GATE CS 2021 Set 1 Q28 — Sequential Circuits

From the given circuit diagram, we can derive the input equations for the three $T$ flip-flops as follows:

• $T_P = Q + R$
• $T_Q = \overline{P}$
• $T_R = \overline{Q}$

The next state of a $T$ flip-flop is given by the characteristic equation: $Q_{next} = Q \oplus T$. Let us trace the states starting from the initial state $PQR = 000$:

1. Initial State $PQR = 000$:
   • $T_P = 0 + 0 = 0 \implies P_{next} = 0 \oplus 0 = 0$
   • $T_Q = \overline{0} = 1 \implies Q_{next} = 0 \oplus 1 = 1$
   • $T_R = \overline{0} = 1 \implies R_{next} = 0 \oplus 1 = 1$
   • Next State 1: $011$
2. Current State $PQR = 011$:
   • $T_P = 1 + 1 = 1 \implies P_{next} = 0 \oplus 1 = 1$
   • $T_Q = \overline{0} = 1 \implies Q_{next} = 1 \oplus 1 = 0$
   • $T_R = \overline{1} = 0 \implies R_{next} = 1 \oplus 0 = 1$
   • Next State 2: $101$
3. Current State $PQR = 101$:
   • $T_P = 0 + 1 = 1 \implies P_{next} = 1 \oplus 1 = 0$
   • $T_Q = \overline{1} = 0 \implies Q_{next} = 0 \oplus 0 = 0$
   • $T_R = \overline{0} = 1 \implies R_{next} = 1 \oplus 1 = 0$
   • Next State 3: $000$

Thus, the next three states of the counter are $011, 101, 000$.