VHDL Code Analysis for Digital Logic

The provided VHDL code snippets:

• SUM <= A XOR B;
• Carry <= A AND B;

where A and B are single-bit inputs, represent the fundamental operations of a digital circuit that performs binary addition. Let's break down what each line signifies in the context of digital logic.

Half Adder Circuit Explained

A Half Adder is a combinational digital circuit that adds two single binary digits (bits) and produces two outputs: a Sum (S) and a Carry-out (Cout). It does not take into account a carry-in from a previous stage of addition, which is why it is called a "half" adder.

• Inputs: Two single bits, A and B.
• Outputs:
  • Sum (S): The result of the addition, excluding any carry.
  • Carry (Cout): The carry generated if both input bits are 1.

Logic Gates for Half Adder Outputs

The VHDL code directly implements the logic gates required for a Half Adder:

1. SUM Output using XOR Gate:

   The statement SUM <= A XOR B; calculates the sum bit. An Exclusive OR (XOR) gate outputs a 1 if its inputs are different, and a 0 if its inputs are the same. In binary addition:

   • $0 + 0 = 0$ (Sum = 0)
   • $0 + 1 = 1$ (Sum = 1)
   • $1 + 0 = 1$ (Sum = 1)
   • $1 + 1 = 0$ (Sum = 0, with a Carry of 1)

   The sum bit (0, 1, 1, 0) perfectly matches the output of an XOR gate for the corresponding inputs (A, B).

2. Carry Output using AND Gate:

   The statement Carry <= A AND B; calculates the carry-out bit. An AND gate outputs a 1 only if all its inputs are 1. In binary addition, a carry is generated only when both input bits are 1 ($1 + 1 = 10_2$, where $1_2$ is the carry). This matches the behavior of an AND gate for the carry bit.

Half Adder Truth Table

Below is the truth table for a Half Adder, illustrating the relationship between inputs (A, B) and outputs (SUM, CARRY).

Based on this analysis, the VHDL code faithfully represents the logic of a Half Adder, which is designed to add two single binary digits and produce their sum and carry outputs without considering an incoming carry.