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## Computer organization Notes - Full Adder

Full adder is developed to overcome the drawback of Half Adder circuit. A full-adder is a combinational circuit that forms the arithmetic sum of three input bits. It consists of three inputs and two outputs. Two of the input variables, denoted by A and B, represent the two significant bits to be added. The third input ,Cin, represents the carry from the previous lower significant position. Two outputs are necessary because the arithmetic sum of three binary digits ranges in value from 0 to 3, and binary 2 or 3 needs two digits. The two outputs are designated by the symbols S (for sum) and C (for carry). The binary variable S output represents the least significant bit of the sum. The binary variable C gives the output carry. The block diagram for full adder is

The truth table for Full Adder is

The Circuit Diagram for Full Adder is

The truth table as shown above contains eight rows, under the input variables designate all possible combinations that the binary variables may have. The value of the output variables are determined from the arithmetic sum of the input bits. When all the input bits are 0, the output is 0. The S output is equal to 1 when only one input is equal to 1 or when all three inputs are equal to 1. The C output has a carry of 1 if two or three inputs are equal to 1. The truth table can be used to find algebraic expressions for the two output variables. Boolean functions for sum S and carry C can be obtained as follows : S = A’B’Cin + A’BC’in + AB’C’in + ABCin C = A’BCin + AB’Cin + ABC’in + ABCin Simplifying, these functions using K-maps. For Sum S For Carry C
From K-maps, we conclude that:
1. There is no further simplification possible for sum S.
So, S = A’B’Cin + A’BC’in + AB’C’in + ABCin
2. Solving Carry function we get, C = AB + ACin + BCin
Now we can draw circuit diagram for full adder as shown in figure.

A full adder can be designed using two Half Adders as follows –

Function for sun S and carry C in case of Half Adder are S = A’B + AB’ = A ⊕B and C = AB
And in case of Full adder functions for Sum and Carry are
S = A’B’Cin + A’BC’in + AB’C’in + ABCin
C = AB + ACin + BCin
If we can represent functions of Full adder in the form of functions of Half Adders then we will be able to design a Full Adder using Half Adders. Sum function for Full adder--\ S = A’B’Cin + A’BC’in + AB’C’in + ABCin
= (A’B’ + AB) Cin + (A’B + AB’)C’in
= (A⊕B)’ Cin + (A⊕B)C’in = (A⊕B) ⊕ Cin
and carry C = A’BCin + AB’Cin + ABC’in + ABCin
= (AB’ + A’B) Cin + AB(Cin + C’in) = (A⊕B) Cin + AB
Logical diagram for full adder using two Half Adders can be drawn as shown in figure as follows :

A full adder made of two half adders is not very economic so we construct full adders directly from input and output relations.

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