Animated Solution for Physics - Semiconductors: Identify the logic operation carried out by the given circuit.

From NAND to NOR

Logic gates are the fundamental building blocks of the digital world. Sometimes, they are combined in ways that look complex at first glance but simplify beautifully when you apply the rules of boolean algebra. Let's break down this interesting circuit step-by-step.

When you look at the given circuit, the first thing you should notice is the two gates at the input stage. These are NAND gates, indicated by the D-shape with a small bubble at the output. However, there's a special configuration here: both inputs of each NAND gate are tied together to a single signal ($A$ for the top gate, $B$ for the bottom gate).

What happens when you tie the inputs of a NAND gate together? The boolean expression for a NAND gate is $\overline{A\cdot B}$. If both inputs are the same, say $A$, the expression becomes $\overline{A\cdot A}$. Since $A\cdot A=A$ in boolean algebra, the output simplifies to just $\overline{A}$.

Therefore, a NAND gate with tied inputs acts perfectly as a NOT gate (an inverter).

So, the intermediate signals $X$ and $Y$ can be written as:

Now, let's move to the second stage of the circuit. The inverted signals $X$ and $Y$ are fed into a new gate. This gate has a straight vertical line at the input side and a curved output side, with no bubble. This is the standard symbol for an AND gate.

The boolean expression for the output $Z$ of an AND gate is simply the logical product of its inputs:

Substituting our expressions for $X$ and $Y$ into this equation, we get the raw boolean expression for the entire circuit:

To identify which single standard logic gate this circuit represents, we need to simplify the expression $\overline{A}\cdot \overline{B}$. This is where the magic of boolean algebra comes in, specifically De Morgan's Laws.

De Morgan's Law states that the complement of a logical OR is equal to the logical AND of the complements. Mathematically, it is written as:

By applying this law in reverse to our equation, we can transform it:

The expression $\overline{A+B}$ is the exact boolean definition of a NOR gate (NOT-OR).

Therefore, this combination of two tied NAND gates and one AND gate perfectly replicates the logic operation of a single NOR gate. This is a brilliant demonstration of how universal gates can be used to synthesize other logic functions!