SR equation; mod-5 counter using D flip-flops

Characteristic equation of SR flip-flop

The characteristic equation gives the next state $Q^+$ in terms of the inputs and the present state $Q$.

Grouping the row S = 1 (with the X cells) gives $S$; grouping cells 01 of both rows gives $\overline{R}Q$:

$$Q^+ = S + \overline{R}\,Q, \qquad \text{with the condition } SR = 0$$

Mod-5 synchronous counter using D flip-flops (negative-edge clock)

Step 1: States 000 → 001 → 010 → 011 → 100 → 000. 3 D flip-flops ($Q_2Q_1Q_0$), all clocked by the falling edge. States 101, 110, 111 are don't cares. For a D flip-flop $D = Q^+$.

Step 2: Excitation table

Step 3: K-maps

Step 4: Equations

$$\begin{aligned} &D_2 = Q_1Q_0 \\ &D_1 = Q_1\overline{Q_0} + \overline{Q_1}Q_0 \\ &D_0 = \overline{Q_2}\overline{Q_0} \end{aligned}$$

so $D_1 = Q_1 \oplus Q_0$.

Step 5: Circuit diagram

   +-----+       +-----+       +-----+
D0>|D   Q|-Q0 D1>|D   Q|-Q1 D2>|D   Q|-Q2
   | FF0 |       | FF1 |       | FF2 |
   |     |       |     |       |     |
   +--o--+       +--o--+       +--o--+
      |             |             |
CLK---+-------------+-------------+
D0 = Q2'.Q0'  (NOR of Q2 and Q0)
D1 = Q1 XOR Q0
D2 = Q1.Q0

Step 6: Unused states: 101 → 010, 110 → 010, 111 → 100, so the counter is self-starting.

Timing diagram

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