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Gates – Part 2

Gates – Part 2. Converting English to Boolean Expressions. The air conditioner should be turned on if and only if: - the temperature is greater than 75 , - the time is between 8a.m. and 5 p.m., - and it is not a holiday. 1. Identify phrases.

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Gates – Part 2

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  1. Gates – Part 2 ECEn 224

  2. Converting English to Boolean Expressions ECEn 224

  3. The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday. ECEn 224

  4. 1. Identify phrases The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday. F =air conditioner should be turned on A = temperature is greater than 75 B = time is between 8a.m. and 5 p.m C = it is a holiday ECEn 224

  5. 2. Identify connective words = The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday. AND ECEn 224

  6. 3. Construct a Boolean Expression The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday. F =air conditioner should be turned on A = temperature is greater than 75 B = time is between 8a.m. and 5 p.m C = it is a holiday F = A • B • C’ ECEn 224

  7. 4. Draw the Network The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday. F = A • B • C’ A F B C ECEn 224

  8. Converting English to Boolean 1. Identify phrases 2. Identify connective words 3. Construct a Boolean Expression 4. Draw the Network ECEn 224

  9. Converting English to Boolean • Be careful: Boolean algebra is precise, English is not. The roads will be very slippery if it snows or rains and there is oil on the road. A C B F = A + BC or F = (A + B) C Which is it? ECEn 224

  10. AND/OR vs. OR/AND Logic Forms ECEn 224

  11. A B C f AND/OR Logic from Truth Table 1. Write the SOP by inspection for f f = A’B’C +AB’C + ABC’ + ABC ECEn 224

  12. AND/OR Logic from Truth Table 2. Simplify the equation f = A’B’C +AB’C + ABC’ + ABC f = (A + A’) B’C + AB (C + C’) f = AB + B’C ECEn 224

  13. AND/OR Logic from Truth Table 3. Draw the logic network f = AB + B’C A f B C “AND/OR” ECEn 224

  14. A B C f OR/AND Logic from Truth Table • Write a POS by inspection from f f’= A’B’C’ + A’BC’ + AB’C’ f = (A’B’C’ + A’BC’ + AB’C’)’ f = (A + B + C)(A + B’ + C)(A’ + B + C) ECEn 224

  15. OR/AND Logic from Truth Table 3. Simplify the equation: f = (A + B + C)(A + B’ + C)(A’ + B + C) f = (A + B)(A + B’)(A’ + B) + C f = (A)(A’ + B) + C f = (AA’ + AB) + C f = AB + C f = (A + C)(B + C) A + BCD = A + B(CD) = (A + B)(A + CD) = (A + B)(A + C)(A + D) (X + Y)(X + Y’) = X ECEn 224

  16. OR/AND Logic from Truth Table 4. Draw the logic network f = (A+C)(B+C) A C f B C “OR/AND” ECEn 224

  17. Types of Gates • Gates already studied • AND, NAND • OR, NOR • Inverter (NOT) • XOR (Exclusive-OR) • XNOR (Equivalence) ECEn 224

  18. NAND/NAND and NOR/NOR Logic ECEn 224

  19. AND/OR to NAND/NAND Algebra-based: AB+CD = [AB+CD]’’ = [(AB)’(CD)’]’ Preferred symbol in this context… Schematic-based: ECEn 224

  20. OR/AND to NOR/NOR Algebra-based: (A+B)(C+D) = [(A+B)(C+D)]’’ = [(A+B)’ + (C+D)’]’ Preferred symbol in this context… Schematic-based: ECEn 224

  21. Alternative Gate Symbols Which is easier to understand? A A B B Q = ? Q = ? C C D D A A B B Q = AB + CD Q = (A+B)(C+D) C C D D If you think of the bubbles as canceling each other out… ECEn 224

  22. Bubble Matching How to make schematics readable, understandable, maintainable, … ECEn 224

  23. Bubble Matching Rules • Choose alternative symbols • Match all interior bubbles • More than one solution • Makes reading of the function trivial A A F F B B C C D D F = [(AB)’(C+D)’]’ ??? F = AB + (C+D) = AB+C+D ECEn 224

  24. More Bubble Matching A A F F B B C C D D This doesn’t work – has unmatched bubbles A F B C D This works. F = AB + C’D’ ECEn 224

  25. Yet More Bubble Matching A A F F B B C C D D Same circuit as on previous slide… Alternative solution = convert top-left gate. F’ = (A’+B’)(C+D)F = AB+C’D’ Same result as on previous slide ECEn 224

  26. Can Bubbles Always Be Matched? This is called reconvergent fanout • No… x A y B F E C D Nodes x and y both drive final gate and so both need same polarity (bubble or no bubble). Can’t satisfy that requirement because x also drives y’s input. ECEn 224

  27. Bubble Matching Summary • Convert symbols to match bubbles • Two versions for each circuit • Inverted output • Non-inverted output • Good schematic style similar to good programming style • Convey meaning as well as function • Document the design ECEn 224

  28. Functional Completeness ECEn 224

  29. Functionally Complete • AND, OR, and NOT make a functionally complete set of operators • There is no truth table which cannot be implemented using AND, OR, NOT • Any set of gates which can implement AND, OR and NOT is also functionally complete ECEn 224

  30. Functionally Complete • Is the set {AND, NOT} functionally complete? • If I could just build an OR gate… Success! or… X + Y = (X’Y’)’ ECEn 224

  31. Functionally Complete • Is the set {OR, NOT} functionally complete? • If I could just build an AND gate… Success! or… XY = (X’ + Y’)’ ECEn 224

  32. Functionally Complete • Is the set {AND, OR} functionally complete? • No! Can’t do a NOT ECEn 224

  33. How About NAND Only? NOT AND OR Success! ECEn 224

  34. How About NOR Only? NOT OR AND Success! ECEn 224

  35. Functional Completeness • A few functionally complete sets: ECEn 224

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