Counter; MOD-6 synchronous counter

Counter

A counter is a sequential circuit made of flip-flops that goes through a fixed sequence of states when clock pulses are applied. The number of distinct states is its modulus (MOD-N). Counters are used to count pulses, divide frequency and generate timing signals. They are synchronous (common clock) or asynchronous (ripple).

Design of MOD-6 synchronous counter (T flip-flops)

Step 1: Sequence 000 → 001 → 010 → 011 → 100 → 101 → 000. $2^3 = 8 \ge 6$, so 3 T flip-flops with a common clock. States 110 and 111 are don't cares. For a T flip-flop, $T = Q \oplus Q^+$.

Step 2: Excitation table

Step 3: K-maps

$T_0 = 1$ in every row.

Step 4: Equations

$$\begin{aligned} &T_2 = Q_2Q_0 + Q_1Q_0 \\ &T_1 = \overline{Q_2}Q_0 \\ &T_0 = 1 \end{aligned}$$

Step 5: Circuit diagram

   +-----+       +-----+       +-----+
T0>|T   Q|-Q0 T1>|T   Q|-Q1 T2>|T   Q|-Q2
   | FF0 |       | FF1 |       | FF2 |
   |     |       |     |       |     |
   +--o--+       +--o--+       +--o--+
      |             |             |
CLK---+-------------+-------------+
T0 = 1
T1 = Q2'.Q0
T2 = Q2.Q0 + Q1.Q0

Unused states: 110 → 111 → 010, so the counter is self-starting.

Step 6: Timing diagram

CLK  __|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾
Q2   ________________|‾‾‾‾‾‾‾|_______
Q1   ________|‾‾‾‾‾‾‾|_______________
Q0   ____|‾‾‾|___|‾‾‾|___|‾‾‾|___|‾‾‾
Cnt  0    1   2   3   4   5   0   1

All flip-flops change together on the falling clock edge. $Q_2$ is high for counts 4 and 5, so it divides the clock by 6.