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# Class Note for ENGIN 112 at UMass(15)

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Date Created: 02/06/15

ENGIN 112 Intro to Electrical and Computer Engineering Lecture 7 More Logic Functions NAND NOR XOR quot ELECTRICAL 9 quot COMPUTER ENGINEERING umvsnswv or MASSACHUSETTS AMHERST ENGIN11Z L7 Mum Lngic Fundinns Septemh2r172 3 Overview More 2input logic gates NAND NOR XOR Extensions to 3input gates Converting between sumofproducts and NANDs SOP to NANDs NANDs to SOP Converting between sumofproducts and NORs SOP to NORs NORs to SOP Positive and negative logic We use primarily positive logic in this course ENGN112 L7 More Logic Functions September 17 2003 Logic functions of N variables Each truth table represents one possible function eg AND OR N If there are N inputs there are 22 For example is N is 2 then there are 16 possible truth tables So far we have defined 2 of these functions 14 more are possible Why consider new functions Cheaper hardware more flexibility ENGN112 L7 More Logic Functions September 17 2003 The NAND Gate quot gtc Y quot39 This is a NAND gate It is a combination of an AND gate followed by an inverter Its truth table shows this NAND gates have several interesting properties NANDaaaa a NOTa NAND abab ab ANDab A B NANDa b a b ab ORab 0 0 0 1 1 0 1 1 ENGN112 L7 More Logic Functions September 17 Obs The NAND Gate These three properties show that a NAND gate with both foitrs in uts driven by the same signal is equivalent to a ga e A NAND gate whose out ut is com lemented is equivalent to an AND ga e and a AND gate with complemented inputs acts as an OR gate Therefore we can use a NAND39gate to im lement all three of the elementary operators A DORNO Therefore ANY switching function can be constructed using only NAND gates Such a gate is said to be primitive or functionally complete ENGN112 L7 More Logic Functions September 17 2003 NAND Gates into Other Gates what are these circuits A i Y NOT Gate A B Y A AND Gate gt Y B OR Gate ENGN112 L7 More Logic Functions September 17 2003 The NOR Gate A ha Y B I J This is a NOR ate It is a combination of an OR gate followed y an inverter lt s truth table shows this NOR gates also have several A B interesting properties 0 0 NORaaaa a N0Ta 0 1 NOR abab ab ORab NORa b a b ab ANDab 1 0 1 1 COO ENGN112 L7 More Logic Functions September 17 Obs Functionally Complete Gates Just like the NAND gate the NOR gate is functionally comple ean logic function can be implemented using just N R gates Both NAND and NOR ates are very valuable as any design can be rea ized using either one It is easier to build an IC chip usin all NAND or NOR gates than to combine AND R and NOT gates NANDNOR gates are typically faster at switching and cheaper to produce ENGN112 L7 More Logic Functions September 17 2003 NOR Gates into Other Gates what are these circuits AND Gate ENGN112 L7 More Logic Functions September 17 2003 The XOR Gate ExclusiveOR jD Y This is a XOR gate XOR gates assert their output when exactly one of the inputs cc is asserted hence the name The switching algebra symbol for this operation is 6 Le 1 10and1 01 ENGN112 L7 More Logic Functions September 17 2003 The XNOR Gate A a 3 This is a XNOR gate This functions as an exclusiveNOR gate or simply the complement of the XOR gate The switching algebra symbol for this operation is 6 Le 1 11and1 00 ENGN112 L7 More Logic Functions cc CO September 17 2003 NOR Gate Equivalence NOR Symbol Equivalent Circuit Truth Table 2D39w Denotes a Q mversion b on NOR Ha H A a A e B A B 0 o n 1 n 1 1 o 1 o 1 u l l l o 639 ENGIN11Z L7 Mum Lngic Fundinns Septemh2r172 3 DeMorgan s Theorem A key theorem in simplifyinlgLBoolean al ebra expression is DeMorgan s eorem It s ates a b a b ab a b Complement the expression ab zx a and simplify abzX a a b zx a a b zX a a b z X a a b9Z9 X a a b z X a ENGN112 L7 More Logic Functions September 17 2003 Example Determine the outgmt expression for the below circuit and simpli y it usmg DeMorgan s Theorem OH H M urn 0 ENGIN11Z L7 Mum Lngic Fundinns Septemh2r172 3 Universality of NAND and NOR gates a A I A 1 e c I 2 I39 0 ENGN112 L7 More Logic Functions An gtQ I NVEF39ITER AD 539 AND 233 DR September 17 2003 Universaliy of NOR gate uivalent representations of the AND OR and N Tgates ENGtNttZ L71MereLegthunmens Septemba 112003 ExamBIe m mans quot a a a m 74L532 a 2 A msoa C s D St 3 x A54 CD 3H3 01 0H 32 4H00J 2 AND on my ellmmatmg double mvelsmns m msao A 39 a 5 mson s7 o m 74LSUD C39 15 m u Lb 15 ENGINHZ L7 More Logic Functions September 17 2003 Interpretation of the two NAND gate symbols A A5 Output goes LOW only B 7 when ail inputs are HiGH LDW slate IS activerHiGH the active state a A v A AB Output is HIGH when any inpui is LOW HIGH slate IS the active state b B activeLOW Determine the output expression for circuit via DeMorgan s Theorem ENGiNitZ Li More Logic Functions Septeth 112003 InterEretation of the two OR gate sxmbols A A B Output goes HIGH Wnen B any input is HIGH HiGH state is acuvaHGH active state a A X 7 E A B Output goes LOW orly B when all inputs are LOWt LOW state is activeLOW active state b Determine the output expression for circuit via DeMorgan s Theorem EilGIiiiiZ Li More Lngit Functions Squariterii ZliliCi Summary Basic logic functions can be made from NAND and NOR functions The behavior of digital circuits can be represented with waveforms truth tables or symbols Primitive gates can be combined to form larger circuits Boolean algebra defines how binary variables with NAND NO can be combined DeMorgan s rules are important Allow conversion to NANDNOR representations ENGN112 L7 More Logic Functions September 17 2003

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