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# LOGIC GATES

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### LOGIC GATES

1. 1. Logic Gates 1 Page 1
2. 2. Logic • Formal logic is a branch of mathematics that deals with true and false values instead of numbers. • In 1840’s, George Boole developed many Logic ideas. •A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. 2 Page 2
3. 3. The logic normally performed is Boolean logic and is most commonly found in digital circuits. Logic gates are primarily implemented electronically using diodes or transistors, but can also be constructed using electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. In electronic logic, a logic level is represented by a voltage or current, depending on the type of electronic logic in use. Page 3
4. 4. Logic Signals There are a number of different systems for representing binary information in physical systems. Here are a few. A voltage signal with zero (0) corresponding to 0 volts and one (1) corresponding to five or three volts. A sinusoidal signal with zero corresponding to some frequency, and one corresponding to some other frequency. A current signal with zero corresponding to 4 milliamps and one corresponding to 20 milliamps. And one last way is to use switches, OPEN for "0" and CLOSED for "1". (And there are more ways!) Page 4
5. 5. Boolean algebra is the algebra of two values. These are usually taken to be 0 and 1, as we shall do here, although F and T, false and true, etc. are also in common use. Whereas elementary algebra is based on numeric operations multiplication xy, addition x + y, and negation −x, Boolean algebra is customarily based on logical counterparts to those operations, namely : (1) conjunction x∧y (AND) (2) disjunction x∨y (OR) (3) complement or negation ¬x (NOT). In electronics: AND is represented as a multiplication OR is represented as an addition NOT is represented with an overbar Page 5
6. 6. Basic logic gates • Not • And • Or • Nand • Nor • Xor 6 Page 6
7. 7. Truth Table A truth table is a good way to show the function of a logic gate. It shows the output states for every possible combination of input states. The symbols 0 (false) and 1 (true) are usually used in truth tables. Page 7
8. 8. NOT The output A is true when the input a is NOT true, the output is the inverse of the input: a = NOT A A NOT gate can only have one input. A NOT gate is also called an inverter. Truth Table: A A a 0 A 1 1 0 Page 8
9. 9. AND "If A AND B are both 1, then Q should be 1.“ (All or nothing.) Logic Gate: A A*B B Truth Table: A 0 A Series Circuit: A*B 0 0 1 1 0 0 0 1 B B 0 1 1 A*B Page 9
10. 10. Three Input AND Gate A B C ABC 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 Page 10
11. 11. OR "If A is 1 OR B is 1 (or both are 1), then Q is 1." Logic Gate: A A+B B Truth Table: A 0 B 1 0 1 1 1 Parallel Circuit: A+B 0 0 1 A B 0 1 1 A+B Page 11
12. 12. • Because + and * are binary operations, they can be cascaded together to OR or AND multiple inputs. A B A B C C A+B+C A+B+C A B A B C ABC ABC Page 12
13. 13. NAND and NOR Gates • NAND and NOR gates can greatly simplify circuit diagrams. NAND inverts the output of AND. • NOR inverts the output of OR. A 0 0 1 1 0 1 1 1 0 A NOR 0 A↑ B 1 1 NAND B B 0 0 A↓ B 1 0 1 0 1 0 0 1 1 0 Page 13
14. 14. XOR and XNOR Gates XOR (exclusive OR) :"If either A OR B is 1, but NOT both, Q is 1." A 0 0 A⊕ B 0 0 1 1 1 0 1 1 XOR B 1 0 XNOR (exclusive NOR) : invert output of XOR A XNOR B A B 0 0 1 0 1 0 1 0 0 1 1 1 Page 14
15. 15. • Find the output of the following circuit x+y (x+y)y y __ • Answer: (x+y)y 15 Page 15
16. 16. • Find the output of the following circuit x xy xy y ___ __ • Answer: xy Page 16
17. 17. Give the Boolean expression of the given circuit x+y (x+y)(xy) xy Answer: xy (x+y)(xy) 17 Page 17
18. 18. • Write the circuits for the following Boolean algebraic expressions __ a) x+y x x+y 18 Page 18
19. 19. • Write the circuits for the following Boolean algebraic expressions _______ b) (x+y)x x+y x+y (x+y)x 19 Page 19
20. 20. More about logic gates • To implement a logic gate in hardware, you use a transistor • Transistors are all enclosed in an “IC”, or integrated circuit • The current Intel Pentium IV processors have 55 million transistors! 20 Page 20
21. 21. Flip-flops • Consider the following circuit: • What does it do? 21 Page 21
22. 22. If you arrange the gates correctly, they will remember an input value. MEMORY This simple concept is the basis of RAM (random access memory) in computers, and also makes it possible to create a wide variety of other useful circuits. Memory relies on a concept called feedback. That is, the output of a gate is fed back into the input. • A flip-flop holds a single bit of memory • In reality, flip-flops are a bit more complicated – Have 5 (or so) logic gates (transistors) per flipflop • Consider a 1 Gb memory chip – 1 Gb = 8,589,934,592 bits of memory – That’s about 43 million transistors! • In reality, those transistors are split into 9 ICs of about 5 million transistors each 22 Page 22
23. 23. Exercises: 1.Give the Boolean expression of the given gate. Answer: (A + B)C 2.Give the Boolean expression of the given gate. Answer: A + BC + D 3.Draw a logic circuit for AB + AC. Page 23
24. 24. Exercises: 4.Draw a logic circuit for (A + B)(C + D) C. 5. Give the truth table for a 3-input (A,B & C) OR gate. 6. What type of logic gate's behavior does this truth table represent? 7.Give the Boolean expression of the given gate. Page 24
25. 25. Exercises: 8.Give the output expressions of the given gates. a. b. c. d. e. f. Page 25
26. 26. 3. AB + AC. Answers to Exercises: 5. 3-input OR gate ABC 4. (A + B)(C + D)C. 6. 3-input OR gate 7. Page 26
27. 27. Answers to Exercises: 8. a.) (ABC)(DE). b.) (ABC)+(DE). c.) (R+S+T) (X+Y+Z). d.) (R+S+T)+(X+Y+Z). e.) (JK)(M + N). f.) (AB) (M + N) (X + Y). Page 27
28. 28. ---the end– 8-) Page 28