Digital Logic

 

Digital Logic refers to the foundation of digital electronics, where systems operate using binary values (0 and 1). These binary values represent the two states of a digital system, often associated with "low" and "high" voltage levels.

Digital logic forms the basis for designing and implementing electronic devices such as computers, mobile phones, and other digital systems.

 

Key Concepts in Digital Logic:

1. Binary System:

   • The binary number system is used to represent data in digital logic.

   • Values are either 0 or 1, corresponding to "off" or "on" states.

2. Logic Gates:

   • Basic building blocks of digital circuits.

   • Perform logical operations on binary inputs to produce a binary output.

   • Examples include:

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

     • OR Gate: Outputs 1 if at least one input is 1.

     • NOT Gate: Inverts the input (0 becomes 1, and vice versa).

3. Combinational Logic:

   • Output depends only on the current inputs.

   • Example: Adders, encoders, decoders, and multiplexers.

4. Sequential Logic:

   • Output depends on current inputs and previous states (memory is involved).

   • Example: Flip-flops, counters, and registers.

5. Boolean Algebra:

   • Mathematical framework for analyzing and designing digital logic circuits.

   • Uses operations like AND, OR, NOT, NAND, NOR, XOR, and XNOR.

6. Truth Tables:

   • Tabular representation showing the relationship between input combinations and output.

7. Applications of Digital Logic:

   • Microprocessors and microcontrollers.

   • Digital communication systems.

   • Automation and control systems.

   • Embedded systems in everyday devices.

 

 

                                                                  Binary Systems 

 

The binary system is a numeral system that uses only two digits: 0 and 1. It is the foundational numbering system in digital electronics and computing because digital devices operate using two states, typically represented as low (0) and high (1) voltage levels.

Key Concepts of the Binary System

1. Base of the System

• The binary system is a base-2 numbering system.

• Each digit in a binary number is called a bit (short for "binary digit").

• A binary number is written as a sequence of 0s and 1s, e.g., 10101.

2. Positional Value

• Each position in a binary number represents a power of 2, starting from 202^0 (rightmost bit) and increasing to the left.

Example: The binary number 1011 can be interpreted as:

(1×23)+(0×22)+(1×21)+(1×20)=8+0+2+1=1110​

 

                   

3. Binary Arithmetic

• Binary arithmetic is essential for digital computing.

Addition:

• Follows simple rules:

  • 0+0=00 + 0 = 0

  • 0+1=10 + 1 = 1

  • 1+0=11 + 0 = 1

  • 1+1=101 + 1 = 10 (carry 1 to the next column)

Subtraction:

• Uses the concept of borrowing.

Multiplication and Division:

• Similar to decimal arithmetic, with adjustments for base-2.

4. Binary Conversion

• Decimal to Binary:

  • Divide the decimal number by 2, record the remainder, and continue until the quotient is 0.

  • Read the remainders from bottom to top.

• Binary to Decimal:

  • Multiply each bit by its corresponding power of 2 and sum them up (as shown in the positional value example).

5. Binary Number Types

• Unsigned Binary Numbers:

  • Represent only positive values.

• Signed Binary Numbers:

  • Represent both positive and negative values.

  • Sign-and-Magnitude: Most significant bit (MSB) indicates the sign (0 for positive, 1 for negative).

  • Two's Complement: Common method for representing negative numbers.

6. Binary Codes

Binary numbers are used to encode data in various forms:

• BCD (Binary-Coded Decimal): Encodes each decimal digit separately.

• ASCII: Represents characters as binary values.

• Gray Code: Ensures only one-bit changes between successive numbers, reducing errors.

Advantages of the Binary System

1. Simplicity: Only two states (0 and 1) are easy to implement in hardware using switches or transistors.

2. Error Detection and Correction: Binary arithmetic supports methods for detecting and correcting errors efficiently.

3. Compatibility: Suited for digital logic and computing systems.

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                                                                                    Logic Gates

 

Logic gates are devices that perform logical operations on binary inputs to produce binary outputs. They are the fundamental building blocks of digital circuits and are used in many electronic devices. 

• Logic gates are made of transistors and diodes. 

• They work based on Boolean algebra, where each terminal is either true (1) or false (0). 

• The output of a logic gate depends on the type of gate and the values of its inputs. 

• Logic gates are often represented by symbols that communicate how they make decisions. 

 

Types of Logic Gates:

     1.AND Gate:

Outputs true (1) only if all its inputs are true.

            Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 1 (since both inputs are true)

• A = 1, B = 0 → Output = 0 (since one input is false)

• A = 0, B = 1 → Output = 0 (since one input is false)

• A = 0, B = 0 → Output = 0 (since both inputs are false)

 

      2.OR Gate:

Outputs true if at least one of its inputs is true.

Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 1 (since at least one input is true)

• A = 1, B = 0 → Output = 1 (since at least one input is true)

• A = 0, B = 1 → Output = 1 (since at least one input is true)

• A = 0, B = 0 → Output = 0 (since both inputs are false)

 

       3.NOT Gate:

Outputs the inverse of its input; if the input is true, the output is false, and vice versa.

Example: If you have one input, A:

• A = 1 → Output = 0 (inverts the input)

• A = 0 → Output = 1 (inverts the input)

 

     4.NAND Gate:

Outputs true if at least one input is false. It's the opposite of the AND gate.

Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 0 (opposite of AND gate)

• A = 1, B = 0 → Output = 1 (opposite of AND gate)

• A = 0, B = 1 → Output = 1 (opposite of AND gate)

• A = 0, B = 0 → Output = 1 (opposite of AND gate)

 

    5.NOR Gate:

Outputs true only if all its inputs are false. It's the opposite of the OR gate.

Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 0 (opposite of OR gate)

• A = 1, B = 0 → Output = 0 (opposite of OR gate)

• A = 0, B = 1 → Output = 0 (opposite of OR gate)

• A = 0, B = 0 → Output = 1 (opposite of OR gate)

 

   6.XOR Gate:

Outputs true if an odd number of inputs are true.

Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 0 (true if odd number of true inputs)

• A = 1, B = 0 → Output = 1 (true if odd number of true inputs)

• A = 0, B = 1 → Output = 1 (true if odd number of true inputs)

• A = 0, B = 0 → Output = 0 (true if odd number of true inputs)

 

     

  7.XNOR Gate:

Outputs true if an even number of inputs are true. It's the opposite of the XOR gate.

Example: If you have two inputs, A and B:

• A = 1, B = 1 → Output = 1 (true if even number of true inputs)

• A = 1, B = 0 → Output = 0 (true if even number of true inputs)

• A = 0, B = 1 → Output = 0 (true if even number of true inputs)

• A = 0, B = 0 → Output = 1 (true if even number of true inputs)

 

Application

• Logic gates perform basic logical functions like combining signals or inverting them.

• They are used in electronic devices to perform tasks like data processing, arithmetic operations, and decision making.

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                                                                       Combinational Circuits

Combinational circuits are digital circuits where the output depends solely on the current inputs. They do not have memory elements, which means they do not store past input values.

combinational circuits are like calculators that give you instant results based on what you input

Here are some key points:

Components: Made up of logic gates like AND, OR, NOT, etc.

Examples: Adders (which perform binary addition), multiplexers (which select one input from several inputs), and encoders (which convert input data into binary code).

 

                                                                             Sequential Circuits

Sequential circuits, on the other hand, depend on both the current inputs and the previous state of the circuit (i.e., they have memory). These circuits can store information and have a feedback mechanism.

sequential circuits are more like computers that remember past information and use it along with new inputs to provide an output

Components: Include flip-flops or latches (basic memory elements) alongside logic gates.

Examples: Counters (which count pulses and maintain state), shift registers (which move data in steps), and memory devices.

 

Key Differences

1. Memory:

   - Combinational circuits have no memory.

   - Sequential circuits have memory elements.

 

2. Output:

   - Combinational circuits' output is determined solely by current inputs.

   - Sequential circuits' output is determined by current inputs and the previous state.

 

3. Timing:

   - Combinational circuits are instant (outputs change immediately with inputs).

   - Sequential circuits rely on clock pulses and can take time to change state.

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