Adders

In electronics, an adder or summer is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used not only in the arithmetic logic unit(s), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations.

Although adders can be constructed for many numerical representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require a more complex adder.

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HALF ADDER (HA)

The half adder adds two input bits and generates a carry and sum, which are the two outputs of a half adder. The input variables of a half adder are called the augend and addend bits. The output variables are the sum and carry. The truth table for the half adder is;

INPUT		OUTPUT
A	B	SUM	CARRY
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

From above truth table we can write Boolean Expression as;

S = Sum;  C = Carry

Realization of above Boolean Expressions using gates;

Fig. Half-Adder using XOR & AND Gates

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FULL ADDER (FA)

This circuit  can add two binary numbers, accept a carry and yield a carry. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the previous less significant stage.

Fig. 1-bit Full Adder

Such a circuit can easily be visualized by means of two half adders (HA) and OR gate as shown in figure. A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input. And connect the carry of both adder to OR gate. The outputs are S and Cout.

Fig.  Full-Adder using two Half-Adder and OR gate

Truth Table for full adder is shown below;

INPUT			OUTPUT
A	B	Ci	S	Co
0	0	0	0	0
0	0	1	0	1
0	1	0	0	1
0	1	1	1	0
1	0	0	0	1
1	0	1	1	0
1	1	0	1	0
1	1	1	1	1

From above truth table we can write Boolean Expression as;

Realization of above Boolean Expression using gates;

Fig. Realization of Full-Adder using Boolean Expression