Useful References :: The Full Adder

Simple Logic Gates - Full Adder

	A	B	C_IN	C_OUT	S
To construct a full adder circuit, we'll need three inputs and two outputs. Since we'll have both an input carry and an output carry, we'll designate them as C_IN and C_OUT. At the same time, we'll use S to designate the final Sum output. The resulting truth table is shown to the right. Looking a little more closely, however, we can note that the S output is actually an XOR between the A input and the half-adder SUM output with B and C_IN inputs. Also, the output carry will be true if any two or all three inputs are logic 1. What this suggests is also intuitively logical: we can use two half-adder circuits. The first will add A and B to produce a partial Sum, while the second will add C_IN to that Sum to produce the final S output. If either half-adder produces a carry, there will be an output carry. Thus, C_OUT will be an OR function of the half-adder Carry outputs. The resulting full adder circuit is shown below.	INPUTS			OUTPUTS
	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

The circuit above is really too complicated to be used in larger logic diagrams, so a separate symbol, shown to the right, is used to represent a one-bit full adder. In fact, it is common practice in logic diagrams to represent any complex function as a "black box" with input and output signals designated. It is, after all, the logical function that is important, not the exact method of performing that function.

Now we can add two binary bits together, accounting for a possible carry from the next lower order of magnitude, and sending a carry to the next higher order of magnitude. To perform multibit addition the way a computer would, a full adder must be allocated for each bit to be added simultaneously. Thus, to add two 4-bit numbers to produce a 4-bit sum (with a possible carry), you would need four full adders with carry lines cascaded, as shown to the left. For two 8-bit numbers, you would need eight full adders, which can be formed by cascading two of these 4-bit blocks. By extension, two binary numbers of any size may be added in this manner.

It is also quite possible to use this circuit for binary subtraction. If a negative number is applied to the B inputs, the resulting sum will actually be the difference between the two numbers. In a modern computer, the adder circuitry will include the means of negating one of the input numbers directly, so the circuit can perform either addition or subtraction on demand. Other functions are commonly included in modern implementations of the adder circuit, especially in modern microprocessors.