Full Adder with NAND Gates

In digital electronics, there are different types of logic circuits used to perform different kinds of arithmetic operations. One of them is adder. Adder (or Binary Adder) is a combinational logic circuit that performs the addition of two or more binary numbers and gives an output sum. There are two types of adders present namely, half adder and full-adder.

Since, adder are logic circuits, thus they are implemented using different types of digital logic gates such as OR gate, AND gate, NOT gate, NAND gates, NOR gates, etc. Here, we will discuss the Full Adder Realization using NAND Gates. But before that lets have a look into the basics of full adder.

What is a Full Adder?

A combinational logic circuit that can add two binary digits (bits) and a carry bit, and produces a sum bit and a carry bit as output is known as a full-adder. In other words, a combinational circuit which is designed to add three binary digits and produces two outputs (sum and carry) is known as a full adder. Thus, a full adder circuit adds three binary digits, where two are the inputs and one is the carry forwarded from the previous addition. The block diagram of the full adder is shown in Figure-1.

From the block diagram of the full adder, it is clear that it has three inputs namely A, B, C_in. Where, A and B are the input bits, and C_in is the carry bit from previous stage. It has two output variables namely sum (S) and carry (C_out).

Truth Table of Full Adder

The following is the truth table of the full-adder circuit

Hence, from the truth table, it is clear that the sum output of the full adder is equal to 1 when only 1 input is equal to 1 or when all the inputs are equal to 1. While the carry output has a carry of 1 if two or three inputs are equal to 1.

The output equations of the full adder can be obtained from the truth table of the full adder. These equations are as follows

Sum Output

$$\mathrm{Sum,\, S=A'B'C_{in}+A'BC'_{in}+AB'C'_{in}+ABC_{in}=A\oplus B\oplus C_{in} }$$

Carry Output

$$\mathrm{Carry,\, C_{out}=AB+AC_{in}+BC_{in}}$$

Now, let us discuss the realization of Full Adder with NAND gates.

Full Adder with NAND Gates

The full adder circuit can be realized using the NAND logic gates as shown in Figure-2.

From the logic circuit diagram of the full adder using NAND gates, we can see that the full adder requires 9 NAND gates.

Equation of the sum output for the full adder circuit with NAND gates is obtained as follows

$$\mathrm{S=\overline{\overline{\left ( A\oplus B \right )\cdot \overline{\left ( A\oplus B \right )C_{in}}}\cdot \overline{C_{in}\cdot\overline{\left ( A\oplus B \right )C_{in}}} }=A\oplus B\oplus C_{in}}$$

Where,

$$\mathrm{A\oplus B=\overline{\overline{A\cdot \overline{AB}}\cdot \overline{B\cdot \overline{AB}}}}$$

And equation of the carry output of the full adder circuit with NAND gate is given by,

$$\mathrm{C_{out}=\overline{\overline{C_{in}\left ( A\oplus B \right )}\cdot \overline{AB}}=AB+\left ( A\oplus B \right )C_{in}}$$

In this way, we may implement the full adder circuit using NAND gates only.