3 Bit Adder Circuit

[Práxedes Jamal]

Anadder, or summer,[1] is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used in the arithmetic logic units (ALUs). They are also used in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators and similar operations.

A full adder can also be constructed from two half adders by connecting A \displaystyle A and B \displaystyle B to the input of one half adder, then taking its sum-output S \displaystyle S as one of the inputs to the second half adder and C i n \displaystyle C_in as its other input, and finally the carry outputs from the two half-adders are connected to an OR gate. The sum-output from the second half adder is the final sum output ( S \displaystyle S ) of the full adder and the output from the OR gate is the final carry output ( C o u t \displaystyle C_out ). The critical path of a full adder runs through both XOR gates and ends at the sum bit S \displaystyle S . Assumed that an XOR gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to:

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. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 3 (from input to carry in first adder) + 31 2 (for carry propagation in latter adders) = 65 gate delays.[6]The general equation for the worst-case delay for a n-bit carry-ripple adder, accounting for both the sum and carry bits, is:

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 \displaystyle P and G \displaystyle 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. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of lookahead carry units.

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.

Such compressors can be used to speed up the summation of three or more addends. If the number of addends is exactly three, the layout is known as the carry-save adder. If the number of addends is four or more, more than one layer of compressors is necessary, and there are various possible designs for the circuit: the most common are Dadda and Wallace trees. This kind of circuit is most notably used in multiplier circuits, which is why these circuits are also known as Dadda and Wallace multipliers.

Using only the Toffoli and CNOT quantum logic gates, it is possible to produce quantum full- and half-adders.[11][12][13] The same circuits can also be implemented in classical reversible computation, as both CNOT and Toffoli are also classical logic gates.

Just as in Binary adders, combining two input currents effectively adds those currents together. Within the constraints of the hardware, non-binary signals (i.e. with a base higher than 2) can be added together to calculate a sum. Also known as a "summing amplifier",[16] this technique can be used to reduce the number of transistors in an addition circuit.

i've designed a full adder circuit in cadence virtuoso. i wanted to know how to calculate over all delay. i know how to calculate delay between two signals using calculator tool. But dont know to calculate over all delay. Please help.. .

Please help explain what you mean by "over all delay". I don't understand your question, since you already know how to calculate delay between two signals. What's the difference in what you're asking?

This of course depends on how you are varying the inputs. If the input stimuli is under your control, then you know when it changes and so you probably don't even need to use the delay() function. You could use the cross function to look for a crossing event in a particular window of the input or output. Then the "overall delay" would be:

I have tried to remove, change or add some pull down resistors on the gate outputs, but the above circuit is the best attempt I have right now. I have also changed out all components and the breadboard I use, just to make sure it isn't some component that is dead. I already have double/triple checked all resistors and connections, so that I don't just have a misplaced wire or wrong resistor.

The following schematic shows the basic one-BJT form for the NOR gate at the top. It then follows up by showing what a full adder would look like if it were based entirely on these gates. You'd need a total of 9 NPN BJTs here. Half of what you are using now. And it should actually work okay and simulate fine. (You may get glitching. But if you observe reasonable setup and hold times, it should work okay.)

Sometimes people call normal adder circuits "modular adder circuits", distinguishing them from circuits where the number of output bits is larger than the number of input bits so that there is never an overflow. I don't really like that naming, since it's not reflective of what's cheap vs what's expensive.

Note that the paper "Factoring with n+2 clean qubits and n-1 dirty qubits" has diagrams and working code implementing modular multiplications. The constructions are a bit convoluted because they go out of their way to use minimal workspace, but they are also built entirely out of classical gets (NOT, CNOT, TOFFOLI); no Hadamards or Fourier transforms. That makes them easier to check.

The two control qubits (the top qubits) have no role in implementing modular addition -- they are required for implementing modular multiplication and exponentiation in the general Shor's circuit (in Shor's circuit one control belongs to the register that carries the number we wish to multiply and the other belongs to the register that carries the power in exponentiation).

In Classiq's library git-repo you can find full implementations of Modular addition, multiplication, exponentiation and Shor's circuits similar to Beauregard's paper (excluding the fact that the exponentiation and Shor are implemented on a full $2n$ register and do not use the 'one controlling qubit trick' as in the paper) Shor - Classiq Library

The inputs of the first half adder are two single binary digits A and B. The output of the first half adder sum S is fed to the input of the second half adder terminal 1 on K. The sum output of the second half adder is obtained across X.

The main reason to call these binary adders like Half Adders is, that there is no range to include the carry bit using an earlier bit. So, this is a main limitation of HAs once used like binary adder particularly in real-time situations which involve adding several bits. So this limitation can be overcome by using the full adders.

This generates SUM and C-OUT is true only when either two of three inputs are HIGH, then the C-OUT will be HIGH. So, we can implement a full adder circuit with the help of two half adder circuits. Initially, the half adder will be used to add A and B to produce a partial Sum and a second-half adder logic can be used to add C-IN to the Sum produced by the first half adder to get the final S output.

Even the sum and carry outputs for half adder can also be obtained with the method of Karnaugh map (K-map). The half adder and full adder boolean expression can be obtained through K-map. So, the K-map for these adders is discussed below.

In the concept of ripple carry adder circuits, the bits that are necessary for addition are immediately available. Whereas every adder section needs to hold its time for the arrival of carry from the previous adder block. Because of this, it takes more time to produce SUM and CARRY as each section in the circuit waits for the arrival of input.

To overcome the delay in ripple carries adder, a carry-lookahead adder was introduced. Here, by using complicated hardware, the propagation delay can be minimized. The below diagram shows a carry-lookahead adder using full adders.

The implementation of a FA can be done through two half adders which are connected logically. The block diagram of this can be shown below which tells the connection of a FA using two half adders.

The sum and carry equations from previous calculations are

Depending on the above two sums & carry equations, the FA circuit can be implemented with the help of two HAs & an OR gate. The circuit diagram of a full adder with two half adders is illustrated above.

Thus, whenever the addition of two binary numbers is done then the digits are added at first the least bits. This process can be performed through a half adder because the simplest n/w that allows adding two 1-bit numbers. The inputs of this adder are the binary digits whereas the outputs are the sum (S) & the carry (C).

Whenever the number of digits is included, then the HA network is utilized simply to connect the least digits, as the HA cannot add the carry number from the earlier class. A full adder can be defined as the base of all digital arithmetic devices. This is used for adding three 1-digit numbers. This adder includes three inputs like A, B, and Cin whereas the outputs are Sum and Cout.

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