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Transcript
Week 6: Gates and Circuits: PART I • READING: Chapter 4 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits What is a “gate”? • A gate is a device that performs a basic operation on electrical signals What is “circuit”? • Gates are combined to form different “circuits” to perform more complicated tasks 2 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits Three notational methods to describe the behavior of gates: 1. Boolean expressions: A form of algebra in which variables and functions take on only one of two possible values (0 and 1) 2. Logic diagrams: graphical representation of a circuit 3. Truth tables: defines the function of a gate by listing all possible input combination and the corresponding output. 3 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits • A gate or logic gate performs only one logical function. Each gate accepts one or more input values and produces a single output value. A0 A1 ... Logic Gate X An Six types of logic gates: 1. 2. 3. 4. 5. 6. NOT AND OR XOR NAND NOR 4 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: NOT Gate • Also referred to as an “inverter” • If the input value is 1, the output is 0; if the input value is 0, the output is 1 Boolean Expression Logic diagram Symbol Truth Table • Sometimes the “ ’ ” mark is replaced by horizontal bar placed over the value: X A' A 5 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: AND Gate • If the two input values are both 1, the output is 1; otherwise, the output is 0 Boolean Expression Logic diagram Symbol Truth Table • Sometimes the “ . ” mark is replaced by the asterisk symbol “ * ” 6 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: OR Gate • If both input values are both 0, the output is 0; otherwise, the output is 1 Boolean Expression Logic diagram Symbol Truth Table 7 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: XOR or exclusive OR Gate • If the two inputs are the same, the output is 0; otherwise, the output is 1 Boolean Expression Logic diagram Symbol Truth Table • Not the difference between the XOR gate and the OR gate; they only differ in one input situation: • When both input signals are 1, OR gate produces a 1 and the XOR gate produces a 0 8 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: NOR Gate • The NOR gate is essentially the opposite of the OR gate. That is, the output of a NOR gate is the same as if you took the output of an OR gate and put it through a NOT gate Boolean Expression Logic diagram Symbol Truth Table 9 EECS 1520 -- Computer Use: Fundamentals Gates and Circuits: NAND Gate • The NAND gate is the opposite of the AND gate. Boolean Expression Logic diagram Symbol Truth Table 10 EECS 1520 -- Computer Use: Fundamentals Transistors How do we implement the “gates”? • A gate uses one or more transistors to establish how the input values map to the output value • A transistor acts like a “switch”. • It either turns on to conduct electricity or turns off to block the flow of electricity 11 EECS 1520 -- Computer Use: Fundamentals Transistors • A transistor has three terminals: source, base and emitter source output base emitter • When an electrical signal is grounded, it has 0 volts! • If the source signal is pulled to ground, the output signal is low • If the source signal remains high, the output signal is high output is “0” output is “1” 12 EECS 1520 -- Computer Use: Fundamentals Transistors – NOT Gate • The output is determined by the base electrical signal. source Vout Vin base Vin Vout 1 0 0 1 emitter • If Vin is high, the source is pulled to ground and Vout is low (i.e. 0) • If Vin is low, the source is not grounded and Vout is high (i.e. 1) “NOT Gate” needs 1 transistor 13 EECS 1520 -- Computer Use: Fundamentals Transistors – NAND Gate source Vout Vin1 Vin2 Vin1 Vin2 Vout 0 0 1 0 1 1 1 0 1 1 1 0 emitter • If Vin1 and Vin2 are high, the source is pulled to ground and Vout is low (i.e. 0) • If Vin1 and Vin2 are low, the source is not grounded and Vout is high (i.e. 1) • If either Vin1 or Vin2 is low, the source is not grounded and Vout is high (i.e. 1) “NAND Gate” needs 2 transistors 14 EECS 1520 -- Computer Use: Fundamentals Transistors – NOR Gate source Vout Vin1 Vin2 emitter emitter Vin1 Vin2 Vout 0 0 1 0 1 0 1 0 0 1 1 0 • If Vin1 and Vin2 are high, the source is pulled to ground and Vout is low (i.e. 0) • If Vin1 and Vin2 are low, the source is not grounded and Vout is high (i.e. 1) • If either Vin1 or Vin2 is low, the source is grounded and Vout is low (i.e. 0) “NOR Gate” needs 2 transistors 15 EECS 1520 -- Computer Use: Fundamentals Transistors – OR Gate • Since OR gate is the opposite of NOR gate, how many transistors would you think will be required to implement the “OR” gate? “OR Gate” needs 3 transistors 16 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • Gates are combined into circuits by using the output of one gate as the input for another gate. • For example: 17 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • For example: Logic diagram Symbol Truth Table • Since there are 3 inputs, there are 8 possible outcomes 18 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • For example: Logic diagram Symbol Boolean expression • D=AB • E = AC • X = AB + AC 19 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • Now, we want to investigate the following Boolean expression: X = A(B+C) • How do we want to create the logic diagram (called circuit 2) of the above Boolean expression? - We have an inner function which consists of an “OR” gate between B and C - We then have an outer function which is an “AND” gate between A and (B+C) Logic diagram Symbol: (circuit 2) A(B+C) B+C 20 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • We have the following: Boolean expression: X = A(B+C) Logic diagram Symbol: A(B+C) B+C Truth table: A B C B+C A(B+C) 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 21 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • Circuit 1: • Circuit 2: A(B+C) B+C A B C D E X A B C B+C A(B+C) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 • Their results are identical! 22 EECS 1520 -- Computer Use: Fundamentals Combinational Circuits • We have therefore demonstrated circuit equivalence • That is, both circuits produce the same results for each input combination • Boolean algebra allows us to apply provable mathematical principles to help us design logical circuits • From the previous example: X = AB + AC = A(B+C) 23 EECS 1520 -- Computer Use: Fundamentals Properties of Boolean Algebra • DeMorgan’s law, in particular, is very useful in Boolean algebra. • For instance, it means that: ___ ___ ___ 1 NAND gate is equivalent to 2 NOT gates with an OR gate 24