Sync vs async; mod-6 synchronous counter (JK)

Synchronous vs asynchronous counters

Design of mod-6 synchronous counter (JK flip-flops)

Step 1: States. 000 → 001 → 010 → 011 → 100 → 101 → 000. Since $2^3 = 8 \ge 6$, three JK flip-flops ($Q_2Q_1Q_0$) with a common clock are used. 110 and 111 are don't cares.

Step 2: Excitation table (JK: 0→0: 0X, 0→1: 1X, 1→0: X1, 1→1: X0)

Step 3: K-maps

$J_0 = K_0 = 1$ since $Q_0$ toggles every clock.

Step 4: Equations

$$\begin{aligned} &J_2 = Q_1Q_0,\quad K_2 = Q_0 \\ &J_1 = \overline{Q_2}Q_0,\quad K_1 = Q_0 \\ &J_0 = 1,\quad K_0 = 1 \end{aligned}$$

Step 5: Circuit diagram

   +-----+       +-----+       +-----+
J0>|J   Q|-Q0 J1>|J   Q|-Q1 J2>|J   Q|-Q2
   | FF0 |       | FF1 |       | FF2 |
K0>|K    |    K1>|K    |    K2>|K    |
   +--o--+       +--o--+       +--o--+
      |             |             |
CLK---+-------------+-------------+
J0 = K0 = 1
J1 = Q2'.Q0    K1 = Q0
J2 = Q1.Q0     K2 = Q0

Step 6: Unused states: 110 → 111 → 000, so the counter is self-starting.

Step 7: Timing diagram

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