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Logic Gates

Logic Gates. Transistors as Switches. V BB voltage controls whether the transistor conducts in a common base configuration. Logic circuits can be built. AND. In order for current to flow, both switches must be closed Logic notation A B = C (Sometimes AB = C). OR.

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Logic Gates

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  1. Logic Gates

  2. Transistors as Switches • VBB voltage controls whether the transistor conducts in a common base configuration. • Logic circuits can be built

  3. AND • In order for current to flow, both switches must be closed • Logic notation AB = C (Sometimes AB = C)

  4. OR • Current flows if either switch is closed • Logic notation A + B = C

  5. Properties of AND and OR • Commutation • A + B = B + A • A  B = B  A Same as Same as

  6. Properties of AND and OR • Associative Property • A + (B + C) = (A + B) + C • A  (B  C) = (A  B)  C =

  7. Properties of AND and OR • Distributive Property • A + B  C = (A + B)  (A + C) • A + B  C

  8. Distributive Property • (A + B)  (A + C)

  9. Binary Addition Notice that the carry results are the same as AND C = A  B

  10. Inversion (NOT) Logic:

  11. Exclusive OR (XOR) Either A or B, but not both This is sometimes called the inequality detector, because the result will be 0 when the inputs are the same and 1 when they are different. The truth table is the same as for S on Binary Addition. S = A  B

  12. Getting the XOR Two ways of getting S = 1

  13. Circuit for XOR Accumulating our results: Binary addition is the result of XOR plus AND

  14. Half Adder Called a half adder because we haven’t allowed for any carry bit on input. In elementary addition of numbers, we always need to allow for a carry from one column to the next. 18 25 3 (plus a carry) 4

  15. Full Adder

  16. Full Adder Circuit

  17. Chaining the Full Adder Possible to use the same scheme for subtraction by noting that A – B = A + (-B)

  18. Binary Counting Use 1 for ON Use 0 for OFF = 00101011 So our example has 25 + 23 + 21 + 20 = 32 + 8 + 2 + 1 = 43 Binary Counter

  19. Counting in Binary

  20. NAND (NOT AND)

  21. NOR (NOT OR)

  22. Exclusive NOR Equality Detector

  23. Summary

  24. Number Systems • Decimal (base 10) {0 1 2 3 4 5 6 7 8 9} • Place value gives a logarithmic representation of the number • Ex. 4378 means • 4 X 103 = 4000 • 3 X 102 = 300 • 7 X 101 = 70 • 8 X 100 = 8 • The place also gives the exponent of the base

  25. 100 105 101 104 102 103 Example • 432,600 4 3 2 6 0 0 Powers of ten: 100 = 1 102 = 100 104 = 10000 101 = 10 103 = 1000 105 = 100000

  26. Binary (base 2) {0 1}

  27. 20 27 21 26 22 25 23 24 Example 1 1 0 1 1 0 0 1

  28. Decimal Equivalent • 1101 1001 1 X 27 = 128 + 1 X 26= 64 + 0 X 25= 0 + 1 X 24= 16 + 1 X 23= 8 + 0 X 22= 0 + 0 X 21= 0 + 1 X 20= 1 217 Notice how powers of two stand out: 20 = 1 21 = 10 22 = 100 23 = 1000

  29. Decimal to Binary Conversion • Ex. 575 • Find the largest power of two less than the number • 29 = 512 • Subtract that power of two from the number • 575 – 512 = 63 • Repeat steps 1 and 2 for the new result until you reach zero. • 25 = 32 63 – 32 = 31 • 24 = 16 31 – 16 = 15 • 23 = 8 15 – 8 = 7 • 22 = 4 7 – 4 = 3 • 21 = 2 3 – 2 = 1 • 20 = 1 1 – 1 = 0 • Construct the number • 1000111111

  30. Another Example • 144 • 27 = 128 144 – 128 = 16 • 24 = 16 16 – 16 = 0 • Result 10010000

  31. Hexadecimal (base 16) • {0 1 2 3 4 5 6 7 8 9 A B C D E F} • Assignments

  32. 3 B 6 E 160 163 161 162 Example 3 X 163 = 12288 11 X 162 = 2816 6 X 161 = 96 14 X 160 = 14  15214

  33. Hexadecimal is Convenient for Binary Conversion

  34. Binary to Hex Conversion • Group binary number by fours (nibbles) • 1101 1001 0110 • Convert each nibble into hex equivalent • 1101 1001 0110 D 9 6

  35. Decimal to Hex Conversion • Ex. 284 • 162 = 256 284 – 256 = 28 • 161 = 16 28 - 16 = 12 (Hex C) • Result 1 1 C

  36. Another Example with an Extension • 1054 • 162 = 256 • But we have several multiples of 256 in 1054 • 1054/256 = 4.12 take integer part • This eliminates 4*256 = 1024 • 1054 – 1024 = 30 • 161 = 16 30 – 16 = 14 (Hex E) • Result 4 1 E

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