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In electronics, an '''adder''' or '''summer''' is a [[digital circuit]] that performs [[addition]] of numbers. | |||
In many [[computer]]s and other kinds of processors, adders are used not only in the [[arithmetic logic unit]](s), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations. | |||
Although adders can be constructed for many numerical representations, such as [[binary-coded decimal]] or [[excess-3]], the most common adders operate on [[binary numeral system|binary]] numbers. | |||
In cases where [[two's complement]] or [[ones' complement]] is being used to represent negative numbers, it is trivial to modify an adder into an [[adder–subtractor]]. | |||
Other [[signed number representations]] require a more complex adder. | |||
==Half adder== | |||
[[File:half Adder.svg|right|thumb|Half adder logic diagram]] The '''half adder''' adds two single binary digits ''A'' and ''B''. It has two outputs, sum (''S'') and carry (''C''). The carry signal represents an overflow into the next digit of a multi-digit addition. The value of the sum is {{nobreak|2''C'' + ''S''}}. The simplest half-adder design, pictured on the right, incorporates an [[XOR gate]] for ''S'' and an [[AND gate]] for ''C''. With the addition of an OR gate to combine their carry outputs, two half adders can be combined to make a full adder.<ref> | |||
{{cite book | |||
| title = Excel HSC Software Design and Development''' | |||
| author = Geoffrey A. Lancaster | |||
| publisher = Pascal Press | |||
| year = 2004 | |||
| isbn = 9781741251753 | |||
| page = 180 | |||
| url = http://books.google.com/books?id=PZkDpS4m0fMC&pg=PA180 | |||
}}</ref> | |||
The half-adder adds two input bits and generate carry and sum which are the two outputs of half-adder.The input variables of a Half adder are called the Augend and addend bits.The output variables are the Sum and Carry.The Truth table for the Half adder is : | |||
{| class="wikitable" style="text-align:center" | |||
|- | |||
!colspan="2"| Inputs !!colspan="2"| Outputs | |||
|- | |||
! ''A'' !! ''B'' !! ''S'' !! ''C'' | |||
|- | |||
| 0 || 0 || 0 || 0 | |||
|- | |||
| 1 || 0 || 1 || 0 | |||
|- | |||
| 0 || 1 || 1 || 0 | |||
|- | |||
| 1 || 1 || 0 || 1 | |||
|- | |||
|} | |||
==Full adder== | |||
[[File:1-bit full-adder.svg|thumb|right|Schematic symbol for a 1-bit full adder with ''C''<sub>in</sub> and ''C''<sub>out</sub> drawn on sides of block to emphasize their use in a multi-bit adder]] | |||
A '''full adder''' adds binary numbers and accounts for values carried in as well as out. A one-bit full adder adds three one-bit numbers, often written as ''A'', ''B'', and ''C''<sub>in</sub>; ''A'' and ''B'' are the operands, and ''C''<sub>in</sub> is a bit carried in from the next less significant stage.<ref name=Mano79>M. Morris Mano, ''Digital Logic and Computer Design'', Prentice-Hall 1979, ISBN 0-13-21450-3 pp.119-123</ref> The full-adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. The circuit produces a two-bit output, output carry and sum typically represented by the signals ''C''<sub>out</sub> and ''S'', where <math>\mathrm{sum} = 2 \times C_{out} + S</math>. The one-bit full adder's [[truth table]] is: | |||
[[File:Full Adder.svg|thumb|right|Full-adder logic diagram]] | |||
{| class="wikitable" style="text-align:center" | |||
|- | |||
!colspan="3"| Inputs !!colspan="2"| Outputs | |||
|- | |||
! ''A'' !! ''B'' !! ''C''<sub>in</sub> !! ''C''<sub>out</sub> !! ''S'' | |||
|- | |||
| 0 || 0 || 0 || 0 || 0 | |||
|- | |||
| 1 || 0 || 0 || 0 || 1 | |||
|- | |||
| 0 || 1 || 0 || 0 || 1 | |||
|- | |||
| 1 || 1 || 0 || 1 || 0 | |||
|- | |||
| 0 || 0 || 1 || 0 || 1 | |||
|- | |||
| 1 || 0 || 1 || 1 || 0 | |||
|- | |||
| 0 || 1 || 1 || 1 || 0 | |||
|- | |||
| 1 || 1 || 1 || 1 || 1 | |||
|} | |||
A full adder can be implemented in many different ways such as with a custom [[transistor]]-level circuit or composed of other gates. One example implementation is with <math>S = A \oplus B \oplus C_{in}</math> and <math>C_{out} = (A \cdot B) + (C_{in} \cdot (A + B))</math>. | |||
In this implementation, the final [[OR gate]] before the carry-out output may be replaced by an [[XOR gate]] without altering the resulting logic. 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. In this light, C<sub>out</sub> can be implemented as <math>C_{out} = (A \cdot B) + (C_{in} \cdot (A \oplus B))</math>. | |||
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 ''C<sub>i</sub>'' to the other input and OR the two carry outputs. Equivalently, ''S'' could be made the three-bit XOR of ''A'', ''B'', and ''C<sub>i</sub>'', and ''C<sub>out</sub>'' could be made the three-bit [[majority function]] of ''A'', ''B'', and ''C<sub>in</sub>''. | |||
==More complex adders== | |||
===Ripple-carry adder=== | |||
[[File:4-bit ripple carry adder.svg|thumb|right|4-bit adder with logic gates shown]] | |||
It is possible to create a logical circuit using multiple full adders to add ''N''-bit numbers. Each full adder inputs a ''C<sub>in</sub>'', which is the ''C<sub>out</sub>'' of the previous adder. This kind of adder is called 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. | |||
Each full adder requires three levels of logic. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 2 (from input to carry in first adder) + 31 * 2 (for carry propagation in later adders) = 64 gate delays. A design with alternating carry polarities and optimized [[AND-OR-Invert]] gates can be about twice as fast.<ref>{{cite conference | |||
| conference = 20th IEEE Symposium on Computer Arithmetic | |||
| title = Fast Ripple-Carry Adders in Standard-Cell CMOS VLSI | |||
| author = Burgess, N. | |||
| pages = 103–111 | |||
| year = 2011 | |||
| url = http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F5991607%2F5992089%2F05992115.pdf%3Farnumber%3D5992115&authDecision=-203 | |||
}}</ref> | |||
===Carry-lookahead adders=== | |||
{{main|Carry-lookahead adder}} | |||
[[File:4-bit carry lookahead adder.svg|thumb|right|4-bit adder with carry lookahead]] | |||
To reduce the computation time, engineers devised faster ways to add two binary numbers by using [[carry-lookahead adder]]s. | |||
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 advanced carry-lookahead architectures are the [[Manchester carry chain]], [[Brent–Kung adder]], and the [[Kogge–Stone 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-skip adder|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. | |||
Other adder designs include the [[carry-select adder]], [[conditional sum adder]], [[carry-skip adder]], and [[carry-complete adder]]. | |||
===Lookahead carry unit=== | |||
[[File:64-bit lookahead carry unit.svg|thumb|right|A 64-bit adder]] | |||
By combining multiple carry lookahead 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 four 16-bit CLAs with two levels of LCUs. | |||
===Carry-save Adders=== | |||
{{main|Carry-save 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: a sum and a carry. 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. | |||
==3:2 compressors== | |||
We can view a full adder as a ''3:2 lossy compressor'': it sums three one-bit inputs, and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. Thus, for example, a binary input of ''101'' results in an output of ''1+0+1=10'' (decimal number '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 lossy compressor'', compressing four possible inputs into three possible outputs. | |||
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: the most common are [[Dadda tree|Dadda]] and [[Wallace tree]]s. This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers. | |||
==References== | |||
<references/> | |||
==External links== | |||
* [http://www.aoki.ecei.tohoku.ac.jp/arith/mg/algorithm.html Hardware algorithms for arithmetic modules], includes description of several adder layouts with figures. | |||
* [http://dev.code.ultimater.net/electronics/8-bit-full-adder-and-subtractor/ 8-bit Full Adder and Subtractor], a demonstration of an interactive Full Adder built in JavaScript solely for learning purposes. | |||
* [http://teahlab.com/Full_Adder/ Interactive Full Adder Simulation], Interactive Full Adder circuit constructed with Teahlab's online circuit simulator. | |||
* [http://teahlab.com/Half_Adder/ Interactive Half Adder Simulation], Half Adder circuit built with Teahlab's circuit simulator. | |||
* [http://www.edaplayground.com/s/example/368 4-bit Full Adder Simulation] built in Verilog, and the accompanying [http://www.youtube.com/watch?v=bL3ihMA8_Gs&hd=1 Ripple Carry Full Adder Video Tutorial] | |||
{{DEFAULTSORT:Adder (Electronics)}} | |||
[[Category:Computer arithmetic]] | |||
[[Category:Adders| ]] | |||
[[Category:Binary logic]] | |||
Revision as of 02:53, 1 January 2014
In electronics, an adder or summer is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used not only in the arithmetic logic unit(s), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations.
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 or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require a more complex adder.
Half adder
The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C). The carry signal represents an overflow into the next digit of a multi-digit addition. The value of the sum is Analyst Programmer Alfonzo Crosser from Newcastle, usually spends time with interests which include frisbee golf - frolf, property developers properties for sale in singapore singapore and collecting music albums. Ended up in recent past visiting Puerto-Princesa Subterranean River National Park.. The simplest half-adder design, pictured on the right, incorporates an XOR gate for S and an AND gate for C. With the addition of an OR gate to combine their carry outputs, two half adders can be combined to make a full adder.[1]
The half-adder adds two input bits and generate carry and sum which are the two outputs of half-adder.The input variables of a Half adder are called the Augend and addend bits.The output variables are the Sum and Carry.The Truth table for the Half adder is :
| Inputs | Outputs | ||
|---|---|---|---|
| A | B | S | C |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Full adder
A full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the next less significant stage.[2] The full-adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. The circuit produces a two-bit output, output carry and sum typically represented by the signals Cout and S, where . The one-bit full adder's truth table is:
| Inputs | Outputs | |||
|---|---|---|---|---|
| A | B | Cin | Cout | S |
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 |
A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. One example implementation is with and .
In this implementation, the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. 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. In this light, Cout can be implemented as .
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 Cout could be made the three-bit majority function of A, B, and Cin.
More complex adders
Ripple-carry adder
It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is called 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. Each full adder requires three levels of logic. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 2 (from input to carry in first adder) + 31 * 2 (for carry propagation in later adders) = 64 gate delays. A design with alternating carry polarities and optimized AND-OR-Invert gates can be about twice as fast.[3]
Carry-lookahead adders
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To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders. 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 advanced carry-lookahead architectures are the Manchester carry chain, Brent–Kung adder, and the Kogge–Stone 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-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.
Other adder designs include the carry-select adder, conditional sum adder, carry-skip adder, and carry-complete adder.
Lookahead carry unit
By combining multiple carry lookahead 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 four 16-bit CLAs with two levels of LCUs.
Carry-save Adders
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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: a sum and a carry. 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.
3:2 compressors
We can view a full adder as a 3:2 lossy compressor: it sums three one-bit inputs, and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. Thus, for example, a binary input of 101 results in an output of 1+0+1=10 (decimal number '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 lossy compressor, compressing four possible inputs into three possible outputs.
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: 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.
References
- ↑
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ M. Morris Mano, Digital Logic and Computer Design, Prentice-Hall 1979, ISBN 0-13-21450-3 pp.119-123
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External links
- Hardware algorithms for arithmetic modules, includes description of several adder layouts with figures.
- 8-bit Full Adder and Subtractor, a demonstration of an interactive Full Adder built in JavaScript solely for learning purposes.
- Interactive Full Adder Simulation, Interactive Full Adder circuit constructed with Teahlab's online circuit simulator.
- Interactive Half Adder Simulation, Half Adder circuit built with Teahlab's circuit simulator.
- 4-bit Full Adder Simulation built in Verilog, and the accompanying Ripple Carry Full Adder Video Tutorial