Digital Logic Design: Logic Gates - Lecture 2

Digital Logic Design Lecture 2 Logic Gates Dr. Muhammad Martuza Ch. 2.1 to 2.4 1 Learning Objectives Students will be able to: § Understand Boolean Variables and Functions § Understand Logical Operation § AN...

Digital Logic Design Lecture 2 Logic Gates Dr. Muhammad Martuza Ch. 2.1 to 2.4 1 Learning Objectives Students will be able to: § Understand Boolean Variables and Functions § Understand Logical Operation § AND, OR, and Inversion Operations § Understand XOR and XNOR operations § Analyze a logic circuit, set-up its truth table and describe its functionality. § Synthesize Boolean Functions 2 Variable and Function Binary Switch § A switch has three parts § Source input and output § Current wants to flow from source input to output § Control input § Voltage that controls whether that current can flow § Control input x=0, switch open § Control input x=1, switch closed 3 Variable and Functions Switch Application § Light bulb application § Build a circuit where switch controls whether light bulb on or off § Light bulb glows when current passes through its filament § When switch is closed (x=1) § Current flows § Light bulb glows § When switch is opened (x=0) § Current doesn’t flow § Light bulb off 4 Variable and Functions Logic Expression § We can describe the system as a logic expression § Input: switch control x § Output: light bulb L § Light L is a function of the input variable x § L = 1 if x = 1 § L =0 if x = 0 § Thus, L(x) = x 5 Variable and Functions Two Switch Application-Series Connection § What if we use two switches to control the light? § x1, x2 are control inputs to the switches § We first connect switches in series § When will the light be on? 6 Variable and Functions: Logical AND Function § We observed § Light on only when both switches are closed § If either switch open, light is off § Logical expression to describe behaviour § L(x1, x2) = x1 · x2 § Function evaluates as follows § L = 1, if x1 = 1 and x2 = 1 § L = 0, otherwise L(x1,x2) = x1 · x2 “.” symbol is called AND operator Implements the logical AND function 7 Variable and Functions Two Switch Application-Parallel Connection § We again use two switches to control the light § x1, x2 are control inputs to the switches § Connect switches in parallel § When will the light be on? 8  Variable and Functions Logical OR Function § We observed § Light on when either switch is closed § Light on when both switches closed § Light off only when both switches are open § Logical expression to describe “+” symbol is called OR operator behaviour Implements the logical OR function § L(x1, x2) = x1 + x2 § Function evaluates as follows § L = 0, if x1 = 0 and x2 = 0 § L = 1, otherwise L(x1,x2) = x1 + x2 9 Variable and Functions Three Switch Application § AND and OR functions § Building blocks for larger circuits § Example with three switches § x1, x2, and x3 are control inputs to the switches § Series-parallel connection § When will the light be on? 10  Variable and Functions AND OR Logic Function § We observed 1 § Light on when x3 is closed while at the same time x1 and/or x2 are closed § Logical expression to describe behaviour 1 § L(x1, x2, x3) = (x1 + x2) · x3 1 11  Variable and Functions Switch Application- Inversion § What about a positive action when the switch is opened? § Switch connected in parallel with light § Closed switch short circuits the light, light is off § Open switch, light turns on § Logical expression to describe behaviour Overbar indicates complement § L(x) = x´ Implements the NOT function § Function evaluates as follows § L = 1, if x = 0 § L = 0, if x = 1 L(x1) = x1 12  Inversion NOT Operation Notation § Different names for NOT operation § Complement, Invert, Inverse § Different notations for NOT operation § x = x´ = !x = ~x § NOT operation can be applied to a single variable or multiple variables § F(x) = x´ § F(x) = x´+ a § F(x) = x1 + x2 = (x1 + x2)´ = !(x1 + x2) = ~(x1 + x2) § NOT operation can be applied to a function § If F(x) = x1 + x2 + x3 Complement of F(x) is F´(x) = (x1 + x2 + x3)´ 13  Truth Table Introduction § Introduced AND, OR, and NOT functions by relating them to simple circuits built with switches § Same operations can be defined in the form of a truth table No of combinations = 2n where n = No of inputs 14 Truth Table Varied Truth Table Formats § Truth tables can have single or multiple outputs § Truth tables can be formatted in different ways 15  Truth Table Introduction § Advantages § Only one truth table § Intuitive to read How many § Disadvantages § Size explosion lines are needed if you have 8 inputs? 5-input truth table? 16 Truth Table Introduction § AND & OR functions can be extended to n-inputs § AND function with n-inputs is equal to 1 only if all n inputs are 1 § OR function with n-input is equal to 1 if at least one, or more n inputs are 1 17 Logic Gates and Networks Basic Logic Gates § AND, OR, & NOT operations can be implemented electronically with transistors § Resulting circuit is a logic gate § One or more inputs § One output, function of its inputs § Logic circuit often described graphically in a circuit diagram or schematic § Consists of graphical symbols for the AND, OR, & NOT gates 19 Logic Gates and Networks Basic Logic Gates … Cont Also Exclusive OR (XOR) and XNOR operations can be implemented electronically with transistors XOR XNOR Inputs Gate Symbol & Function Coincidence Operator A A B A F F B F = AÅ B B F = AÅ B 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 The output of the XOR gate is equal to 1 if an odd number of inputs have a value of 1 and 0 otherwise. è Odd Function 20  Logic Gates and Networks Network of Gates § Larger circuits are implemented by a network of gates § Logic function previously constructed using switches can be implemented by a network of gates. 21 Logic Gates and Networks Logic Network Gates Cost & Complexity § Circuit complexity directly correlates to cost § Always desirable to reduce cost, thus important to determine how to implement circuit as inexpensively as possible § Network of gates § Also called logic network, logic circuit, circuit 22 Logic Gates and Networks Digital System Designer Concern § Digital system designer faced with two basic issues § Determine function of an existing logic network ® Analysis § Designing a logic network to implement the desired function ® Synthesis 23 Logic Gates and Networks Analysis of Logic Network § Convert logic network into logic expression f= f=a+b f = x1´ + b f = x1´ + (x1 · x2) 24 Logic Gates and Networks Analysis of Logic Network § Convert logic network into logic expression f= f = a´ f = (x1 + b)´ f = (x1 + (x1 · x2))´ 25 Logic Gates and Networks Analysis of Logic Network § Convert logic network into logic expression 26 Logic Gates and Networks Analysis of Logic Network § Another way to specify circuit functionality – truth table § Consider what happens to output f for all possible inputs 27 Logic Gates and Networks Analysis of Logic Network – second example § Let’s analyze another logic circuit § What is the corresponding truth table? 28  Logic Gates and Networks Functional Equivalence § You may have noticed we got the same truth table for both examples § Logic circuits are functionally equivalent § Logic function can be implemented in a variety of ways § Designer wants to use the simpler one – lower cost § How do we determine the best implementation? § Manipulate function using a set of rules (Boolean Algebra) § Other techniques discussed in Chapter 4 29  Logic Gates and Networks Timing Diagram § Yet another way to describe logic circuit behaviour – timing diagram § Time runs from left to right § Each input value is held for a fixed period § Waveform shown indicating inputs, output, and internal signal values 30 Logic Gates and Networks Synthesis of Logic Network § Second design issue – Synthesis § How do I go from a logic expression to a logic circuit? 31  Logic Gates and Networks Synthesis of Logic Network § Convert f= x1´ + (x1 · x2) to a logic circuit § What are your inputs? § x1, x2 § What are/is your output(s)? § F § How is the function evaluated? § Parentheses § NOT operation § OR operation 32  Logic Gates and Networks Synthesis of Logic Network § Convert H = a · (b + c´)´ to a logic circuit § Expression inside parentheses first § OR operation or NOT operation? § NOT operation § AND operation 33  Logic Gates and Networks Synthesis of Logic Network § Convert R = (a · b´) · (b + c´) to a logic circuit § Expressions inside parentheses first § AND operation 34  Logic Gates and Networks Synthesis of Logic Network § Convert Fout = (a1 · bw´ + c)´ to a logic circuit § Expression inside parentheses first § NOT, AND, or OR operation? § NOT operation 35 Logic Gates and Networks Synthesis of Logic Network § Convert Fout = (a · b · c) to a logic circuit § How do we implement a 3-input gate? § Using 2-input AND gates § Using 3-input AND gates – Same for OR gates – J=k+l+m+n 36 Practice Problems from the Textbook § P2.8 § P2.9 37