The Binary Adder

The Binary Adder

Another common and very useful combinational logic circuit is that of the Binary Adder circuit. The Binary Adder is made up from standard AND and Ex-OR gates and allow us to "add" single bits of data together to produce two outputs, the SUM ("S") of the addition and a CARRY ("C"). One of the main uses for the Binary Adder is in arithmetic and counting circuits.

Consider the addition of two denary (base 10) numbers below.

Each column is added together starting from the right hand side. As each column is added together a carry is generated if the result is greater or equal to ten, the base number. This carry is then added to the result of the addition of the next column to the left and so on, simple school math's addition. Binary addition is based on similar principals but a carry is only generated when the result in any column is greater or equal to "2", the base number of binary.

Binary Addition

Binary Addition follows the same basic rules as for the denary addition above except in binary there are only two digits and the largest digit is "1", so any "SUM" greater than 1 will result in a "CARRY". This carry 1 is passed over to the next column for addition and so on. Consider the single bit addition below.

The single bits are added together and "0 + 0", "0 + 1", or "1 + 0" results in a sum of "0" or "1" until you get to "1 + 1" then the sum is equal to "2". For a simple 1-bit addition problem like this, the resulting carry bit could be ignored which would result in an output truth table resembling that of an Ex-OR Gate as seen in the Logic Gates section and whose result is the sum of the two bits but without the carry. An Ex-OR gate only produces an output "1" when either input is at logic "1", but not both. However, all microprocessors and electronic calculators require the carry bit to correctly calculate the equations so we need to rewrite them to include 2 bits of output data as shown below.

From the above equations we know that an Ex-OR gate will only produce an output "1" when "EITHER" input is at logic "1", so we need an additional output to produce a carry output, "1" when "BOTH" inputs "A" and "B" are at logic "1" and a standard AND Gate fits the bill nicely. By combining the Ex-OR gate with the AND gate results in a simple digital binary adder circuit known commonly as the "Half-Adder" circuit.

The Half-Adder Circuit

1-bit Adder with Carry-Out

Symbol	Truth Table
	A	B	SUM	CARRY
	0	0	0	0
	0	1	1	0
	1	0	1	0
	1	1	0	1
Boolean Expression: Sum = A ⊕ B     Carry = A . B

From the truth table we can see that the SUM (S) output is the result of the Ex-OR gate and the Carry-out (CO) is the result of the AND gate. One major disadvantage of the Half-Adder circuit when used as a binary adder, is that there is no provision for a "Carry-in" from the previous circuit when adding together multiple data bits. For example, suppose we want to add together two 8-bit bytes of data, any resulting carry bit would need to be able to "ripple" or move across thebit patterns starting from the least significant bit (LSB). As the Half-Adder has no carry input the resultant added value would be incorrect. One simple way to overcome this problem is to use a "Full-Adder" type binary adder circuit.

The Full-Adder Circuit

The main difference between the "Full-Adder" and the previous seen "Half-Adder" is that a Full-Adder has 3-inputs, the two same data inputs "A" and "B" as before plus an additional "Carry-In" (C-in) input as shown below.

Full-Adder with Carry-In

Symbol	Truth Table
	A	B	C-in	Sum	C-out
	0	0	0	0	0
	0	1	0	1	0
	1	0	0	1	0
	1	1	0	0	1
	0	0	1	1	0
	0	1	1	0	1
	1	0	1	0	1
	1	1	1	1	1
Boolean Expression: Sum = A ⊕ B ⊕ C-in

The Full-Adder circuit above consists of three Ex-OR gates, two AND gates and an OR gate. The truth table for the Full-Adder includes an additional column to take into account the Carry-in input as well as the summed output and Carry-out. 4-bit Full-Adder circuits are available as standard IC packages in the form of the TTL 74LS83 or the 74LS283 which can add together two 4-bit binary numbers and generate a SUM and a CARRY output.