GATE2025_CS1_Q60

Consider the given sequential circuit designed using D-Flip-flops. The circuit is initialized with some value (initial state). The number of distinct states the circuit will go through before returning back to the initial state is ______. (Answer in integer)

✅ Final Answer:

The correct answer is 8.

🔹 Short Answer:

The circuit shown is a 4-bit Johnson Counter (also known as a twisted-ring counter). In this configuration, the output of each flip-flop is connected to the input of the next, but the inverted output of the last flip-flop (Q₃) is fed back to the input of the first (D₀). The number of distinct states (cycle length) in an n-bit Johnson counter is 2n. For this 4-bit counter, the cycle length is 2 × 4 = 8.

🔸 Long Answer:

To determine the number of distinct states, we must first derive the next-state equations from the circuit diagram and then trace the sequence of states.

Step 1: Analyzing the Circuit Diagram

The circuit consists of four D-flip-flops, labeled from left to right as FF₀, FF₁, FF₂, and FF₃, with outputs Q₀, Q₁, Q₂, and Q₃ respectively. Let's trace the connections:

• The input to FF₁ is D₁ = Q₀.
• The input to FF₂ is D₂ = Q₁.
• The input to FF₃ is D₃ = Q₂.

This forms a standard 4-bit shift register. The key is the feedback loop to FF₀. The inverted output of the last flip-flop, Q₃_bar, is connected to the input of the first, D₀. The small bubble on the output of Q₃ in such diagrams, or the context of the problem, typically indicates inversion.

This specific configuration is known as a Johnson Counter or twisted-ring counter. The next state equations are:

Next State Equations
Q₀(t+1) = Q₃(t)_bar
Q₁(t+1) = Q₀(t)
Q₂(t+1) = Q₁(t)
Q₃(t+1) = Q₂(t)

Step 2: Trace the State Sequence

Let's assume the circuit starts in the state 0000 and trace the sequence of states (Q₃Q₂Q₁Q₀).

Clock Pulse	Current State (Q₃Q₂Q₁Q₀)	Q₃_bar	Next State (Q₃⁺Q₂⁺Q₁⁺Q₀⁺)
0 (Initial)	0000	1	0001
1	0001	1	0011
2	0011	1	0111
3	0111	1	1111
4	1111	0	1110
5	1110	0	1100
6	1100	0	1000
7	1000	0	0000
8	0000	1	0001 (Cycle repeats)

Step 3: Determine the Cycle Length

As seen from the table, the circuit passes through 8 distinct states before returning to the initial state `0000`. The sequence of states is:

0000 → 0001 → 0011 → 0111 → 1111 → 1110 → 1100 → 1000 → 0000

The number of distinct states in this cycle is 8. This matches the general formula for an n-bit Johnson counter, which has a cycle length of 2n.

Bibliography:
- Mano, M. M., & Ciletti, M. D. (2017). Digital Design. Pearson. (Chapter 6: Registers and Counters).

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