1 How gates work NOT ( ) NAND ( ) o - PDF document

The digital logic level Bits and Gates ... The digital logic level Level 2 Conventional Machine Level Level 1 Micro Programming Level Computer Science Ground Floor Digitial Logic Level CS240 Computer Organization Department of Computer Science Electronic Circuits Basement Electrical Engineering Wellesley College Subbasement Technology of Implementation Solid State Physics Digital logic 12-2 Digital circuits Gates o A digital circuit is a circuit in which only two logical values o Tiny electronic devices called gates compute various are present. functions on these two-valued signals. 5 v High (logic 1; true; asserted) 2 v Undefined 1 v Low (logic 0; false; unasserted) *Exact voltage levels and their interpretation varies with technology. Digital logic 12-3 Digital logic 12-4 1

How gates work NOT ( ) NAND ( ) o Transistors can be made to behave like very fast binary switches. + V cc + V cc +V cc (Supply Voltage) V out V out V 1 V in Collector V 2 Base Emitter (Circuit Ground) Digital logic 12-5 Digital logic 12-6 Integrated Circuits Wafer, chip, gate o Gates are manufactured in units called integrated circuits. Digital logic 12-7 Digital logic 12-8 2

Wafer, chip, gate The finished product Light Mask (a) Expose photoresist (b) Wash away unhardened photoresist (c) Etch away uncovered copper (d) Wash away photoresist Digital logic 12-9 Digital logic 12-10 Five basic gates Design new gates from old o The exclusive or is often NOT NAND NOR used as a one-bit comparator. XOR AND OR *Gates with more then two inputs are possible. Who am I ? Digital logic 12-11 Digital logic 20-12 3

Boolean Algebra Boolean expressions & circuit equivalence A + B A A A A A + B A B A + B = B B B B 1 0 0 + A = A A 0 A A A A A A A = A A 0 *We think of inputs to wires as variables and gates as simple functions. Digital logic 12-13 Digital logic 12-14 Name that rule Name that rule A B A + A A = A A A + B B + A = B A AB A(BC) A 0 A 0 A (AB)C 0 B = A BC A = B C C Digital logic 12-15 Digital logic 12-16 4

Double bubble, toil and trouble Take care A A A + 1 A = A + B = A + B 1 B B A = A B B A + AB A = A + B A = B B AB Digital logic 12-17 Digital logic 12-18 A universal gate* Circuit simplification o Can we find a simpler circuit that performs the same function.* o Start with an equivalent Boolean expression F(A, B, C) = *All things being equal, smaller circuits are cheaper, faster, cooler, *Are there others? and easier to design. Digital logic 12-19 Digital logic 12-20 5

Voting machines Code detectors o A majority circuit outputs A four input AND gate o 1 whenever a majority of recognizes exactly one input code. its inputs equal 1. o We design a simple majority circuit for three What would a code detector inputs o that recognizes 1001 look like? What would a code detector o the recognizes either 1001 or 1111 look like? Digital logic 12-21 Digital logic 12-22 Could be worse* *Could it be better? Digital logic 12-23 6