Information about Adder (electronics)
In electronics, an adder or summer is a digital circuit that performs addition of numbers.
In modern computers adders reside in the arithmetic logic unit (ALU) where other operations are performed.
Although adders can be constructed for many numerical representations, such as Binary-coded decimal or excess-3, the most common adders operate on binary numbers.
In cases where two's complement is being used to represent negative numbers it is trivial to modify an adder into an adder-subtracter.
A half adder has two inputs, generally labelled A and B, and two outputs, the sum S and carry C. S is the two-bit XOR of A and B, and C is the AND of A and B. Essentially the output of a half adder is the sum of two one-bit numbers, with C being the most significant of these two outputs.
The second type of single bit adder is the full adder. The full adder takes into account a carry input such that multiple adders can be used to add larger numbers. To remove ambiguity between the input and output carry lines, the carry in is labelled Ci or Cin while the carry out is labelled Co or Cout.
A half adder is a logical circuit that performs an addition operation on two binary digits. The half adder produces a sum and a carry value which are both binary digits.
Following is the logic table for a half adder:
A full adder is a logical circuit that performs an addition operation on three binary digits. The full adder produces a sum and carry value, which are both binary digits. It can be combined with other full adders (see below) or work on its own.
Note that the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. This is because the only discrepancy between OR and XOR gates occurs when both inputs are 1; for the adder shown here, this is never possible. Using only two types of gates is convenient if one desires to implement the adder directly using common IC chips.
A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input and or the two carry outputs. Equivalently, S could be made the three-bit xor of A, B, and Ci and Co could be made the three-bit majority function of A, B, and Ci. The output of the full adder is the two-bit arithmetic sum of three one-bit numbers.
It is possible to create a logical circuit using several full adders to add multiple-bit numbers. Each full adder inputs a
, which is the 
of the previous adder. This kind of adder is a ripple carry adder, since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder.
The layout of a ripple carry adder is simple, which allows for 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. Following the path from to shows 2 gates that must be passed through.
Ergo, a 32-bit adder requires 31 carry computations and the final sum calculation for a total of 
gate delays.
To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry lookahead adders. They work by creating Propagate and Generate 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'), a carry is generated in that bit position (both inputs are '1'), or if a carry is 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 advanced carry lookahead architectures are the Manchester carry chain and the Brent-Kung adder.
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 bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates sum and carry values for either possible carry input to the block.
Other adder designs include the conditional sum adder, carry skip adder, and carry complete adder.
By combining multiple carry look-ahead adders even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses 16 4-bit CLAs with two levels of LCUs.
3:2 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: the most common are Dadda and Wallace trees. This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers.
Types of adders
For single bit adders, there are two general types.A half adder has two inputs, generally labelled A and B, and two outputs, the sum S and carry C. S is the two-bit XOR of A and B, and C is the AND of A and B. Essentially the output of a half adder is the sum of two one-bit numbers, with C being the most significant of these two outputs.
The second type of single bit adder is the full adder. The full adder takes into account a carry input such that multiple adders can be used to add larger numbers. To remove ambiguity between the input and output carry lines, the carry in is labelled Ci or Cin while the carry out is labelled Co or Cout.
Half adder
)
Half adder circuit diagram
A half adder is a logical circuit that performs an addition operation on two binary digits. The half adder produces a sum and a carry value which are both binary digits.
Following is the logic table for a half adder:
| Input | Output | ||
|---|---|---|---|
| A | B | C | S |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |
Full adder
)
Full adder circuit diagram
Inputs: {A, B, CarryIn} → Outputs: {Sum, CarryOut}
Inputs: {A, B, CarryIn} → Outputs: {Sum, CarryOut}
)
Schematic symbol for a 1-bit full adder
A full adder is a logical circuit that performs an addition operation on three binary digits. The full adder produces a sum and carry value, which are both binary digits. It can be combined with other full adders (see below) or work on its own.
| Input | Output | |||
|---|---|---|---|---|
 |
 |
 |
 |
|
| 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 |
Note that the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. This is because the only discrepancy between OR and XOR gates occurs when both inputs are 1; for the adder shown here, this is never possible. Using only two types of gates is convenient if one desires to implement the adder directly using common IC chips.
A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input and or the two carry outputs. Equivalently, S could be made the three-bit xor of A, B, and Ci and Co could be made the three-bit majority function of A, B, and Ci. The output of the full adder is the two-bit arithmetic sum of three one-bit numbers.
Multiple-bit adders
Ripple carry adder
When multiple full adders are used with the carry ins and carry outs chained together then this is called a ripple carry adder because the correct value of the carry bit ripples from one bit to the next.It is possible to create a logical circuit using several full adders to add multiple-bit numbers. Each full adder inputs a
, which is the of the previous adder. This kind of adder is a ripple carry adder, since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder.
)
4-bit ripple carry adder circuit diagram
The layout of a ripple carry adder is simple, which allows for 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. Following the path from
to shows 2 gates that must be passed through.
Ergo, a 32-bit adder requires 31 carry computations and the final sum calculation for a total of gate delays.
Carry look-ahead adders
To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry lookahead adders. They work by creating Propagate and Generate 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'), a carry is generated in that bit position (both inputs are '1'), or if a carry is 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 advanced carry lookahead architectures are the Manchester carry chain and the Brent-Kung adder.
)
4-bit adder with Carry Look Ahead
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 bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates sum and carry values for either possible carry input to the block.
Other adder designs include the conditional sum adder, carry skip adder, and carry complete adder.
Lookahead Carry Unit
By combining multiple carry look-ahead adders even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses 16 4-bit CLAs with two levels of LCUs.
)
A 64-bit adder
3:2 compressors
We can view a full adder as a 3:2 compressor: it sums three one-bit inputs, and returns the result as a single two-bit number. Thus, for example, an input of 101 results in an output of 1+0+1=10 (2). The carry-out represents bit one of the result, while the sum represents bit zero. Likewise, a half adder can be used as a 2:2 compressor.3:2 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: the most common are Dadda and Wallace trees. This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers.
See also
External links
- Hardware algorithms for arithmetic modules, includes description of several adder layouts with figures.
Digital electronics are electronics systems that use digital signals. Digital electronics are representations of Boolean algebra and are used in computers, mobile phones, and other consumer products.
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Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
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computer is a machine which manipulates data according to a list of instructions.
Computers take numerous physical forms. The first devices that resemble modern computers date to the mid-20th century (around 1940 - 1941), although the computer concept and various machines
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Computers take numerous physical forms. The first devices that resemble modern computers date to the mid-20th century (around 1940 - 1941), although the computer concept and various machines
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arithmetic logic unit (ALU) is a digital circuit that performs arithmetic and logical operations. The ALU is a fundamental building block of the central processing unit of a computer, and even the simplest microprocessors contain one for purposes such as maintaining timers.
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In computing and electronic systems, binary-coded decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. Its main virtue is that it allows easy conversion to decimal digits for printing or display and faster decimal
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Excess-3 binary coded decimal (XS-3), also called biased representation or Excess-N, is a numeral system used on some older computers that uses a pre-specified number N as a biasing value.
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binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
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The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit two's complement).
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In digital circuits, an adder-subtracter is a circuit that is capable of adding or subtracting numbers (in particular, binary). Below is a circuit that does adding or subtracting'' depending on a control signal.
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Summation is the addition of a set of numbers; the result is their sum. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a subtle procedure known as a series.
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Carry or carrying can mean:
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- Carry (arithmetic), when a digit become bigger than limit and the extra is moved to the left
- Carry flag, the equivalent in calculation in a computer
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XOR gate is a digital logic gate that implements exclusive disjunction - it behaves according to the truth table to the right. A HIGH output (1) results if one, and only one, of the inputs to the gate is HIGH (1).
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The AND gate is a digital logic gate that implements logical conjunction - it behaves according to the truth table to the right. A HIGH output (1) results only if both the inputs to the AND gate are HIGH (1).
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or, also known as logical disjunction or inclusive disjunction is a logical operator that results in true whenever one or more of its operands are true. In grammar, or is a coordinating conjunction.
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In Boolean logic, the majority function (also called the median operator) is a function from n inputs to one output. The value of the operation is false when n/2 or fewer arguments are false, and true otherwise.
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propagation delay, or gate delay, is the length of time starting from when the input to a logic gate becomes stable and valid, to the time that the output of that logic gate is stable and valid.
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A carry look-ahead adder is one type of adder used in digital logic. It can be contrasted with the simpler, but usually slower, ripple carry adder (see adder for detail on ripple carry adders).
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A carry look-ahead adder is one type of adder used in digital logic. It can be contrasted with the simpler, but usually slower, ripple carry adder (see adder for detail on ripple carry adders).
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A carry look-ahead adder is one type of adder used in digital logic. It can be contrasted with the simpler, but usually slower, ripple carry adder (see adder for detail on ripple carry adders).
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In electronics, an adder is a combinatorial or sequential logic element which computes the n-bit sum of two n-bit numbers. The Brent-Kung adder is one of the more advanced designs, having a gate level depth of .
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A carry bypass adder improves the delay of a ripple-carry adder.
The two addends are split in blocks of n bits. The output carry of each block is dependent on the input carry only if, for each of the n bits in the block, at least one addend has a
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The two addends are split in blocks of n bits. The output carry of each block is dependent on the input carry only if, for each of the n bits in the block, at least one addend has a
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In electronics, an adder is a combinatorial or sequential logic element which computes the n-bit sum of two n-bit numbers. The carry-select adder is simple but rather fast, having a gate level depth of .
Adding two n-bit numbers is performed by breaking the input into blocks.
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Adding two n-bit numbers is performed by breaking the input into blocks.
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A Lookahead Carry Unit (LCU) is a logical unit in digital circuit design used to decrease calculation time in adder units and used in conjunction with carry look-ahead adders (CLAs).
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A carry-save adder is a type of digital adder, used in computer microarchitecture to compute the sum of three or more n-bit numbers in binary. It differs from other digital adders in that it outputs two numbers of the same dimensions as the inputs, one which is a sequence
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Dadda multiplier is a hardware multiplier design invented by computer scientist Luigi Dadda in 1965. It is similar to the Wallace multiplier, but it is slightly faster (for all operand sizes) and requires fewer gates (for all but the smallest operand sizes).
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Wallace tree is an efficient hardware implementation of a digital circuit that multiplies two integers.
The Wallace tree has three steps:
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The Wallace tree has three steps:
- Multiply (that is - AND) each bit of one of the arguments, by each bit of the other, yielding results.
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adding machine is a type of calculator, usually specialized for bookkeeping calculations. In the United States, very old adding machines were usually built to read in dollars and cents.
It was invented by the French mathematician Blaise Pascal in 1642.
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It was invented by the French mathematician Blaise Pascal in 1642.
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In digital circuits, an adder-subtracter is a circuit that is capable of adding or subtracting numbers (in particular, binary). Below is a circuit that does adding or subtracting'' depending on a control signal.
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In electronics, a subtractor can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each
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A Binary multiplier is an electronic hardware device used in digital electronics or a computer or other electronic device to perform rapid multiplication of two numbers in binary representation.
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