GATE CS 2011 Q.51 (CS Q.51): Sequential Circuits, Counters and Registers

Common data questions

Common Data for Questions 50 and 51:

Consider the following circuit involving three D-type flip-flops used in a certain type of counter configuration.

Synchronous counter circuit with three D flip-flops P, Q, R, NOR gate generating D of Q from P and R, and AND gate generating D of R from Q and R-bar

If all the flip-flops were reset to 0 at power on, what is the total number of distinct outputs (states) represented by $PQR$ generated by the counter?

Official answer and solution

Official answer: B
From: Official GATE 2011 answer key

Solution

Trace the state transition sequence starting from reset state 000 until a state repeats.

Next-state equations: $P^+ = R$ , $Q^+ = \overline{P + R}$ , $R^+ = Q \cdot \overline{R}$ .

Step 0 (initial reset): $PQR = 000$

$P^+ = 0$
$Q^+ = \overline{0 + 0} = 1$
$R^+ = 0 \cdot 1 = 0$
Next state is $010$ .

Step 1: $PQR = 010$

$P^+ = 0$
$Q^+ = \overline{0 + 0} = 1$
$R^+ = 1 \cdot 1 = 1$
Next state is $011$ .

Step 2: $PQR = 011$

$P^+ = 1$
$Q^+ = \overline{0 + 1} = 0$
$R^+ = 1 \cdot 0 = 0$
Next state is $100$ .

Step 3: $PQR = 100$

$P^+ = 0$
$Q^+ = \overline{1 + 0} = 0$
$R^+ = 0 \cdot 1 = 0$
Next state is $000$ (cycle repeats).

The sequence of states visited is $000 \to 010 \to 011 \to 100 \to 000$ . There are exactly 4 distinct states in the cycle.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2011 Computer Science and Information Technology (CS)
Question: 51 of 65, printed as CS Q.51
Section: Computer Science and Information Technology

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