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Understanding Logic Circuits: Truth Tables, Gates, and Boolean Algebra Fundamentals

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  1. Chapter 2 Introduction to Logic Circuits

  2. Objectives • Know what Truth Tables are • Know the Truth Tables for the Basic Gates • AND, OR, NOT, NAND, NOR • Know how to Analyze Simple Logic Circuits • Via Timing Diagrams and Truth Tables • Be able to use Boolean Algebra to manipulate simple digital logic equations • Know what a Venn diagram is and how they apply to digital logic • Know how to implement a function in both Sum of Products and Product of Sums form • Know how to use Quartus II’s schematic entry tool to describe simple logic circuits • Know how to use Quartus II to simulate a logic circuit

  3. Variables and Functions • Consider a Flashlight • L is the Function that represents the Flashlight • L = 0 when light is off, L = 1 when light is on • x is the Variable that represents the switch • Switch open x = 0, switch closed x = 1 • x is an input Variable • When x = 0 L = 0 • When x = 1 L = 1 • L(x) = x

  4. AND & OR • Flashlight controlled by two switches in series = AND • L(x1,x2) = x1.x2 • Flashlight controlled by two switches in parallel = OR • L(x1,x2) = x1+x2 • AND and OR two very important building blocks

  5. Inversion (NOT) • Light on if switch open • L = 1 if x = 0 • Light off if switch closed • L = 0 if x = 1 • L(x) = x • x = x’ = !x = ~x = NOTx

  6. Truth Tables x1 x2 x1.x2 x1+x2 !x1 0 0 0 0 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 AND OR NOT

  7. Logic Gates AND OR NOT (Inverter)

  8. Logic Networks or Logic Circuits • Using AND, OR, and NOT logic functions of any complexity can be implemented • Number of gates/complexity of logic network is directly related to its cost • Reducing cost is always desirable • Reducing complexity is always desirable

  9. Analysis of Logic Circuit • Analysis • Determine how an existing Logic Circuit functions • Synthesis • Design a new network that implements a desired Logic Function

  10. Logic Analysis • Truth Table • Timing Diagram

  11. Truth Table X1 X2 A B C D E f(X1,X2) 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 f(X1,X2) = !X1.!X2 + X1.X2 + X1.!X2

  12. Timing Diagram X1 X2 A B C = A.B D = X1.X2 E = X1.B f(X1,X2) 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  13. Truth Table X1 X2 A B C D f(X1,X2) 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 f(X1,X2) = !X1.!X2 + X1

  14. Timing Diagram X1 X2 A B C = A.B f(X1,X2) 1 0 1 0 1 0 1 0 1 0 1 0

  15. Functionally Equivalent Circuits = !X1.!X2 + X1.X2 + X1.!X2 = !X1.!X2 + X1 !X1.!X2 + X1.X2 + X1.!X2 = !X1.!X2 + X1 Two equations result in same output – Which is better?

  16. Boolean Algebra • In Boolean Algebra elements can take on one of two values, 0 and 1 • Axioms of Boolean Algebra • 1a. 0 . 0 = 0 • 1b. 1 + 1 = 1 • 2a. 1 . 1 = 1 • 2b. 0 + 0 = 0 • 3a. 0 . 1 = 1 . 0 = 0 • 3b. 1 + 0 = 0 + 1 = 1 • 4a. If x = 0 then !x = 1 • 4b. If x = 1 then !x = 0

  17. Single-Variable Theorems 5a. x . 0 = 0 5b. x + 1 = 1 6a. x . 1 = x 6b. x + 0 = 0 7a. x . x = x 7b. x + x = x 8a. x . !x = 0 8b. x + !x = 1 9 !!x = x Validity easy to prove by substituting x = 0 and x = 1 and using the Axioms

  18. Duality • Dual of an expression is achieved by • Replacing all ANDs with ORs and all ORs with ANDs • Replacing all 1s with 0s and all 0s with 1s

  19. Two and Three Variable Properties Combining 14a. X . Y + X . !Y = X 14b. (X + Y) . (X + !Y) = X DeMorgan’s Theorem 15a. !(X . Y) = !X + !Y 15b. !(X + Y) = !X . !Y 16a. X + !X . Y = X + Y 16b. X . (!X + Y) = X . Y Commutative 10a. X . Y = Y . X 10b. X + Y = Y + X Associative 11a. X . (Y . Z) = (X . Y) . Z 11b. X + (Y + Z) = (X + Y) + Z Distributive 12a. X . (Y + Z) = X . Y + X . Z 12b. X + Y . Z = (X + Y) . (X + Z) Absorption 13a. X + X . Y = X 13b. X . (X + Y) = X Consensus 17a. X . Y + Y . Z + !X . Z = X . Y + !X . Z 17b. (X + Y) . (Y + Z) . (!X + Z) = (X + Y) . (!X + Z)

  20. Algebraic Manipulation • !X1 . !X2 + X1 . X2 + X1 . !X2 =? !X2 + X1 • !X1 . !X2 + X1 . (X2 + !X2) • Via 12a. Distributive (X . Y) + (X . Z) = X . (Y + Z) • !X1 . !X2 + X1 . (1) • Via 8b. X + !X = 1 • !X1 . !X2 + X1 = !X1 . !X2 + X1 • Via 6a. X . 1 = X • !X2 +X1 • Via 16a. X + !X . Y = X + Y

  21. Venn Diagram

  22. !X2 + X1 !X1 . !X2 + X1 . X2 + X1 . !X2

  23. Notation, Terminology, & Precedence • + versus + called sum • . versus . called product • = AND • = OR • NOT, AND, OR Order of Precedence

  24. Synthesis or Design • If the alarm is enabled and the window is open the alarm should sound • Assign variable names and functions • Create truth table • Write Sum of Products or Product of Sums • Create Schematic or HDL

  25. Assign Variable Names and Functions • Alarm Enabled = En • Window Open = Wo • Alarm Sound = Al

  26. Create Truth Table

  27. Write Sum of Products or Product of Sums f = En . Wo = Al En Al Wo

  28. Cost • Total number of gates • Plus • Total number of inputs to all gates

  29. Sum of Products • Uses minterms to express the function • f(X1,X2) = Σ(m0,m2,m3) = !X1 . !X2 + X1 . !X2 + X1 .X2 • Canonical Sum of Products • Note: Brown uses Σm(0,2,3) as a simple form • Using Boolean logic theorems and properties the Canonical Sum of Products expression can be manipulated to produce • f(X1,X2) = !X2 + X1 • Which is a Minimum Cost Sum of Products expression of f Row # X1 X2 f(X1,X2) 00 0 0 1 01 0 1 0 10 1 0 1 11 1 1 1 !X1 . !X2 - minterm X1 . !X2 - minterm X1 . X2 - minterm

  30. Product of Sums • Uses Maxterms to express function • f(X1,X2) = П(M1) = (X1 + !X2). (!X1 + X2) • Canonical Product of Sums • Note: Brown uses П M(1) as a simple form • Also Minimum Cost Product of Sums in this case Row # X1 X2 f(X1,X2) 00 0 0 1 01 0 1 0 10 1 0 0 11 1 1 1 X1 + !X2 – maxterm !X1 + X2 – maxterm

  31. Using De Morgan’s Theoremto generate Product of Sums • !f(X1,X2) = (!X1 . X2) + (X1 . !X2) • !(!f(X1,X2)) = !((!X1 . X2) + (X1 . !X2)) • f(X1,X2) = !(!X1 . X2) . !(X1 . !X2) • f(X1,X2) = (X1 + !X2) . (!X1 + X2) Row # X1 X2 f(X1,X2) 00 0 0 1 01 0 1 0 10 1 0 0 11 1 1 1

  32. Minterm vs Maxterm • Σ = Minterm – rows where f(x) = 1 • П = Maxterm – rows where f(x) = 0 Row # X1 X2 f(X1,X2) 00 0 0 1 01 0 1 0 10 1 0 1 11 1 1 1 !X1 . !X2 - minterm X1 + !X2 - maxterm X1 . !X2 - minterm X1 . X2 - minterm f(X1,X2) = Σ(m0,m2,m3) = !X1 . !X2 + X1 . !X2 + X1 .X2 f(X1,X2) = П(M1) = X1 + !X2

  33. Further Examples • f(m0,m2,m3,m7) • Truth table • Sum of Products • Product of Sums • Minimization • Schematic • Timing Diagram

  34. Sum of Products !X1!X2!X3 + !X1X2!X3 + !X1X2X3 + X1X2X3 !X1!X3(!X2 + X2) + X2X3(!X1 + X1) !X1!X3 + X2X3

  35. Product of Sums (X1+X2+!X3)(!X1+X2+X3)(!X1+X2+!X3)(!X1+!X2+X3) ((X1 +!X3)+X2)((!X1+X3)+X2)((!X1+X3)+X2)((!X1+!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+!X2) A B B C C D A B C D A C B D ((X1 +!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+X2)((!X1+X3)+!X2) ((X1 +!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+X2!X2)

  36. NAND and NOR Gatesvia DeMorgan’s Theorem !(X1.X2) = !X1 + !X2 = !X1 + !X2 !(X1 + X2) = !X1.!X2 = !X1.!X2

  37. NAND gates in Sum of ProductsNOR gate in Product of Sums Sum of Products Product of Sums

  38. Design Example • Three-Way Light Control • Did you understand the book’s example? • Multiplexer Circuit

  39. Reading Laboratory Preparation • Chapter 2 – Omit section 2.10 • Quartus II Introduction Using Schematic Design • ftp://ftp.altera.com/up/pub/Altera_Material/QII_9.0/Digital_Logic/DE2/Tutorials/tut_quartus_intro_schem.pdf

  40. Homework Problems • 2.1-2.4, 2.6, 2.8, 2.9, 2.10 • 2.28-2.30 Create Truth Table, attempt to minimize, and write SOP & POS • 2.31-2.34 Create Truth Table, attempt to minimize, and write required form • 2.35 Create Truth Table • 2.44, 2.45 Print or save as jpg and email Schematic and Simulator Waveform

  41. Schematic Design Poor Good

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