 0001 1 1 2 0010 0000 0010 2 2 3 0011 0000 0011 3 3 4 0100 0000 0100 4 4 5 0101 0000 0101 5 5 6 0110 0000 0110 6 6 7 0111 0000 0111 7 7 8 1000 0000 1000 10 8 9 1001 0000 1001 11 9 10 1010 0001 0000 12 A 11 1011 0001 0001 13 B 12 1100 0001 0010 14 C 13 1101 0001 0011 15 D 14 1110 0001 0100 16 E 15 1111 0001 0101 17 F
5. 5. LOGIC GATES
6. 6. AND, OR , NOT
7. 7. NAND , NOR, EXOR
8. 8. Boolean Algebra • An algebraic system that describes the logic circuit, in which the variables are limited to two values, usually 0 and 1. • George Boole developed an algebra for values for the systematic treatment of logic. • Boolean algebra deals with variables that take on two discrete values, 0 and 1 , and with operations that assume logical meaning. • Situations involving “yes-no, true –false,on-off” can be represented by Boolean Logical operations.
9. 9. OR Boolean Algebra Laws 1) A + 1= 1 2) A + 0 = A 3) A.0 = 0 4) A.1 = A 5) A + A =A elec 6) A.A = A elec 7)A.A = 0 elec 8) A + A = 1 elec 9) A + B = B + A elec OR AND OR AND AND OR 10) AB + AC= A(B + C)elec elec 11) A + BC =(A+B)(A+C) 12) A + B = A.B elec 13) A.B = A + B 14) AΦ B= A.B + A.B elec 15) A + AB = A + B NAND (exor) elec
10. 10. Boolean Algebra Laws 1) Anything Ored to itself is equal to itself. A + A =A 2) Anything ANDed to itself is equal to itself. A . A =A 3) It does not matter in which order we consider inputs for OR and AND gates. A+B=B+A A.B=B.A 4) We can use truth table to show we can treat bracketed terms in the same way as the ordinary algebra. A. (B +C)=A.B + A.C A +(B.C) =( A+B) . (A+C)
11. 11. Boolean Algebra Laws 5) Anything ORed with its own inverse equals 1. A +A =1 6) Anything ANDed with its own inverse equals =0 A.A=0 7) Anything Ored with a zero is equal to itself. Anything Ored with a 1 is equal to 1. A + 0 =A ; A + 1= 1 8) Anything ANDed with a 0 is equal to zeo; anything ended with 1 is equal to itself. A.0 = 0 A.1 = A
12. 12. Six Axioms on Properties of Boolean Algebra Commutative Axiom: A.B=B.A A+B=B+A Distributive Axiom: A.(B+C)=(A.B) +(A.C) A+(B.C)=(A+B ).(A+C) Idempotency Axiom: A.A=A A+A=A Absorption Axiom A.(A +B)=A A +(A.B)=A Complementation Axiom A.A=0 A+A= 1 A=A De Morgan’s theorem A.B= A + B A+B= A. B
13. 13. De Morgan’s Law
14. 14. Application of Logic Gates Application No.1
15. 15. Application No: 2 • A system uses 3 switches: A,B and C. A combination of the three switches determines whether an alarm ,X, will make a sound. • If switch A or B are in the ON position, and if switch C is in the OFF position then a signal to sound an alarm X, is produced.
16. 16. • Solution • 1)Construct a Truth Table for 3 inputs , A,B,C(23=8) A B C 1 0 0 0 2 0 0 1 3 0 1 0 4 0 1 1 5 1 0 0 6 1 0 1 Output X 0 0 1 0 1 0 1 0
17. 17. • 2) Get the value of P and Q, form the logic of A,B,and C based on the logic circuit. A B P C Q=c 1 0 0 0 0 1 2 0 0 0 1 0 0 1 3 0 1 1 1 0 4 0 1 1 0 1 5 1 0 1 1 0 0 1 6 1 1 0 0 1
18. 18. P and Q= X P Q X 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 (A+B).C=X
19. 19. Karnaugh Maps
20. 20. Three Variable Map
21. 21. Four Variable Map
22. 22. Four Variable Map
23. 23. Application of Karnaugh Map
24. 24. Karnaugh Map
25. 25. K- Map Overlapping Group
26. 26. The End