Digital Systems performs a variety of operations. Among various information processing tasks are the arithmetic operations which includes binary addition, subtraction, multiplication and division. The most common and basic arithmetic operation is addition of two binary digits. The simplest digital circuit which performs the binary addition operation on two binary digits is called Half Adder. The addition of three binary digits is performed by Full Adder.

**What is Full Adder:**

A Full Adder is the digital Circuit which implements addition operation on three binary digits. Two of the three binary digits are significant digits A and B and one is the carry input (C-In) bit carried from the previous-less significant stage. Thus the Full Adder operates on these three binary digits to generate two binary digits at its output referred to as Sum (SUM) and Carry-Out (C-out). The truth table of the Full Adder is as shown in the following figure:

** Full Adder Truth Table:**

The output variable Sum (S) and Carry (C-Out) are obtained by the arithmetic sum of inputs A, B and C-In. The binary variables A and B represent the significant inputs of the Full adder whereas the binary variable C-in represents the carry bit carried from the lower position stage. The Sum (S) of the full adder will be 1 if only one of the three inputs are 1 or all are one otherwise the Sum (S) variable will be 0; as the sum of two 1s in the binary number system is represented by two binary digits with 0 on the lower position and 1 carry out to the higher significant position. Thus variable Carry-Out represents that output of the Full Adder which is carried out on the addition of the two or three binary digits. Thus the Carry-Out (C-Out) bit is 1 when any two of the inputs are 1 or all of the inputs of the full adder are 1.

**Introduction to Full Adder:**

The full adder is generally is used as a component in a cascade of adders where the circuit performs the arithmetic sum of eight, sixteen or thirty two bit binary numbers.

**Full Adder Boolean Expression:**

Boolean Expression of the digital combinational circuit represents the input and output relationship of the circuit. Boolean Expression of the digital circuit can be used to assess the number and type of basic gates used to design the circuit. The Boolean Expression also represents the topology in which the digital gates are combined to create the final output. The Boolean Expression of the Full Adder along with its gate level realization is as shown in the following figure:

The Boolean Expression for the full adder circuit represented in the above image is written in Sum of Product form. Note that both output bits S and C are written in Sum of Product form. It is important to note in the above circuit for the full Adder that the inputs A, B and C are applied at the inputs of the AND Gate and the output of the AND Gates are then applied at input of the OR Gate to generate the final output. It can be seen the circuit of the full adder is actually designed using the Boolean Expressions that is product of the inputs is formed by the AND Gate and sum is produced by the OR gate thus yielding the gate level realization of the Sum of Product representation of the Boolean Expression.

**Full Adder using NAND gate:**

A Full Adder can be designed in a number of ways. As the Boolean Expression that is represented in the Sum of Product form can also be expressed as Product of Sum form. Thus the similar circuit can also be designed using the Product of Sum representation of the Boolean Expression. The Circuit that realizes the Boolean Expression written in Product of Sum form is similar in functionality to the circuit that realizes the same Boolean Expression written in Sum of Product form. Thus we can have two different circuits with identical input and output relationship. Similarly the Boolean Expressions can also be exploited and manipulated in a number of ways in order to design the circuit that represents the similar functionality. For example the De Morganâ€™s theorem can be used in order to derive multiple solutions to the same problem. The NAND and NOR Gates are classified as Universal Gates that these gates can be used implement any possible Boolean Expression. The Full Adder can also be designed using only NAND gate or NOR Gate. Due to this universality of the NAND Gates one does not need any other gate thus eliminating the use of multiple ICs. The Full Adder circuit using the NAND Gates and the Boolean Expression are as shown in the following figure:

**Full Adder Applications:**

Full Adders finds applications in a lot of circuits which are comparatively complex and carry out complex operations. Some of the common applications of the Full Adder are listed as follows:

- Full Adders are used in the ALU (Arithmetic Logic Unit) of the microprocessor.
- Full Adders are used in Ripple Carry Adders where it is employed to add n-bits at a time.
- The Full Adders are also used to calculate the addresses of memory locations inside the processor.
- In some parts of the processor the full adders are also used to calculate the table indices and increment and decrement of the operators.

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