GATE CS 2014 Set 3 Q45 — Combinational Circuits

By analyzing the connections in the given synchronous sequential circuit, we find the inputs to the JK flip-flops are:

• For $FF_2$: $J_2 = \overline{Q_0}$ and $K_2 = Q_0$
• For $FF_1$: $J_1 = Q_2$ and $K_1 = \overline{Q_2}$
• For $FF_0$: $J_0 = Q_1$ and $K_0 = \overline{Q_1}$

Using the characteristic equation of a JK flip-flop, $Q^+ = J\overline{Q} + \overline{K}Q$, we obtain the next-state equations:

This configuration represents a 3-bit Johnson counter. Starting with the initial state $Q_2Q_1Q_0 = 000$, we trace the state transitions for the next 3 clock cycles:

1. Clock Cycle 1: $$Q_2^+ = \overline{0} = 1, \quad Q_1^+ = 0, \quad Q_0^+ = 0 \implies Q_2Q_1Q_0 = 100$$
2. Clock Cycle 2: $$Q_2^+ = \overline{0} = 1, \quad Q_1^+ = 1, \quad Q_0^+ = 0 \implies Q_2Q_1Q_0 = 110$$
3. Clock Cycle 3: $$Q_2^+ = \overline{0} = 1, \quad Q_1^+ = 1, \quad Q_0^+ = 1 \implies Q_2Q_1Q_0 = 111$$

Thus, the state sequence for the next 3 clock cycles is $100, 110, 111$.