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# Digitale Schaltungstechnik/ Zähler/ Synchron/ Umwandlung/ D-JK/ Bsp. 2

### Beispiel: beliebige Zählfolge

#### Aufgabe

2 12 8 3 6 7 0


#### Wahrheitstabelle

Eingang Ausgang
${\displaystyle Q_{n}}$ ${\displaystyle Q_{n+1}}$
dez ${\displaystyle Q_{3}}$ ${\displaystyle Q_{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{0}}$ ${\displaystyle Q_{3}}$ ${\displaystyle Q_{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{0}}$
0 0 0 0 0 0 0 1 0
1 0 0 0 1 x x x x
2 0 0 1 0 1 1 0 0
3 0 0 1 1 0 1 1 0
4 0 1 0 0 x x x x
5 0 1 0 1 x x x x
6 0 1 1 0 0 1 1 1
7 0 1 1 1 0 0 0 0
8 1 0 0 0 0 0 1 1
9 1 0 0 1 x x x x
10 1 0 1 0 x x x x
11 1 0 1 1 x x x x
12 1 1 0 0 1 0 0 0
13 1 1 0 1 x x x x
14 1 1 1 0 x x x x
15 1 1 1 1 x x x x

#### KV-Diagramme

${\displaystyle Q_{3n+1}}$ ${\displaystyle Q_{3}Q_{2}}$ ${\displaystyle Q_{3}{\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ {\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ Q_{2}}$
${\displaystyle Q_{1}\ Q_{0}}$ 15   X 11  X 3  0 7  0
${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$ 14  X 10  X 2   1 6  0
${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ 12  1  8  0 0   0 4  X
${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ 13  X  9  X 1  X 5   X
 ${\displaystyle Q_{3n+1}=Q_{3}Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$

${\displaystyle Q_{2n+1}}$ ${\displaystyle Q_{3}Q_{2}}$ ${\displaystyle Q_{3}{\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ {\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ Q_{2}}$
${\displaystyle Q_{1}\ Q_{0}}$ 15   X 11  X 3  1 7  0
${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$ 14  X 10  X 2   1 6  1
${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ 12  0  8  0 0   0 4  X
${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ 13  X  9  X 1  X 5   X
 ${\displaystyle Q_{2n+1}=Q_{1}{\overline {Q_{0}}}\lor {\overline {Q_{2}}}Q_{1}}$

${\displaystyle Q_{1n+1}}$ ${\displaystyle Q_{3}Q_{2}}$ ${\displaystyle Q_{3}{\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ {\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ Q_{2}}$
${\displaystyle Q_{1}\ Q_{0}}$ 15   X 11  X 3  1 7  0
${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$ 14  X 10  X 2   0 6  1
${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ 12  0  8  1 0   1 4  X
${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ 13  X  9  X 1  X 5   X
 ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}{\overline {Q_{2}}}\lor {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$

${\displaystyle Q_{0n+1}}$ ${\displaystyle Q_{3}Q_{2}}$ ${\displaystyle Q_{3}{\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ {\overline {Q_{2}}}}$ ${\displaystyle {\overline {Q_{3}}}\ Q_{2}}$
${\displaystyle Q_{1}\ Q_{0}}$ 15   X 11  X 3  0 7  0
${\displaystyle Q_{1}\ {\overline {Q_{0}}}}$ 14  X 10  X 2   0 6  1
${\displaystyle {\overline {Q_{1}}}\ {\overline {Q_{0}}}}$ 12  0  8  1 0   0 4  X
${\displaystyle {\overline {Q_{1}}}\ Q_{0}}$ 13  X  9  X 1  X 5   X
 ${\displaystyle Q_{0n+1}=Q_{3}{\overline {Q_{2}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$


#### Gleichung 3

${\displaystyle Q_{3n+1}=Q_{3}Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$

${\displaystyle Q_{3n+1}=J{\overline {Q_{3n}}}\lor {\overline {K}}Q_{3n}}$


Der Ausdruck ${\displaystyle {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ lässt sich nicht direkt zuordnen. Um es dennoch zuzuordnen verwenden wir einen Trick:

${\displaystyle Q_{3}\lor {\overline {Q_{3}}}=1}$

${\displaystyle 1\land {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$

${\displaystyle 1\land ({\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})}$

${\displaystyle (Q_{3}\lor {\overline {Q_{3}}})({\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})}$

${\displaystyle Q_{3}{\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}\lor {\overline {Q_{3}}}\ {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$

${\displaystyle Q_{3}}$ ausklammern ${\displaystyle Q_{3n+1}=Q_{3}Q_{2}\lor Q_{3}{\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}\lor {\overline {Q_{3}}}\ {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle Q_{3n+1}=Q_{3}(Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})\lor {\overline {Q_{3}}}\ {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle Q_{3n+1}=Q_{3}(Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})\lor {\overline {Q_{3}}}({\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})}$ ${\displaystyle Q_{3n+1}=Q_{3}{\color {Red}(Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})}\lor {\overline {Q_{3}}}{\color {blue}({\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}})}}$ ${\displaystyle Q_{3n+1}={\overline {Q_{3n}}}{\color {Red}J_{3}}\lor Q_{3n}{\color {Blue}{\overline {K_{3}}}}}$ ${\displaystyle J_{3}=Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle {\overline {K_{3}}}={\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle J_{3}=Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$ ${\displaystyle K_{3}=Q_{2}\lor {\overline {Q_{1}}}\lor Q_{0}}$
${\displaystyle J_{3}=Q_{2}\lor {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}}$
${\displaystyle K_{3}=Q_{2}\lor {\overline {Q_{1}}}\lor Q_{0}}$


#### Gleichung 2

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

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


Beim Teilausdruck

${\displaystyle Q_{1}{\overline {Q_{0}}}}$


müssen wir Analog dem Oberen Beispiel vorgehen:

${\displaystyle {\overline {Q_{2}}}Q_{1}{\overline {Q_{0}}}\lor Q_{2}Q_{1}{\overline {Q_{0}}}}$

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

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

#### Gleichung 1

${\displaystyle Q_{1n+1}={\overline {Q_{1}}}\ {\overline {Q_{2}}}\lor {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$


Analog dem vorherigen Beispiel;

${\displaystyle Q_{1n+1}={\overline {Q_{1}}}\ {\overline {Q_{2}}}\lor Q_{1}{\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{1}}}\ {\overline {Q_{2}}}Q_{0}\lor Q_{1}{\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}\lor {\overline {Q_{1}}}\ {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$

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

${\displaystyle Q_{1}}$ ausklammern ${\displaystyle Q_{1n+1}={\overline {Q_{1}}}\ {\overline {Q_{2}}}\lor Q_{1}{\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{1}}}\ {\overline {Q_{2}}}Q_{0}\lor Q_{1}{\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}\lor {\overline {Q_{1}}}\ {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle Q_{1n+1}=Q_{1}({\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}})\lor {\overline {Q_{1}}}\ {\overline {Q_{2}}}\lor {\overline {Q_{1}}}\ {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{1}}}\ {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle Q_{1n+1}=Q_{1}({\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}})\lor {\overline {Q_{1}}}({\overline {Q_{2}}}\lor {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}})}$ ${\displaystyle Q_{1n+1}=Q_{1}({\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}})\lor {\overline {Q_{1}}}({\overline {Q_{2}}}\lor {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}})}$ ${\displaystyle Q_{1n+1}=J{\overline {Q_{1n}}}\lor {\overline {K}}Q_{1n}}$ ${\displaystyle J_{1}={\overline {Q_{2}}}\lor {\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle {\overline {K_{1}}}={\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle J_{1}={\overline {Q_{2}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle {\overline {K_{1}}}={\overline {Q_{2}}}Q_{0}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$
${\displaystyle J_{1}=}$
${\displaystyle K_{1}=}$


#### Gleichung 0

${\displaystyle Q_{0n+1}=Q_{3}{\overline {Q_{2}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$

${\displaystyle Q_{0n+1}=Q_{3}{\overline {Q_{2}}}Q_{0}\lor Q_{3}{\overline {Q_{2}}}\ {\overline {Q_{0}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$

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

${\displaystyle Q_{0}}$ ausklammern ${\displaystyle Q_{0n+1}=Q_{3}{\overline {Q_{2}}}Q_{0}\lor Q_{3}{\overline {Q_{2}}}\ {\overline {Q_{0}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle Q_{0n+1}=Q_{0}(Q_{3}{\overline {Q_{2}}})\lor Q_{3}{\overline {Q_{2}}}\ {\overline {Q_{0}}}\lor {\overline {Q_{3}}}Q_{2}{\overline {Q_{0}}}}$ ${\displaystyle Q_{0n+1}=Q_{0}(Q_{3}{\overline {Q_{2}}})\lor {\overline {Q_{0}}}(Q_{3}{\overline {Q_{2}}}\ \lor {\overline {Q_{3}}}Q_{2})}$ ${\displaystyle Q_{0n+1}=Q_{0}(Q_{3}{\overline {Q_{2}}})\lor {\overline {Q_{0}}}(Q_{3}{\overline {Q_{2}}}\ \lor {\overline {Q_{3}}}Q_{2})}$ ${\displaystyle Q_{0n+1}=J{\overline {Q_{0n}}}\lor {\overline {K}}Q_{0n}}$ ${\displaystyle J_{0}=Q_{3}{\overline {Q_{2}}}\ \lor {\overline {Q_{3}}}Q_{2}}$ ${\displaystyle {\overline {K_{0}}}=Q_{3}{\overline {Q_{2}}}}$ ${\displaystyle J_{0}=Q_{3}{\overline {Q_{2}}}\ \lor {\overline {Q_{3}}}Q_{2}}$ ${\displaystyle K_{0}={\overline {Q_{3}}}\lor Q_{2}}$
${\displaystyle J_{0}=Q_{3}{\overline {Q_{2}}}\ \lor {\overline {Q_{3}}}Q_{2}}$
${\displaystyle K_{0}={\overline {Q_{3}}}\lor Q_{2}}$