1 Dr. D. J. Jackson Lecture 1-1Electrical & Computer Engineering ECE380 Digital Logic Introduction Dr. D. J. Jackson Lecture 1-2Electrical & Computer Engineering Digital hardware • Logic circuits are used to build computer hardware as well as other products (digital hardware) • Late 1960’s and early 1970’s saw a revolution in digital capability – Smaller transistors – Larger chip size • More transistors/chip gives greater functionality, but requires more complexity in the design process 2 Dr. D. J. Jackson Lecture 1-3Electrical & Computer Engineering 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 1-4Electrical & Computer Engineering How complex is a digital design? • Complexity can, and generally does, surpass human capability – 16 million transistors/cm 2 now – 100 million transistors/cm 2 in 10 years (?) • Provides motivation for computer-based design techniques • Most engineering work is done with CAD packages 3 Dr. D. J. Jackson Lecture 1-5Electrical & Computer Engineering 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 – Software relies on mathematical model and analytical approach – Transparent to user – Many details are abstracted – Useful/required for real problems Dr. D. J. Jackson Lecture 1-6Electrical & Computer Engineering 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 4 Dr. D. J. Jackson Lecture 1-7Electrical & Computer Engineering Types of chips •Standard chips – Contain a small amount of circuitry (<100 transistors) – Performs simple functions – 7400 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 1-8Electrical & Computer Engineering 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 5 Dr. D. J. Jackson Lecture 1-9Electrical & Computer Engineering A field-programmable gate array Memory block Group of 8 logic cells Interconnection wires Dr. D. J. Jackson Lecture 1-10Electrical & Computer Engineering The development process (1) Design Correct? Required Product Define specifications Initial design Simulation Redesign no 6 Dr. D. J. Jackson Lecture 1-11Electrical & Computer Engineering The development process (2) Prototype implementation Testing Make corrections Meets specs? Minor errors? Redesign Finished product no no yes yes Dr. D. J. Jackson Lecture 1-12Electrical & Computer Engineering 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 1 Dr. D. J. Jackson Lecture 2-1Electrical & Computer Engineering ECE380 Digital Logic Introduction to Logic Circuits: Variables, functions, truth tables, gates and networks Dr. D. J. Jackson Lecture 2-2Electrical & Computer Engineering Logic circuits • Logic circuits perform operations on digital signals – Implemented as electronic circuits where signal values are restricted to a few discrete values •Inbinary logic circuits there are only two values, 0 and 1 • The general form of a logic circuit is a switching network Switching Network X 1 X 2 X 3 X m Y 1 Y 2 Y 3 Y n discrete values 2 Dr. D. J. Jackson Lecture 2-3Electrical & Computer Engineering Boolean algebra • Direct application to switching networks – Work with 2-state devices Æ 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 (0, 1) –X=0, X=1 – 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 2-4Electrical & Computer Engineering 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 =0 and closed if x =1 x 1 = x 0 = (a) Two states of a switch S x (b) Symbol for a switch 3 Dr. D. J. Jackson Lecture 2-5Electrical & Computer Engineering 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=1 • If the light is off -> L=0 • The state of L, as function of x is – L(x)=x •L(x) is a logic function • x is an input variable (a) Simple connection to a battery S x (b) Using a ground connection as the return path L Battery Light x Power supply S L Dr. D. J. Jackson Lecture 2-6Electrical & Computer Engineering 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 1 , x 2 )= x 1 · x 2 – L=1 iff (if and only if) x 1 AND x 2 are 1 The logical AND function (series connection) S x 1 L Power supply S x 2 Light “·” AND operator The circuit implements a logical AND function x 1 · x 2 =x 1 x 2 4 Dr. D. J. Jackson Lecture 2-7Electrical & Computer Engineering Variables and functions (OR) • Using a parallel connection, the light will be on only if either or both switches are closed –L(x 1 , x 2 )= x 1 + x 2 –L=1 if x 1 OR x 2 is 1 (or both) S x 1 L Power supply S x 2 The logical OR function (parallel connection) Light “+” OR operator The circuit implements a logical OR function Dr. D. J. Jackson Lecture 2-8Electrical & Computer Engineering Variables and functions • Various series-parallel connections would realize various logic functions –L(x 1 , x 2 , x 3 )= (x 1 + x 2 )· x 3 S x 1 L Power supply S x 2 Light S x 3 [...]... Computer Engineering f’= ΠM Dr D J Jackson Lecture 4-16 8 ECE380 Digital Logic Introduction to Logic Circuits: Design Examples Electrical & Computer Engineering Dr D J Jackson Lecture 5-1 Design examples • Logic circuits provide a solution to a problem • Some may be complex and difficult to design • Regardless of the complexity, the same basic design issues must be addressed 1 Specify the desired behavior... f(a,b,c)=(a’+b)c – f(a,b,c)=a’b+c Electrical & Computer Engineering Dr D J Jackson Lecture 3-17 9 ECE380 Digital Logic Introduction to Logic Circuits: Synthesis using AND, OR, and NOT gates Electrical & Computer Engineering Dr D J Jackson Lecture 4-1 Example logic circuit design • Assume we want to design a logic circuit with three inputs x, y, and z • The circuit output should be 1 only when x=1 and either... 1 Dr D J Jackson Lecture 2-14 7 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 x1 x2 x1 x2 x1 ⋅ x2 ⋅ … ⋅ xn x1 ⋅ x2 xn AND gates Electrical & Computer Engineering Dr D J Jackson Lecture 2-15 Logic gates and networks x1 x2 x1... Electrical & Computer Engineering Dr D J Jackson Lecture 2-16 8 Logic gates and networks • A larger circuit is implemented by a network of gates – Called a logic network or logic circuit x x x 1 2 f = (x + x ) ⋅ x 1 2 3 3 Electrical & Computer Engineering Dr D J Jackson Lecture 2-17 Logic gates and networks • Draw the truth table and the logic circuit for the following function – F(a,b,c) = ac+bc’ a... function could be written as – f(x,y,z)=xy’z+xyz’+xyz • sum of products form Electrical & Computer Engineering Dr D J Jackson Lecture 4-2 1 Example logic circuit design Electrical & Computer Engineering Dr D J Jackson Lecture 4-3 Example logic circuit design • Implements f correctly, f(x,y,z)=xy’z+xyz’+xyz but is not the simplest such network xy’z+xyz’+xyz xy’z+xy from 14a y z f x 0 0 0 0 x(y’z+y)... Jackson Lecture 2-18 9 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 0→ 0→1→ 1 1→ 1→0→ 0 1 A 0→ 1→0→ 1 0→ 0→0→ 1 1→1→ 0→ 1 f B 2 Network that implements = x1 + x1 ⋅ x2 f Electrical & Computer Engineering Dr D J Jackson Lecture 2-19 Analysis of a logic network • The function of a logic network can also be described... manipulation can be used to simplify Boolean expressions – Simpler 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 Electrical & Computer Engineering Dr... by a timing diagram (gives dynamic behavior of the network) 1 x 1 0 x 1 2 0 A 1 0 B 1 0 f 1 0 Time Timing diagram Electrical & Computer Engineering Dr D J Jackson Lecture 2-20 10 ECE380 Digital Logic Introduction to Logic Circuits: Boolean algebra Electrical & Computer Engineering Dr D J Jackson Lecture 3-1 Axioms of Boolean algebra • Boolean algebra: based on a set of rules derived from a small number... xy’z+xy from 14a y z f x 0 0 0 0 x(y’z+y) from 12a 0 1 0 0 x(y+z) from 16a 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Electrical & Computer Engineering Dr D J Jackson Lecture 4-4 2 Example logic circuit design • Obviously, the cost (in terms of gates and connections) of this network is much less than the initial network • The process of generating a circuit from a stated desired functional behavior... 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 Electrical & Computer Engineering Dr D J Jackson Lecture 3-15 Precedence of operations • In the absence of parentheses, operations in a logical expression are performed in the order . Engineering ECE380 Digital Logic Introduction to Logic Circuits: Variables, functions, truth tables, gates and networks Dr. D. J. Jackson Lecture 2-2Electrical & Computer Engineering Logic circuits • Logic. 1-1Electrical & Computer Engineering ECE380 Digital Logic Introduction Dr. D. J. Jackson Lecture 1-2Electrical & Computer Engineering Digital hardware • Logic circuits are used to build computer. & Computer Engineering 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