# L2: Combinational Logic Design (Construction and Boolean Algebra)

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1 L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson Education, 25. Some lecture material adapted from J. Rabaey, A. Chandrakasan, B. Nikolic, Digital Integrated Circuits: A Design Perspective Copyright 23 Prentice Hall/Pearson. Lecture Based on Notes by Professor Anantha Chandrakasan

2 Review: Noise Margin Truth Table IN OUT IN OUT "" V OH V IH V(y) V OH Slope = - Undefined Region NM L = V IL -V OL NM H = V OH -V IH V IL Slope = - "" V OL V OL V OL V IL V IH V OH V(x) Large noise margins protect against various noise sources 2

3 TTL Logic Style (97 s-early 8 s) C + B + v BE - E v CE - Q Q2 Q3 74LS4 (courtesy TI) 3

4 MOS Technology: The NMOS Switch source gate drain D N+ N+ G V T =.5V P-substrate S V s Switch Model OFF R NMOS V GS < V T ON R NMOS V GS > V T NMOS ON when Switch Input is High 4

5 NMOS Device Characteristics Polysilicon Aluminum 6 x -4 VGS= 2.5 V 5 4 Resistive Saturation VGS= 2. V I D (A) 3 2 VGS=.5 V VGS=. V D V DS (V) G + V GS - S I D V T =.5V MOS is a very non-linear. Switch-resistor model sufficient for first order analysis. 5

6 PMOS: The Complementary Switch source gate drain S P+ P+ G V T = -.5V N-substrate D V DD Switch OFF R PMOS ON R PMOS Model V GS > V T V GS < V T PMOS ON when Switch Input is Low 6

7 The CMOS Inverter V DD Switch Model V DD G S R PMOS IN D D OUT IN OUT G R NMOS S IN Rail-to-rail Swing in CMOS 7

8 Inverter VTC: Load Line Analysis V DD G S D IN OUT D G S CMOS gates have: Rail-to-rail swing (V to V DD ) Large noise margins zero static power dissipation V (V) in 8 V out (V).5.5

9 Possible Function of Two Inputs There are 6 possible functions of 2 input variables: X Y F X Y 6 possible functions (F F 5 ) X AND Y X Y X XOR Y X OR Y X = Y X NOR Y NOT (X OR Y) NOT Y NOT X X NAND Y NOT (X AND Y) In general, there are 2 (2^n) functions of n inputs 9

10 Common Logic Gates Gate Symbol Truth-Table Expression X Y Z NAND X Y Z Z = X Y X Y Z AND X Y Z Z = X Y X Y Z NOR X Y Z Z = X + Y X Y Z OR X Y Z Z = X + Y

11 Exclusive (N)OR Gate XOR (X Y) X Y Z X Y Z Z = X Y + X Y X or Y but not both ("inequality", "difference") XNOR (X Y) X Y Z X Y Z Z = X Y + X Y X and Y the same ("equality") Widely used in arithmetic structures such as adders and multipl

12 Generic CMOS Recipe V dd... A F(A,,A n ) A n pullup: make this connection when we want F(A,,A n ) = pulldown: make this connection when we want F(A,,A n ) = Note: CMOS gates result in inverting functions! (easier to build NAND vs. AND) A B A A PDN B B PUN C L O A B PDN PUN O ff n ff n ff n n ff How do you build a 2-input NOR Gate? 2

13 Theorems of Boolean Algebra (I) Elementary. X + = X D. X = X 2. X + = 2D. X = 3. X + X = X 3D. X X = X 4. (X) = X 5. X + X = 5D. X X = Commutativity: 6. X + Y = Y + X 6D. X Y = Y X Associativity: 7. (X + Y) + Z = X + (Y + Z) 7D. (X Y) Z = X (Y Z) Distributivity: 8. X (Y + Z) = (X Y) + (X Z) 8D. X + (Y Z) = (X + Y) (X + Z) Uniting: 9. X Y + X Y = X 9D. (X + Y) (X + Y) = X Absorption:. X + X Y = X D. X (X + Y) = X. (X + Y) Y = X Y D. (X Y) + Y = X + Y 3

14 Theorems of Boolean Algebra (II) Factoring: 2. (X Y) + (X Z) = 2D. (X + Y) (X + Z) = X (Y + Z) X + (Y Z) Consensus: 3. (X Y) + (Y Z) + (X Z) = 3D. (X + Y) (Y + Z) (X + Z) = X Y + X Z (X + Y) (X + Z) De Morgan's: 4. (X + Y +...) = X Y... 4D. (X Y...) = X + Y +... Generalized De Morgan's: 5. f(x,x2,...,xn,,,+, ) = f(x,x2,...,xn,,,,+) Duality Dual of a Boolean expression is derived by replacing by +, + by, by, and by, and leaving variables unchanged f (X,X2,...,Xn,,,+, ) f(x,x2,...,xn,,,,+) 4

15 Simple Example: One Bit Adder -bit binary adder inputs: A, B, Carry-in outputs: Sum, Carry-out A B Cin S Cout A B Cin S Cout Product term (or minterm) Sum-of-Products Canonical Form S = A B Cin + A B Cin + A B Cin + A B Cin Cout = A B Cin + A B Cin + A B Cin + A B Cin ANDed product of literals input combination for which output is true Each variable appears exactly once, in true or inverted form (but not both) 5

16 Simplify Boolean Expressions Cout = A B Cin + A B Cin + A B Cin + A B Cin = A B Cin + A B Cin + A B Cin + A B Cin + A B Cin + A B Cin = (A + A) B Cin + A (B + B) Cin + A B (Cin + Cin) = B Cin + A Cin + A B = (B + A) Cin + A B S = A B Cin + A B Cin + A B Cin + A B Cin =( A B + A B )Cin + (A B + A B) Cin =(A B) Cin + (A B) Cin = A B Cin 6

17 Sum-of-Products & Product-of-Sum Product term (or minterm): ANDed product of literals input combination for which output is true A B C minterms A B C m A B C m A B C m2 A B C m3 A B C m4 A B C m5 A B C m6 A B C m7 short-hand notation form in terms of 3 variables F in canonical form: F(A, B, C) = Σm(,3,5,6,7) = m + m3 + m5 + m6 + m7 F = A B C + A B C+ A B C + A B C + ABC canonical form minimal form F(A, B, C) = A B C + A B C + AB C + ABC + ABC = (A B + A B + AB + AB)C + ABC = ((A + A)(B + B))C + ABC = C + ABC = ABC + C = AB + C Sum term (or maxterm) - ORed sum of literals input combination for which output is false A B C maxterms A + B + C M F in canonical form: A + B + C M F(A, B, C) = ΠM(,2,4) A + B + C M2 = M M2 M4 A + B + C M3 = (A + B + C) (A + B + C) (A + B + C) A + B + C M4 canonical form minimal form A + B+ C M5 F(A, B, C) = (A + B + C) (A + B + C) (A + B + C) A + B +C M6 = (A + B + C) (A + B + C) A +B + C M7 (A + B + C) (A + B + C) = (A + C) (B + C) short-hand notation for maxterms of 3 variables 7

18 The Uniting Theorem Key tool to simplification: A (B + B) = A Essence of simplification of two-level logic Find two element subsets of the ON-set where only one variable changes its value this single varying variable can be eliminated and a single product term used to represent both elements A B F F = A B +AB = (A +A)B = B B has the same value in both on-set rows B remains A has a different value in the two rows A is eliminated 8

19 Boolean Cubes Just another way to represent truth table Visual technique for identifying when the uniting theorem can be applied n input variables = n-dimensional "cube" -cube X Y X XY 2-cube Y Z XYZ X 3-cube Y Z W X 4-cube WXYZ 9

20 Mapping Truth Tables onto Boolean Cubes Uniting theorem Circled group of the on-set is called the A B F B F A adjacency plane. Each adjacency plane corresponds to a product term. ON-set = solid nodes OFF-set = empty nodes A varies within face, B does not this face represents the literal B Three variable example: Binary full-adder carry-out logic A B Cin Cout (A+A)BCin B C A AB(Cin+Cin) Cout = BCin+AB+ACin A(B+B)Cin The on-set is completely covered by the combination (OR) of the subcubes of lower dimensionality - note that is covered three times 2

21 Higher Dimension Cubes F(A,B,C) = Σm(4,5,6,7) B C A on-set forms a square i.e., a cube of dimension 2 (2-D adjacency plane) represents an expression in one variable i.e., 3 dimensions 2 dimensions A is asserted (true) and unchanged B and C vary This subcube represents the literal A In a 3-cube (three variables): -cube, i.e., a single node, yields a term in 3 literals -cube, i.e., a line of two nodes, yields a term in 2 literals 2-cube, i.e., a plane of four nodes, yields a term in literal 3-cube, i.e., a cube of eight nodes, yields a constant term "" In general, m-subcube within an n-cube (m < n) yields a term with n m literals 2

22 Karnaugh Maps Alternative to truth-tables to help visualize adjacencies Guide to applying the uniting theorem - On-set elements with only one variable changing value are adjacent unlike in a linear truth-table A B 2 3 A B F Numbering scheme based on Gray code e.g.,,,, (only a single bit changes in code for adjacent map cells) 2-variable K-map 3-variable K-map A B C AB A C B 22 AB CD A B D 4-variable K-map

23 K-Map Examples Cout = F(A,B,C) = F(A,B,C) = Σm(,4,5,7) F = F' simply replace 's with 's and vice versa F'(A,B,C) = Σm(,2,3,6) 23 F' =

24 Four Variable Karnaugh Map F(A,B,C,D) = Σm(,2,3,5,6,7,8,,,4,5) F = C + A B D + B D Find the smallest number of the largest possible subcubes that cover the ON-set C D A B K-map Corner Adjacency Illustrated in the 4-Cube 24

25 K-Map Example: Don t Cares Don't Cares can be treated as 's or 's if it is advantageous to do so F(A,B,C,D) = Σm(,3,5,7,9) + Σd(6,2,3) F = A D + B C D w/o don't cares F = C D + A D w/ don't cares By treating this DC as a "", a 2-cube can be formed rather than one -cube In PoS form: F = D (A + C) Equivalent answer as above, but fewer literals 25

26 Hazards 26

27 Fixing Hazards In general, it is difficult to avoid hazards need a robust design methodology to deal with hazards. 27

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