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Logic Gates

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Transistors as Switches ¡V BB voltage controls whether the transistor conducts in a common base configuration. ¡Logic circuits can be built

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AND ¡In order for current to flow, both switches must be closed ¤Logic notation A B = C (Sometimes AB = C) ABC 000 010 100 111

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OR ¡Current flows if either switch is closed ¤Logic notation A + B = C ABC 000 011 101 111

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Properties of AND and OR ¡Commutation ¤A + B = B + A ¤A B = B A Same as

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Properties of AND and OR ¡Associative Property ¤A + (B + C) = (A + B) + C ¤A (B C) = (A B) C =

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Properties of AND and OR ¡Distributive Property ¤A + B C = (A + B) (A + C) ¤A + B C ABCQ 0000 0010 0100 1001 1011 1101 1111

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Distributive Property ¡(A + B) (A + C) ABCQ 0000 0010 0100 1001 1011 1101 1111

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Binary Addition ABSC(arry) 0000 1010 0110 1101 Notice that the carry results are the same as AND C = A B

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Inversion (NOT) AQ 01 10 Logic:

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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 ABS 000 101 011 110

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Getting the XOR ABS 000 101 011 110 Two ways of getting S = 1

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

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Half Adder Called a half adder because we havent 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 4 3 (plus a carry)

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Full Adder INPUTSOUTPUTS ABC IN C OUT S 00000 00101 01001 01110 10001 10110 11010 11111

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Full Adder Circuit

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

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Binary Counting Use 1 for ON Use 0 for OFF = 00101011 Binary Counter So our example has 2 5 + 2 3 + 2 1 + 2 0 = 32 + 8 + 2 + 1 = 43

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Counting in Binary 111110112110101 2101211002210110 3111311012310111 41001411102411000 51011511112511001 611016100002611010 711117100012711011 8100018100102811100 9100119100112911101 10101020101003011110

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NAND (NOT AND) ABQ 001 011 101 110

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NOR (NOT OR) ABQ 001 010 100 110

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Exclusive NOR ABQ 001 010 100 111 Equality Detector

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Summary Summary for all 2-input gates InputsOutput of each gate A B ANDNAND OR NORXORXNOR 00010101 01011010 10011010 11101001

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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 10 3 = 4000 ۞ 3 X 10 2 = 300 ۞ 7 X 10 1 = 70 ۞ 8 X 10 0 = 8 ¤The place also gives the exponent of the base

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Example ¡432,600 4 3 2 6 0 0 10 5 10 4 10 3 10 0 10 1 10 2 Powers of ten: 10 0 = 110 2 = 10010 4 = 10000 10 1 = 1010 3 = 100010 5 = 100000

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Binary (base 2) {0 1} BinaryDecimal 00 11 102 113 1004 1015 1106 1117 10008 10019 101010

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Example 1 1 0 1 1 0 0 1 2727 2626 2525 2020 2121 2 2424 2323

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Decimal Equivalent ¡1101 1001 1 X 2 7 = 128 + 1 X 2 6 = 64 + 0 X 2 5 = 0 + 1 X 2 4 = 16 + 1 X 2 3 = 8 + 0 X 2 2 = 0 + 0 X 2 1 = 0 + 1 X 2 0 = 1 217 Notice how powers of two stand out: 2 0 = 1 2 1 = 10 2 2 = 100 2 3 = 1000

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Decimal to Binary Conversion ¡Ex. 575 ¤Find the largest power of two less than the number ۞ 2 9 = 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. ۞ 2 5 = 32 63 – 32 = 31 ۞ 2 4 = 16 31 – 16 = 15 ۞ 2 3 = 8 15 – 8 = 7 ۞ 2 2 = 4 7 – 4 = 3 ۞ 2 1 = 2 3 – 2 = 1 ۞ 2 0 = 1 1 – 1 = 0 ¤Construct the number ۞ 1000111111

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Another Example ¡144 ¤2 7 = 128 144 – 128 = 16 ¤2 4 = 16 16 – 16 = 0 ¡Result 10010000

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Hexadecimal (base 16) ¡ {0 1 2 3 4 5 6 7 8 9 A B C D E F} ¡ Assignments DecHexDecHex 0088 1199 2210A 3311B 4412C 5513D 6614E 7715F

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Example 16 3 16 2 16 0 16 1 3 B 6 E 3 X 16 3 = 12288 11 X 16 2 = 2816 6 X 16 1 = 96 14 X 16 0 = 14 15214

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Hexadecimal is Convenient for Binary Conversion BinaryHexBinaryHex 00100129 111010A 1021011B 1131100C 10041101D 10151110E 11061111F 11171 000010 10008 Nibble

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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

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Decimal to Hex Conversion ¡ Ex. 284 ¤16 2 = 256 284 – 256 = 28 ¤16 1 = 16 28 - 16 = 12 (Hex C) ¤Result 1 1 C

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Another Example with an Extension ¡ 1054 ¤16 2 = 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 ¤16 1 = 16 30 – 16 = 14 (Hex E) ¤Result 4 1 E

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