Synchronous decade counter with timing

A synchronous decade (BCD / mod-10) counter counts from 0000 to 1001 (0 to 9) and returns to 0000 on the 10th clock pulse. All flip-flops receive the same clock, so they change state together.

Step 1: Number of flip-flops and states

• 10 states are needed, so $2^n \ge 10 \Rightarrow n = 4$ T flip-flops ($Q_3Q_2Q_1Q_0$, $Q_0$ = LSB).
• Used states: 0000 to 1001. Unused states 1010 to 1111 are taken as don't cares (X).
• T flip-flop excitation: $T = Q \oplus Q^+$ (T = 1 whenever the flip-flop must change).

Step 2: Excitation table

Step 3: K-maps (X = don't care)

$T_0 = 1$ for every state. The other inputs:

Step 4: Simplified equations

$$\begin{aligned} &T_3 = Q_3Q_0 + Q_2Q_1Q_0 \\ &T_2 = Q_1Q_0 \\ &T_1 = \overline{Q_3}Q_0 \\ &T_0 = 1 \end{aligned}$$

So three 2-input AND gates ($\overline{Q_3}Q_0$, $Q_1Q_0$ and $Q_3Q_0$), one 3-input AND gate ($Q_2Q_1Q_0$) and one 2-input OR gate (for $T_3$) are needed.

Step 5: Circuit diagram

   +-----+       +-----+       +-----+       +-----+
T0>|T   Q|-Q0 T1>|T   Q|-Q1 T2>|T   Q|-Q2 T3>|T   Q|-Q3
   | FF0 |       | FF1 |       | FF2 |       | FF3 |
   |     |       |     |       |     |       |     |
   +--o--+       +--o--+       +--o--+       +--o--+
      |             |             |             |
CLK---+-------------+-------------+-------------+
T0 = 1          T1 = Q3'.Q0
T2 = Q1.Q0      T3 = Q3.Q0 + Q2.Q1.Q0

Step 6: Timing diagram (negative-edge clock)

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

$Q_0$ toggles on every clock edge, $Q_1$ toggles when $Q_0 = 1$ (except at count 9), $Q_2$ toggles when $Q_1Q_0 = 1$, and $Q_3$ goes high at count 8 and returns low after count 9. The counter therefore divides the clock frequency by 10 ($f_{Q_3} = f_{clk}/10$).

Self-starting check: the unused states go back into the count (1010→1011→0110, 1100→1101→0100, 1110→1111→0010), so the counter cannot lock out.