KARNAUGH MAPS

Objectives1. Given a function (completely or in completely specified) of threeto five variable, plot it on a Karnaugh map.The function may be given in minterm, maxterm, or algebraic form.2. Determine the essential prime implicants of a function from a map.3. Obtain the minimum sum-of-products or minimumproduct-of-sums form of a function from the map.4. Determine all of the prime implicants of a function from a map.5. Understand the relation between operations performed usingthe map and the corresponding algebraic operation.

5.1 Minimum Forms of Switching FunctionsExample: Find a minimum sum-of-productsF=F ( a,b,c)= ∑ m(0,1,2,5,6,7)a'b'c'+ a'b'c + a'bc'+ ab'c + abc'+ abc=a'b'+b'c+bc'+abF= a'b'c'+ a'b'c+a'bc'+ ab'c+abc'+ abc=a'b'+bc'+ac

5.1 Minimum Forms of Switching FunctionsExample: Find a minimum product-of-sums)'')('()'(')'(')'()'()''(')'(')'()'')(')(''')(''')(''')('(DCDBADCCBADBADCBDCBCBADBADCBADCBADCBADCBADCBADCBA+++=+++++=++++++++=++++++++++++++++++Eliminate by consensus

5.2 Two- and Three-Variable Karnaugh MapsA 2-variable Karnaugh Map

5.2 Two- and Three-Variable Karnaugh MapsTruth Table for a function FA B0 00 11 01 1F1100

5.2 Two- and Three-Variable Karnaugh MapsTruth Table and Karnaugh Map for Three-Variable FunctionA B C0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1F01010101

5.2 Two- and Three-Variable Karnaugh MapsLocation of Minterms on a Three-Variable Karnaugh Map

5.2 Two- and Three-Variable Karnaugh MapsKarnaugh Map of F(a, b, c) = ∑ m(1, 3, 5) = (0, 2, 4, 6, 7)F(a,b,c)=m1+m3+m5=M0+M2+M4+M6+M7

5.2 Two- and Three-Variable Karnaugh MapsKarnaugh Maps for Product Terms

5.2 Two- and Three-Variable Karnaugh MapsGiven Functionf ( a,b,c)= abc'+ b'c + a'

5.2 Two- and Three-Variable Karnaugh MapsSimplification of a Three-Variable FunctionF= T1 + T2= a'c +b'c

5.2 Two- and Three-Variable Karnaugh MapsComplement of Map in Figure 5-6(a)F = T + T = c'+ ab12

5.2 Two- and Three-Variable Karnaugh MapsKarnaugh Maps Which Illustrate the Consensus TheoremConsensus term is redundant

5.2 Two- and Three-Variable Karnaugh MapsFunction with Two Minimal FormsF= ∑ m(0,1,2,5,6,7 )

5.3 Four-Variable Karnaugh MapsLocation of Minterms on Four-Variable Karnaugh Map

5.3 Four-Variable Karnaugh MapsPlot of acd + a’b + d’f ( a,b,c,d)= acd + a'b + d'

5.3 Four-Variable Karnaugh MapsSimplification of Four-Variable Functions

5.3 Four-Variable Karnaugh MapsSimplification of an Incompletely Specified FunctionDon’t care term

5.3 Four-Variable Karnaugh Maps1's offf=x'z'+ wyz+w'y'z'+ x'y0's of ff ' = y'z+wxz'+ w'xyf = ( y + z ' )( w'+ x'+ z )( w + x'+minimum product of sum fory ' )f

5.4 Determination of Minimum ExpressionsUsing Essential Prime Implicants- Implicants of F : Any single ‘1’ or any group of “1’s which can be combinedtogether on a Map- prime Implicants of F : A product term if it can not be combined with other terms toeliminate variableIt is not Prime implicantssince it can be combinedwith other termsPrime implicantsPrime implicants

5.4 Determination of Minimum ExpressionsUsing Essential Prime ImplicantsDetermination of All Prime Implicants

5.4 Determination of Minimum ExpressionsUsing Essential Prime ImplicantsBecause all of the prime implicants of a function are generally not needed informing the minimum sum of products, selecting prime implicants is needed.-CDis not needed to cover for minimum expression-B’C , AC , BD are “essential” prime implicants- CD is not an “essential “ prime implicants

5.4 Determination of Minimum ExpressionsUsing Essential Prime Implicants1. First, find essential prime implicants2. If minterms are not covered by essential prime implicants only, more primeimplicants must be added to form minimum expression.A'C + A'B'D'+ ACD+A'BDorBCD

5.4 Determination of Minimum ExpressionsUsing Essential Prime ImplicantsFlowchart for Determining a Minimum Sumof Products Using a Karnaugh Map

5.4 Determination of Minimum ExpressionsUsing Essential Prime Implicants1) A’B covers I 6and its adjacent essential PI2) AB’D’ covers I 10and its adjacent essential PI3) AC’D is chosen for minimal cover AC’D is not an essential PI’’’’

5.5 Five-Variable Karnaugh MapsFive-Variable Karnaugh Map

5.5 Five-Variable Karnaugh Maps

5.5 Five-Variable Karnaugh Maps∑F( A,B,C,D,E)= m(0,1,4,5,13,15,20,21,22,23,24,26,28,30,31)F=Resulting minimum solutionAB'CA'B'D'+ ABE'+ ACD + A'BCE + orP1 P2P3P4B'CD'

5.5 Five-Variable Karnaugh Maps∑F( A,B,C,D,E)= m(0,1,3,8,9,14,15,16,17,19,25,27,31)Final solutionC'D'EF=B'C'D'+ B'C'E+A'C'D'+ A'BCD+ABDE+orP1P2P3P4P5AC'E

5.6 Other Uses of Karnaugh Mapsminturmexpansion ofmaxtermexpansion offfisisff= ∑m(0,2,3,4,8,10,11,15)= ∏ M (1,5,6,7,9,12,13,14)same

5.6 Other Uses of Karnaugh MapsFigure 5-26F = ABCD + B'CDE + A'B'+ BCE'Using the consensus theorem :F = ABCD + B'CDE + A'B'+ BCE'+ ACDEminimum solution :F = A'B'+ BCE'+ ACDE

5.7 Other Forms of Karnaugh MapsFigure 5-27. Veitch Diagrams

5.7 Other Forms of Karnaugh MapsFigure 5-28. Other Forms of Five-Variable Karnaugh Maps