Adder-subtracter

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.However, it is possible to construct a circuit that performs both addition and subtraction at the same time.

Construction

Construction of a binary adder-subtracter is quite easy.Assume you have an "n"-bit adder for $A$ and $B$, then $S\; =\; A\; +\; B$.Also assume the numbers are in two's complement.Then to perform $B\; -\; A$, two's complement theory says to invert each bit with a NOT gate then add one.This yields $S\; =\; B\; +\; overline\{A\}\; +\; 1$, which is easy to do with a slightly modified adder.

By preceding each $A$ input bit on the adder with a 2-to-1 multiplexer where:
* Input 0 ($I\_0$) is straight through ($A\_i$)
* Input 1 ($I\_1$) is negated ($overline\{A\_i\}$)that has control input $D$ and the initial carry connect is also connected to $D$ then:
* when $D=0$ the modified adder performs addition
* when $D=1$ the modified adder performs subtraction

This works because when $D=1$ the $A$ input to the adder is really $overline\{A\}$ and the carry in is $1$. Adding $B$ to $overline\{A\}$ and $1$ yields the desired subtraction of $B-A$.

Role in the arithmetic logic unit

Adders are a part of the core of an arithmetic logic unit (ALU).The control unit decides which operations an ALU should perform (based on the op code being executed) and sets the ALU operation.The $D$ input to the adder-subtracter above would be one such control line from the control unit.

The adder-subtracter above could easily be extended to include more functions.For example, a 2-to-1 multiplexer could be introduced on each $B\_i$ that would switch between zero and $B\_i$; this could be used (in conjunction with $D=1$) to yield the two's complement of $A$ since $-A\; =\; overline\{A\}\; +\; 1$.

A further step would be to change the 2-to-1 mux on $A$ to a 4-to-1 with the third input being zero, then replicating this on $B\_i$ thus yielding the following output functions:
* $0$ (with the both $A\_i$ and $B\_i$ input set to zero and $D=0$)
* $1$ (with the both $A\_i$ and $B\_i$ input set to zero and $D=1$)
* $A$ (with the $B\_i$ input set to zero)
* $B$ (with the $A\_i$ input set to zero)
* $A+1$ (with the $B\_i$ input set to zero and $D=1$)
* $B+1$ (with the $A\_i$ input set to zero and $D=1$)
* $A+B$
* $A-B$
* $B-A$
* $overline\{A\}$ (with $A\_i$ set to invert; $B\_i$ set to zero; and $D=0$)
* $-A$ (with $A\_i$ set to invert; $B\_i$ set to zero; and $D=1$)
* $overline\{B\}$ (with $B\_i$ set to invert; $A\_i$ set to zero; and $D=0$)
* $-B$ (with $B\_i$ set to invert; $A\_i$ set to zero; and $D=1$)

By adding more logic in front of the adder, a single adder can be converted into much more than just an adder — an ALU.

ee also

* Adder (electronics)
* Carry look-ahead adder
* Carry save adder
* Adding machine
* Subtractor