Merge branch 'master' of github.com:TUBAF-IfI-LiaScript/VL_EingebetteteSysteme

Author: SebastianZug

08_StandardSchaltwerke.md

@@ -531,14 +531,53 @@ DI &= \overline{I} \\
 \end{aligned}
 $$
 
-![Bild](./images/08_StandardSchaltwerke/Zaehler.svg.png)<!-- style="width: 90%; max-width: 800px;" -->
+**Alternative Herleitung:**
 
-Das nachfolgende Beispiel realisiert eine alternative Umsetzung und integriert zu Demonstrationszwecken einen Reset-Kanal. Aktivieren Sie zunächst die Clock-Leitung einmal. Für den Zählmechanismus muss dann $\overline{RESET}$ gesetzt sein.
+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.)
+
+Es kann passieren, dass carryOut kurz auf HIGH springt, obwohl es LOW sein sollte. Das liegt daran, dass die Ausgänge der Gates nicht sofort anliegen, sondern einige Zeit brauchen. Wegen der Taktung ist das aber Kein Problem, außer die Taktrate ist zu hoch.
+
+Am Anfang `count` und `~reset` auf LOW lassen, clock puls geben und dann `count` und `~reset` auf HIGH setzen um bei dem nächten clock Puls das Zählen zu beginnen.
 
 ```json @DigiSim.evalJson
 {"devices":{"and5":{"label":"anew and ~reset","type":"And","propagation":1,"bits":1,"position":{"x":355,"y":5}},"and6":{"label":"bnew and ~reset","type":"And","propagation":1,"bits":1,"position":{"x":355,"y":80}},"and7":{"label":"cnew and ~reset","type":"And","propagation":1,"bits":1,"position":{"x":355,"y":145}},"and8":{"label":"dnew and ~reset","type":"And","propagation":1,"bits":1,"position":{"x":355,"y":215}},"resetButton":{"label":"~reset","type":"Button","propagation":0,"position":{"x":25,"y":265}},"countButton":{"label":"count  (dflip)","type":"Button","propagation":0,"position":{"x":-95,"y":245}},"clkButton":{"label":"clk","type":"Button","propagation":0,"position":{"x":195,"y":345}},"lampa":{"label":"a","type":"Lamp","propagation":1,"position":{"x":655,"y":5}},"lampb":{"label":"b","type":"Lamp","propagation":1,"position":{"x":655,"y":70}},"lampc":{"label":"c","type":"Lamp","propagation":1,"position":{"x":655,"y":140}},"lampd":{"label":"d","type":"Lamp","propagation":1,"position":{"x":655,"y":210}},"lampco":{"label":"carry out","type":"Lamp","propagation":1,"position":{"x":630,"y":-60}},"dffa":{"label":"dff a","type":"Dff","propagation":1,"polarity":{"clock":true},"bits":1,"initial":"x","position":{"x":505,"y":5}},"dffb":{"label":"dff b","type":"Dff","propagation":1,"polarity":{"clock":true},"bits":1,"initial":"x","position":{"x":505,"y":90}},"dffc":{"label":"dff c","type":"Dff","propagation":1,"polarity":{"clock":true},"bits":1,"initial":"x","position":{"x":505,"y":160}},"dffd":{"label":"dff d","type":"Dff","propagation":1,"polarity":{"clock":true},"bits":1,"initial":"x","position":{"x":510,"y":235}},"and1":{"label":"aflip and a  (carryOut)","type":"And","propagation":1,"bits":1,"position":{"x":135,"y":-65}},"and2":{"label":"bflip and b  (aflip)","type":"And","propagation":1,"bits":1,"position":{"x":60,"y":0}},"and3":{"label":"cflip and c  (bflip)","type":"And","propagation":1,"bits":1,"position":{"x":35,"y":80}},"and4":{"label":"dflip and d  (cflip)","type":"And","propagation":1,"bits":1,"position":{"x":-30,"y":150}},"xor1":{"label":"a xor aflip","type":"Xor","propagation":1,"bits":1,"position":{"x":210,"y":5}},"xor2":{"label":"b xor bflip","type":"Xor","propagation":1,"bits":1,"position":{"x":215,"y":70}},"xor3":{"label":"c xor cflip","type":"Xor","propagation":1,"bits":1,"position":{"x":210,"y":130}},"xor4":{"label":"d xor dflip","type":"Xor","propagation":1,"bits":1,"position":{"x":210,"y":200}}},"connectors":[{"from":{"id":"countButton","port":"out"},"to":{"id":"xor4","port":"in2"}},{"from":{"id":"countButton","port":"out"},"to":{"id":"and4","port":"in1"},"vertices":[{"x":-80,"y":190}]},{"from":{"id":"and3","port":"out"},"to":{"id":"and2","port":"in1"},"vertices":[{"x":25,"y":65},{"x":5,"y":65}]},{"from":{"id":"and4","port":"out"},"to":{"id":"and3","port":"in1"},"vertices":[{"x":-15,"y":125}]},{"from":{"id":"and4","port":"out"},"to":{"id":"xor3","port":"in2"}},{"from":{"id":"and3","port":"out"},"to":{"id":"xor2","port":"in2"}},{"from":{"id":"and2","port":"out"},"to":{"id":"xor1","port":"in2"}},{"from":{"id":"dffa","port":"out"},"to":{"id":"xor1","port":"in1"},"vertices":[{"x":315,"y":-5}]},{"from":{"id":"and2","port":"out"},"to":{"id":"and1","port":"in1"},"vertices":[{"x":95,"y":-10}]},{"from":{"id":"and1","port":"out"},"to":{"id":"lampco","port":"in"}},{"from":{"id":"clkButton","port":"out"},"to":{"id":"dffd","port":"clk"},"vertices":[{"x":435,"y":360}]},{"from":{"id":"clkButton","port":"out"},"to":{"id":"dffc","port":"clk"},"vertices":[{"x":435,"y":360}]},{"from":{"id":"clkButton","port":"out"},"to":{"id":"dffb","port":"clk"},"vertices":[{"x":455,"y":340}]},{"from":{"id":"clkButton","port":"out"},"to":{"id":"dffa","port":"clk"}},{"from":{"id":"and8","port":"out"},"to":{"id":"dffd","port":"in"}},{"from":{"id":"xor1","port":"out"},"to":{"id":"and5","port":"in1"}},{"from":{"id":"xor2","port":"out"},"to":{"id":"and6","port":"in1"}},{"from":{"id":"xor3","port":"out"},"to":{"id":"and7","port":"in1"}},{"from":{"id":"xor4","port":"out"},"to":{"id":"and8","port":"in1"}},{"from":{"id":"and7","port":"out"},"to":{"id":"dffc","port":"in"}},{"from":{"id":"and6","port":"out"},"to":{"id":"dffb","port":"in"}},{"from":{"id":"and5","port":"out"},"to":{"id":"dffa","port":"in"}},{"from":{"id":"resetButton","port":"out"},"to":{"id":"and8","port":"in2"}},{"from":{"id":"resetButton","port":"out"},"to":{"id":"and7","port":"in2"},"vertices":[{"x":305,"y":255}]},{"from":{"id":"resetButton","port":"out"},"to":{"id":"and6","port":"in2"},"vertices":[{"x":305,"y":255}]},{"from":{"id":"resetButton","port":"out"},"to":{"id":"and5","port":"in2"},"vertices":[{"x":305,"y":255}]},{"from":{"id":"dffd","port":"out"},"to":{"id":"lampd","port":"in"}},{"from":{"id":"dffc","port":"out"},"to":{"id":"lampc","port":"in"}},{"from":{"id":"dffb","port":"out"},"to":{"id":"lampb","port":"in"}},{"from":{"id":"dffa","port":"out"},"to":{"id":"lampa","port":"in"}},{"from":{"id":"dffd","port":"out"},"to":{"id":"xor4","port":"in1"},"vertices":[{"x":310,"y":210},{"x":190,"y":185}]},{"from":{"id":"dffd","port":"out"},"to":{"id":"and4","port":"in2"},"vertices":[{"x":185,"y":180},{"x":15,"y":200}]},{"from":{"id":"dffc","port":"out"},"to":{"id":"xor3","port":"in1"},"vertices":[{"x":545,"y":140},{"x":285,"y":110}]},{"from":{"id":"dffc","port":"out"},"to":{"id":"and3","port":"in2"},"vertices":[{"x":545,"y":140},{"x":285,"y":110},{"x":130,"y":130}]},{"from":{"id":"dffb","port":"out"},"to":{"id":"and2","port":"in2"},"vertices":[{"x":340,"y":60}]},{"from":{"id":"dffb","port":"out"},"to":{"id":"xor2","port":"in1"},"vertices":[{"x":510,"y":60}]},{"from":{"id":"dffa","port":"out"},"to":{"id":"and1","port":"in2"},"vertices":[{"x":310,"y":-5}]}],"subcircuits":{}}
 ```
 
+Nachteil dieser Schaltung: höhere Laufzeiten
+
 ## Übungsaufgaben
 
 + Entwerfen Sie einen Zähler, der eine Zählrichtungsvorgabe ermöglicht und evaluieren Sie Ihre Lösung mit einem Simulator.