Understanding the 3-bit Johnson Counter

A Johnson counter, also known as a creeping counter, is a type of digital counter where the output of the last flip-flop is connected to the input of the first flip-flop, but in an inverted form. Unlike a standard ripple counter or a binary counter that cycles through all possible states, a Johnson counter produces a sequence of states that is half the size of the total possible states for a given number of bits.

In this case, we are considering a 3-bit Johnson counter. A 3-bit counter has a total of $2^3 = 8$ possible states, which range from 000 to 111 in binary.

Generating the Johnson Counter States Sequence

Let the three flip-flops be represented by $Q_2$, $Q_1$, and $Q_0$. The input to the first flip-flop ($Q_2$) is the inverted output of the last flip-flop ($ \overline{Q_0} $).

Let's trace the sequence, assuming an initial state of 000:

• State 1: Initial state = 000
• State 2: Input to $Q_2$ is $ \overline{Q_0} = \overline{0} = 1 $. The previous $Q_2$ shifts to $Q_1$, and $Q_1$ shifts to $Q_0$. New state = 100.
• State 3: Input to $Q_2$ is $ \overline{Q_0} = \overline{0} = 1 $. New state = 110.
• State 4: Input to $Q_2$ is $ \overline{Q_0} = \overline{0} = 1 $. New state = 111.
• State 5: Input to $Q_2$ is $ \overline{Q_0} = \overline{1} = 0 $. New state = 011.
• State 6: Input to $Q_2$ is $ \overline{Q_0} = \overline{1} = 0 $. New state = 001.
• State 7: Input to $Q_2$ is $ \overline{Q_0} = \overline{1} = 0 $. New state = 000. The counter returns to the initial state, completing the cycle.

The sequence of states generated by a 3-bit Johnson counter is: 000, 100, 110, 111, 011, 001.

Calculating Unused States

A 3-bit counter can theoretically represent $2^3 = 8$ distinct states:

• 000
• 001
• 010
• 011
• 100
• 101
• 110
• 111

However, the Johnson counter sequence only utilizes 6 of these states:

• 000
• 100
• 110
• 111
• 011
• 001

To find the number of unused states, we subtract the number of generated states from the total possible states:

Number of unused states = Total possible states - Number of generated states

Number of unused states = $ 8 - 6 = 2 $.

The specific states that are not reached in this sequence are 010 and 101.

Conclusion

Therefore, the number of unused states in a 3-bit Johnson counter is 2.