Implementation of XOR Gate from NOR Gate

An Overview

To carry out numerous logical processes, logic gates are crucial elements in the design of digital circuits. One such gate that generates a high output when the inputs are different from one another is the XOR (Exclusive OR) gate. Using NOR gates to create an XOR gate is an intriguing strategy that will be discussed in this article. Understanding this implementation helps us better grasp the flexibility and relationships between logic gates.

XOR Gate

Another fundamental logic gate that is frequently utilised in digital circuits is the XOR (Exclusive OR) gate. When there are an odd number of HIGH inputs, it generates a HIGH output. To put it another way, the output is only HIGH when the inputs are different from one another.

A two-input XOR gate's truth table is as follows −

Input A	Input B	Output
0	0	0
0	1	1
1	0	1
1	1	0

The Boolean expression for a two-input XOR gate is:

$\mathrm{Output\:=\:A\:\oplus\:B}$

where '$\mathrm{\oplus}$' represents the XOR operation.

NOR Gate

A NOR gate is a type of logic gate that executes a logical disjunction (OR) and then a logical negation (NOT). It generates a single output signal from two or more input signals. Only when all of a NOR gate's inputs are LOW (0) will the output be HIGH (1), and for all other input configurations, the output will be LOW (0).

A two-input NOR gate's truth table is as follows −

Input A	Input B	Output
0	0	1
0	1	0
1	0	0
1	1	0

The Boolean expression for a two-input NOR gate is −

Output = ~ (A + B)

where '~' represents logical negation (NOT) and '+' represents logical disjunction (OR).

Numerous more logic gates and intricate digital circuits can be implemented using NOR gates, which are frequently used in digital circuits. Since they are regarded as universal gates, any other logic gate or circuit can be built only out of NOR gates.

Implementation of XOR Gate from NOR Gate

It takes at least five NOR gates to create the circuit diagram for an XOR gate using only NOR gates. More than five NOR gates can also contain an XOR gate. The schematic for an XOR gate employing five NOR gates is shown in the accompanying figure.

$\mathrm{Y\:=\:(A\:\underline{B}\:+\:\underline{A}\:B)}$

Here is how to obtain the output of XOR gate from the above circuit −

The leftmost NOR gate has inputs A and B and its output is $\mathrm{\underline{A+B}}$

Inputs for the upper NOR gate are A and $\mathrm{\underline{A+B}}$ and the output is $\mathrm{\underline{A\:+\:\underline{A+B}}}$

Again, the inputs for the lower NOR gate are B and $\mathrm{\underline{A+B}}$ and its output is $\mathrm{\underline{B\:+\:\underline{A+B}}}$

The inputs for the 4th NOR gate are the outputs of the upper and lower NOR gates i.e. $\mathrm{\underline{A\:+\:\underline{A+B}}}$ and $\mathrm{\underline{B\:+\:\underline{A+B}}}$

The output of the 4th NOR gate = $\mathrm{\underline{\underline{A\:+\:\underline{A+B}}\:+\:\underline{B\:+\:\underline{A+B}}}}$

$\mathrm{=\:\underline{\underline{A.}\:\underline{(\underline{A\:+\:B})}\:+\:\underline{B.}\:\underline{(\underline{A\:+\:B})}}}$

$\mathrm{\underline{\underline{A.}\:(A\:+\:B)\:+\:\underline{B.}\:(A\:+\:B)}}$

$\mathrm{\underline{\underline{A.}\:A\:+\:\underline{A.}\:B\:+\:\underline{B.}\:A\:+\:\underline{B.}\:B}}$

$\mathrm{\underline{\underline{A.}\:B\:+\:\underline{B.}\:A}}$

Now, this is the input for the last or 5th NOR gate. This NOR gate gives the output as the inversion of its input. Hence the final output of the above circuit is, $\mathrm{\underline{\underline{B.\underline{A}\:+\:\underline{B.}\:A}}}$ or, $\mathrm{Y\:=\:(A.\underline{B}\:+\:\underline{A.}\:B)}$

This is the output of the XOR gate. Hence the above NOR gate-based circuit is the circuit of XOR gate.

Thus the output of the above circuit is the same as the output of an XOR gate. Hence the above circuit represents the circuit diagram of Exclusive OR gate using NAND gates.