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Digital Logic Circuits

Explore the fundamentals of digital logic circuits, binary logic, Boolean algebra, NAND/NOR and XOR gates, and decoder principles. Learn about analog vs. digital systems, quantization, representations of digital designs, and the benefits of digital technology. Discover the importance of digital circuits in various applications like electronic controllers and digital watches. Delve into the design process, truth tables, Boolean equations, and gate representations for combinational logic. Gain insights into deriving Boolean equations and standard logic gate representations for digital designs.

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Digital Logic Circuits

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  1. Digital Logic Circuits • Binary Logic and Gates • Logic Simulation • Boolean Algebra • NAND/NOR and XOR gates • Decoder fundamentals • Half Adder, Full Adder, Ripple Carry Adder

  2. Analog vs Digital • Analog • Continuous • Time • Every time has a value associated with it, not just some times • Magnitude • A variable can take on any value within a range • e.g. • temperature, voltage, current, weight, length, brightness, color

  3. Digital Systems Digital vs. Analog Waveforms Digital: only assumes discrete values Analog: values vary over a broad range continuously

  4. Quantization

  5. Analog vs Digital • Digital • Discontinuous • Time (discretized) • The variable is only defined at certain times • Magnitude (quantized) • The variable can only take on values from a finite set • e.g. • Switch position, digital logic, Dow-Jones Industrial, lottery, batting-average

  6. Analog to Digital • A Continuous Signal is Sampled at Some Time and Converted to a Quantized Representation of its Magnitude at that Time • Samples are usually taken at regular intervals and controlled by a clock signal • The magnitude of the signal is stored as a sequence of binary valued (0,1) bits according to some encoding scheme

  7. Digital to Analog • A Binary Valued, B = { 0, 1 }, Code Word can be Converted to its Analog Value • Output of D/A Usually Passed Through Analog Low Pass Filter to Approximate a Continuous Signal • Many Applications Construct a Signal Digitally and then D/A • e.g., RF Transmitters, Signal Generators

  8. Digital is Ubiquitous • Electronic Circuits based on Digital Principles are Widely Used • Automotive Engine/Speed Controllers • Microwave Oven Controllers • Heating Duct Controls • Digital Watches • Cellular Phones • Video Games

  9. Why Digital? • Increased Noise Immunity • Reliable • Inexpensive • Programmable • Easy to Compute Nonlinear Functions • Reproducible • Small

  10. Digital Design Process • Computer Aided Design Tools • Design entry • Synthesis • Verification and simulation • Physical design • Fabrication • Testing

  11. Definition

  12. Representations for combinational logic • Exclusive-or (XOR, EXOR, not-equivalence, ring-OR) • Algebraic symbol: • Gate symbol: • Truth tables • Graphical (logic gates) • Algebraic equations (Boolean)

  13. Boolean algebra & logic circuits

  14. Representations of a Digital Design Truth Tables tabulate all possible input combinations and their associated output values Example: half adder adds two binary digits to form Sum and Carry Example: full adder adds two binary digits and Carry in to form Sum and Carry Out NOTE: 1 plus 1 is 0 with a carry of 1 in binary

  15. Representations of Digital Design: Boolean Algebra values: 0, 1 variables: A, B, C, . . ., X, Y, Z operations: NOT, AND, OR, . . . NOT X is written as X X AND Y is written as X & Y, or sometimes X Y X OR Y is written as X + Y Deriving Boolean equations from truth tables: Sum = A B + A B Carry 0 0 0 1 A 0 0 1 1 B 0 1 0 1 Sum 0 1 1 0 OR'd together product terms for each truth table row where the function is 1 if input variable is 0, it appears in complemented form; if 1, it appears uncomplemented Carry = A B

  16. Representations of a Digital Design: Boolean Algebra Another example: Sum = A B Cin + A B Cin + A B Cin + A B Cin Sum 0 1 1 0 1 0 0 1 Cout 0 0 0 1 0 1 1 1 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 Cin 0 1 0 1 0 1 0 1 Cout = A B Cin + A B Cin + A B Cin + A B Cin

  17. Gate Representations of a Digital Design most widely used primitive building block in digital system design Standard Logic Gate Representation Half Adder Schematic Net: electrically connected collection of wires Netlist: tabulation of gate inputs & outputs and the nets they are connected to

  18. Design methodology

  19. Top-down vs. bottom-up design

  20. Analysis procedures

  21. Schematic for 4 Bit ALU Invertor ANDGate EXORGate ORGate

  22. 4 2 4 A D B S Simulation of 4 Bit ALU if S=0 then D=B-A if S=1 then D=A-B if S=2 then D=A+B if S=3 then D=-A

  23. Elementary Binary Logic Functions • Digital circuits represent information using two voltage levels. • binary variables are used to denote these values • by convention, the values are called “1” and “0” and we often think of them as meaning “True” and “False” • Functions of binary variables are called logic functions. • AND(A,B) = 1 if A=1 and B=1, else it is zero. • AND is generally written in the shorthand A×B (or A&B or AÙB) • OR(A,B) = 1 if A=1 or B=1, else it is zero. • OR is generally written in the shorthand form A+B (or A|B or AÚB) • NOT(A) = 1 if A=0 else it is zero. • NOT is generally written in the shorthand form (or ØA or A) • AND, OR and NOT can be used to express all other logic functions.

  24. EQUAL NAND NOR OR BÞA EXOR A ONE ZERO AND (BÞA) B A (AÞB) AÞB B A B 1111 0001 0000 1000 1100 0100 0010 0111 0101 1010 1110 1101 1011 0011 1001 0110 0011 0101 Two Variable Binary Logic Functions • Can make similar truth tables for 3 variable or 4 variable functions, but gets big (256 & 65,536 columns). • Representing functions in terms of AND, OR, NOT. • NAND(A,B) = (A×B) • EXOR(A,B) = (A×B) + (A×B)

  25. Timing Diagram X X AND Gate X×Y Y Y X X×Y OR Gate X+Y Y X+Y Inverter X X’ X’ Basic Logic Gates • Logic gates “compute” elementary binary functions. • output of an AND gate is “1” when both of its inputs are “1”, otherwise the output is zero • similarly for OR gate and inverter • Timing diagram shows how output values change over time as input values change

  26. A A B B C C D E F Multivariable Gates 6 input OR Gate 3 input AND Gate A+B+C+D+E+F • AND function on n variables is “1” if and only if ALL its arguments are “1”. • n input AND gate output is “1” if all inputs are “1” • OR function on n variables is “1” if and only if at least one of its arguments is “1”. • n input OR gate output is “1” if any inputs are “1” • Can construct “large” gates from 2 input gates. • however, large gates can be less expensive than required number of 2 input gates A×B×C

  27. A B C B×C A+B×C 00001111 00110011 01010101 00100010 00101111 A B A+B×C C Elements of Boolean Algebra • Boolean algebra defines rules for manipulating symbolic binary logic expressions. • a symbolic binary logic expression consists of binary variables and the operators AND, OR and NOT (e.g. A+B×C) • The possible values for any Boolean expression can be tabulated in a truth table. • Can define circuit forexpression by combininggates.

  28. Schematic Capture & Logic Simulation wires advancesimulation gates signalwaveforms terminals schematicentry tools signalnames

  29. A A+B×C (B×(C)) Boolean Functions to Logic Circuits • Any Boolean expression can be converted to a logic circuit made up of AND, OR and NOT gates. step 1: add parentheses to expression to fully define order of operations - A+(B×(C)) step 2: create gate for “last” operation in expression gate’s output is value of expression gate’s inputs are expressions combined by operation • step 3: repeat for sub-expressions and continue until done • Number of simple gates needed to implement expression equals number of operations in expression. • so, simpler equivalent expression yields less expensive circuit • Boolean algebra provides rules for simplifying expressions

  30. Basic Identities of Boolean Algebra 2. X×1 =X 4. X×0 = 0 6. X×X=X 8. X×X’ = 0 11. X×Y=Y×X 13. X×(Y×Z) = (X×Y)×Z 15. X+(Y×Z) =(X+Y)×(X+Z) 17. (X×Y)’ = X+Y 1. X + 0 =X 3. X + 1 = 1 5. X + X=X 7. X + X’ = 1 9. (X’)’ =X 10. X + Y=Y + X 12. X+(Y+Z) = (X+Y)+Z 14. X(Y+Z) =X×Y + X×Z 16. (X+ Y)=X×Y commutative associative distributive DeMorgan’s • Identities define intrinsic properties of Boolean algebra. • Useful in simplifying Boolean expressions • Note: 15-17 have no counterpart in ordinary algebra. • Parallel columns illustrate duality principle.

  31. X+(Y×Z) =(X+Y)×(X+Z) (X+ Y)=X×Y XYZ Y×Z X+(Y×Z) X+Y X+Z (X+Y)×(X+Z) XY (X+ Y) X×Y 00 1 1 000 0 0 0 0 0 0 01 0 0 1 0 0 0 001 0 10 0 0 0 010 0 1 0 0 11 0 1 011 1 1 1 1 1 1 1 100 0 1 1 101 0 1 1 1 0 110 1 1 1 1 1 111 1 1 1 1 Verifying Identities Using Truth Tables • Can verify any logical equation with small number of variables using truth tables. • Break large expressions into parts, as needed.

  32. DeMorgan’s Law

  33. DeMorgan’s Laws for n Variables • We can extend DeMorgan’s laws to 3 variables by applying the laws for two variables. (X + Y + Z)=(X+ (Y + Z))- by associative law =X×(Y + Z)- by DeMorgan’s law =X×(Y×Z) - by DeMorgan’s law = X×Y×Z- by associative law (X×Y×Z)=(X×(Y×Z))- by associative law =X+ (Y×Z) - by DeMorgan’s law =X+ (Y+ Z)- by DeMorgan’s law = X+ Y+ Z- by associative law • Generalization to n variables. • (X1 + X2 + × × × + Xn)=X1×X2× × × Xn • (X1×X2× × × Xn)=X1 + X2 + × × × + Xn

  34. X Y Z by identity 14 F=XY(Z +Z)+XZ by identity 7 X Y F=XY×1+XZ =XY +XZ by identity 2 X Z Y Z Simplification of Boolean Expressions F=XYZ+XYZ+XZ

  35. The Duality Principle • The dual of a Boolean expression is obtained by interchanging all ANDs and ORs, and all 0s and 1s. • example: the dual of A+(B×C)+0 is A×(B+C)×1 • The duality principle states that if E1 and E2 are Boolean expressions then E1= E2 dual(E1)=dual(E2) where dual(E) is the dual of E. For example, A+(B×C)+0 = (B×C)+D  A×(B+C)×1 = (B+C)×D Consequently, the pairs of identities (1,2), (3,4), (5,6), (7,8), (10,11), (12,13), (14,15) and (16,17) all follow from each other through the duality principle.

  36. The Consensus Theorem Theorem. XY + XZ +YZ = XY + XZ Proof. XY + XZ +YZ = XY + XZ + YZ(X + X) 2,7 = XY + XZ + XYZ + XYZ 14 = XY + XYZ + XZ + XYZ 10 = XY(1 + Z)+ XZ(1 + Y) 2,14 = XY + XZ 3,2 Example. (A + B)(A+ C) = AA+ AC + AB + BC = AC + AB + BC = AC + AB Dual. (X + Y)(X + Z)(Y + Z) = (X + Y)(X + Z)

  37. Taking the Complement of a Function Method 1. Apply DeMorgan’s Theorem repeatedly. (X(YZ+ YZ)) = X+ (YZ+ YZ) = X + (YZ)(YZ) = X+ (Y + Z)(Y+ Z) Method 2. Complement literals and take dual (X(YZ+ YZ))= dual(X(YZ + YZ)) = X+ (Y + Z)(Y+ Z)

  38. Sum of Products Form • The sum of products is one of two standard forms for Boolean expressions. sum-of-products-expression = term+term ... +term term = literal×literal×××× ×literal Example. XYZ + XZ + XY + XYZ • A minterm is a term that contains every variable, in either complemented or uncomplemented form. Example. in expression above, XYZ is minterm, but XZ is not • A sum of minterms expression is a sum of products expression in which every term is a minterm Example.XYZ + XYZ + XYZ + XYZ is sum of minterms expression that is equivalent to expression above

  39. Product of Sums Form • The product of sums is the second standard form for Boolean expressions. product-of-sums-expression = s-term×s-term ... ×s-term s-term = literal+literal+××× +literal Example. (X+Y+Z )(X+Z)(X+Y)(X+Y+Z) • A maxterm is a sum term that contains every variable, in complemented or uncomplemented form. Example. in exp. above, X+Y+Z is a maxterm, but X+Z is not • A product of maxterms expression is a product of sums expression in which every term is a maxterm Example.(X+Y+Z )(X+Y+Z)(X+Y+Z)(X+Y+Z)is product of maxterms expression that is equivalent to expression above

  40. X X NAND Gate NOR Gate (X+Y) (X×Y) Y Y = = = = NAND and NOR Gates • In certain technologies (including CMOS), a NAND (NOR) gate is simpler & faster than an AND (OR) gate. • Consequently circuits are often constructed using NANDs and NORs directly, instead of ANDs and ORs. • Alternative gate representations makes this easier.

  41. Alternative Implementation A EXOR gate A AB+AB B B Exclusive Or and Odd Function • The odd function on n variables is 1 when an odd number of its variables are 1. • odd(X,Y,Z) = XYZ+ XYZ + XYZ + XYZ = X Y Z • similarly for 4 or more variables • Parity checking circuits use the odd function to provide a simple integrity check to verify correctness of data. • any erroneous single bit change will alter value of odd function, allowing detection of the change • The EXOR function is defined by AB = AB + AB.

  42. X X X+Y X×Y Y Y Positive and Negative Logic • In positive logic systems, a high voltage is associated with a logic 1, and a low voltage with a logic 0. • positive logic is just one of two conventions that can be used to associate a logic value with a voltage • sometimes it is more convenient to use the opposite convention • In logic diagrams that use negative logic, a polarity indicator is used to indicate the correct logical interpretation for a signal. • Circuits commonly use a combination of positive and negative logic.

  43. Analysis example

  44. Truth tables from logic diagram

  45. Logic simulation

  46. Decoder Fundamentals • Route data to one specific output line. • Selection of devices, resources • Code conversions. • Arbitrary switching functions • implements the AND plane • Asserts one-of-many signal; at most one output will be asserted for any input combination

  47. Encoding Binary Decimal Unencoded Encoded 0 0001 00 1 0010 01 2 0100 10 3 1000 11 Note: Finite state machines may be unencoded ("one-hot") or binary encoded. If the all 0's state is used, then one less bit is needed and it is called modified one-hot coding.

  48. Why Encode?A Logarithmic Relationship

  49. A B 1 1 1 0 0 1 00 Y Y Y Y E Q 3 E Q 2 E Q 1 E Q 0 AND 2 A B AND 2 A A B AND 2 A D 0 A B AND 2 B D 1 2:4 Decoder What happens when the inputs goes from 01 to 10?

  50. A B C A B C A B C A B C 1 1 1 0 0 1 00 Y Y Y Y E Q 3 E Q 2 E Q 1 E Q 0 AND 3 AND 3 A AND 3 A D 0 D 1 ENABLE AND 3 B 2:4 Decoder with Enable 1 1 1 0 0 1 00

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