GATE CS 2017 Set 1 Q.33 (CS Q.33): Sequential Circuits, Counters and Registers

Consider a combination of T and D flip-flops connected as shown below. The output of the D flip-flop is connected to the input of the T flip-flop and the output of the T flip-flop is connected to the input of the D flip-flop.

Sequential circuit with a T flip-flop on the left with output Q1 and a D flip-flop on the right with output Q0. The output Q0 of the D flip-flop feeds back to the T input of the T flip-flop. The output Q1 of the T flip-flop connects directly to the D input of the D flip-flop. A common clock line drives both flip-flops.

Initially, both $Q_0$ and $Q_1$ are set to 1 (before the 1st clock cycle). The outputs

$Q_1 Q_0$ after the 3rd cycle are 11 and after the 4th cycle are 00 respectively

$Q_1 Q_0$ after the 3rd cycle are 11 and after the 4th cycle are 01 respectively

$Q_1 Q_0$ after the 3rd cycle are 00 and after the 4th cycle are 11 respectively

$Q_1 Q_0$ after the 3rd cycle are 01 and after the 4th cycle are 01 respectively

Official answer and solution

Official answer: B
From: Official GATE 2017 answer key

Solution

Derive the next-state excitation equations for the interconnected T and D flip-flops and trace the states over 4 clock cycles.

From the circuit diagram:

The input of the T flip-flop is fed by $Q_0$ : $T = Q_0$ .
The next state of the T flip-flop is: $Q_1^+ = T \oplus Q_1 = Q_0 \oplus Q_1$ .
The input of the D flip-flop is fed by $Q_1$ : $D = Q_1$ .
The next state of the D flip-flop is: $Q_0^+ = D = Q_1$ .

The initial state before the 1st clock cycle is $Q_1 Q_0 = 11$ .

Trace the circuit state clock by clock:

Cycle 1: $Q_1^+ = 1 \oplus 1 = 0$ , and $Q_0^+ = 1 \implies Q_1 Q_0 = 01$ .
Cycle 2: $Q_1^+ = 1 \oplus 0 = 1$ , and $Q_0^+ = 0 \implies Q_1 Q_0 = 10$ .
Cycle 3: $Q_1^+ = 0 \oplus 1 = 1$ , and $Q_0^+ = 1 \implies Q_1 Q_0 = 11$ .
Cycle 4: $Q_1^+ = 1 \oplus 1 = 0$ , and $Q_0^+ = 1 \implies Q_1 Q_0 = 01$ .

Thus, after the 3rd cycle $Q_1 Q_0 = 11$ , and after the 4th cycle $Q_1 Q_0 = 01$ . This corresponds to Option B.

Final answer

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

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 1
Question: 33 of 65, printed as CS Q.33
Section: Computer Science and Information Technology

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