Table of Contents

## Binary Adder

The binary adder is a combinational circuit that can perform summation of the input binary numbers. The most common or basic arithmetic operation is the addition of binary digits. A combination circuit which performs the additions of two bits is a called a half adder while that performs the addition of three bits is a full adder.

## Half Adder

The Half adder is the simplest of all adder circuits. Half adder is a combinational arithmetic circuit that adds two numbers and produces a sum bit (S) and carry bit (C) both as the output. Let us consider two inputs bits A and B, then sum bit (S) is the X-OR of A and B and the carry bit (C) will be the AND of A and B. It is evident from the function of a half adder that it requires one X-OR gate and one AND gate for its construction. Let us first consider the addition of single bits.

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

These are the least possible single-bit combinations. But it is noticed that the output as a result of the addition of 1 + 1 results in 10. This can be solved using an EXOR Gate, or the sum result must be re-written as a 2-bit output. Thus, the above equations can be written as

0 + 0 = 00

0 + 1 = 01

1 + 0 = 01

1 + 1 = 10

## Half Adder Truth Table

Here the output “1” of “10” becomes the carry-out. The “SUM” is the normal output and “CARRY” is the carry-out. Though the half adder is the simplest adder circuit, it has a major disadvantage. A half adder can add only two input bits (A and B) and is not affected by the carry of the input. As a result, if the input that is given to a half adder has a carry, then it will be neglected and it adds only the bits A and B. But this results in an incomplete binary addition and hence it gets its name a half adder. The truth table of half adder is shown in the above table.

## Full Adder

The full adder is a much complex adder circuit compared to the half adder. The major difference between a half adder and a full adder is the number of input terminals that are fed to the adder circuit. The full adder has three inputs and two outputs. The first two inputs are A and B and the third input is an input carry designated as C_{IN}. The full adder is designed in such a way that it can take in eight bits together to create a byte-wide adder and cascade the carry bit from one adder to the next. The output carry is designated as C_{OUT} and the normal output is designated as S. Consider the following truth table for a full adder.

## Full Adder Truth Table

The output S is an XOR between the input A and the half adder sum output with B and C_{IN} inputs. The C_{OUT} will only be true if any of the two inputs out of the three are HIGH. Thus, a full adder circuit can be implemented with the help of two half adder circuits. The function of the first half adder is to add the inputs A and B to produce a partial sum. The second half adder adds the output of the first adder and the C_{IN} to get the final output (S). If there will be an output carry. Thus C_{OUT} will be an OR function of the half adder carry outputs. The sum of the products (SOP) for the above truth table with sum S_{n} and carry C_{n} is given as:

The full adder circuit is shown in figure below. It comprises of two XOR gates and two AND gates and one OR gate. It can also be constructed using a NAND gate by employing double complement method.

Though the implementation of larger logic diagrams is possible with the above full adder logic, a simpler symbol is mostly used to represent the operation. The simple one bit symbol of a full adder is as shown in figure below.

The Block diagram of full adder implementation via a pair of half adders is as shown in figure below.

## Multi-bit Adder using Full Adder

The above-mentioned adder is used to sum up to 2 bits together taking a carry from the next lower order of magnitude and sending a carry to the next higher order of magnitude. For a multi-bit operation, each bit must be represented by a full adder and all the bits must be added simultaneously. Thus, to add two 8-bit numbers, we need 8 full adders which can be formed by cascading two of the 4-bit blocks. The addition of two 4-bit numbers is shown below.

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