GATE CS 1993 Q6.2 — Sequential Circuits

The sequential circuit consists of a feedback loop with a NOR gate and an XOR gate:

• The NOR gate has inputs $a$ and $Y$ (feedback), producing output $X = ext{NOR}(a, Y) = \overline{a + Y}$.
• The XOR gate has inputs $b$ and $X$, producing the feedback output $Y = b \oplus X$.

Substituting $X$ into the expression for $Y$ gives the feedback equation:

We analyze the stability of the system for different input combinations of $(a, b)$:

1. Case 1: $a = 1$
   $X = \overline{1 + Y} = 0$
   $Y = b \oplus 0 = b$
   Since $b$ is a constant input, $Y$ is stable at $b$. Thus, the system is stable.
2. Case 2: $a = 0$
   $X = \overline{0 + Y} = \bar{Y}$
   $Y = b \oplus \bar{Y}$
   • If $b = 1$: $Y = 1 \oplus \bar{Y} = Y$ (stable state).
   • If $b = 0$: $Y = 0 \oplus \bar{Y} = \bar{Y}$ (unstable/oscillating state).

To ensure a stable state, we must avoid the unstable condition $a = 0$ and $b = 0$. This restriction is represented by: