DIGITAL LOGIC CIRCUITS - Engineering

1 BINARY NUMBER SYSTEM electronic CIRCUITS that handle informationencoded in binary form (deal with signalsthat have only two values, 0and 1) DIGITAL .. computers, watches, controllers, telephones, cameras, .. DIGITAL LOGIC circuitsNumber ..inwhatever base Decimal value of the given number_____Decimal: 1,998= 1x103 +9x102 +9x101 +8x100=1,000+900+90+8 =1,998 Binary:11111001110 = 1x210 +1x29 +1x28 +1x27 +1x26 +1x23 +1x22 +1x21 =1,024+512+258+128+64+8+4+2 = 1,998 Emil M. PetriuProf. Emil M. Petriu,School of Electrical Engineering and Computer Science, University of OttawaDIGITAL LOGIC CIRCUITS_____N2 NComments_____011224384165326647 1288 256951210 1,024 Kilo as 210 is the closest power of 2 to 1,000 (decimal)112, ,768 215Hz often used as clock crystal frequency in DIGITAL 1,048,576 Mega as 220is the closest power of 2 to 1,000,000 (decimal).

2 30 1,073,741,824 Giga as 230 is the closest power of 2 to 1,000,000,000(decimal)_____Powers of 2 Emil M. Petriu_____N <02N_____-12-1 = Powers of 2 Binary numbers less than 1= 1x2-1+1x2-3+ 1x2-4+1x2-6= Decimal Emil M. PetriuHexadecimal: 7CE71214 <== Decimal1111100111011111001110 Binary:= 7x162+12x161+14x160= 1998------------------------------------ ---------------------------------------- ------------------------Binary Decimal Hexadecimal----------------------------- ---------------------------------------- -------------------------------0000 0 00001 1 10010 2 20011 3 30100 4 40101 5 50110 6 60111 7 71000 8 81001 9 91010 10 A1011 11 B1100 12 C1101 13 D1110 14 E1111 15

3 F--------------------------------------- ---------------------------------------- ---------------------HEXADECIMAL Emil M. PetriuLOGIC OPERATIONS AND TRUTH TABLESD igital LOGIC CIRCUITS handle data encoded in binary form, signalsthat have only two values, 0and LOGIC dealing with true and false comes in handy to describethe behaviour of these CIRCUITS : 0is usually associated with false and1with true. Quite complex DIGITAL LOGIC CIRCUITS ( entire computers) can be built using a few types of basic circuitscalled gates, each performing a single elementary LOGIC operation : NOT, AND, OR, NAND, NOR, Boole s binary algebra is used as a formal / mathematical tool to describe and design complex binary LOGIC CIRCUITS . Emil M. PetriuA B A + B_____0 0 00 1 11 0 11 1 1 _____ABF = A + BORAF =AA A_____0 11 0 NOTABF =A.

4 BA B A . B_____0 0 00 1 01 0 01 1 1 _____ANDGATES Emil M. PetriuNANDA B A . B_____0 0 10 1 11 0 11 1 0 _____NORA B A + B_____0 0 10 1 01 0 01 1 0 _____ABF =A . BABF = A + more GATES Emil M. PetriuXORABF = A B A B A B_____0 0 00 1 11 0 11 1 0 _____ EQU orXNORABF = A B A B A B_____0 0 10 1 01 0 01 1 1 _____ .. and more GATES Emil M. PetriuGATES .. with more inputsA B C A. 0 0 0 0 01 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1_____F = = = A+B+CABCF = A+B+CABCA+B+ +B+C_____10000000_____ANDORNANDNOREXAMPL ES OF GATES WITH THREE INPUTS Emil M.

5 PetriuLogic Gate Array that Produces an Arbitrarily Chosen OutputA B C F_____0 0 0 0 0 01 00 1 0 10 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1_____ ABCABCABCA . B . CA . B . CA . B . CFF = +A . B . CA . B . C+A . B . C+A . B . CA . B . C Sum-of-products form of the LOGIC circuit . Emil M. PetriuBOOLEAN ALGEBRAAND rulesA . A = AA . A = 00 . A = 01 . A = AA . B = B . AA . (B . C) =(A . B) . CA . (B + C) = A . B + A . CA . B = A + BA B C A. (B+C) + 0 0 0 00 0 1 0 00 1 0 0 00 1 1 0 01 0 0 0 01 0 1 1 11 1 0 1 11 1 1 1 1 Proof : Emil M.

6 PetriuOR rulesA + A = AA + A = 10 + A = A1 + A = 1A + B = B + AA +(B + C) =(A + B) + CA + B . C = (A + B) . (A +C)A + B = B C A + (A+B) . (A+C)_____0 0 0 0 00 0 1 0 00 1 0 0 00 1 1 1 11 0 0 1 11 0 1 1 11 1 0 1 11 1 1 1 1 BOOLEAN ALGEBRA .. continued Emil M. PetriuA. B = A + BA + B = A .BDeMorgan s TheoremA B A .B A + B0 0 1 10 1 0 01 0 0 01 1 0 0 A + B A. B_____1 11 11 10 0 Emil M. PetriuABCABCABCA . B . CA . B . CA . B . CFA . B . CF=ABC + ABC + ABC + ABCSum-of-productsform of the LOGIC function:Simplifying LOGIC functions using Boolean algebra rules Emil M.

7 PetriuF = ABC + ABC + ABC + ABCF = (ABC + ABC) + (ABC + ABC)F = A(BC + BC) + A ( BC + BC)F = AB( C + C) + AC ( B + B)11 Simplifying LOGIC functions using Boolean algebra rules .. continuedABCAAA . CFA . B F = AB + AC Emil M. PetriuSimplifying LOGIC functions using Karnaugh mapsA B C F(0) 0 0 0 ..(1)0 0 1 ..(2)0 1 0 ..(3) 0 1 1 ..(4)1 0 0 ..(5)1 0 1 ..(6)1 1 0 ..(7)1 1 1 ..Each line in the truth table corresponds to a square in the Karnaugh map=> graphical representation of a truth table for a LOGIC Karnaugh map squares are labeled so that horizontally or vertically adjacent squaresdiffer only in one variable. (Each square in the top row is considered to be adjacent to a corresponding square in the bottom row. Each square in the left most column is considered to be adjacent to a corresponding square in theright most column.)

8 C A B00 01 11 1000 2 6 41 3 7 51 Karnaugh map Emil M. PetriuC DA B00 01 11 10000 4 12 81 5 13 93 7 15 112 6 14 10011110A B C D F(0) 0 0 0 0 ..(1)0 0 0 1 ..(2)0 0 1 0 ..(3) 0 0 1 1 ..(4)0 1 0 0 ..(5)0 1 0 1 ..(6)0 1 1 0 ..(7)0 1 1 1 ..(8) 1 0 0 0 ..(9) 1 0 0 1 ..(10) 1 0 1 0 ..(11) 1 0 1 1 ..(12) 1 1 0 0 ..(13) 1 1 0 1 ..(14) 1 1 1 0 ..(15) 1 1 1 1 ..Simplifying LOGIC functions of 4 variables using Karnaugh maps Emil M. PetriuA B C F(0) 0 0 0 0(1)0 0 1 0(2)0 1 0 1(3) 0 1 1 1(4)1 0 0 0(5)1 0 1 1(6)1 1 0 0(7)1 1 1 1 Simplifying LOGIC functions using Karnaugh maps.

9 LoopingThe LOGIC expressions for an output can be simplified by properlycombining squares (looping) in the Karnaugh maps which contain a pair of adjacent 1s eliminates the variable that appears in both direct and complemented A B00 01 11 10010 10 00 1 1 1F = AB + AC Emil M. PetriuLooping aquad of adjacent 1s eliminates the two variables that appears in both direct and complemented B C D F(0) 0 0 0 0 1(1)0 0 0 1 0(2)0 0 1 0 1(3) 0 0 1 1 0(4)0 1 0 0 0(5)0 1 0 1 1(6)0 1 1 0 0(7)0 1 1 1 1(8) 1 0 0 0 0(9) 1 0 0 1 0(10) 1 0 1 0 0(11) 1 0 1 1 0(12) 1 1 0 0 1(13) 1 1 0 1 1(14) 1 1 1 0 1(15) 1 1 1 1 1 Simplifying LOGIC functions using Karnaugh maps.

10 More loopingC DA B00 01 11 10000111101 0 1 00 1 1 00 1 1 01 0 1 0 BDA B DABF = A B D + AB + BD Emil M. PetriuABA+BA . B AB=A+BABA. BAB=DeMorgan s TheoremA. B = A + BA + B = A .BEquivalent Gate Symbols Emil M. PetriuNAND gate implementation of the sum-of-product LOGIC functionsNAND gates are faster than ANDs and ORs in most technologiesABCA AFA B CFAF = AB + ACX = ( X ) Emil M. PetriuADDING BINARY NUMBERS0+1_____11+0_____11+1_____10 The binary number 10is equivalent to the decimal2 Carry(over)Sum0+0_____0 Inputs OutputsA B Carry Sum0 0 0 00 1 011 0 0 11 1 10 Truth tableABSumCarrySum = A+BCarry = A . BHalf-AddercircuitAdding two bits: Emil M. PetriuAdding multi-bit numbers:1 0 1 1 0 1 0_____100010010101111000108D +90D 198D1 1 0 1 1 0 0 +SumCarryA BCOCISA BCOCISA BCOCISA BCOCISA BCOCISA BCOCISA BCOCISA BCOCISS7S6 S5 S4 S3 S2S1 S0A7B7A6B6A5 B5 A4B4A3B3A2B2A1B1A0 B0 Carry OutCarry In =0A7A6A5A4A3A2A1A0+B7B6 B5B4B3 B2B1B0 S7S6 S5 S4 S3 S2S1 S0 Emil M.