unit 1 COA

Digital Design Chapter 2: Combinational Logic Design

Global Context

  • Created by Frank Vahid, 2006

  • Designed to accompany "Digital Design, First Edition"

  • Instructors can modify and use for course activities

  • Slides must retain copyright notice

  • No public posting of animated versions or PowerPoint source

Introduction to Combinational Circuits

  • Definition: Digital circuits whose outputs depend only on current input values.

  • Example: Switching logic states from inputs to outputs.

Switches

  • Function: Basis of binary digital circuits.

  • Key Terms:

    • Voltage: Electric potential difference between two points.

    • Current: Flow of charged particles.

    • Resistance: Tendency to resist current flow (Ohm’s Law: V = I * R).

Components of a Switch

  • Parts: Source input, control input, output.

    • Source Input: Where current flows from.

    • Control Input: Voltage determining if current flows (on/off).

CMOS Transistor

  • Role: Fundamental switch in modern integrated circuits.

  • Operation: Positive voltage attracts electrons, creating conduction path between source and drain.

  • Types:

    • nMOS: conducts when gate is 1.

    • pMOS: Conducts when gate is 0.

Logic Gates as Building Blocks

  • Importance: Logic gates simplify the design process over switches.

  • Basic Gates:

    • AND: Outputs 1 only if all inputs are 1.

    • OR: Outputs 1 if any input is 1.

    • NOT: Inverts input values.

Boolean Algebra and Digital Circuits

  • Concept: Boolean variables can only be 0 or 1.

  • Operators return outputs as 0 or 1:

    • AND: F = a AND b = 1 if both a and b are 1.

    • OR: F = a OR b = 1 if either a or b is 1.

    • NOT: F = NOT a = 1 if a is 0.

Converting Expressions to Boolean Equations

  • Examples: Converting English to equations:

    • Q1: a is 1 and b is 1 → F = a AND b

    • Q2: either a or b is 1 → F = a OR b

    • Q3: both a and b are not 0 → F = NOT(a) AND NOT(b) (Option 1) or F = a OR b (Option 2).

Implementing Boolean Logic with Transistors

  • Transistors are used to create logic gates based on Boolean operations.

  • Truth Tables: Used for defining behavior of logic functions.

Timing Diagrams and Logic Gates

  • Timing Diagrams illustrate state transitions over time for inputs and outputs.

Example: Seat Belt Warning Light System

  • Design Requirements:

    • Inputs: Seatbelt fastened (s), Key inserted (k), Person in seat (p).

    • Boolean Equation: w = p AND NOT(s) AND k (to activate warning light).

Boolean Algebra Terminology

  • Variables: Represent values (0 or 1).

  • Literals: Instances of variables in truth or complemented form.

  • Product Term: Combination of literals.

  • Sum-of-Products: Representation as OR of product terms.

Properties of Boolean Algebra

  • Commutative: a + b = b + a, a * b = b * a.

  • Associative: (a + b) + c = a + (b + c).

  • Distributive: a * (b + c) = a * b + a * c.

  • Identity: 0 + a = a, 1 * a = a.

  • Complement: a + a’ = 1, a * a’ = 0.

Additional Properties and Laws

  • DeMorgan's Laws: Useful for simplifying and implementing Boolean functions.

Representing Functions with Truth Tables

  • Truth tables define output for all input combinations for a function.

  • Example function F(a, b, c) defined for binary values.

Standard Representation and Canonical Form

  • Standard Representation: Only one truth table representation for a function.

  • Canonical Form: Representation using sum of minterms.

Multiple-Output Circuits

  • Circuits can have multiple outputs using shared gates or separate circuits.

Multiplexers (Muxes) and Decoders

  • Decoder: Converts binary input to one high output.

  • Multiplexer (Mux): Routes one input based on selected binary values.

Schematic Capture and Simulation

  • Computer tools for capturing logic graphically and simulating outputs.

  • Delay Considerations: Real gates experience delay in output changes.

Chapter Summary

  • Combinational circuits output is determined by current inputs without memory.

  • Overview of gates, Boolean algebra, and combinational logic design processes.

  • Advanced gates and components like multiplexers and decoders enhance digital design.