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Week 6: Gates and Circuits: PART I READING: Chapter 4

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Gates and Circuits 2 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 EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits 3 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. EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits 4 Six types of logic gates: 1.NOT 2.AND 3.OR 4.XOR 5.NAND 6.NOR A gate or logic gate performs only one logical function. Each gate accepts one or more input values and produces a single output value. EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: NOT Gate 5 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 Logic diagram SymbolTruth Table Boolean Expression Sometimes the “ ’ ” mark is replaced by horizontal bar placed over the value: EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: AND Gate 6 If the two input values are both 1, the output is 1; otherwise, the output is 0 Logic diagram SymbolTruth Table Boolean Expression Sometimes the “. ” mark is replaced by the asterisk symbol “ * ” EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: OR Gate 7 Logic diagram SymbolTruth Table Boolean Expression If both input values are both 0, the output is 0; otherwise, the output is 1 EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: XOR or exclusive OR Gate 8 Logic diagram SymbolTruth Table Boolean Expression If the two inputs are the same, the output is 0; otherwise, the output is 1 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 EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: NOR Gate 9 Logic diagram SymbolTruth Table Boolean Expression 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 EECS 1520 -- Computer Use: Fundamentals

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Gates and Circuits: NAND Gate 10 Logic diagram SymbolTruth Table Boolean Expression The NAND gate is the opposite of the AND gate. EECS 1520 -- Computer Use: Fundamentals

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Transistors 11 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 EECS 1520 -- Computer Use: Fundamentals

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Transistors 12 A transistor has three terminals: source, base and emitter emitter base source When an electrical signal is grounded, it has 0 volts! If the source signal is pulled to ground, the output signal is low output is “0” If the source signal remains high, the output signal is high output is “1” output EECS 1520 -- Computer Use: Fundamentals

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Transistors – NOT Gate 13 If V in is high, the source is pulled to ground and V out is low (i.e. 0) If V in is low, the source is not grounded and V out is high (i.e. 1) emitter base source The output is determined by the base electrical signal. V in V out “NOT Gate” needs 1 transistor V in V out 10 01 EECS 1520 -- Computer Use: Fundamentals

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14 Transistors – NAND Gate V in1 V in2 source V out emitter If V in1 and V in2 are high, the source is pulled to ground and V out is low (i.e. 0) If V in1 and V in2 are low, the source is not grounded and V out is high (i.e. 1) If either V in1 or V in2 is low, the source is not grounded and V out is high (i.e. 1) “NAND Gate” needs 2 transistors V in1 V in2 V out 001 011 101 110 EECS 1520 -- Computer Use: Fundamentals

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15 Transistors – NOR Gate emitter source V in1 V out emitter V in2 If V in1 and V in2 are high, the source is pulled to ground and V out is low (i.e. 0) If V in1 and V in2 are low, the source is not grounded and V out is high (i.e. 1) If either V in1 or V in2 is low, the source is grounded and V out is low (i.e. 0) “NOR Gate” needs 2 transistors V in1 V in2 V out 001 010 100 110 EECS 1520 -- Computer Use: Fundamentals

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16 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 EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 17 Gates are combined into circuits by using the output of one gate as the input for another gate. For example: EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 18 For example: Logic diagram Symbol Truth Table Since there are 3 inputs, there are 8 possible outcomes EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 19 For example: Logic diagram Symbol Boolean expression D = A B E = AC X = AB + AC EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 20 Now, we want to investigate the following Boolean expression: X = A(B+C) Logic diagram Symbol: (circuit 2) B+C 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) EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 21 We have the following: X = A(B+C)Boolean expression: Logic diagram Symbol: Truth table: B+C A(B+C) ABCB+CA(B+C) 000 001 010 011 100 101 110 111 ABCB+CA(B+C) 0000 0011 0101 0111 1000 1011 1101 1111 ABCB+CA(B+C) 00000 00110 01010 01110 10000 10111 11011 11111 EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 22 Circuit 1: Their results are identical! Circuit 2: ABCB+CA(B+C) 00000 00110 01010 01110 10000 10111 11011 11111 B+C A(B+C) ABCDEX 000000 001000 010000 011000 100000 101011 110101 111111 EECS 1520 -- Computer Use: Fundamentals

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Combinational Circuits 23 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 X = AB + AC = A(B+C) From the previous example: EECS 1520 -- Computer Use: Fundamentals

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Properties of Boolean Algebra 24 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 EECS 1520 -- Computer Use: Fundamentals

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