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Digital Logic1.Digital Logic
2. Number Systems 3. Logic Gates
4. Boolean Algebra 5. Karnaugh Maps
PE-4030Weeks 11 & 12
Chapter 8:
Digital LogicProfessor Charlton S. InaoProfessor , Mechatronics System DesignDefence UniversityCollege of EngineeringBishoftu, Ethiopia
Instructional Objectives
To explain and understand the following concept:
1. Digital Logic 2. Number Systems 3. Logic Gates
4. Boolean Algebra 5. Karnaugh Maps
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.
Number System In MechatronicsDecimal Binary BCD Octal Hexadecimal
0 0000 00000000 0 0
1 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
LOGIC GATES
AND, OR , NOT
NAND , NOR, EXOR
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.
Boolean Algebra Laws1) A + 1= 12) A + 0 = A3) A.0 = 04) A.1 = A5) A + A =A6) A.A = A7)A.A = 08) A + A = 19) A + B = B + A
10) AB + AC= A(B + C)11) A + BC =(A+B)(A+C)12) A + B = A.B13) A.B = A + B14) AΦ B= A.B + A.B(exor)
15) A + AB = A + Belec
elec
elec
elec
elec
elec
elec
elec
NAND
elec
elec
OR
OR
OR
OR
AND
AND
AND
Boolean Algebra Laws1) Anything Ored to itself is equal to itself.2) Anything ANDed to itself is equal to itself. 3) It does not matter in which order we consider
inputs for OR and AND gates.
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)
A + A =A
A . A =A
A + B = B + A A . B = B . A
Boolean Algebra Laws5) Anything ORed with its own inverse equals 1. A +A =16) Anything ANDed with its own inverse equals =0 A.A=07) Anything Ored with a zero is equal to itself. Anything
Ored with a 1 is equal to 1. A + 0 =A ; A + 1= 18) Anything ANDed with a 0 is equal to zeo; anything
ended with 1 is equal to itself. A.0 = 0 A.1 = A
Six Axioms on Properties of Boolean Algebra
Commutative Axiom:A.B=B.AA+B=B+ADistributive Axiom:A.(B+C)=(A.B) +(A.C)A+(B.C)=(A+B ).(A+C)Idempotency Axiom:A.A=AA+A=A
Absorption AxiomA.(A +B)=AA +(A.B)=AComplementation Axiom
A.A=0
A+A= 1A = A
De Morgan’s theoremA.B= A + BA+B= A. B
De Morgan’s Law
Application of Logic GatesApplication No.1
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.
• 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
7 1 1 0
8 1 1 1
Output
X
0
0
1
0
1
0
1
0
• 2) Get the value of P and Q, form the logic of A,B,and C based on the logic circuit.
A B P1 0 0 0
2 0 0 0
3 0 1 1
4 0 1 1
5 1 0 1
6 1 0 1
7 1 1 1
8 1 1 1
C Q=c0 1
1 0
0 1
1 0
0 1
1 0
0 1
1 0
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
P and Q= X
(A+B).C=X
Karnaugh Maps
Three Variable Map
Four Variable Map
Four Variable Map
Application of Karnaugh Map
Karnaugh Map
K- Map Overlapping Group
The End