Problem 7 - Digital Design

Design a Mod-11 up-down counter using D flip-flop in the following steps.

1.

Draw a state diagram.

2.

Obtain a state table.

3.

Obtain an excitation table.

4.

Obtain logic equations and draw a logic diagram.

1.

$$\includegraphics{mod-11-up-down-counter-state-diagram.ps}$$

2.

$$\begin{tabular}{c\vert cc} & \multicolumn{2}{\vert c}{Next state}... ... 1001 & 0111 \\ 1001 & 1010 & 1000 \\ 1010 & 0000 & 1001 \\ \end{tabular}$$

3.

$$\begin{tabular}{ccc\vert c} \multicolumn{3}{c\vert}{Inputs of the... ...\ 1 0 1 0 & 1 & 0 & 0 0 0 0 \\ 1 0 1 0 & 0 & 1 & 1 0 0 1 \\ \end{tabular}$$

4.

1011XX, 1100XX, 1101XX, 1110XX, 1111XX are don't-care conditions.

$$\includegraphics{karnaugh-maps-mod-11-up-down-counter.ps}$$

$$D = A'B'C'D'u'd + ABCud' + A'BDdu' + B'C'Dud' + ADu'd$$

$$= (A'B'C'D' + A'BD + AD)u'd + (ABC + B'C'D)ud'$$

$$= (A'(B'C'D' + BD) + AD)u'd + (ABC + B'C'D)ud'$$

$$C = A'B'C'Du'd + A'BCu'd + ABC'ud' + A'Cud' + B'Cud' + ACu'd$$

$$= (A'B'C'D + A'BC + AC)u'd + (ABC' + A'C + B'C)ud'$$

$$= (A'(B'C'D + BC) + AC)u'd + (ABC' + (A'+ B')C)ud'$$

$$= (A'(B'C'D + BC) + AC)u'd + (ABC' + (AB)'C)ud'$$

$$B = A'B'C'u'd + A'B'Cu'd + A'BD'ud' + AB'C'ud' + AB'Cud' + ABD'u'd$$

$$= (A'B'C' + ABD' + A'B'C)u'd + (A'BD' + AB'C' + AB'C)ud'$$

$$= (A'B'(C'+C) + ABD')u'd + (A'BD' + AB'(C'+C))ud'$$

$$= (A'B' + ABD')u'd + (A'BD' + AB')ud'$$

$$A = A'Ddu' + A'Bdu' + A'B'Cdu' + A'D'ud' + A'B'C'ud'$$

$$= (D + B + B'C)A'du' + (D'+ B'C')A'ud'$$

$$= (D + (B + B')(B + C))A'du' + (D'+ B'C')A'ud'$$

$$= ((D + B + C)du' + (D'+ B'C')ud')A'$$