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# Beispiel: 0 bis 7 Zähler

## Erzeugen der Gleichung

${\displaystyle t_{n}}$ ${\displaystyle t_{n+1}}$
n ${\displaystyle Q_{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{0}}$ ${\displaystyle Q_{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{0}}$
0 0 0 0 0 0 1
1 0 0 1 0 1 0
2 0 1 0 0 1 1
3 0 1 1 1 0 0
4 1 0 0 1 0 1
5 1 0 1 1 1 0
6 1 1 0 1 1 1
7 1 1 1 0 0 0

${\displaystyle Q_{2n+1}}$ ${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ ${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ ${\displaystyle Q_{1}\ Q_{0}}$ ${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$
${\displaystyle {\overline {Q_{2}}}}$ 0 0 1 0 3 1 2 0
${\displaystyle Q_{2}}$ 4 1 5 1 7 0 6 1
${\displaystyle Q_{2n+1}={\overline {Q_{2}}}Q_{1}Q_{0}\lor Q_{2}{\overline {Q_{1}}}\lor Q_{2}{\overline {Q_{0}}}}$

${\displaystyle Q_{1n+1}}$ ${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ ${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ ${\displaystyle Q_{1}\ Q_{0}}$ ${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$
${\displaystyle {\overline {Q_{2}}}}$ 0 0 1 1 3 0 2 1
${\displaystyle Q_{2}}$ 4 0 5 1 7 0 6 1
${\displaystyle Q_{1n+1}={\overline {Q_{1}}}Q_{0}\lor Q_{1}{\overline {Q_{0}}}}$

${\displaystyle Q_{0n+1}}$ ${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ ${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ ${\displaystyle Q_{1}\ Q_{0}}$ ${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$
${\displaystyle {\overline {Q_{2}}}}$ 0 1 1 0 3 0 2 1
${\displaystyle Q_{2}}$ 4 1 5 0 7 0 6 1
${\displaystyle Q_{0n+1}={\overline {Q_{0}}}}$


## Umformen der Gleichung

### Gleichung 2

Wir haben diese Gleichung für die Schaltung:

${\displaystyle Q_{2n+1}={\overline {Q_{2}}}Q_{1}Q_{0}\lor Q_{2}{\overline {Q_{1}}}\lor Q_{2}{\overline {Q_{0}}}}$


und diese für das JK-Flipflop:

${\displaystyle Q_{n+1}=J{\overline {Q_{n}}}\lor {\overline {K}}Q_{n}}$


Anpasste auf Q2n+1 gilt:

${\displaystyle Q_{2n+1}=J{\overline {Q_{2n}}}\lor {\overline {K}}Q_{2n}}$


Wir müssen nun die Gleichung unsere Schaltung in die gleiche Form wie die des JK Flipflops bringen

${\displaystyle Q_{2}}$ ausklammern ${\displaystyle Q_{2n+1}={\overline {Q_{2}}}Q_{1}Q_{0}\lor Q_{2}{\overline {Q_{1}}}\lor Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle Q_{2n+1}={\overline {Q_{2}}}Q_{1}Q_{0}\lor Q_{2}({\overline {Q_{1}}}\lor {\overline {Q_{0}}})}$ ${\displaystyle Q_{2n+1}={\overline {Q_{2}}}(Q_{1}Q_{0})\lor Q_{2}({\overline {Q_{1}}}\lor {\overline {Q_{0}}})}$ ${\displaystyle Q_{2n+1}={\overline {Q_{2}}}{\color {Red}(Q_{1}Q_{0})}\lor Q_{2}{\color {Blue}({\overline {Q_{1}}}\lor {\overline {Q_{0}}})}}$ ${\displaystyle Q_{2n+1}={\overline {Q_{2n}}}{\color {Red}J_{2}}\lor Q_{2n}{\color {Blue}{\overline {K_{2}}}}}$ ${\displaystyle {\color {Red}J_{2}=Q_{1}Q_{0}}}$ ${\displaystyle {\color {Blue}{\overline {K_{2}}}={\overline {Q_{1}}}\lor {\overline {Q_{0}}}}}$ ${\displaystyle J_{2}=Q_{1}Q_{0}}$ ${\displaystyle K_{2}=Q_{1}Q_{0}}$ ${\displaystyle J_{2}=K_{2}=Q_{1}Q_{0}}$

### Gleichung 1

Gleichung für D-Flipflop:

${\displaystyle Q_{1n+1}={\overline {Q_{1}}}Q_{0}\lor Q_{1}{\overline {Q_{0}}}}$

${\displaystyle Q_{1}}$ ausklammern ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}Q_{0}\lor Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}Q_{0}\lor Q_{1}({\overline {Q_{0}}})}$ ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}(Q_{0})\lor Q_{1}({\overline {Q_{0}}})}$ ${\displaystyle Q_{1n+1}={\overline {Q_{1n}}}{\color {Red}J_{1}}\lor Q_{1n}{\color {Blue}{\overline {K_{1}}}}}$ ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}{\color {Red}(Q_{0})}\lor Q_{1}{\color {Blue}({\overline {Q_{0}}})}}$ ${\displaystyle {\color {Red}J_{1}=Q_{0}}}$ ${\displaystyle {\color {Blue}{\overline {K_{1}}}={\overline {Q_{0}}}}}$ ${\displaystyle J_{1}=Q_{0}}$ ${\displaystyle K_{1}=Q_{0}}$ ${\displaystyle J_{1}=K_{1}=Q_{0}}$

### Gleichung 0

${\displaystyle Q_{0n+1}={\overline {Q0}}}$

${\displaystyle J_{0}=K_{0}=1}$