Logic Circuit - Engineering Electrical Circuits - Lecture Slides, Slides of Electrical Circuit Analysis

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Logic Circuit Synthesis Docsity.com ReCall the Basic Gates A AND A <p NAND A——|>0—— NOT Bo) > OR A ep) oa NOR A pe) >——aen EOR A B A®B ENOR Docsity.com Boolean Algebraic Properties • Commutative: A.B = B.A and A+B = B+A • Distributive: – A.(B+C) = (A.B) + (A.C) – A+(B.C) = (A+B).(A+C) • Identity Elements: 1.A = A and 0 + A = A • Inverse: A.A = 0 and A + A = 1  Associative: • A.(B.C) = (A.B).C and A+(B+C) = (A+B)+C  DeMorgan's Laws: • A.B = A + B and • A+B = A.B Docsity.com Prove Distributive Law • ReCall Bolean Distributive Law • The Circuits at Right Implement The Sides of this Identity • If the Identity is TRUE, then the TruthTable for the two circuits must be Identical ( ) ACABCBA +≡+ D D Docsity.com Prove Distributive Law • Constructing a Truth Table that Includes and Expands [AB+AC] & [A(B+C)] A B C B+C AB AC AB+AC A(B+C) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 IDENTICAL Docsity.com DeMorgan in MATLAB • DeMorgan → Case-b >> % case-b >> y = 3 y = 3 >> b1 = ~((y == 2)|(y > 5)) b1 = 1 >> b2 = (~(y == 2))&(~(y >5)) b2 = 1 0 0 1 1 ( ) BABA ∗=+ ? Docsity.com Multiple-Input Gates • The Flexibility of Boolean Algebra permits straightforward Extension to Gates with Many Inputs • The Number of Entries in the TT is 2N, where: – N ≡ No. of INputs Docsity.com Gates  Boolean (WhiteBoard) • Write the Boolean Expression Equivalent to the Logic Circuit Analyzed Last Lecture Q QDC DC DBA =⋅+⋅+⋅⋅ Docsity.com MinTerms  Sum of Products • Consider an Example Boolean function of three variables by Truth Table • Function Notation – X, Y, Z→ Inputs – Q → Output • The only non-zero OutPuts are at: – X = 0, Y = 1, Z = 0 – X = 1, Y = 1, Z = 0 • The Fcn is 1 for those 2 input-sets and 0 for all other input conditions. X Y Z Q 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 FIX: R(101)=1 R(110)=0 Docsity.com MinTerms  Sum of Products • The Function Operation Requires – The output to be 1 whenever • X=0 AND Y=1 AND Z=0 – OR when • X=1 AND Y=0 AND Z=1 • This Description can written using Boolean Algebra: • Function read as: (NOT-X AND Y AND NOT-Z) OR (X AND NOT-Y AND Z) • By Way of – NOT-X = 0 – NOT-Y = 0 – Etc. ZYXZYXQ ⋅⋅+⋅⋅= X Y Z Q 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 Docsity.com MinTerms  Sum of Products • The Ckt Below Implements Fcn: • The function is composed of two groups of three. • Each group of three is a minterm. – minterm implies that each of the groups of 3 in the expression takes on a value of 1 only for one of the 8 possible combos of X, Y and Z and their inverses ZYXZYXQ ⋅⋅+⋅⋅= Q X Y Z Q 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 Docsity.com MinTerms & SOP Sumarized • We can sum up the MinTerm Construiction with the following: • A truth table gives a unique sum-of-products function that follows directly from expanding the ones in the truth table as minterms. Docsity.com MinTerms Example • Construct Fcn for • ID the Rows with ONES for minterms • Thus This Fuction has a Total of FOUR MinTerms A B C V 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 A B C V 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Docsity.com MinTerms Example • T1 product is the First MinTerm by Multiplication • Similarly Construct Terms 2-4 by Multiplication • To Write the V(ABC) Function Simply OR (add) the MinTerms A B C V 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 CBAT ⋅⋅=1 CBAT ⋅⋅=2 CBAT ⋅⋅=3 CBAT ⋅⋅=4 ( ) ( ) ( ) ( )CBACBA CBACBA V ⋅⋅+⋅⋅ +⋅⋅+⋅⋅ = Docsity.com Sum-of-Products Summarized • One or more AND gates feeding a single OR gate at the output • Example: '''' EACECDAB ++ A 'B C 'D E A 'C 'E Docsity.com PRODUCT of SUMS summarized • One or more OR gates feeding a single AND gate at the output – The DUAL of Sum of Products ))()(( '''' ECAEDCBA +++++ A 'B C 'D E A 'C 'E Docsity.com Prod-of-Sums  MaxTerms • Use MaxTerms to Write the Boolean Equation for • ID rows with OutPut of ZERO A B C Q 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 A B C Q 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 Docsity.com Prod-of-Sums  MaxTerms • The Logic Circuit for the Example A B C Q 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 Q Docsity.com SOP & MINterms ShortHand • Consider the Fcn • Written in SOP Form • Notice the function consists of minterms from rows 0, 2, and 6 • Thus the ShortHand • In Even ShorterHand – Quickly ID the minterms Row A B C Q minterm 0 0 0 0 1 → A’∙B’∙C’ 1 0 0 1 0 2 0 1 0 1 → A’∙B∙C’ 3 0 1 1 0 4 1 0 0 0 5 1 0 1 0 6 1 1 0 1 → A∙B∙C’ 7 1 1 1 0 CABCBACBAQ ++= 620 mmmQ ++= ( )∑= 6,2,0mQ Docsity.com POS & MAXTerms ShortHand • Consider the Fcn • Written POS Form • Notice the function consists of maxterms from rows 3, 4, and 7 • Thus the ShortHand • In Even ShorterHand – Quickly ID the maxterms ( )( )( )CBACBACBAQ ++++++= 743 MMMQ ++= ( )∏= 7,4,3MQ Row A B C Q maxterm 0 0 0 0 1 1 0 0 1 1 2 0 1 0 1 3 0 1 1 0 → A+B’+C’ 4 1 0 0 0 → A’+B+C 5 1 0 1 1 6 1 1 0 1 7 1 1 1 0 → A’+B’+C’ Docsity.com SOP  Canonical • Any SOP expression can be forced into canonical form by ANDing the incomplete terms with terms of the Form (X+X’) where X is the name of the missing variable, e.g.: • This operation produce the minterms ( ) ( ) ( ) BCACABABC BCAABCCABABC BCAACCAB BCABCBAQ ++= +++= +++= +=,, Docsity.com MINIMUM: SOP & POS  The minimum sum of products (MSOP) of a function, f, is a SOP representation of f that contains the fewest number of product terms and fewest number of literals (Variable-Instances) of any SOP representation of f.  Example -- f(a,b,c,d) = ∑m(3,7,11,12,13,14,15) = ab + a′cd + acd = ab + cd  The minimum product of sums (MPOS) of a function, f, is a POS representation of f that contains the fewest number of sum terms and the fewest number of literals of any POS representation of f.  Example -- f(a,b,c,d) = ∏M(0,1,2,4,5,6,8,9,10) = (a + c)(a + d)(a′ + b + d)(b + c′ + d) = (a +c)(a + d)(b + c)(b + d) Docsity.com Karnaugh Maps • Karnaugh maps (K-maps) -- convenient tool for representing switching functions of up to six variables. • K-maps form the basis of useful heuristics (algorithms) for finding MSOP and MPOS representations. • An n-variable K-map has 2n cells with each cell corresponding to a row of an n-variable truth table. Docsity.com What is GRAY code sequence? • Gray code sequence only changes one bit as we go from one number to the next in the sequence, unlike binary. • Adjacent cells vary by only one bit because a Gray code sequence varies by only one bit. Docsity.com Karnaugh Map MiniMization • Write the Boolean expression in SOP form • For each product term, write a 1 in all the squares which are included in the term, 0 elsewhere – canonical form: one square – one term missing: two adjacent squares – two terms missing: 4 adjacent squares Docsity.com Karnaugh Map MiniMization • Example: • Then the K-map • What the 1’s mean ABCCABCBABCAQ +++= A\BC 00 01 11 10 0 0 0 A’BC 0 1 0 AB’C ABC ABC’ A\BC 00 01 11 10 0 0 0 1 0 1 0 1 1 1 Docsity.com Adjacent Pairs • Again: • “Cover” all the 1’s with maximum grouping: • The simplified Eqn is one that Sums All Terms corresponding to each of the groups: ABCCABCBABCAQ +++= A\BC 00 01 11 10 0 0 0 1 0 1 0 1 1 1 Varies Only by “A” (BC group) Varies Only by “B” (AC group) Varies Only by “C” (AB group) BCACABQ ++= Docsity.com Wrap-Around Adjacency • The Top/Bot and Left/Right Edges of the K- maps are Adjacent as well – Think of Map being “Rolled Up” Only C’ does not change Docsity.com Karnaugh Map Simplification • The K-Map uses the following rules for simplification of expressions by grouping together adjacent cells containing ones 1. Groups may not include any cell containing a zero Docsity.com Karnaugh Rules 4. Each group should be as large as possible 5. Each cell containing a one must be in at least one group Docsity.com Karnaugh Rules 6. Groups May OverLap 7. Groups May “Wrap-Around” – Top↔Bot – Left↔Right Docsity.com Karnaugh Rules 8. There should be as few groups as possible, as long as this does not contradict any of the previous rules. Docsity.com Even More Complicated Examples  Eliminate the letters that CHANGE Across the Group (if the group is >1) Docsity.com “don’t care” Conditions • In certain cases some of the minterms may never occur or it may not matter what happens if they do • In such cases we fill in the Karnaugh map with an X (or d) to mean “don't care” • When minimizing an X is like a "joker“ – X can be 0 or 1; Which Ever Helps Minimization • e.g. this – With d=1 Simplifies to Q = B A\BC 00 01 10 11 00 0 0 1 d 01 0 0 1 1 Docsity.com Don’t Care Examples • “Don’t care” conditions should be changed to either 0 or 1 to produce K-map Grouping that yields the simplest expression. Docsity.com