ECE380 Digital Logic

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1 ECE38 Digital Logic Introduction Dr. D. J. Jackson Lecture - Digital hardware Logic circuits are used to build computer hardware as well as other products (digital hardware) Late 96 s and early 97 s saw a revolution in digital capability maller transistors Larger chip size More transistors/chip gives greater functionality, but requires more complexity in the design process Dr. D. J. Jackson Lecture -2

Digital hardware Integrated circuits are fabricated on silicon wafers Wafers are cut & packaged to form individual chips Chips have from tens to millions of transistors Dr. D. J.

2 Digital hardware Integrated circuits are fabricated on silicon wafers Wafers are cut & packaged to form individual chips Chips have from tens to millions of transistors Dr. D. J. Jackson Lecture -3 How complex is a digital design? Complexity can, and generally does, surpass human capability 6 million transistors/cm 2 now million transistors/cm 2 in years (?) Provides motivation for computer-based design techniques Most engineering work is done with CAD packages Dr. D. J. Jackson Lecture -4

3 Two design approaches Traditional Relies on mathematical models Analytical approaches Provides insight and understanding of problem Useful for small problems Inadequate for large (real) problems CAD oftware relies on mathematical model and analytical approach Transparent to user Many details are abstracted Useful/required for real problems Dr. D. J. Jackson Lecture -5 Traditional versus CAD design CAD tool usage is essential Insight and basic understanding offered by traditional approach is still important Initial conceptualization is still traditional Effective use of CAD tools requires some understanding of what the tools are doing Use of design options requires insight Dr. D. J. Jackson Lecture -6

4 Types of chips tandard chips Contain a small amount of circuitry (< transistors) Performs simple functions 74 series devices Programmable logic devices (PLD) Collection of gates with programmable interconnections Function is configurable by designer/user Design with PLD is via a CAD tool Dr. D. J. Jackson Lecture -7 Types of chips Custom-designed chips Optimized for a specific task better performance Larger amount of logic circuitry Cost of production is high Large volume required to justify cost Dr. D. J. Jackson Lecture -8

5 A field-programmable gate array Group of 8 logic cells Memory block Interconnection wires Dr. D. J. Jackson Lecture -9 The development process () Required Product Define specifications Initial design imulation Redesign Design Correct? no Dr. D. J. Jackson Lecture -

6 The development process (2) Prototype implementation Make corrections yes Testing Minor errors? no Redesign Meets specs? no yes Finished product Dr. D. J. Jackson Lecture - What should you expect to gain from this course? Understanding of concepts, models, algorithms and processes for digital logic design Relevance of the material to subsequent courses and to your career Problem solving skills Formulating and attacking new problems Need to struggle with problems evolve your problem solving skills Communicate solutions in a clear, concise manner Dr. D. J. Jackson Lecture -2

7 ECE38 Digital Logic Introduction to Logic Circuits: Variables, functions, truth tables, gates and networks Dr. D. J. Jackson Lecture 2- Logic circuits Logic circuits perform operations on digital signals Implemented as electronic circuits where signal values are restricted to a few discrete values In binary logic circuits there are only two values, and The general form of a logic circuit is a switching network X X 2 X 3 X m witching Network discrete values Y Y 2 Y 3 Y n Dr. D. J. Jackson Lecture 2-2

8 Boolean algebra Direct application to switching networks Work with 2-state devices 2-valued Boolean algebra (switching algebra) Use a Boolean variable (X, Y, etc.) to represent an input or output of a switching network Variable may take on only two values (, ) X=, X= These symbols are not binary numbers, they simply represent the 2 states of a Boolean variable They are not voltage levels, although they commonly refer to the low or high voltage input/output of some circuit element Dr. D. J. Jackson Lecture 2-3 Variables and functions The simplest binary element is a switch that has two states If the switch is controlled by x, we say the switch is open if x = and closed if x = x = x = (a) Two states of a switch x (b) ymbol for a switch Dr. D. J. Jackson Lecture 2-4

9 Variables and functions Assume the switch controls a lightbulb as shown The output is defined as the state of the light L If the light is on -> L= If the light is off -> L= The state of L, as function of x is L(x)=x L(x) is a logic function x is an input variable Battery Power supply x (a) imple connection to a battery x (b) Using a ground connection as the return path L L Light Dr. D. J. Jackson Lecture 2-5 Variables and functions (AND) Consider the possibility of two switches controlling the state of the light Using a series connection, the light will be on only if both switches are closed L(x, x 2 )= x x 2 L= iff (if and only if) x AND x 2 are Power supply x The logical AND function (series connection) x 2 L Light AND operator x x 2 =x x 2 The circuit implements a logical AND function Dr. D. J. Jackson Lecture 2-6

10 Variables and functions (OR) Using a parallel connection, the light will be on only if either or both switches are closed L(x, x 2 )= x + x 2 L= if x OR x 2 is (or both) x Power supply L Light + OR operator x 2 The logical OR function (parallel connection) The circuit implements a logical OR function Dr. D. J. Jackson Lecture 2-7 Variables and functions Various series-parallel connections would realize various logic functions L(x, x 2, x 3 )= (x + x 2 ) x 3 x Power supply x 3 L Light x 2 Dr. D. J. Jackson Lecture 2-8

11 Variables and functions What would the following logic function look like if implemented via switches? L(x, x 2, x 3, x 4 )= (x x 2 )+ (x 3 x 4 ) x x 2 Power supply L Light x 3 x 4 Dr. D. J. Jackson Lecture 2-9 Inversion Before, actions occur when a switch is closed. What about the possibility of an action occurring when a switch is opened? L(x)= x Where L= if x= and L= if x= L(x) is the inverse (or complement) of x R Power supply x L x, x, NOT x The circuit implements a logical NOT function Dr. D. J. Jackson Lecture 2-

12 Inversion of a function If a function is defined as f(x, x 2 )= x + x 2 Then the complement of f is f(x, x 2 )= x + x 2 = (x + x 2 ) imilarily, if f(x, x 2 )= x x 2 Then the complement of f is f(x, x 2 )= x x 2 = (x x 2 ) Dr. D. J. Jackson Lecture 2- Truth tables Tabular listing that fully describes a logic function Output value for all input combinations (valuations) x x 2 x x 2 x x 2 x + x 2 x x NOT AND OR Dr. D. J. Jackson Lecture 2-2

13 Truth tables Truth table for AND and OR functions of three variables Dr. D. J. Jackson Lecture 2-3 Truth tables of functions If L(x,y,z)=x+yz, then the truth table for L is: + x y z yz x+yz Dr. D. J. Jackson Lecture 2-4

14 Logic gates and networks Each basic logic operation (AND, OR, NOT) can be implemented resulting in a circuit element called a logic gate A logic gate has one or more inputs and one output that is a function of its inputs x x x x x 2 2 x 2 x x 2 x n x n AND gates Dr. D. J. Jackson Lecture 2-5 Logic gates and networks x x x x + x 2 2 x 2 x + x x n x n OR gates x x NOT gate Dr. D. J. Jackson Lecture 2-6

15 Logic gates and networks A larger circuit is implemented by a network of gates Called a logic network or logic circuit x x 2 f = ( x + x ) x x Dr. D. J. Jackson Lecture 2-7 Logic gates and networks Draw the truth table and the logic circuit for the following function F(a,b,c) = ac+bc a b c ac bc' ac+bc' a c b Dr. D. J. Jackson Lecture 2-8

16 Analysis of a logic network To determine the functional behavior of a logic network, we can apply all possible input signals to it x x 2 A B f Network that implementsf = x + x x 2 Dr. D. J. Jackson Lecture 2-9 Analysis of a logic network The function of a logic network can also be described by a timing diagram (gives dynamic behavior of the network) x x 2 A B f Timing diagram Time Dr. D. J. Jackson Lecture 2-2

17 ECE38 Digital Logic Introduction to Logic Circuits: Boolean algebra Dr. D. J. Jackson Lecture 3- Axioms of Boolean algebra Boolean algebra: based on a set of rules derived from a small number of basic assumptions (axioms) a = b += 2a = 2b += 3a = = 3b +=+= 4a If x= then x = 4b If x= then x = Dr. D. J. Jackson Lecture 3-2

18 ingle-variable theorems From the axioms are derived some rules for dealing with single variables 5a x = 5b x+= 6a x =x 6b x+=x 7a x x=x 7b x+x=x 8a x x = 8b x+x = 9 x =x ingle-variable theorems can be proven by perfect induction ubstitute the values x= and x= into the expressions and verify using the basic axioms Dr. D. J. Jackson Lecture 3-3 Duality Axioms and single-variable theorems are expressed in pairs Reflects the importance of duality Given any logic expression, its dual is formed by replacing all + with, and vice versa and replacing all s with s and vice versa f(a,b)=a+b dual of f(a,b)=a b f(x)=x+ dual of f(x)=x The dual of any true statement is also true Dr. D. J. Jackson Lecture 3-4

19 Two & three variable properties a. x y=y x Commutative b. x+y=y+x a. x (y z)=(x y) z Associative b. x+(y+z)=(x+y)+z 2a. x (y+z)=x y+x z Distributive 2b. x+y z=(x+y) (x+z) 3a. x+x y=x Absorption 3b. x (x+y)=x Dr. D. J. Jackson Lecture 3-5 Two & three variable properties 4a. x y+x y =x Combining 4b. (x+y) (x+y )=x 5a. (x y) =x +y DeMorgan s 5b. (x+y) =x y Theorem 6a. 6b. x+x y=x+y x (x +y)=x y Dr. D. J. Jackson Lecture 3-6

20 Induction proof of x+x y=x+y Use perfect induction to prove x+x y=x+y x y x y x+x y x+y equivalent Dr. D. J. Jackson Lecture 3-7 Perfect induction example Use perfect induction to prove (xy) =x +y x y xy (xy) x y x +y equivalent Dr. D. J. Jackson Lecture 3-8

21 Proof (algebraic manipulation) Prove (X+A)(X +A)(A+C)(A+D)X = AX (X+A)(X +A)(A+C)(A+D)X (X+A)(X +A)(A+CD)X (using 2b) (X+A)(X +A)(A+CD)X (A)(A+CD)X (using 4b) (A)(A+CD)X AX (using 3b) Dr. D. J. Jackson Lecture 3-9 Algebraic manipulation Algebraic manipulation can be used to simplify Boolean expressions impler expression => simpler logic circuit Not practical to deal with complex expressions in this way However, the theorems & properties provide the basis for automating the synthesis of logic circuits in CAD tools To understand the CAD tools the designer should be aware of the fundamental concepts Dr. D. J. Jackson Lecture 3-

22 Venn diagrams Venn diagram: graphical illustration of various operations and relations in an algebra of sets A set s is a collection of elements that are members of s (for us this would be a collection of Boolean variables and/or constants) Elements of the set are represented by the area enclosed by a contour (usually a circle) Dr. D. J. Jackson Lecture 3- Venn diagrams (a) Constant (b) Constant X X X X (c) Variable X (d) X Dr. D. J. Jackson Lecture 3-2

23 Venn diagrams X Y X Y (e) XY (f) X+Y X Y X Y Z (g) XY (h) XY+Z Dr. D. J. Jackson Lecture 3-3 Venn diagrams (x+y) = x y X Y X Y X Y X X Y Y DeMorgan s Theorem (X+Y) X Y X Y X Y Equivalent Venn diagrams imply equivalent functions Dr. D. J. Jackson Lecture 3-4

24 Notation and terminology Because of the similarity with arithmetic addition and multiplication operations, the OR and AND operations are often called the logical sum and product operations The expression ABC+A BD+ACE Is a sum of three product terms The expression (A+B+C)(A +B+D)(A+C+E ) Is a product of three sum terms Dr. D. J. Jackson Lecture 3-5 Precedence of operations In the absence of parentheses, operations in a logical expression are performed in the order NOT, AND, OR Thus in the expression AB+A B, the variables in the second term are complemented before being ANDed together. That term is then ORed with the ANDed combination of A and B (the AB term) Dr. D. J. Jackson Lecture 3-6

25 Precedence of operations Draw the circuit diagrams for the following f(a,b,c)=(a +b)c f(a,b,c)=a b+c Dr. D. J. Jackson Lecture 3-7

Chapter 3 Describing Logic Circuits Dr. Xu

Chapter 3 Describing Logic Circuits Dr. Xu Chapter 3 Objectives Selected areas covered in this chapter: Operation of truth tables for AND, NAND, OR, and NOR gates, and the NOT (INVERTER) circuit. Boolean