Switching Circuits & Logic Design


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1 Switching Circuits & Logic Design JieHong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 23 5 Karnaugh Maps Kmap Walks and Gray Codes 2
2 Outline Minimum forms of switching functions Two and threevariable Karnaugh maps Fourvariable Karnaugh maps Determination of minimum expressions using essential prime implicants Fivevariable Karnaugh maps Other uses of Karnaugh maps Other forms of Karnaugh maps 3 Limitations of lgebraic Simplification Two problems of algebraic simplification. Not systematic 2. Difficult to check if a minimum solution is achieved The Karnaugh map method overcomes these limitations Typically for Boolean functions with 5 variables The QuineMcCluskey method can deal with even larger functions (Subject of Unit 6, skipped) 4
3 Minimum Forms of Switching Functions Correspondence between Boolean expressions and logic circuits SOP (POS) can be implemented with twolevel NDOR (ORND) gate circuits Reducing the number of terms and literals of an SOP expression corresponds to reducing the number of gates and gate inputs Combine terms by XY'+XY=X Eliminate redundant terms by consensus theorem Minimum SOP is not necessarily unique n SOP may be minimal (locally) but not minimum (globally) E.g., F = a'b'c'+a'b'c+a'bc'+ab'c+abc'+abc = a'b'+b'c+bc'+ab (minimal but not minimum) = a'b'+bc'+ac (minimum) 5 TwoVariable Karnaugh Maps 2variable Kmap B F(m ) F(m 2 ) =, B= =, B= F(m ) F(m 3 ) =, B= =, B= minterm locations B 'B' 'B ' B F F = 'B'+ 'B = '(B'+B) = ' 6
4 ThreeVariable Karnaugh Maps 3variable Kmap F 2 6 minterm locations F = ''+ '+ B'C'+ ' = 'B+ C'+ ' = 'B+ C' 7 ThreeVariable Karnaugh Maps 3variable Kmap (zeros omitted) B ' C' 8
5 ThreeVariable Karnaugh Maps 3variable Kmap bc a f(a,b,c) = abc'+b'c+a' 9 ThreeVariable Karnaugh Maps 3variable Kmap F = m +m 3 +m 5 = M M 2 M 4 M 6 M 7 F = a'c+b'c bc a bc a simplify
6 ThreeVariable Karnaugh Maps 3variable Kmap G = (m +m 3 +m 5 )' = (M M 2 M 4 M 6 M 7 )' G = ab+c' bc a bc a simplify ThreeVariable Karnaugh Maps 3variable Kmap yz x yz x simplify xy+x'z+yz = xy+x'z (consensus theorem) 2
7 ThreeVariable Karnaugh Maps 3variable Kmap F = a'b'+bc'+ac = a'c'+b'c+ab bc a bc a 3 FourVariable Karnaugh Maps 4variable Kmap F = acd+a'b+d' CD B cd ab minterm locations 4
8 FourVariable Karnaugh Maps 4variable Kmap cd ab cd ab simplify f = m(,3,4,5,,2,3) f = ab'cd'+a'b'd+bc' 5 FourVariable Karnaugh Maps 4variable Kmap cd ab cd ab simplify f 2 = m(,2,3,5,6,7,8,,,4,5) f 2 = c+b'd'+a'bd 6
9 FourVariable Karnaugh Maps Simplify incompletely specified function ll the s must be covered, but X s are optional and are set to s only if they will simplify the expression cd ab cd ab X X X X simplify X X f= m(,3,5,7,9)+ d(6,2,3) f = a'd+c'd 7 FourVariable Karnaugh Maps Simplify productofsums Circle s instead of s pply De Morgan s law converting SOP to POS yz wx yz wx simplify f = x'z'+wyz+w'y'z'+x'y f' = y'z + wxz'+w'xy f = (y+z')(w'+x'+z)(w+x'+y') 8
10 Determination of Minimum Expressions Using Essential Prime Implicants Implicant product term of a function ny single or any group of s on a Kmap combined together forms a product term Prime implicant maximal implicant n implicant that cannot be combined with another term to eliminate a variable ll of the prime implicants of a function can be obtained from a Kmap by expanding the s as much as possible in every possible way 9 Determination of Minimum Expressions Using Essential Prime Implicants Example cd ab a'b'c'd' ac' a'b'c, a'cd', ac' are prime implicants a'b'c'd', abc', ab'c' are implicants (but not prime implicants) a'b'c ab'c' abc' a'cd' 2
11 Determination of Minimum Expressions Using Essential Prime Implicants Determine all prime implicants In finding prime implicants, don t cares are treated as s. However, a prime implicant composed entirely of don t cares can never be part of the minimum solution Not all prime implicants are needed in forming the minimum SOP ab cd Example ll prime implicants: a'b'd, bc', ac, a'c'd, ab, b'cd (composed entirely of don t cares) Minimum solution: F = a'b'd+bc'+ac X X 2 Determination of Minimum Expressions Using Essential Prime Implicants Essential prime implicant (EPI) prime implicant that covers some minterm not covered by any other prime implicant If a single term covers some minterm and all of its adjacent s and X s, then the term is an EPI Must be present in the minimum SOP B B CD CD EPIs already cover all s Minimum SOP! f = CD+BD+B'C+C f= BD+B'C+C 22
12 Determination of Minimum Expressions Using Essential Prime Implicants SOP minimization. Select all essential prime implicants 2. Find a minimum set of prime implicants which cover the minterms not covered by the essential prime implicants There may be freedom left after all essential prime implicants are selected (it affects optimality especially for functions with more variables) CD B 'C' 'B'D' F = 'C'+ 'B'D'+CD+ 'BD or D CD 23 Determination of Minimum Expressions Using Essential Prime Implicants Flowchart for determining a minimum SOP using Kmap Choose a which has not been covered That term is an essential prime implicant. Loop it Find all adjacent s and X s ll uncovered s checked? NO re the chosen and its adjacent s and X s covered by a single term? YES YES Find a minimum set of prime implicants which cover the remaining s on the map Note: ll essential prime implicants have been determined at this point NO STOP 24
13 Determination of Minimum Expressions Using Essential Prime Implicants Example B CD X Step : 4 checked Step 2: 5 checked Step 3: 6 checked EPI 'B selected Step 4: 8 checked Step 5: 9 checked Step 6: checked EPI B'D' selected Step 7: 3 checked X 7 6 X 5 (up to this point all EPIs determined) Step 8: C'D selected to cover remaining s 25 FiveVariable Karnaugh Maps 5var Kmap These two terms do not combine DE DE / / minterm locations B'DE' CDE BD' 26
14 FiveVariable Karnaugh Maps 5var Kmap Expand to eliminate DE Expand to eliminate E / Expand to eliminate B Expand to eliminate D Expand to eliminate C 27 FiveVariable Karnaugh Maps 5var Kmap P / DE B'C F = 'B'D'+BE'+CD+'E+ or P P 2 P 3 P 4 B'CD' Shaded s show prime implicants P 4 P 3 P 2 28
15 FiveVariable Karnaugh Maps 5var Kmap / DE P2 P C'E F = B'C'D'+B'C'E+'C'D'+'D+BDE+ or P P 2 P 3 P 4 P 5 C'D'E P 3 P 5 P 4 29 Other Uses of Karnaugh Maps Use Kmap to prove the equivalence of two Boolean expressions Kmaps are canonical representations of Boolean functions, similar to truth tables Use Kmap to perform Boolean operations ND, OR, NOT operations can be done over K maps (truth tables) 3
16 Other Uses of Karnaugh Maps Use Kmap to facilitate factoring Identify common literals among product terms CD B F = 'B'C'+'B'D+CB+CD' = 'B'(C'+D)+C(B+D') 3 Other Uses of Karnaugh Maps Use Kmap to guide simplification DE F = D+B'CDE+'B'+E' = D+B'CDE+'B'+E'+CDE = 'B'+E'+CDE / dd this term Then these two terms can be removed 32
17 Other Forms of Karnaugh Maps Other conventions (Veitch diagrams) C D B C B 33 Other Forms of Karnaugh Maps Other conventions (5var Kmap) F = D'E'+B'C'D'+E+''E'+CDE DE DE = = 34
18 Other Forms of Karnaugh Maps Other conventions (5var Veitch diagram) F = D'E'+B'C'D'+E+''E'+CDE B E D C C 35
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