Sync vs async; MOD-10 synchronous counter

Definitions

• Synchronous counter: a counter in which all flip-flops are driven by the same clock pulse, so all outputs change at the same time. Logic gates decide which flip-flops toggle.
• Asynchronous (ripple) counter: a counter in which only the first flip-flop gets the external clock; each later flip-flop is clocked by the output of the previous one, so the outputs change one after another.

Design of MOD-10 synchronous counter (JK flip-flops)

Step 1: 10 states (0000–1001) need 4 JK flip-flops ($Q_3Q_2Q_1Q_0$). States 1010–1111 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 (only the non-trivial ones)

$K_1 = Q_0$ and $J_2 = K_2 = Q_1Q_0$ are read the same way.

Step 4: Equations

$$\begin{aligned} &J_3 = Q_2Q_1Q_0,\quad K_3 = Q_0 \\ &J_2 = Q_1Q_0,\quad K_2 = Q_1Q_0 \\ &J_1 = \overline{Q_3}Q_0,\quad K_1 = Q_0 \\ &J_0 = 1,\quad K_0 = 1 \end{aligned}$$

Step 5: Circuit diagram

   +-----+       +-----+       +-----+       +-----+
J0>|J   Q|-Q0 J1>|J   Q|-Q1 J2>|J   Q|-Q2 J3>|J   Q|-Q3
   | FF0 |       | FF1 |       | FF2 |       | FF3 |
K0>|K    |    K1>|K    |    K2>|K    |    K3>|K    |
   +--o--+       +--o--+       +--o--+       +--o--+
      |             |             |             |
CLK---+-------------+-------------+-------------+
J0 = K0 = 1        J1 = Q3'.Q0   K1 = Q0
J2 = K2 = Q1.Q0    J3 = Q2.Q1.Q0  K3 = Q0

Step 6: Timing diagram

CLK  __|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾
Q3   ________________________________|‾‾‾‾‾‾‾|_______
Q2   ________________|‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾|_______________
Q1   ________|‾‾‾‾‾‾‾|_______|‾‾‾‾‾‾‾|_______________
Q0   ____|‾‾‾|___|‾‾‾|___|‾‾‾|___|‾‾‾|___|‾‾‾|___|‾‾‾
Cnt  0    1   2   3   4   5   6   7   8   9   0   1

All flip-flops change together on the falling clock edge. $Q_3$ is high for counts 8 and 9, giving $f_{Q_3} = f_{clk}/10$. The unused states return to the main count within two clocks (1010→1011→0100, 1100→1101→0100, 1110→1111→0000), so the counter is self-starting.