1EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu1Lecture #16OUTLINE Logic Binary representations Combinatorial logic circuitsReading Chap 7-7.5EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu2Digital Circuits – Introduction Analog: signal amplitude is continuous with time. Digital: signal amplitude is represented by a restricted set of discrete numbers. Binary: only two values are allowed to represent the signal: High or low (i.e. logic 1 or 0). Digital word: Each binary digit is called a bit A series of bits form a word Byte is a word consisting of 8-bitsAdvantages of digital signal Digital signal is more resilient to noiseÆ can more easily differentiate high (1) and low (0) Transmission Parallel transmission over a bus containing n wires. Faster but short distance (internal to a computer or chip) Serial transmission (transmit bits sequentially) Longer distance2EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu3Binary Representation N bit can represent 2Nvalues: typically from 0 to 2N-1 3-bit word can represent 8 values: e.g. 0, 1, 2, 3, 4, 5, 6, 7 Conversion Integer to binary Fraction to binary (13.510=1101.12and 0.39210=0.0110012) Octal and hexadecimalEE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu4Logic Gates and Memories Logic gates Combine several logic variable inputs to produce a logic variable output Memory Memoryless: output at a given instant depends the input values of that instant. Momory: output depends on previous and present input values.3EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu5Boolean algebras are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoreticoperations intersection, union and complement. They are named after George Boole, an English mathematician at University College Cork, who first defined them as part of a system of logic in the mid 19th century. Specifically, Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus. Today, Boolean algebras find many applications in electronic design. They were first applied to switching by Claude Shannon in the 20th century.EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu6Boolean algebras The operators of Boolean algebra may be represented in various ways. Often they are simply written as AND, OR and NOT. In describing circuits, NAND (NOT AND), NOR (NOT OR) and XOR (eXclusive OR) may also be used. Mathematicians often use + for OR and · for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures) and represent NOT by a line drawn above the expression being negated.4EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu7Boolean Algebra NOT operation (inverter) AND operation OR operation01AAAA=+=g100() ()AA AAAAAB BAABC ABC=====ggggggg gg110() ()AAAAAAABBAABCABC+=+=+=+=+++=+ +EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu8Graphic RepresentationAA01AAAA=+=gFull square = complete set =1Yellow part = NOT(A) =AWhite circle = A5EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu9Graphic RepresentationABABAB+()()ABABABABABABAB⊕= + = + + = ++ggExclusive OR=yellow and blue part –intersection/overlap part=exactly when only one of the input is true EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu10Boolean Algebra Distributive Property De Morgan’s laws An excellent web site to visit http://en.wikipedia.org/wiki/Boolean_algebra()()()()AB C AB ACABC A B A C+= ++=+ +gggggABABABAB+==+gg6EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu11F = A•B•C + A•B•C + (C+D)•(D+E)F = C•(A+D+E) + D•EExamplesEE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu12Logic Functions, Symbols, & Notation“NOT”F = ATRUTHNAME SYMBOL NOTATION TABLEFA111001010000FBA“OR”F = A+BFAB0110FA111101110000FBA“AND”F = A•BFAB7EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu13Logic Functions, Symbols, & Notation 2“NOR”F = A+B011101110000FBA“NAND”F = A•BFAB011101110100FBA“XOR”(exclusive OR)F = A + BFABFAB011001010100FBAEE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu14Circuit RealizationAB⊕()()ABABABABABABAB⊕= + = + + = ++ggAAABABBB8EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu15Fan in/Fan out Complex digital operations are formed with a variety of gates interconnected to yield the desired logic function. Sometimes a number of inputs are connected to one gate input and output of a gate may be connected to a number of gates. Fan-in: the maximum number of logic gates that can be connected at the input of a gate without altering its performance. Fan-out: the maximum number of logic gates that can be connected to the output of a gate without altering its performance. Typical fan-in and fan-out numbers are 3.EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu16Inverter = NOT GateVoutVinVinVoutVV/2Ideal Transfer Characteristics9EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu17VDD/RDVDDNMOS Resistor Pull-UpvDSiD0vOUTvIN0VDDRD+vDS= vOUT–iD+vIN–VDDRD+vDS= vOUT–iD+vIN–Circuit:Voltage-Transfer CharacteristicVDDVT0110FAAFincreasingvGS= vIN> VTvGS= vin≤ VT vIN= VDDVDDEE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu18Disadvantages of NMOS Logic Gates Large values of RDare required in order to achieve a low value of VOL keep power consumption lowÆ Large resistors are needed, but these take up a lot of space.• One solution is to replace the resistor with an NMOSFET that is always on.10EE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu19The CMOS Inverter: Intuitive PerspectiveVDDRnVIN= VDDCIRCUITSWITCH MODELSVDDRpVIN= 0 VVOUTVOUTVOL= 0 V VOH= VDDLow static power consumption, sinceone MOSFET is always off in steady stateVDDVINVOUTSDGGSDEE40 Summer 2005: Lecture 16 Instructor: Octavian Florescu20CMOS Inverter Voltage Transfer CharacteristicVINVOUTVDDVDD00N: offP: linN: linP: offN: linP: satN: satP: linN: satP: satVDDVINVOUTSDGGSDVDDVINVOUTVDDVINVOUTSDGGSDABDEC11EE40 Summer 2005: Lecture 16
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