6 bit ripple carry adder with overflow
Using only two types of gates is convenient if the circuit is being implemented using simple IC chips which contain only one gate type per chip. Assumed that an XOR-gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to. It is possible to create a logical circuit using multiple full adders to add N -bit numbers. Each full adder inputs a C in , which is the C out of the previous adder.
This kind of adder is called a ripple-carry adder RCA , since each carry bit "ripples" to the next full adder. The layout of a ripple-carry adder is simple, which allows fast design time; however, the ripple-carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder.
The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. The carry-in must travel through n XOR-gates in adders and n carry-generator blocks to have an effect on the carry-out. To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders CLA.
They work by creating two signals P and G for each bit position, based on whether a carry is propagated through from a less significant bit position at least one input is a 1 , generated in that bit position both inputs are 1 , or killed in that bit position both inputs are 0.
In most cases, P is simply the sum output of a half adder and G is the carry output of the same adder. After P and G are generated, the carries for every bit position are created. Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry-skip or carry-bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates the sum and carry values for either possible carry input 0 or 1 to the block, using multiplexers to select the appropriate result when the carry bit is known.
By combining multiple carry-lookahead adders, even larger adders can be created. This can be used at multiple levels to make even larger adders. Other adder designs include the carry-select adder , conditional sum adder , carry-skip adder , and carry-complete adder. If an adding circuit is to compute the sum of three or more numbers, it can be advantageous to not propagate the carry result. Instead, three-input adders are used, generating two results: The sum and the carry may be fed into two inputs of the subsequent 3-number adder without having to wait for propagation of a carry signal.
After all stages of addition, however, a conventional adder such as the ripple-carry or the lookahead must be used to combine the final sum and carry results. A full adder can be viewed as a 3: The carry-out represents bit one of the result, while the sum represents bit zero.
Likewise, a half adder can be used as a 2: Such compressors can be used to speed up the summation of three or more addends. If the addends are exactly three, the layout is known as the carry-save adder.
When multi-bit unsigned quantities are added, overflow occurrs if there is a carry out from the leftmost most significant bit. Consider overflow detection when adding two one-bit signed quntities. Although one bit is required to represent the data , another bit has to represent the sign.
The truth table includes five columns with three inputs and two outputs. The full adder knows nothing about the difference between signed and unsigned numbers. In 2's complement binary representation, the sign bit is simply the leftmost, or most significant , bit of the data type. Full adder truth table for the sign bit can be extended to include new output which indicates if overfow condition has occured.
Two-bit signed data type: Notice that when operands have opposite signs , their sum will never overflow:. Specific overflow detection requires knowing the operation and the representation. Too often people mistake overflow condition for unsigned overflow, when the carry out is 1. One way to detect it is to XOR the carry in and the carry out.
Recall that to represent 2's complement negative number, we must Flip all bits Add 1.