adder

An adder (sometimes called a summer) is a digital circuit that adds two N-bit numbers and generates an N-bit number. In addition to generating a sum, adders often also generate an overflow flag and a carry flag. Adders are used in many parts of the microprocessor such as the ALU, PC, counters, calculating effective addresses and table indices, multipliers, filters, and in various other components.

Basic design

A 1-bit adder adds two single-bit values together. There are four such possible operations. All but the 1+1 operation result in a single-digit sum. The 1+1 operation produces a sum with two digits. The higher significant bit of that value is known as a carry. The digital component that performs the addition of two bits is called a half adder. When two multi-bit numbers are added together, the carry out from the lower bit must be accounted for in the higher addition of the higher bits. When a half adder accounts for a carry in, it becomes a full adder.

Half Adders (HA)

Main article: Half adder

Half Adder
Input		C_out	S	Q₁₀
0	0	0	0	0
0	1	0	1	1
1	0	0	1	1
1	1	1	0	2

A half adder is a simple device that adds two single bit inputs. The result of a half adder (in base 10) is either 0, 1, or 2. Two bits are required to represent that output; they are called the sum S and carry-out C_out. The carry-out of one half adder is typically used as the carry-in of the next half adder. For that reason it is said to have double the weight of the other bit.

The sum is 1 only when one of the operands is 1, otherwise it's 0. This can be realized by simply XORing them together. The carry out bit is one only when both addends are one; ANDing the two bits will generate the desired output.

Full Adder (FA)

Main article: full adder

A major drawback of a half adder is that it lacks the ability to add two bits and account for a carry-in that might have been brought from the previous digit. As previously stated, the carry-out of one half adder is the carry-in of the next half adder. A full adder is a simple device that can receive a carry-in bit input in addition to adding two single bit inputs. A full adder has three inputs A, B, and C_in and two outputs S and C_out. Full adders are typically combined together in a cascading way (C_in to _out), creating N-bit adders (16, 32, 64, etc..).

The sum output can be expressed as:

BCD Adders

Main article: BCD Adder

Most adders typically use the binary numeral system, however they can use any other numerical representation such as binary-coded decimal. Binary adders are typically simpler to design when compared to a BCD adder where roughly 20 percent more circuitry is required.

Advanced Designs

Due to the adder's central role in so many digital circuits, it has been the subject of many researches. Various different designs have been developed over the years to meet a broad range of requirements (e.g. complexity, cost, space, and time trade-offs).