3a. State the following i. Distributive Law ii. Absorptive Law iii. Associative Law b. Design a digital counter to sequence through the following numbers using JK flip flop 0, 2, 4, 3, 6, 7, 0 c. Briefly explain what is meant by flip flop

We list the present states, next states, and determine the required JK inputs for each flip-flop using the JK flip-flop excitation table ($Q_n \to Q_{n+1}$: $0 \to 0 \implies J=0, K=x$; $0 \to 1 \implies J=1, K=x$; $1 \to 0 \implies J=x, K=1$; $1 \to 1 \implies J=x, K=0$).

Present State ($Q_C Q_B Q_A$)	Next State ($Q_C^+ Q_B^+ Q_A^+$)	$J_C$	$K_C$	$J_B$	$K_B$	$J_A$	$K_A$
000	010	0	x	1	x	0	x
010	100	1	x	x	1	0	x
100	011	x	1	1	x	1	x
011	110	1	x	x	0	x	1
110	111	x	0	x	0	1	x
111	000	x	1	x	1	x	1

We use K-maps to simplify the expressions for each JK input. Unused states (001, 101) are treated as don't cares (x).

$J_C$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & 0 & x & 1 & 1 \\ \hline 1 & x & x & x & x \\ \hline \end{array}$$

Grouping: $J_C = \overline{Q_C} Q_B$

$K_C$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & x & x & x & x \\ \hline 1 & 1 & x & 1 & 0 \\ \hline \end{array}$$

Grouping: $K_C = Q_C \overline{Q_B} + Q_C Q_A = Q_C (\overline{Q_B} + Q_A)$

$J_B$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & 1 & x & x & x \\ \hline 1 & 1 & x & x & x \\ \hline \end{array}$$

Grouping: $J_B = \overline{Q_A}$

$K_B$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & x & x & 0 & 1 \\ \hline 1 & x & x & 1 & 0 \\ \hline \end{array}$$

Grouping: $K_B = Q_C Q_A + \overline{Q_C} \overline{Q_A}$

$J_A$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & 0 & x & x & 0 \\ \hline 1 & 1 & x & x & 1 \\ \hline \end{array}$$

Grouping: $J_A = Q_C$

$K_A$ K-map:

$$\begin{array}{|c|c|c|c|c|} \hline Q_C Q_B \setminus Q_A & 00 & 01 & 11 & 10 \\ \hline 0 & x & x & 1 & x \\ \hline 1 & x & x & 1 & x \\ \hline \end{array}$$

Grouping: $K_A = Q_B$

The simplified equations for the JK inputs are:

$$J_C = \overline{Q_C} Q_B$$ $$K_C = Q_C (\overline{Q_B} + Q_A)$$ $$J_B = \overline{Q_A}$$ $$K_B = Q_C Q_A + \overline{Q_C} \overline{Q_A}$$ $$J_A = Q_C$$ $$K_A = Q_B$$

The counter consists of three JK flip-flops ($FF_C, FF_B, FF_A$) connected to a common clock signal.

• $J_C$ is connected to the output of an AND gate with inputs $\overline{Q_C}$ and $Q_B$.
• $K_C$ is connected to the output of an AND gate with inputs $Q_C$ and the output of an OR gate. The OR gate's inputs are $\overline{Q_B}$ and $Q_A$.
• $J_B$ is connected to $\overline{Q_A}$.
• $K_B$ is connected to the output of an XNOR gate with inputs $Q_C$ and $Q_A$. (Alternatively, two AND gates and one OR gate: $(Q_C \cdot Q_A) + (\overline{Q_C} \cdot \overline{Q_A})$).
• $J_A$ is connected to $Q_C$.
• $K_A$ is connected to $Q_B$.