Logic Gates: Computer Logic Design

• Lecture covers logic gates, computer logic design, and logic gates.

Logic Gates Overview

• Logic gates serve as the fundamental building blocks of digital circuits.
• They manage the control flow of information
• They are made of transistors
• They implement logical operations or functions.

Fundamental Logic Gates

• The fundamental logic gates include AND, OR, and NOT gates

Characteristics of Logic Gates

• Logic gates are characterized by their:
  • Operation of the gate
  • Function or truth table
  • Timing diagram
  • Application examples

Boolean Algebra Terms

• Variable: A symbol representing a logical quantity that can have a value of 0 or 1. Example is X.
• Complement: The inverse of a variable, indicated by a bar over the variable.
• Literal: An instance of a variable or its complement. Example is X.

Boolean Addition: OR Operation

• A sum term is a sum of literals, like A + B or A + B + C.
• A sum term equals 1 if any literal equals 1 and zero if all literals = 0.
• In logic circuits, an OR operation produces a sum term without AND operations.

Boolean Multiplication: AND Operation

• Product Term: a product of literals, such as A.B.C
• A product term equals 1 if all literals equal 1, otherwise is 0 if one is 0.
• An AND operation in logic circuits produces a product term with no OR operations.

Boolean Inversion: NOT Operation

• Complements the literal i.e. B or X.
• An inverted term = 1 if literal = 0.
• An inverted term = 0 if literal = 1.
• NOT operations produce inversion in logic circuits.

Boolean Logic vs. Binary Arithmetic

• They are similar but different, using symbols and arithmetic/logic variables.

AND Gate

• AND gates perform logical multiplication and the output is "high" only when all inputs are "high".
• Standard symbol is defined by ANSI/IEEE Standard 91-1984.
• Inputs are A, B, …, N and the output is F = A * B.

AND Gate: Truth Table

• The truth table outlines the output for each input combination:
  • If A = 0 and B = 0, then F = 0
  • If A = 0 and B = 1, then F = 0
  • If A = 1 and B = 0, then F = 0
  • If A = 1 and B = 1, then F = 1

AND Gate: Timing Diagram

• A timing diagram displays the relationship between waveforms over time.

AND Gate: Application Example

• Used in enable/disable switches, such as a stop-watch
• If input B is HIGH (enabled), the clock signal is sent to the counter.

OR Gate

• Produces a standard logical symbol.
• Performs Boolean addition: F = A + B + C +...+ N.
• Output is HIGH if one or more inputs are HIGH.

OR Gate: Truth Table

• Truth table:
  • A = 0, B = 0 gives F = 0
  • A = 0, B = 1 gives F = 1
  • A = 1, B = 0 gives F = 1
  • A = 1, B = 1 gives F = 1

OR Gate: Timing Diagram

• A timing diagram of the OR-Gate is shown.

OR Gate: Application Example

• Used in car door alarm system.
• With all doors closed (0000 input), the output of the OR gate is 0.
• Open doors result in a HIGH output (1), which activates the alarm.

NOT Gate

• Symbol shows inversion functionality.
• Inverts or complements input and changes one logic level to the opposite.
• The bubble on the symbol indicates negation, with output F = A inverted.

NOT Gate: Truth Table

• Truth table:
  • Input A = 0 yields output F = 1
  • Input A = 1 yields output F = 0

NOT Gate: Timing Diagram

• Timing diagram shows inversion of input signal A.

NOT Gate: Application Example

• Used in 1's complement operations.

NAND Gate

• NAND Gate contracts NOT-AND.
• Output is LOW only when all inputs are HIGH, the output level is opposite of an AND gate.
• A active low output is indicated by bubble.
• The equations are:
  • F = A•B
  • F = A•B•C• ..... N

NAND Gate: Truth Table

• Two-input NAND gate truth table:
  • If A = 0 and B = 0, then middle output E = 0 and final output F = 1
  • If A = 0 and B = 1, then middle output E = 0 and final output F = 1
  • If A = 1 and B = 0, then middle output E = 0 and final output F = 1
  • If A = 1 and B = 1, then middle output E = 1 and final output F = 0

NAND Gate: Timing Diagram

• Shows timing diagram of the NAND gate.

NAND Gate: Application Example

• Can be used in Device Failure Alarm Systems.
• NAND gate inputs are 111 of working fans resulting in a 0 output.
• Failing fans gives the NAND gate a 1 output that activates the alarm.

NAND Gate as Negative-OR

• NAND truth table mirrors an OR gate with inverted inputs.

NAND Gate as a Universal Gate

• NAND gates can perform all fundamental gate operations (NOT, AND, OR).

NAND as NOT Gate

• Connect all the input pins of a NAND gate together to create a NOT gate.
• Truth table:
• If A = 0 and B = 0, then F = 1
• If A = 1 and B = 1, then F = 0

NAND as AND Gate

• AND gate operation is achieved by inverting using NOT-(NOT-AND) = AND
• Apply a NOT operation (implemented via a NAND gate) to the output of a NAND gate.

NAND as OR Gate

• NAND gates can be configured to perform OR gate functions.

NOR Gate

• NOR Gate Definition: A contraction of NOT-OR
• Symbol: Standard logic symbol that shows the NOR function.
• Formula: F =A+B, F =A+B+C+..+N
• Output: Produces a LOW output when one or more of its input is HIGH
• Output Type: Bubble indicates ACTIVE LOW output

NOR Gate: Truth Table

• If A = 0 and B = 0, then F = 1.
• If A = 0 and B = 1, then F = 0.
• If A = 1 and B = 0, then F = 0.
• If A = 1 and B = 1, then F = 0.

NOR Gate: Timing Diagram

• Timing diagram shows input-output relationship for NOR gate.

NOR Gate: Application

• Used for designing Washing Machine Controllers to monitor the lid, water level, and load.
• If the lid is open, water level low, or machine overloaded, the NOR gate switches the machine off.

NOR Gate as Negative-AND

• Truth Table: the truth table for a two-input NOR gate looks awfully similar to an AND gate with the inputs inverted.

NOR Gate as Universal Gate

• NOR can be used to perform logic operations like; NOT, AND, OR

NOR as NOT Gate

• Solution is to join all the input pins of the NOR gate together.

NOR as OR Gate

• Using NOR gate to perform the OR gate operation is achieved by applying a NOT operation implemented through NOR gate to the output of the NOR gate.

NOR as AND Gate

• NOR gate can be used to form and AND gate.

XOR Gate

• Exclusive OR is designated XOR for short.
• Standard logic symbol.
• Unique Properties: this gate has only two inputs; has a logical expression.
• Output Condition: the output is HIGH only when the two inputs are at opposite logic levels.

XOR Gate: Truth Table

• Input A: 0, Input B: 0, Output F: 0
• Input A: 0, Input B: 1, Output F: 1
• Input A: 1, Input B: 0, Output F: 1
• Input A: 1, Input B: 1, Output F: 0

XOR Gate: Timing Diagram

• Timing diagram displays XOR gate behavior.

XOR Gate: Application

• Used for detecting odd parity.
• Applying 0011 at the input respectively generates 0 at the outputs, indicating that number of 1s are not odd Applying 1011 at the input generates a 1 at the output, indicating an odd number of 1s.

XNOR Gate

• XNOR gate is short for exclusive-NOR
• Logic Symbol: Has a standard logic symbol.
• Unique Properties: This gate has only two inputs and a logical expression.
• Output Condition: The output is LOW only when the two inputs are at opposite logic levels.

XNOR Gate: Truth Table

• Truth table:
  • With A = 0 and B = 0, F = 1.
  • With A = 0 and B = 1, F = 0.
  • With A = 1 and B = 0, F = 0.
  • With A = 1 and B = 1, F = 1.

XNOR Gate: Timing Diagram

• Timing diagram illustrates the gate's behavior over time.

XNOR Gate: Application Example

• Used in detecting even parity.
• For input 1011, XNOR gate outputs indicate the number of 1s is not even.
• The application 0011 XNOR gate results: 1 indicating even number of 1s.

Summary of Logic Gates

• Fundamental logic gates: AND, OR, NOT
• Universal gates: NAND, NOR
• Other gates: XOR, XNOR
• Elements:
  • Standard logic symbol
  • Logical/Boolean expression
  • Truth table
  • Application example

Review Questions

• What are the gates that are used to activate an alarm (active HIGH) whenever at least one of its inputs is LOW?
• What are some other gate configurations?