Understanding the Concept:
A ring counter is a synchronous circular shift register where the true output ($Q$) of the final stage flip-flop is fed directly back into the synchronous input ($D$ or $J$) of the very first stage flip-flop. For an $N$-stage shift register configuration, the count sequence forms a circulating loop of binary patterns.
Step 1: State Space Comparison of Counters.
Let us analyze the capacities of various structures utilizing $N$ flip-flops:
• Total Possible Arbitrary States: $2^N$
• Standard Binary Counter (Asynchronous/Synchronous): $2^N$ unique states.
• Johnson (Twisted Ring) Counter: $2N$ unique states.
• Standard Ring Counter: Exactly $N$ unique states.
Step 2: Enumeration of States for $N=4$.
Given a 4-stage ring counter ($N = 4$), the circuit transitions linearly through a loop of exactly 4 unique states. If initialized to a standard one-hot encoded state (e.g., $1000$), the sequence behaves as follows:
\[
\text{State 1: } 1000
\]
\[
\text{State 2: } 0100
\]
\[
\text{State 3: } 0010
\]
\[
\text{State 4: } 0001
\]
On the next clock pulse, the trailing $1$ recirculates back to the input, regenerating State 1 ($1000$). Thus, the number of distinct, valid operating states equals $N = 4$.