Understanding n-bit Johnson Counters and States

A Johnson counter is a type of digital counter. It is a modified ring counter where the inverted output of the last flip-flop is fed back as the input to the first flip-flop. This creates a sequence of states. An n-bit Johnson counter uses n flip-flops.

Total Possible States in an n-bit System

In any digital system with n binary bits (like a counter made of n flip-flops), each bit can be either 0 or 1. The total number of unique combinations or states possible with n bits is given by $2^n$. This represents all the states that the flip-flops could potentially hold.

States Used by an n-bit Johnson Counter

Unlike a standard binary counter that sequences through all $2^n$ states (or a simple ring counter that sequences through n states), an n-bit Johnson counter sequences through a specific set of states. It is also known as a 2n-bit counter because it cycles through exactly $2n$ unique states before repeating its sequence. These states follow a specific pattern where bits 'walk' through the counter, first as 0s and then as 1s.

For example, a 3-bit Johnson counter (n=3) cycles through $2 \times 3 = 6$ states:

• 000
• 100
• 110
• 111
• 011
• 001
• (Back to 000)

A 4-bit Johnson counter (n=4) cycles through $2 \times 4 = 8$ states:

• 0000
• 1000
• 1100
• 1110
• 1111
• 0111
• 0011
• 0001
• (Back to 0000)

Calculating Unused States in an n-bit Johnson Counter

The number of unused states in an n-bit Johnson counter is the difference between the total possible states an n-bit system can have and the number of states the n-bit Johnson counter actually uses.

Total possible states = $2^n$

States used by n-bit Johnson counter = $2n$

Therefore, the number of unused states is calculated as:

Unused States = Total Possible States - States Used

Unused States = $2^n - 2n$

Formula for Unused States in Johnson Counter

The formula for the number of unused states in an n-bit Johnson counter is $2^n - 2n$. This formula gives the count of states that are not part of the standard Johnson counter sequence.

Example Calculation

Let's verify this formula with an example:

• For a 3-bit Johnson counter (n=3):
• Total possible states = $2^3 = 8$
• States used = $2 \times 3 = 6$
• Unused states = $8 - 6 = 2$.
• Using the formula $2^n - 2n = 2^3 - (2 \times 3) = 8 - 6 = 2$. The formula matches the example.
• For a 4-bit Johnson counter (n=4):
• Total possible states = $2^4 = 16$
• States used = $2 \times 4 = 8$
• Unused states = $16 - 8 = 8$.
• Using the formula $2^n - 2n = 2^4 - (2 \times 4) = 16 - 8 = 8$. The formula matches the example.

Revision Table: Johnson Counter Key Properties

Additional Information on Digital Counters

Counters are fundamental building blocks in digital electronics, used for sequencing, frequency division, and timing. Besides the Johnson counter, other types of counters include:

• Ring Counter: Similar to a Johnson counter but the output of the last flip-flop is directly fed back to the first, not inverted. An n-bit ring counter sequences through only n states.
• Ripple Counter: Asynchronous counter where flip-flops are toggled by the output of the previous flip-flop.
• Synchronous Counter: All flip-flops are clocked simultaneously, leading to faster operation and avoiding timing issues found in ripple counters.

Understanding the state sequences of different counters like the Johnson counter is important for designing digital systems that require specific timing and control signals. Unused states are often avoided or handled specifically in circuit design to prevent unexpected behavior.