Alternative Zähler Erläuterung

Author: fjangfaragesh

08_StandardSchaltwerke.md

@@ -491,7 +491,44 @@ DI &= \overline{I} \\
 \end{aligned}
 $$
 
-> Bitte hier die Simulation einfügen
+**Alternative Herleitung:**
+
+Der Zustand eines Zählerbits wechselt, wenn das nächst kleinere von 1 zu 0 springen wird.
+
+Das bedingte wechseln eines Bits lässt sich mit XOR realisieren:
+$ x \bigoplus 0 = x $;
+$ x \bigoplus 1 = \overline{x} $;
+Wenn man möchte, dass $ x $ wechselt, wenn $ x_{flip} $ high ist, dann kann $ x \bigoplus x_{flip} $ verwendet werden.
+
+Aus der Bedingung oben ergibt sich:
+
+$ a_{flip} = b_{flip} \cdot b $, (a wird wechseln, wenn b wechseln wird und b 1 ist, wobei b das nächst niederwertigste Bit von a ist.)
+
+Der neue Zustand von a: $ a' = a \bigoplus a_{flip} $
+
+$ b_{flip} $ lässt sich wie $ a_{flip} $ berechnen, wenn es nicht das niederwertigste Bit ist. Das niederwertigste Bit wechselt dann wenn der Eingang $ count $ (carryIn) auf High ist.
+
+Der Ausgang $ carryOut $ lässt sich berechnen mit $ a_{flip} \cdot a $.
+
+Für einen 4 Bit Zähler mit den bits a,b,c,d (a höchstwertig, d niederwertigste) ergeben sich diese Formeln:
+
+$$
+\begin{aligned}
+d_{flip} &= count\\
+d' &= d_{flip} \bigoplus d\\
+c_{flip} &= d_{flip} \cdot d\\
+c' &= c \bigoplus c_{flip}\\
+b_{flip} &= c_{flip} \cdot c\\
+b' &= b \bigoplus b_{flip}\\
+a_{flip} &= b_{flip} \cdot b\\
+a' &= a \bigoplus a_{flip}\\
+carryOut &= a_{flip} \cdot a\\
+\end{aligned}
+$$
+
+Aus technischen Gründen hat die Folgende Simmulation noch eine Reset Schaltung (jeder neue Zustand wird noch mit $ \overline{reset} $ UND genommen.)
+
+Am Anfang `count` und `~reset` auf LOW lassen, clock puls geben und dann `count` und `~reset` auf HIGH um bei dem nächten clock Puls das Zählen zu beginnen.
 
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