# Full Adder in Digital Logic

Full Adder is the adder that adds three inputs and produces two outputs. The first two inputs are A and B and the third input is an input carry as C-IN. The output carry is designated as C-OUT and the normal output is designated as S which is SUM. A full adder logic is designed in such a manner that can take eight inputs together to create a byte-wide adder and cascade the carry bit from one adder to another. we use a full adder because when a carry-in bit is available, another 1-bit adder must be used since a 1-bit half-adder does not take a carry-in bit. A 1-bit full adder adds three operands and generates 2-bit results.

**Full Adder Truth Table:**

**Logical Expression for SUM:** = A’ B’ C-IN + A’ B C-IN’ + A B’ C-IN’ + A B C-IN = C-IN (A’ B’ + A B) + C-IN’ (A’ B + A B’) = C-IN XOR (A XOR B) = (1,2,4,7)

**Logical Expression for C-OUT:** = A’ B C-IN + A B’ C-IN + A B C-IN’ + A B C-IN = A B + B C-IN + A C-IN = (3,5,6,7)

**Another form in which C-OUT can be implemented:** = A B + A C-IN + B C-IN (A + A’) = A B C-IN + A B + A C-IN + A’ B C-IN = A B (1 +C-IN) + A C-IN + A’ B C-IN = A B + A C-IN + A’ B C-IN = A B + A C-IN (B + B’) + A’ B C-IN = A B C-IN + A B + A B’ C-IN + A’ B C-IN = A B (C-IN + 1) + A B’ C-IN + A’ B C-IN = A B + A B’ C-IN + A’ B C-IN = AB + C-IN (A’ B + A B’)

Therefore COUT = AB + C-IN (A EX – OR B)

Full Adder logic circuit.

### Implementation of Full Adder using Half Adders:

2 Half Adders and an OR gate is required to implement a Full Adder.

With this logic circuit, two bits can be added together, taking a carry from the next lower order of magnitude, and sending a carry to the next higher order of magnitude.

### Implementation of Full Adder using NAND gates: **Implementation of Full Adder using NOR gates:**

Total 9 NOR gates are required to implement a Full Adder. In the logic expression above, one would recognize the logic expressions of a 1-bit half-adder. A 1-bit full adder can be accomplished by cascading two 1-bit half adders.