Lecture on Half Adder

Introduction

• A half adder is a small circuit fundamental to building larger circuits like ripple carry adders.
• It is used to add two single-bit numbers.

Conceptual Overview

• Two single-bit inputs allow for four combinations:
  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 10 (binary for 2)
• This forms a truth table:
  • Inputs: A, B
  • Output: Sum, Carry
  • 00 -> 0, 0
  • 01 -> 1, 0
  • 10 -> 1, 0
  • 11 -> 0, 1

Truth Table Analysis

• Truth table does not match standard operations like AND, OR, NOT.
• Resembles the exclusive OR (XOR) operation:
  • Output is 1 if inputs are different.
  • XOR Gate gives 1 for different inputs, 0 if the same.

XOR Gate Construction

• XOR can be simulated using AND, OR, and NOT gates:
  • Use NOT gates to invert inputs and outputs appropriately.
  • Combine using AND and OR to simulate XOR behavior.
• XOR Gate symbol: A swoosh behind the OR gate indicates XOR.

Completing the Half Adder

• Adds two outputs: Sum (XOR output) and Carry (AND output)
• Carry: Only 1 when both inputs are 1 (A AND B).

Final Construction

• Connect both inputs to XOR for SUM output.
• Connect both inputs to AND for CARRY output.
• Add both outputs to form the final half adder circuit.

Representation

• Textbook representation often uses a compact form with XOR and AND gates linked to input pins.

Future Topics

• Full Adder: Integrates multiple half adders to handle carry from previous sums.
• Next steps involve understanding how to use multiple half adders for full addition.

Summary

• A half adder is essential in digital circuits for binary addition.
• It consists of an XOR gate for the SUM and an AND gate for the CARRY.
• The construction of half adders is a step towards building complex processing units.