Counting circuits: Binary counter BCD counter Johnson counter.

The idea

Counting circuits are sequential logic systems built from flip-flops that advance through a defined sequence of output states each time a clock pulse arrives. A binary counter counts in natural binary, so an n-bit counter has 2ⁿ states and recycles after reaching its maximum. A BCD counter counts only from 0000 to 1001 (0 to 9) and then resets, making it suitable for driving decimal displays. A Johnson counter uses a shift register with the inverted last output fed back to the first input, producing a sequence of 2n states for n flip-flops. You analyse each circuit by writing the next-state table, predicting outputs after successive clock edges, and identifying the modulus.

For example: A 3-bit binary counter cycles through 000, 001, 010, 011, 100, 101, 110, 111 before repeating, giving a modulus of 8.

What earns the marks

• States that a binary counter with n flip-flops has 2ⁿ distinct output states and a modulus of 2ⁿ.
• Describes a BCD counter as one that counts from 0000 to 1001 and then returns to 0000, giving a modulus of 10.
• Explains that a Johnson counter feeds the inverse of the last flip-flop output back to the first input, producing 2n states for n flip-flops.
• Constructs or completes a next-state table showing the output after each clock pulse for a given counting circuit.
• Identifies the modulus of a counter from its sequence or from the number of flip-flops and circuit type.
• Distinguishes between free-running binary counting and truncated sequences such as BCD or Johnson counting.

Common mistakes and how to put them right

• Assuming a BCD counter has a modulus of 16 because it uses four flip-flops.

  Put it right: The correct modulus is 10 because it is truncated at 1001.

• Confusing a Johnson counter with a ring counter.

  Put it right: A ring counter recirculates a single 1 and has n states, whereas a Johnson counter inverts the feedback and has 2n states.

• Forgetting that the count advances on the active clock edge, so the output listed for a state is the value after that edge, not before it.

Exam tips

• Draw the full state sequence before answering, then count the states to confirm the modulus.
• For BCD questions, explicitly state the reset condition, for example that 1010 is detected and forces the counter back to 0000.
• When comparing counters, tabulate the number of flip-flops, number of states and feedback arrangement side by side.