Lecture No 8 Boolean Algebra and Logic Simplification

Lecture No. 8 Boolean Algebra and Logic Simplification

Digital Logic & Design Dr. Waseem Ikram Lecture 08

Recap n Operational Characteristics n DC Supply Voltage n Noise Margin n Power Dissipation n Frequency Response n Fan Out

TTL Series 74 74 S 74 LS 74 ALS 74 F Propagation Delay (ns) 9 3 9. 5 1. 7 4 3 Power Dissipation (m. W) 10 20 2 8 1. 2 6 Speed-Power (p. J) 90 60 19 13. 6 4. 8 18 Max. Clock Rate (MHz) 35 125 45 200 70 100 Fan-out (same series) 10 20 20 40 20 33 Performance Rating product

CMOS Series 74 HC 74 AHC Propagation Delay (ns) 18 5 3. 7 Power Dissipation (m. W) Static 0. 00275 0. 0055 0. 00275 Power Dissipation (m. W) at 100 KHz 0. 0625 0. 08 0. 0625 Speed-Power product (p. J) at 100 KHz 1. 125 0. 4 0. 23 Max. Clock Rate (MHz) 50 160 170 74 LVC 74 ALVC Propagation Delay (ns) 9 4. 3 3 Power Dissipation (m. W) Static 0. 0016 0. 0008 Max. Clock Rate (MHz) 90 100 150

Boolean Algebra n n n Variable Complement Literal

Boolean Addition & Multiplication n n Boolean Addition performed by OR gate Sum Term describes Boolean Addition Boolean Multiplication performed by AND gate Product Term describes Boolean Multiplication

Boolean Addition n Sum of literals Sum term = 1 if any literal = 1 Sum term = 0 if all literals = 0

Boolean Multiplication n Product of literals Product term = 1 if all literals = 1 Product term = 0 if any one literal = 0

Laws, Rules & Theorems of Boolean Algebra n n n Commutative Law for addition and multiplication Associative Law for addition and multiplication Distributive Law Rules of Boolean Algebra Demorgan’s Theorems

Commutative Law n n Commutative Law for Addition A+B=B+A Commutative Law for Multiplication A. B = B. A

Associative Law n Associative Law for Addition A + (B + C) = (A + B) + C

Associative Law n Associative Law for Multiplication A. (B. C) = (A. B). C

Distributive Law A. (B + C) = A. B + A. C

Rules of Boolean Algebra 1. 2. 3. 4. 5. 6. A+0=A A+1=1 A. 0 = 0 A. 1 = A A+A=A A+ =1 7. 8. 9. 10. 11. 12. A. A = A A. = 0 =A A + A. B = A A+ =A+ B (A+B). (A+C) = A+B. C

Demorgan’s Theorems n First Theorem n Second Theorem

Demorgan’s Theorems n Any number of variables n Combination of variables

Boolean Analysis of Logic Circuits n n Boolean Algebra provides concise way to represent operation of a logic circuit Complete function of a logic circuit can be determined by evaluating the Boolean expression using different input combinations

Boolean Analysis of Logic Circuits n n From the expression, the output is a 1 if variable D = 1 and =1 =1 if AB=1 or C=0

Boolean Analysis of Logic Circuits Inputs Output A 0 0 B 0 0 C 0 0 D 0 1 F 0 1 A 1 1 B 0 0 C 0 0 D 0 1 F 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1

Simplification using Boolean Algebra

Simplification using Boolean Algebra n AB + A(B+C) + B(B+C) = AB + AC + BB +BC = AB + AC + B = B + AC

Simplified Circuit

Standard forms of Boolean Expressions n n Sum-of-Products form Product-of-Sums form

Standard forms of Boolean Expressions n n Sum-of-Products form AB + ABC + CDE + Product-of-Sums form

Implementation of SOP expression

Implementation of POS expression

Conversion of general expression to SOP form