Boolean Algebra and Circuit Simplification Techniques

1. Chapter 5 Boolean Algebra and Reduction Techniques 1

2. Objectives • You should be able to: • Write Boolean equations for combinational logic applications. • Use Boolean algebra laws and rules to simplify combinational logic circuits. • Apply DeMorgan’s theorem to complex Boolean equations to arrive at simplified equivalent equations. 2

3. Objectives (Continued) • Design single-gate logic circuits by using the universal capability of NAND and NOR gates. • Troubleshoot combinational logic circuits. • Implement sum-of-products expressions using AND-OR-INVERT gates. 3

4. Objectives (Continued) • Use the Karnaugh mapping procedure to systematically reduce complex Boolean equations to their simplest form. • Describe the steps involved in solving a complete system design application. 4

5. Combinational Logic • Using two or more logic gates to perform a more useful, complex function • A combination of logic functions B = KD + HD • Boolean Reduction B = D(K+H) 5

6. Discussion Point • Write the Boolean equation for the circuit below: 6

7. VHDL Proof • A circuit reduction can easily be proved using VHDL • Write a program for each equation • Run a simulation for all possible input conditions • Compare the results

8. Boolean Laws and Rules • Commutative laws of addition and multiplication: The order of the variables does not matter. • A + B = B + A • AB = BA 7

9. Boolean Laws and Rules • Associative laws of addition and multiplication • A + (B + C) = (A + B) + C • A(BC) = (AB)C 8

10. Boolean Laws and Rules • Distributive laws • A(B + C) = AB + AC • (A + B)(C + D) = AC + AD + BC + BD 9

11. Boolean Laws and Rules • Rule 1: Anything ANDed with a 0 equals 0 • A • 0 = 0 • Rule 2: Anything ANDed with a 1 equals itself • A • 1 = A 10

12. Boolean Laws and Rules • Rule 3: Anything ORed with a 0 equals itself • A + 0 = A • Rule 4: Anything ORed with a 1 is equal to 1 • A + 1 = 1 11

13. Boolean Laws and Rules • Rule 5: Anything ANDed with itself is equal to itself • A • A = A • Rule 6: Anything ORed with itself is equal to itself • A + A = A 12

14. Boolean Laws and Rules • Rule 7: Anything ANDed with its complement equals 0 • A • A = 0 • Rule 8: Anything ORed with its complement equals 1 • A + A = 1 13

15. Boolean Laws and Rules • Rule 9: Anything complemented twice will return to its original logic level • A = A 14

16. Boolean Laws and Rules • Rule 10: • A + Ā B = A + B • Ā + AB = Ā + B 15

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18. Discussion Point • Which Boolean laws are illustrated below? • B + (D + E) = (B + D) + E • AB = BA • A + B + C = B + C + A • A(C + D) = AC + AD • What are some strategies for remembering the 10 Boolean rules? 17

19. Simplification of Combinational Logic Circuits Using Boolean Algebra • Equivalent circuits can be formed with fewer gates • Cost is reduced • Reliability is improved • Use laws and rules of Boolean Algebra 18

20. Simplification of Combinational Logic Circuits Using Boolean Algebra • Simplify the logic circuit shown by using the appropriate laws and rules. 19

21. Simplification of Combinational Logic Circuits Using Boolean Algebra • Simplify the logic circuit shown by using the appropriate laws and rules. 20

22. Using Quartus II to Simplify Equations • Quartus II software determines the simplest form of an equation during compiling. • It eliminates unnecessary gates • Minimizes the number of gates in the FPGA

23. DeMorgan’s Theorem • Used to simplify circuits containing NAND and NOR gates • A B = A + B • A + B = A B 21

24. DeMorgan’s Theorem • Break the bar over the variables and change the sign between them • Inversion bubbles - used instead of inverters to show inversion. • Use parentheses to maintain proper groupings • Results in Sum-of-Products (SOP) form 22

25. DeMorgan’s Theorem • Bubble Pushing 23

26. DeMorgan’s Theorem • Bubble Pushing • Change the logic gate • (AND to OR or OR to AND) • Add bubbles to the inputs and outputs where there were none and remove original bubbles 24

27. Truth Tables in VHDL using Vector Signals • Define inputs as an internal signal • Inputs are grouped as vectors • Assign values to the elements of the vector • Assign outputs for each input combination

28. The Universal Capability of NAND and NOR Gates • The NAND as an inverter. 25

29. The Universal Capability of NAND and NOR Gates • Forming an AND with two NANDs 26

30. The Universal Capability of NAND and NOR Gates • Forming an OR with three NANDs 27

31. The Universal Capability of NAND and NOR Gates • Forming a NOR with four NANDs 28

32. Discussion Point • The technique used to form all gates from NANDs can also be used with NOR gates. • Here is an inverter: • Form the other logic gates using only NORs. 29

33. AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions • Product-of-sums (POS) form 30

34. AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions • Sum-of-products (SOP) form 30

35. Karnaugh Mapping • Used to minimize the number of gates • Reduce circuit cost • Reduce physical size • Reduce gate failures • Requires SOP form 31

36. Karnaugh Mapping • Graphically shows output level for all possible input combinations • Moving from one cell to an adjacent cell, only one variable changes 32

37. Karnaugh Mapping • Steps for K-map reduction: • Transform the Boolean equation into SOP form • Fill in the appropriate cells of the K-map • Encircle adjacent cells in groups of 2, 4 or 8 • Watch for the wraparound • Find terms by determining which variables remain constant within circles 33

38. Discussion Point • Use a K-map to simplify the circuit. 34

39. System Design Applications • Use Karnaugh Mapping to reduce equations • Use AND-OR-INVERT gates to implement logic 35

40. System Design Applications • Use K-maps to simplify less complex circuits before implementing the circuit using an AOI. • More complex circuits are better suited for implementation using PLDs. 36

41. Summary • Several logic gates can be connected together to form combinational logic. • There are several Boolean laws and rules that provide the means to form equivalent circuits. • Boolean algebra is used to reduce logic circuits to simpler equivalent circuits that function identically to the original circuit. 41

42. Summary • DeMorgan’s theorem is required in the reduction process whenever inversion bars cover more than one variable in the original Boolean equation. • NAND and NOR gates are sometimes referred to as universal gates, because they can be used to form any of the other gates. 42

43. Summary • AND-OR-INVERT (AOI) gates are often used to implement sum-of-products (SOP) equations. • Karnaugh mapping provides a systematic method of reducing logic circuits. • Combinational logic designs can be entered into a computer using schematic block design software or VHDL. 43

44. Summary • Using vectors in VHDL is a convenient way to group like signals together similar to an array. • Truth tables can be implemented in VHDL using vector signals with the selected signal assignment statement. • Quartus II can be used to determine the simplified equation of combinational circuits. 43