Chapter 2 Study Notes: Digital System Functions

Chapter 2: Digital System Functions

The Binary Concept

• Many systems can be modeled with binary concepts, having only two states.

  • These two states may be defined as:

  • “High” or “Low”

  • “On” or “Off”

  • “Yes” or “No”

  • Binary values of “1” or “0”

• A common voltage level associated with binary states:

  • 5V represents high (1)

  • 0V represents low (0)

Binary Signals

• The binary signal encapsulates the concept of having two states:

  • High corresponds to the presence of a signal (e.g., 5V)

  • Low corresponds to the absence of a signal (e.g., 0V)

Gate Decision Making

Definition of a Gate

• A gate allows one or more binary inputs to produce a logical output based on those inputs.

• The gate evaluates conditions and produces a result at its output.

Logical AND Gate

• High Beam Light Example:

  • The automotive high beam light can be activated only when:

  • The light switch is ON

  • The high beam switch is also ON

• Logical Representation:

  • This scenario exemplifies the Logical AND where both conditions must be satisfied for the output to be high.

Logical OR Gate

• Dome Light Example:

  • The automotive dome light will turn ON if:

  • The passenger door switch is activated OR

  • The driver door switch is activated.

• Logical Representation:

  • The output is high if at least one of the inputs is high.

AND Gate Function

Input and Output Definitions

• Let:

  • Output = Y

  • Inputs = A, B, C, etc.

• Binary Representation:

  • Binary 1 indicates the presence of an event or signal.

  • Binary 0 indicates the absence of that event or signal.

AND Gate Rules

• If all inputs (A, B, C…) = 1, then the output Y = 1.

• If any input = 0, then Y = 0.

• This can model functionalities like a series circuit, where devices must be appropriately closed to turn on a light.

OR Function

• Input and Output Definitions:

  • An OR gate can have any number of inputs but only one output.

• Boolean Logic:

  • The output will be 1 if:

  • At least one input is 1.

• If all inputs = 0, the output will always be 0.

OR Gate Function Examples

1. Example 1:

   • If switch A or switch B is closed, the output is high (light ON).

2. Example 2:

   • An illustration of a parallel circuit where multiple switches can control a single output.

NOT Function

• Functionality:

  • The NOT function has a single input and a single output.

  • It inverts the input value:

  • If input = 0, output = 1

  • If input = 1, output = 0

• Because it reverses the input, it is also known as an inverter.

NOT Gate Applications

1. Example 1:

   • When a pushbutton is NOT pressed (0), light is ON.

   • Pushbutton pressed (1) will turn the light OFF.

2. Example 2:

   • In a low-pressure indicating circuit:

     • Power ON (1) and pressure switch NOT closed (0) will turn the warning indicator ON.

     • When the pressure rises to close the switch, the indicator switches OFF.

NAND Function

• Definition:

  • The NAND gate operates like an AND gate with an inverter connected to its output.

• Behavior:

  • The output is 0 only when all inputs are 1.

NOR Function

• Definition:

  • The NOR gate operates like an OR gate with an inverter connected to its output.

• Behavior:

  • The output is 1 only when all inputs are 0.

XOR (Exclusive-OR) Function

• Characteristics:

  • The output goes HIGH if exactly one of the inputs is HIGH.

  • The XOR function typically has two inputs and one output.

  • Frequently used for comparing two binary numbers.

Boolean Equations from Gates

• AND Gate Boolean Representation:

  • Formula: $Y = A \bullet B$

• OR Gate Boolean Representation:

  • Formula: $Y = A + B$

• NOT Gate Boolean Representation:

  • Formula: $Y = <br>eg A$

Boolean Equation Examples

1. Example 1:

   • Each logic function can be expressed through a Boolean expression.

   • NOT, AND, NAND, OR, NOR transformations illustrated with symbols.

2. Example 2:

   • Any combination can be represented as:

   • $Y = AB + C$

   • $Y = (A + B)C$

Circuit Development Using Boolean Expressions

1. Example 1:

   • Circuit consisting of an AND gate with inputs A and B, followed by an OR gate taking C and prior AND gate output.

2. Example 2:

   • Another AND gate example with inputs B and C.

Creating Boolean Expressions from Circuits

• Example 1: Given a circuit:

  • Inputs: A, B, C, D

  • Output: Y

  • Final Equation: $Y = CD(A + B)$

Hard Wired vs. Programmed Logic

1. Definition of Hardwired Logic:

   • Refers to logic control determined by the physical interconnections of devices.

   • Implemented using relays; fixed unless reconfigured.

2. Control Circuit Example: A typical hardwired stop/start motor control circuit represented in ladder diagrams.

Programmed Logic Details

• Programmed Logic Description:

  • Utilizes input configurations corresponding to instructions such as Normally Open (NO) or Normally Closed (NC).

  • A rung consists of input conditions leading to an output.

Selecting Word-Level Logic Instructions

• Utilize different instructions based on the desired logic functionality:

  • AND checks if matching bits are both ON.

  • OR checks if one or both matching bits are ON.

  • XOR checks for differences in matching bits.

  • NOT inverts the status of bits in a word.

Programmed Instruction Examples

• AND Instruction: Set to 1 at B3:10 when both Source A and B bits are high and input A is true.

• OR Instruction: Set to 1 at B3:20 when any of Source A or B are high.

• XOR Instruction: Produces output only when Source A and B bits differ.

  • Illustrated with binary inputs.

• NOT Instruction: Inverts bits from B3:9 to B3:10 when input A is true.