Combinational Logic & Circuits

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1 Week-I Combinational Logic & Circuits Spring' Logic Design Page

2 Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and characteristics Read Chapter-! Spring' Logic Design Page 2

3 Combinational Logic One or more digital signal inputs One or more digital signal outputs Outputs are only functions of current input values (ideal) plus logic propagation delays I I m O t t Combinational Logic F I t,...i m t O O n O n t t F n I t,...i m Spring' Logic Design Page 3 t

4 Combinational Logic (cont.) Combinational logic has no memory! Outputs are only function of current input combination Nothing is known about past events Repeating a sequence of inputs always gives the same output sequence Sequential logic (to be covered later) does have memory Repeating a sequence of inputs can result in an entirely different output sequence Spring' Logic Design Page 4

5 Combinational Logic Example A circuit controlling the level of fluid in a tank inputs are: HI - if fluid level is too high, otherwise LO - if fluid level is too low, otherwise outputs are: Pump - to pump fluid into tank, for pump off Drain - to open tank drain, for drain closed input to output relationship is described by a truth table Spring' Logic Design Page 5

6 Combinational Logic Example (cont.) HI LO Pump Drain Truth Table Representation x x Tank level is OK Low level, pump more in High level, drain some out Inputs cannot occur Schematic Representation HI LO Pump Drain Spring' Logic Design Page 6

7 Switching Algebra Based on Boolean Algebra Developed by George Boole in 854 Formal way to describe logic statements and determine truth of statements Only has two-values domain ( and ) Huntington s Postulates define underlying assumptions Spring' Logic Design Page 7

8 Huntington s Postulates Closure If X and Y are in set (,) then operations X+Y and X Y are also in set (,) Identity X + = X X = X Commutativity X + Y = Y + X X Y = Y X Spring' Logic Design Page 8

9 Huntington s Postulates (cont.) Distributive X (Y + Z) = (X Y) + (X Z) X + (Y Z) = (X + Y) (X + Z) Complement X X X X Duality Principle: Note that for each property, one form is the dual of the other; (s to s, s to s, s to +s, +s to s) Spring' Logic Design Page 9

10 Switching Algebra Operations - Not Unary complement or inversion operation Usually shown as overbar (X), other forms are ~ X, X X X X X X X ~ Spring' Logic Design Page

11 Switching Algebra Operations - AND Also known as the conjunction operation; output is true () only if all inputs are true Algebraic operators are, &, X Y X Y & Spring' Logic Design Page

12 Switching Algebra Operations - OR Also known as the disjunction operation; output is true () if any input is true Algebraic operators are +,, X Y X+Y Spring' Logic Design Page 2

13 Logic Expressions Terms and Definitions Logic Expression - a mathematical formula consisting of logical operators and variables Logic Operator - a function that gives a well defined output according to switching algebra Logic Variable - a symbol representing the two possible switching algebra values of and Logic Literal - the values and or a logic variable or it s complement Z = X.Y + X.W logic expression X, Y, W, Z logic variables +,. operators Spring' Logic Design Page 3

14 Logic Expressions - Precedence Like standard algebra, switching algebra operators have a precedence of evaluation NOT operations have the highest precedence AND operations are next OR operations are lowest Parentheses explicitly define the order of operator evaluation If in doubt, USE PARENTHESES! Spring' Logic Design Page 4

15 Spring' Logic Design Page 5

16 Logic Expressions vs. Logic Circuits Each expression can be realized by a logic circuit Multiple expressions for a single function Simpler the expression is, faster and cheaper the circuit is So we try to simplify logic expressions if possible We call it minimization Spring' Logic Design Page 6

17 Logic Expression Minimization Goal is to find an equivalent of an original logic expression that: a) has fewer variables per term b) has fewer terms c) needs less logic to implement Example: E = A.B.C.D + A.B.C.D = A.B.C There are three main manual methods Algebraic minimization Karnaugh Map minimization Quine-McCluskey (tabular) minimization [Assignment: Read this from the book] Spring' Logic Design Page 7

18 Algebraic Minimization Process is to apply the switching algebra postulates, laws, and theorems(that will follow) to transform the original expression Hard to recognize when a particular law can be applied Difficult to know if resulting expression is truly minimal Very easy to make a mistake Incorrect complementation Dropped variables Spring' Logic Design Page 8

19 Switching Algebra Laws and Theorems Involution: X X Spring' Logic Design Page 9

20 Identity: Switching Algebra Laws and Theorems X X X X X X Spring' Logic Design Page 2

21 Switching Algebra Laws and Idempotence: Theorems X X X X X X Spring' Logic Design Page 2

22 Switching Algebra Laws and Associativity: Theorems X (Y Z) (X Y) Z X (Y Z) (X Y) Z Spring' Logic Design Page 22

23 Switching Algebra Laws and Adjacency: Theorems X Y X Y X X Y X Y X Spring' Logic Design Page 23

24 Switching Algebra Laws and Theorems Absorption: X X Y X X X Y X Spring' Logic Design Page 24

25 Switching Algebra Laws and Simplification: Theorems X X Y X Y X X Y X Y Spring' Logic Design Page 25

26 How to Prove? Either use the postulates and simplify X+(X.Y) = X+Y () Use distributive postulate X+YZ = (X+Y).(X+Z) So () can be simplified as: (X+X ).(X+Y) =.(X+Y) = X+Y Spring' Logic Design Page 26

27 How to Prove? Truth Table Approach X Y X+(X.Y) X+Y Two output columns are the same! Spring' Logic Design Page 27

28 Switching Algebra Laws and Theorems Consensus: X Y X Z Y Z X Y X Z X Y X Z Y Z X Y X Z Spring' Logic Design Page 28

29 Switching Algebra Laws and Theorems DeMorgan s Theorem: General form: X Y X Y X Y X Y F(,, X,... X ) G(,, X,... X ) n n Spring' Logic Design Page 29

30 DeMorgan s Theorem Very useful for complementing function expressions: e.g. F X Y Z; F X Y Z F X Y Z F X Y Z F X Y X Z Spring' Logic Design Page 3

31 Minimization via Adjacency Adjacency is easy to use and very powerful Look for two terms that are identical except for one variable Application removes one term and one variable from the remaining term A B C D A B C D A B C A B C D A B C D A B C A B C D D e.g. A B C D A B C D A B C A B C Spring' Logic Design Page 3

32 Example of Adjacency Minimization x 3 b 3 b 2 b b b 3 b 2 b b b 3 b 2 b b b 3 b 2 b b b 3 b 2 b b b b b b b b b b b b b b b b b b b b b b b b b b x Adjacencies Duplicate 3rd. term and rearrange (X+X=X) Apply adjacency on term pairs b b b b b b b b b x Spring' Page Logic Design

33 Another Algebraic Simplification Example W = X Y Z+X Y Z +XYZ+X YZ+X YZ = X Y +YZ +X Z How? Now we study logic circuits closely and show some other easier simplification method(s). Spring' Logic Design Page 33

34 Combinational Circuit Analysis Combinational circuit analysis starts with a schematic and answers the following questions: What is the truth table(s) for the circuit output function(s)? What is the logic expression(s) for the circuit output function(s)? A C Z B Spring' Logic Design Page 34

35 Literal Analysis Literal analysis is process of manually assigning a set of values to the inputs, tracing the results, and recording the output values For n inputs there are 2 n possible input combinations From input values, gate outputs are evaluated to form next set of gate inputs Evaluation continues until gate outputs are circuit outputs Literal analysis only gives us the truth table Spring' Logic Design Page 35

36 Literal Analysis - Example x x x x x x x A B Z C A C B Z Assign input values Determine gate outputs and propagate Repeat until we reach output Spring' Page Logic Design

37 Symbolic Analysis Like literal analysis we start with the circuit diagram Instead of assigning values, we determine gate output expressions instead Intermediate expressions are combined in following gates to form complex expressions We repeat until we have the output function and expression Symbolic analysis gives both the truth table and logic expression Spring' Logic Design Page 37

38 Symbolic Analysis (cont.) Note that we are constructing the truth table as we go truth table has a column for each intermediate gate output intermediate outputs are combined in the truth table to generate the complex columns Symbolic analysis is more work but gives us complete information Spring' Logic Design Page 38

39 Symbolic Analysis - Example A A C Generate intermediate expression C C A C Z + B C Create associated TT column B B C Repeat till output reached A B C C A C B C Z = A C + B C Spring' Logic Design Page 39

40 Standard Expression Forms What if we have only the truth table? Can we obtain the simplified expressions from the truth table? We need it when we do design instead of analysis Consider the Liquid Tank example: HI LO Pump Drain x x By observing the pump column, can we write an expression for pump? Spring' Logic Design Page 4

41 Standard Expression Forms Two standard (canonical) expression forms Canonical sum form a.k.a disjunctive normal form or sum-of-products OR of AND terms Canonical product form a.k.a conjunctive normal form or product-of-sums AND or OR terms In both forms, each st-level operator corresponds to one row of truth table 2nd-level operator combines st-level results Spring' Logic Design Page 4

42 F[A,B,C] A B C F[A,B,C] A B C Standard Forms (cont.) Standard Sum Form Sum of Products (OR of AND terms) A B C Minterms A B C A B C A B C A B C Standard Product Form Product of Sums (AND of OR terms) Maxterms A B C Spring' Logic Design Page 42

43 Standard Sum Form Each product (AND) term is a Minterm ANDed product of literals in which each variable appears exactly once, in true or complemented form (but not both!) Each minterm has exactly one in the truth table When minterms are ORed together each minterm contributes a to the final function NOTE: Not all product terms are minterms! Spring' Logic Design Page 43

44 Minterms and Standard Sum Form C B A Minterms m = m = m 2 = m 3 = m 4 = m 5 = m 6 = m 7 = C B A C B A C B A C B A C B A C B A C B A C B A m m 3 m 6 m 7 F F A B C A B C A B C A B C F A, B, C m m 3 m 6 m 7 F A, B, C m, 3, 6, 7 Spring' Page Logic Design

45 Standard Product Form Each OR (sum) term is a Maxterm ORed product of literals in which each variable appears exactly once, in true or complemented form (but not both!) Each maxterm has exactly one in the truth table When maxterms are ANDed together each maxterm contributes a to the final function NOTE: Not all sum terms are maxterms! Spring' Logic Design Page 45

46 Maxterms and Standard Product Form C B A Maxterms M = M = M 2 = M 3 = M 4 = M 5 = M 6 = M 7 = C B A C B A C B A C B A C B A C B A C B A C B A M M 2 M 4 M 5 F F A B C A B C A B C A B C F A,B,C M M 2 M 4 M 5 F A,B,C M, 2, 4, 5 Spring' Page Logic Design

47 Karnaugh Map Minimization Given a truth table Karnaugh Map (or K-map) minimization is a visual minimization technique Is an application of adjacency Procedure guarantees a minimal expression Easy to use; fast Problems include: Applicable to limited number of variables (4 ~ 8) Errors in translation from TT to K-map Not grouping cells correctly Errors in reading final expression Spring' Logic Design Page 47

48 K-map Minimization (cont.) Basic K-map is a 2-D rectangular array of cells Each K-map represents one bit column of output Each cell contains one bit of output function Arrangement of cells in array facilitates recognition of adjacent terms Adjacent terms differ in one variable value; equivalent to difference of one bit of input row values e.g. m6 () and m7 () Spring' Logic Design Page 48

49 Truth Table Rows and Adjacency Standard TT ordering doesn t show adjacency A B C D minterm m m m 2 m 3 m 4 m 5 m 6 m 7 m 8 m 9 m m m 2 m 3 m 4 m 5 Key is to use gray code for row order A B C D minterm m m m 3 m 2 m 6 m 7 m 5 m 4 m 2 m 3 m 5 m 4 m m m 9 m 8 This helps but it s still hard to see all possible adjacencies. Spring' Logic Design Page 49

50 Folding of Gray Code Truth Table ABCD into K-map AB CD Spring' Logic Design Page 5

51 K-map Minimization (cont.) For any cell in 2-D array, there are four direct neighbors (top, bottom, left, right) 2-D array can therefore show adjacencies of up to four variables. CD AB A Four variable K-map AB C A C D Three variable K-map B C B Don t forget that cells are adjacent top to bottom and side to side. It also wraps around! Spring' Logic Design Page 5

52 Truth Table to K-map Number of TT rows MUST match number of K-map cells A B C D F x x x x x x x x x x x x x x x x AB CD C A m m2 m5 m3 m9 m7 m5 m2 B D Note different ways K-map is labeled Spring' Logic Design Page 52

53 K-map Minimization Example b3 b2 C b b A D B b3 b2 b b x3 Entry of TT data into K-map Watch out for ordering of and rows and columns! Use s for now Spring' Page Logic Design

54 Grouping - Applying Adjacency C CD AB A B K-map wraps around!cdab= cell is adjacent to CDAB= cell D ABCD ABC ABCD If two cells have the same value ( here) and are next to each other, the terms are adjacent. This adjacency is shown by enclosing them. Groups can have common cells. Group size is a power of 2 and groups are rectangular (,2,4,8 rectangular cells). You can group s or s. Spring' Logic Design Page 54

55 Reading the Groups AB CD C A D ABC If s grouped, the expression is a product term, s - sum term. Within group, note when variable values change as you go cell to cell. This determines how the term expression is formed by the following table B Grouping s Grouping s Variable changes Exclude Exclude Variable constant Inc. comp. Inc. true Variable constant Inc. true Inc. comp. Spring' Logic Design Page 55

56 Reading the Groups (cont.) When reading the term expression If the associated variable value changes within the group, the variable is dropped from the term If reading s, a constant value indicates that the associated variable is true in the AND term If reading s, a constant value indicates that the associated variable is true in the OR term Spring' Logic Design Page 56

57 Implicants and Prime Implicants CD AB A Single cells or groups that could be part of a larger group are known as implicants C B D Implicants A group that is as large as possible is a prime implicant (It cannot be grouped by other implicants to eliminate a variable) Prime Implicants Single cells can be prime implicants if they cannot be grouped with any other cell Spring' Logic Design Page 57

58 Implicants and Minimal Expressions Any set of implicants that encloses (covers) all values is sufficient ; i.e. the associated logical expression represents the desired function. All minterms or maxterms are sufficient. The smallest set of prime implicants that covers all values forms a minimal expression for the desired function. There may be more than one minimal set. Spring' Logic Design Page 58

59 Essential and Secondary Prime Implicants If a prime implicant has any cell that is not covered by any other prime implicant, it is an essential prime implicant If a prime implicant is not essential is is a secondary prime implicant or nonessential P.I. A minimal set includes ALL essential prime implicants and the minimum number of secondary PIs as needed to cover all values. Spring' Logic Design Page 59

60 K-map Minimization Method Technique is valid for either s or s A) Find all prime implicants (largest groups of s or s in order of largest to smallest) B) Identify minimal set of PIs ) Find all essential PIs 2) Find smallest set of secondary PIs The resulting expression is minimal. Spring' Logic Design Page 6

61 K-map Minimization (Contd.) b3 b2 b3 b b b3 b2 b* b3 b2 b* b3 b2 b* b b b2 We want a sum of products expression so we circle s. * PIs are essential; no implicants remain (no secondary PIs). The minimal expression is: X 3 b3 b2 b b3 b2 b b3 b2 b Spring' Logic Design Page 6

62 Another K-map Minimization Example CD AB A A C * We want a sum of products expression so we circle s. C A C D * B B C D D A B D * PIs are essential; and we have 2 secondary PIs. The minimal expressions are: F A C A C D B C D F A C A C D A B D Spring' Logic Design Page 62

63 A 3 rd K-map Minimization Example A C * A D CD C AB A B A B C * D C D * We want a product of sums expression so we circle s. * PIs are essential; and we have secondary PI which is redundant. The minimal expression is: F (A C) (C D) (A B C) Study this Spring' Logic Design Page 63

64 5 Variable K-Maps Uses two 4 variable maps side-by-side groups spanning both maps occupy the same place in both maps CD AB A CD AB A C D C D B E = E = ƒ(a,b,c,d,e) = m (3,4,7,,,4,5,6,7,2,26,27,3 3) Spring' Logic Design Page 64 B

65 5 Variable K-Maps CD AB A CD AB A C D C D B B E = ƒ(a,b,c,d,e) = m(3,4,7,,, E = 4,5,6,7,2,26,27,3 3) F (A,B,C,D,E) B D A D E A B C D B C D E Spring' Logic Design Page 65

66 Don t Cares For expression minimization, don t care values ( - or x ) can be assigned either or Hard to use in algebraic simplification; must evaluate all possible combinations K-map minimization easily handles don t cares Basic don t care rule for K-maps is include the dc ( - or x ) in group if it helps to form a larger group; else leave it out Spring' Logic Design Page 66

67 K-map Minimization with Don t Cares AB A CD BD* x A* We want a sum of products expression so we circle s and x s (don t cares) BC* C B x x x x x D * PIs are essential; no other implicants remain ( no secondary PIs). The minimal expression is: X 3 A BC BD Spring' Logic Design Page 67

68 K-Map Minimization with Don t Cares A C D B C AB A CD x D We want a product of sums expression so we circle s and x s (don t cares) A B* C B x x x x x B C D * PIs are essential; there are 3 secondary Pis. The minimal expressions are: F (A B) (B C D) F (A B) (A C D) Spring' Logic Design Page 68

69 Cost Criteria K-Maps result with minimized sum-of-products or products-of-sums expressions This corresponds to 2-level circuits called AND- OR networks or OR-AND Networks. It is usually assumed that inputs and their complerments are available 3 or higher input AND, OR gates possible (Fan-in: number of inputs to a gate) Spring' Logic Design Page 69

70 Cost Criteria Consider F=XY+XZ+YZ () Which can be simplified to: F=X(Y+Z)+YZ (2) Cost of these circuits may be defined as: Literal Cost: Number of Literal appearances in the expression () has a cost of : 6 literals (2) has a cost of :5 literals Gate input Cost: Number of inputs to the gates (): 9 (2): 8 So, (2) is less costly (but slower since 3 gate delays!) Spring' Logic Design Page 7

71 Additional Logic Operations For two inputs, there are 6 ways we can assign output values Besides AND and OR, five others are useful The unary Buffer operation is useful in the real world X Z=X X Z=X X Z=X Spring' Logic Design Page 7

72 Additional Logic Operations - NAND NAND (NOT - AND) is the complement of the AND operation X Y X Y & Spring' Logic Design Page 72

73 Additional Logic Operations - NOR NOR (NOT - OR) is the complement of the OR operation X Y X+Y Spring' Logic Design Page 73

74 Additional Logic Operations -XOR Exclusive OR is similar to the inclusive OR except output is for, inputs Alternatively the output is when modulo 2 input sum is equal to X Y X+Y = Spring' Logic Design Page 74

75 Additional Logic Operations - XNOR Exclusive NOR is the complement of the XOR operation Alternatively the output is when modulo 2 input sum is not equal to X Y X+Y = Spring' Logic Design Page 75

76 Minimal Logic Operator Sets AND, OR, NOT are all that s needed to express any combinational logic function as switching algebra expression operators are all that were originally defined Two other minimal logic operator sets exist Just NAND gates Just NOR gates We can demonstrate how just NANDs or NORs can do AND, OR, NOT operations Spring' Logic Design Page 76

77 NAND as a Minimal Set X X (X.X) =X X XY (XY) XY Y X (X Y ) =X+Y Y Spring' Logic Design Page 77

78 NOR as a Minimal Set Spring' Logic Design Page 78

79 Convert from AND-OR to Its easy to do the conversion:. NAND-NAND Similar conversion is possible for OR-AND to NOR-NOR Do it yourself! Spring' Logic Design Page 79

80 Odd-Function Remember XOR function X Y when X and Y are different Consider 3-input XOR: (X Y) Z= X ( Y Z)= X Y Z K-Map for 3-input XOR: Z XY =(XY +X Y) Z = (XY +X Y) Z + (XY +X Y) Z =XY Z +X YZ +XY Z+X YZ Spring' Logic Design Page 8

81 Odd-Function Cont. Output is when an odd number of inputs are. Why is it important? Parity bit for even parity in ASCII code will use this function.. For n bits, same structure continues.. A A A2 A3 Parity bit Spring' Logic Design Page 8

82 Three State Buffers IN_H OUT_H IN_H OUT_L EN_H EN_L IN_H OUT_H EN_L IN X EN OUT HI-Z High-impedance state (like an open circuit) Same as input Spring' Logic Design Page 82

83 Three-State Buffers Cont. Three-state buffers are used usually when you want to connect only one of k inputs to a wire IN EN IN2 EN2 IN3 EN3 Only one of IN,IN2, IN3 will be connected, so only one of EN, EN2, EN3 should be at the same time! Otherwise, the circuit will be burned. Spring' Logic Design Page 83

84 Three State Outputs Standard logic gate outputs only have two states; high and low Outputs are effectively either connected to +V or ground (low impedance) Certain applications require a logic output that we can turn off or disable Output is disconnected (high impedance) This is the three-state output May be stand-alone (a buffer) or part of another function output Spring' Logic Design Page 84

85 Documenting Combinational Systems Schematic (circuit) diagrams are a graphical representation of the combinational circuit Best practice is to organize drawing so data flows left to right, control, top to bottom Two conventions exist to denote circuit signal connections Only T intersections are connections; others just cross over Solid dots are placed at connection points (This is preferred) Spring' Logic Design Page 85

86 Timing Diagrams Timing diagrams show how inputs and outputs for a logic circuit change in time For given input timing diagrams, we can draw the output timing diagrams for a given combinational circuit. Example:Consider the following XOR circuit with 2 inputs: X Y Z Spring' Logic Design Page 86

87 Timing Diagrams (Cont d) Timing Diagram for X (given), Y(given), and Z X Y Z Actually, there is a delay at each new rising/falling pulse, but not shown here! (will be discussed Later) Spring' Logic Design Page 87

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