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) ABC

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

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

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

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

Distributive Property ¡(A + B) (A + C) ABCQ

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

Inversion (NOT) AQ Logic:

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

Getting the XOR ABS Two ways of getting S = 1

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

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 (plus a carry)

Full Adder INPUTSOUTPUTS ABC IN C OUT S

Full Adder Circuit

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

Binary Counting Use 1 for ON Use 0 for OFF = Binary Counter So our example has = = 43

Counting in Binary

NAND (NOT AND) ABQ

NOR (NOT OR) ABQ

Exclusive NOR ABQ Equality Detector

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

Number Systems ¡Decimal (base 10) { } ¤Place value gives a logarithmic representation of the number ¤Ex 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

Example ¡432, Powers of ten: 10 0 = = = = = =

Binary (base 2) {0 1} BinaryDecimal

Example

Decimal Equivalent ¡ X 2 7 = X 2 6 = X 2 5 = X 2 4 = X 2 3 = X 2 2 = X 2 1 = X 2 0 = Notice how powers of two stand out: 2 0 = = = = 1000

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 = 31 ۞ 2 4 = – 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 ۞

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

Hexadecimal (base 16) ¡ { A B C D E F} ¡ Assignments DecHexDecHex A 3311B 4412C 5513D 6614E 7715F

Example B 6 E 3 X 16 3 = X 16 2 = X 16 1 = X 16 0 =

Hexadecimal is Convenient for Binary Conversion BinaryHexBinaryHex A B C D E F Nibble

Binary to Hex Conversion ¡ Group binary number by fours (nibbles) ¤ ¡Convert each nibble into hex equivalent ¤ D 9 6

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

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 = 14 (Hex E) ¤Result 4 1 E