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Logic Gates are the basic building blocks of any digital system. They perform logical operations on one or more binary inputs to produce a single binary output.
Here is a breakdown of the common logic gates, including their symbols, Boolean expressions, and truth tables.
1. Basic Logic Gates
These three gates are the fundamental operations from which all other logical functions are derived.
| Gate | Symbol | Boolean Expression | Output Condition | Truth Table (A, B \rightarrow
Y) |
|—|—|—|—|—|
| AND | [Symbol of AND Gate] | Y = A \cdot B (or Y = AB) | Output is HIGH (1) only if all inputs are HIGH. | 0 0 \rightarrow 0
0 1 \rightarrow 0
1 0 \rightarrow 0
1 1 \rightarrow 1 |
| OR | [Symbol of OR Gate] | Y = A + B | Output is HIGH (1) if any input is HIGH. | 0 0 \rightarrow 0
0 1 \rightarrow 1
1 0 \rightarrow 1
1 1 \rightarrow 1 |
| NOT | [Symbol of NOT Gate] | Y = A’ (or Y = \bar{A}) | Output is the INVERSE of the single input. | 0 \rightarrow 1
1 \rightarrow 0 |
2. Universal Gates
NAND and NOR are called Universal Gates because any Boolean function and any other logic gate (AND, OR, NOT, XOR, XNOR) can be implemented using only NAND gates or only NOR gates.
| Gate | Symbol | Boolean Expression | Output Condition | Truth Table (A, B \rightarrow Y) |
|—|—|—|—|—|
| NAND (NOT-AND) | [Symbol of NAND Gate] | Y = (A \cdot B)’ (or Y = \overline{AB}) | Output is LOW (0) only if all inputs are HIGH. | 0 0 \rightarrow 1
0 1 \rightarrow 1
1 0 \rightarrow 1
1 1 \rightarrow 0 |
| NOR (NOT-OR) | [Symbol of NOR Gate] | Y = (A + B)’ (or Y = \overline{A+B}) | Output is HIGH (1) only if all inputs are LOW. | 0 0 \rightarrow 1
0 1 \rightarrow 0
1 0 \rightarrow 0
1 1 \rightarrow 0 |
3. Other Gates (Exclusive Gates)
These gates are used for special purposes like comparison and arithmetic operations (e.G., in adders).
| Gate | Symbol | Boolean Expression | Output Condition | Truth Table (A, B \rightarrow Y) |
|—|—|—|—|—|
| XOR (Exclusive OR) | [Symbol of XOR Gate] | Y = A \oplus B or Y = A\bar{B} + \bar{A}B | Output is HIGH (1) only if the inputs are different. | 0 0 \rightarrow 0
0 1 \rightarrow 1
1 0 \rightarrow 1
1 1 \rightarrow 0 |
| XNOR (Exclusive NOR) | [Symbol of XNOR Gate] | Y = A \odot B or Y = AB + \bar{A}\bar{B} | Output is HIGH (1) only if the inputs are the same (equivalent to an Equality Checker). | 0 0 \rightarrow 1
0 1 \rightarrow 0
1 0 \rightarrow 0
1 1 \rightarrow 1 |
Combinational circuits are the core of digital logic, where the output depends only on the current input states, with no memory or feedback loops.
Here is a breakdown of the design procedure and key combinational circuits:
1. Combinational Circuit Design Procedure
Designing a combinational circuit is a structured process that transforms a problem specification into an optimized logic diagram:
* Problem Specification: Understand the required behavior of the circuit (e.G., “Design a circuit that lights up when an odd number of inputs are high”).
* Input/Output Determination: Determine the number of input and output variables, and assign a symbol (label) to each.
* Truth Table Derivation: Create the truth table that lists all possible input combinations and the corresponding desired output(s).
* Boolean Expression Derivation: Write the canonical Sum-of-Products (SOP) or Product-of-Sums (POS)
expression directly from the truth table.
* Simplification/Minimization: Simplify the Boolean expression using Karnaugh Maps (K-Maps) or Boolean Algebra theorems to reduce the number of logic gates needed.
* Logic Diagram Drawing: Draw the final circuit diagram using the logic gates corresponding to the simplified Boolean expression.
2. Arithmetic Circuits (Adders and Subtractors)
These circuits are the basis for the Arithmetic Logic Unit (ALU) in a CPU.
A. Half Adder (HA)
* Function: Adds two single binary digits (A and B).
* Inputs: 2 (A, B).
* Outputs: 2 (Sum, Carry-out C_{out}).
* Expressions:
* Sum = A \oplus B (XOR Gate)
* C_{out} = A \cdot B (AND Gate)
B. Full Adder (FA)
* Function: Adds three single binary digits (A, B, and a Carry-in C_{in} from a previous stage).
* Inputs: 3 (A, B, C_{in}).
* Outputs: 2 (Sum, Carry-out C_{out}).
* Implementation: Can be implemented using two Half Adders and one OR gate.
* Expressions:
* * C. Half Subtractor (HS)
* Function: Subtracts one single binary digit (B, the subtrahend) from another (A, the minuend).
* Inputs: 2 (A, B).
* Outputs: 2 (Difference, Borrow-out B_{out}).
* Expressions:
* Difference = A \oplus B (XOR Gate)
* D. Full Subtractor (FS)
* Function: Subtracts three single binary digits (A, B, and a Borrow-in B_{in}).
* Inputs: 3 (A, B, B_{in}).
* Outputs: 2 (Difference, Borrow-out B_{out}).
* Implementation: Can be implemented using two Half Subtractors and one OR gate.
3. Data Handling and Selection Circuits
These circuits are used for routing data in digital systems.
A. Multiplexer (MUX) – Data Selector
* Function: Selects one of 2^n input data lines and routes it to a single output line.
* Inputs: 2^n Data Inputs, n Select Lines.
* Output: 1 Data Output.
* Application: Parallel-to-Serial conversion, logic function implementation.
B. Demultiplexer (DEMUX) – Data Distributor
* Function: Takes a single input data line and routes it to one of 2^n possible output lines.
* Inputs: 1 Data Input, n Select Lines.
* Outputs: 2^n Output Lines.
* Application: Serial-to-Parallel conversion, routing a source to multiple destinations.
C. Decoder
* Function: Converts an n-bit input code into 2^n distinct output lines. Only one output line is active (HIGH or LOW) for any given input combination.
* Inputs: n Input Lines.
* Outputs: 2^n Output Lines.
* Application: Converting binary addresses to device enables (e.G., 3-to-8 decoder is often used in memory addressing).
D. Encoder
* Function: Performs the reverse of a decoder. It converts 2^n or fewer active input lines into an n-bit output code.
* Inputs: 2^n Input Lines (typically only one is active at a time).
* Outputs: n Output Lines.
* Application: Converting physical switch/key presses into a binary code. Priority Encoders handle cases where multiple inputs are active simultaneously by giving a higher-indexed input priority.
4. Other Specialized Circuits
A. Comparators
* Function: Compares the magnitudes of two binary numbers (A and B) and determines if they are equal, or which one is greater.
* Outputs: Typically three outputs: A > B, A = B, A < B.
* Implementation: A basic 1-bit comparator uses XNOR gates for the equality check and additional logic gates for the greater/lesser checks. Larger comparators are built by chaining single-bit comparators.
B. Code Converters
* Function: Translates data from one binary code format to another.
* Design: The design procedure follows the general combinational circuit steps:
* Create a truth table mapping the input code (e.G., BCD) to the desired output code (e.G., Excess-3).
* Use K-Maps to simplify the Boolean expression for each output bit.
* Examples: BCD-to-7-Segment display decoder, Binary-to-Gray code converter, BCD-to-Excess-3 converter.