Four bit ripple carry adder diagram
However, the "slow roads" on the way to the faster levels begin to impose a drag on the whole system for instance, a bit adder could have up to 24 gate delays in its carry processing , and the mere physical transmission of signals from one end of a long number to the other begins to be a problem. At these sizes, carry-save adders are preferable, since they spend no time on carry propagation at all.
Carry-lookahead logic uses the concepts of generating and propagating carries. Although in the context of a carry-lookahead adder, it is most natural to think of generating and propagating in the context of binary addition, the concepts can be used more generally than this. In the descriptions below, the word digit can be replaced by bit when referring to binary addition of 2.
The addition of two 1-digit inputs A and B is said to generate if the addition will always carry, regardless of whether there is an input-carry equivalently, regardless of whether any less significant digits in the sum carry. The addition of two 1-digit inputs A and B is said to propagate if the addition will carry whenever there is an input carry equivalently, when the next less significant digit in the sum carries.
Note that propagate and generate are defined with respect to a single digit of addition and do not depend on any other digits in the sum. Sometimes a slightly different definition of propagate is used. Due to the way generate and propagate bits are used by the carry-lookahead logic, it doesn't matter which definition is used.
In the case of binary addition, this definition is expressed by. For binary arithmetic, or is faster than xor and takes fewer transistors to implement. Given these concepts of generate and propagate, a digit of addition carries precisely when either the addition generates or the next less significant bit carries and the addition propagates.
For each bit in a binary sequence to be added, the carry-lookahead logic will determine whether that bit pair will generate a carry or propagate a carry. This allows the circuit to "pre-process" the two numbers being added to determine the carry ahead of time.
Then, when the actual addition is performed, there is no delay from waiting for the ripple-carry effect or time it takes for the carry from the first full adder to be passed down to the last full adder. Below is a simple 4-bit generalized carry-lookahead circuit that combines with the 4-bit ripple-carry adder we used above with some slight adjustments:.
For the example provided, the logic for the generate g and propagate p values are given below. The numeric value determines the signal from the circuit above, starting from 0 on the far left to 3 on the far right:. To determine whether a bit pair will propagate a carry, either of the following logic statements work:.
The XOR is used normally within a basic full adder circuit; the OR is an alternative option for a carry-lookahead only , which is far simpler in transistor-count terms. The carry-lookahead 4-bit adder can also be used in a higher-level circuit by having each CLA logic circuit produce a propagate and generate signal to a higher-level CLA logic circuit.
Putting 4 4-bit CLAs together yields four group propagates and four group generates. The calculation of the gate delay of a bit adder using 4 CLAs and 1 LCU is not as straight forward as the ripple carry adder. The Manchester carry chain is a variation of the carry-lookahead adder  that uses shared logic to lower the transistor count.
As can be seen above in the implementation section, the logic for generating each carry contains all of the logic used to generate the previous carries. A Manchester carry chain generates the intermediate carries by tapping off nodes in the gate that calculates the most significant carry value. However, not all logic families have these internal nodes, CMOS being a major example. Dynamic logic can support shared logic, as can transmission gate logic. One of the major downsides of the Manchester carry chain is that the capacitive load of all of these outputs, together with the resistance of the transistors causes the propagation delay to increase much more quickly than a regular carry lookahead.
A Manchester-carry-chain section generally doesn't exceed 4 bits. 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. If the addends are four or more, more than one layer of compressors is necessary, and there are various possible design for the circuit: This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers.
From Wikipedia, the free encyclopedia. Digital Logic and Computer Design.