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LiaScript

Standardschaltnetze

Parameter Kursinformationen
Veranstaltung: @config.lecture
Semester @config.semester
Hochschule: Technische Universität Freiberg
Inhalte: Realisierunrtragen Sie die Funktion in ein Karnaugh-Veitch-Dig von Schaltnetzen, Standardschaltnetze
Link auf GitHub: https://github.com/TUBAF-IfI-LiaScript/VL_EingebetteteSysteme/blob/master/05_Standardschaltnetze.md
Autoren @author

** Fragen an die Veranstaltung**

  • Unterscheiden Sie Multiplexer und Demultiplexer.
  • Wie lassen sich mit einem Multiplexer beliebige Wahrheitstafeln abbilden? Welche Grenzen hat dieser Ansatz?
  • Welche Aufgaben realisieren Dekoder?
  • Nennen Sie Anwendungsbeispiele für einen Multiplexer.

                Abstraktionsebenen

           +----------------------------+ -.
  Ebene 6  | Problemorientierte Sprache |  |
           +----------------------------+  |
                                           ⎬ Anwendungssoftware
           +----------------------------+  |
  Ebene 5  | Assemblersprache           |  |
           +----------------------------+ -.

           +----------------------------+
  Ebene 4  | Betriebssystem             |     Systemsoftware
           +----------------------------+

           +----------------------------+
  Ebene 3  | Instruktionsset            |     Maschinensprache
           +----------------------------+

           +----------------------------+  -.
  Ebene 2  | Mikroarchitektur           |   |
           +----------------------------+   |
                                            ⎬ Automaten, Speicher, Logik
           +----------------------------+   |       ╔═══════════════╗
  Ebene 1  | Digitale Logik             |   |    ◀══║ HIER SIND WIR!║
           +----------------------------+  -.       ╚═══════════════╝

           +----------------------------+
  Ebene 0  | E-Technik, Physik          |     Analoge Phänomene
           +----------------------------+                                      .

Dekodierer / Kodierer

In der digitalen Elektronik ist ein Binärdecoder eine kombinatorische Logikschaltung, die binäre Informationen von den $n$ codierten Eingängen in maximal $k = 2^n$ eindeutige Ausgänge umwandelt. Sie werden zum Beispiel für die Ansteuerung von Siebensegmentanzeigen und als Adressdecoder für Speicher und Port-mapped I/O genutzt.


        Dekodierer              Kodierer
               +                  +
              /| ...............> |\
             / | ---------------> | \
            +  | ---------------> |  +
Input   --> |  | ---------------> |  | --->  Output
        --> |  | ---------------> |  | --->
        --> |  | ---------------> |  | --->
            +  | ---------------> |  +
             \ | ---------------> | /
              \| ---------------> |/
               +                  +                                            .

So kann z.B. ein abgewandeltes NOT Gatter als 1:2-Binärdecoder mit 1-Eingang und 2-Ausgänge klassifiziert werden, da er mit einem Eingang A zwei Ausgänge $A$ und $\overline{A}$ hat.

n-zu-k Dekodierer

3-8 Dekodierer

$A$ $B$ $C$ $y_0$ $y_1$ $y_2$ $y_3$ $y_4$ ...
0 0 0 1 0 0 0 0 ...
0 0 1 0 1 0 0 0 ...
0 1 0 0 0 1 0 0 ...
0 1 1 0 0 0 1 0 ...
1 0 0 0 0 0 0 1 ...
1 0 1 0 0 0 0 0 ...
1 1 0 0 0 0 0 0 ...
1 1 1 0 0 0 0 0 ...

Für jede Eingangskombination wird genau 1 Ausgang aktiviert.

$$ \begin{aligned} y_0 &= \overline{A} \cdot \overline{B} \cdot \overline{C} \\ y_1 &= \overline{A} \cdot \overline{B} \cdot C \\ y_2 &= \overline{A} \cdot B \cdot \overline{C} \\ y_3 &= \overline{A} \cdot B \cdot C \end{aligned} $$

{"devices":{"a":{"label":"A","type":"Button","propagation":0,"position":{"x":-90,"y":65}},"b":{"label":"B","type":"Button","propagation":0,"position":{"x":-90,"y":115}},"c":{"label":"C","type":"Button","propagation":0,"position":{"x":-100,"y":210}},"y0":{"label":"y0","type":"Lamp","propagation":0,"position":{"x":590,"y":45}},"y1":{"label":"y1","type":"Lamp","propagation":0,"position":{"x":590,"y":105}},"y2":{"label":"y2","type":"Lamp","propagation":0,"position":{"x":595,"y":175}},"y3":{"label":"y3","type":"Lamp","propagation":0,"position":{"x":590,"y":230}},"y4":{"label":"y4","type":"Lamp","propagation":0,"position":{"x":595,"y":285}},"y5":{"label":"y5","type":"Lamp","propagation":0,"position":{"x":590,"y":345}},"y6":{"label":"y6","type":"Lamp","propagation":0,"position":{"x":595,"y":415}},"y7":{"label":"y7","type":"Lamp","propagation":0,"position":{"x":595,"y":480}},"notA":{"label":"NOT a","type":"Not","propagation":0,"bits":1,"position":{"x":55,"y":90}},"notB":{"label":"NOT b","type":"Not","propagation":0,"bits":1,"position":{"x":40,"y":165}},"notC":{"label":"NOT c","type":"Not","propagation":0,"bits":1,"position":{"x":35,"y":375}},"and1":{"label":"~A and ~B","type":"And","propagation":0,"bits":1,"position":{"x":230,"y":95}},"and2":{"label":"~A and B","type":"And","propagation":0,"bits":1,"position":{"x":230,"y":195}},"and3":{"label":"A and ~B","type":"And","propagation":0,"bits":1,"position":{"x":235,"y":290}},"and4":{"label":"A and B","type":"And","propagation":0,"bits":1,"position":{"x":230,"y":410}},"and5":{"label":"~A and ~B and ~C","type":"And","propagation":0,"bits":1,"position":{"x":420,"y":40}},"and6":{"label":"~A and ~B and C","type":"And","propagation":0,"bits":1,"position":{"x":430,"y":100}},"and7":{"label":"~A and B and ~C","type":"And","propagation":0,"bits":1,"position":{"x":435,"y":170}},"and8":{"label":"~A and B and C","type":"And","propagation":0,"bits":1,"position":{"x":440,"y":225}},"and9":{"label":"A and ~B and ~C","type":"And","propagation":0,"bits":1,"position":{"x":430,"y":280}},"and10":{"label":"A and ~B and C","type":"And","propagation":0,"bits":1,"position":{"x":430,"y":340}},"and11":{"label":"A and B and ~C","type":"And","propagation":0,"bits":1,"position":{"x":435,"y":410}},"and12":{"label":"A and B and C","type":"And","propagation":0,"bits":1,"position":{"x":445,"y":475}}},"connectors":[{"from":{"id":"a","port":"out"},"to":{"id":"notA","port":"in"}},{"from":{"id":"b","port":"out"},"to":{"id":"notB","port":"in"}},{"from":{"id":"c","port":"out"},"to":{"id":"notC","port":"in"}},{"from":{"id":"and1","port":"out"},"to":{"id":"and5","port":"in1"}},{"from":{"id":"and1","port":"out"},"to":{"id":"and6","port":"in1"}},{"from":{"id":"and2","port":"out"},"to":{"id":"and7","port":"in1"}},{"from":{"id":"and2","port":"out"},"to":{"id":"and8","port":"in1"}},{"from":{"id":"and3","port":"out"},"to":{"id":"and9","port":"in1"}},{"from":{"id":"and3","port":"out"},"to":{"id":"and10","port":"in1"}},{"from":{"id":"and4","port":"out"},"to":{"id":"and11","port":"in1"}},{"from":{"id":"and4","port":"out"},"to":{"id":"and12","port":"in1"}},{"from":{"id":"notA","port":"out"},"to":{"id":"and1","port":"in1"}},{"from":{"id":"notB","port":"out"},"to":{"id":"and1","port":"in2"}},{"from":{"id":"notA","port":"out"},"to":{"id":"and2","port":"in1"}},{"from":{"id":"b","port":"out"},"to":{"id":"and2","port":"in2"},"vertices":[{"x":130,"y":160}]},{"from":{"id":"a","port":"out"},"to":{"id":"and3","port":"in1"},"vertices":[{"x":10,"y":215}]},{"from":{"id":"notB","port":"out"},"to":{"id":"and3","port":"in2"}},{"from":{"id":"a","port":"out"},"to":{"id":"and4","port":"in1"},"vertices":[{"x":10,"y":225},{"x":155,"y":395}]},{"from":{"id":"b","port":"out"},"to":{"id":"and4","port":"in2"},"vertices":[{"x":130,"y":160}]},{"from":{"id":"c","port":"out"},"to":{"id":"and6","port":"in2"},"vertices":[{"x":295,"y":265}]},{"from":{"id":"notC","port":"out"},"to":{"id":"and5","port":"in2"},"vertices":[{"x":330,"y":335}]},{"from":{"id":"c","port":"out"},"to":{"id":"and8","port":"in2"},"vertices":[{"x":310,"y":275}]},{"from":{"id":"notC","port":"out"},"to":{"id":"and7","port":"in2"},"vertices":[{"x":330,"y":335},{"x":370,"y":295}]},{"from":{"id":"notC","port":"out"},"to":{"id":"and9","port":"in2"},"vertices":[{"x":335,"y":335}]},{"from":{"id":"notC","port":"out"},"to":{"id":"and11","port":"in2"},"vertices":[{"x":320,"y":345}]},{"from":{"id":"c","port":"out"},"to":{"id":"and10","port":"in2"}},{"from":{"id":"c","port":"out"},"to":{"id":"and12","port":"in2"},"vertices":[{"x":360,"y":370}]},{"from":{"id":"and5","port":"out"},"to":{"id":"y0","port":"in"}},{"from":{"id":"and6","port":"out"},"to":{"id":"y1","port":"in"}},{"from":{"id":"and7","port":"out"},"to":{"id":"y2","port":"in"}},{"from":{"id":"and8","port":"out"},"to":{"id":"y3","port":"in"}},{"from":{"id":"and9","port":"out"},"to":{"id":"y4","port":"in"}},{"from":{"id":"and10","port":"out"},"to":{"id":"y5","port":"in"}},{"from":{"id":"and11","port":"out"},"to":{"id":"y6","port":"in"}},{"from":{"id":"and12","port":"out"},"to":{"id":"y7","port":"in"}}],"subcircuits":{}}

Adressdekoder

                         Adressbus [A_2, A_3 ...]
    +---------------------------------------------------------+
    |                                                         |
---------+          +--------+ D_0     +--------------+       |
     A_0 | -------->| A      |-------->|cs   Mem 0    |<------+
         |          |        |         +--------------+       |
     A_1 | -------->| B      |                :               |
         |          |        | D_1     +--------------+       |
         |  Adress- |        |-------->|cs   Mem 1    |<------+
         |  bus     |        |         +--------------+       |
Mikro    |          | 2-zu-4 |                :               |
prozessor|          | Dekoder| D_2     +--------------+       |
         |          |        |-------->|cs   Mem 2    |<------+
         |          |        |         +--------------+       |
         |          |        |                :               |
         |          |        | D_3     +--------------+       |
         |          |        |-------->|cs   Mem 3    |<------+
---------+          +--------+         +--------------+
    :                                         :                 cs = Chip Select
    ...........................................
                          Datenbus

BCD Dekoder für 7 Segmentanzeige

BCD Dekoder

const int PINS[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
const int PAUSE = 1000;
byte segDigits[10][8] = {
  { 1,1,1,1,1,1,1,0 },  // = 0
  { 0,1,1,0,0,0,0,0 },  // = 1
  { 1,1,0,1,1,1,0,1 },  // = 2
  { 1,1,1,1,1,0,0,1 },  // = 3
  { 0,1,1,0,0,0,1,1 },  // = 4rtragen Sie die Funktion in ein Karnaugh-Veitch-Di
};
void setup() {
  for(int i=0;i<8;i++){
    pinMode(PINS[i], OUTPUT);
  }
}

void loop() {
  for (int count = 0; count <=9; ++count) {
    for(int i=0; i<10; i++) {
      digitalWrite(PINS[i], segDigits[count][i]);
    }
    delay(PAUSE);
  }
}

@AVR8js.sketch

Verschaltung von Dekodern

Realisierung eines 4-zu-16 Dekoders auf der Basis von zwei 3-zu-8 Dekodern

               3-zu-8 Dekoder
               +----------+
A ------------>| 2^0     0|---->  0
B ------------>| 2^1     1|---->  1
C ------------>| 2^2     2|---->  2
D --+          |         3|---->  3
    |  |\      |         4|---->  4
    +--+ o---->| EN      5|---->  5
    |  |/      |         6|---->  6
    |          |         7|---->  7
    |          +----------+
    |
    |          +----------+
    |     A  ->| 2^0     0|---->  8
    |     B -->| 2^1     1|---->  9
    |     C--->| 2^2     2|----> 10
    |          |         3|----> 11
    |          |         4|----> 12
    +--------->| EN      5|----> 13
               |         6|----> 14
               |         7|----> 15
               +----------+                                                    .

n-zu-k Kodierer

  • n Ausgänge $y_0$, $y_1$, ... ,$y_{n-1}$
  • $k = 2^n$ Eingänge $x_0$, $x_1$, ... ,$x_{k-1}$
  • nur genau eine Eingangsleitung darf auf 1 sein: $x_i = 1$, $xj&lt;&gt;i = 0$

Jeder Eingangsleitung ist genau eine Kombination der möglichen Belegungen der Ausgangsleitungen zugeordnet, z.B. ihre binäre Repräsentation.

8-3 Kodierer

$x_0$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $x_6$ $x_7$ $y_2$ $y_1$ $y_0$
@blue(1) 0 0 0 0 0 0 0 0 0 0
0 @blue(1) 0 0 0 0 0 0 0 0 1
0 0 @blue(1) 0 0 0 0 0 0 1 0
0 0 0 @blue(1) 0 0 0 0 0 1 1
0 0 0 0 @blue(1) 0 0 0 1 0 0
0 0 0 0 0 @blue(1) 0 0 1 0 1
0 0 0 0 0 0 @blue(1) 0 1 1 0
0 0 0 0 0 0 0 @blue(1) 1 1 1

$$ \begin{aligned} y_0 &= x_1 + x_3 + x_5 + x_7 \\ y_1 &= x_2 + x_3 + x_6 + x_7 \\ y_2 &= x_4 + x_5 + x_6 + x_7 \end{aligned} $$

Achtung: Die Wahrheitstafel ist unvollständig !

  1. Falsche Ausgangszustände sind möglich!
  2. Was passiert wenn alle Pegel 0 sind?
{"devices":{"x0":{"label":"x0","type":"Button","propagation":0,"position":{"x":0,"y":0}},"x1":{"label":"x1","type":"Button","propagation":0,"position":{"x":0,"y":50}},"x2":{"label":"x2","type":"Button","propagation":0,"position":{"x":0,"y":105}},"x3":{"label":"x3","type":"Button","propagation":0,"position":{"x":0,"y":160}},"x4":{"label":"x4","type":"Button","propagation":0,"position":{"x":0,"y":215}},"x5":{"label":"x5","type":"Button","propagation":0,"position":{"x":0,"y":265}},"x6":{"label":"x6","type":"Button","propagation":0,"position":{"x":0,"y":320}},"x7":{"label":"x7","type":"Button","propagation":0,"position":{"x":0,"y":375}},"y0":{"label":"y0","type":"Lamp","propagation":0,"position":{"x":570,"y":185}},"y1":{"label":"y1","type":"Lamp","propagation":0,"position":{"x":530,"y":210}},"y2":{"label":"y2","type":"Lamp","propagation":0,"position":{"x":485,"y":240}},"or1":{"label":"x1 or x3","type":"Or","propagation":0,"bits":1,"position":{"x":190,"y":80}},"or2":{"label":"x5 or x7","type":"Or","propagation":0,"bits":1,"position":{"x":195,"y":140}},"or3":{"label":"x1 or x3 or x5 or x7","type":"Or","propagation":0,"bits":1,"position":{"x":345,"y":120}},"or4":{"label":"x2 or x3","type":"Or","propagation":0,"bits":1,"position":{"x":200,"y":200}},"or5":{"label":"x6 or x7","type":"Or","propagation":0,"bits":1,"position":{"x":205,"y":290}},"or6":{"label":"x2 or x3 or x6 or x7","type":"Or","propagation":0,"bits":1,"position":{"x":350,"y":225}},"or7":{"label":"x4 or x5","type":"Or","propagation":0,"bits":1,"position":{"x":200,"y":360}},"or8":{"label":"x4 or x5 or x6 or x7","type":"Or","propagation":0,"bits":1,"position":{"x":360,"y":320}}},"connectors":[{"from":{"id":"x1","port":"out"},"to":{"id":"or1","port":"in1"}},{"from":{"id":"x3","port":"out"},"to":{"id":"or1","port":"in2"}},{"from":{"id":"x5","port":"out"},"to":{"id":"or2","port":"in1"},"vertices":[{"x":135,"y":170}]},{"from":{"id":"x7","port":"out"},"to":{"id":"or2","port":"in2"},"vertices":[{"x":145,"y":310}]},{"from":{"id":"x2","port":"out"},"to":{"id":"or4","port":"in1"}},{"from":{"id":"x3","port":"out"},"to":{"id":"or4","port":"in2"}},{"from":{"id":"x6","port":"out"},"to":{"id":"or5","port":"in1"},"vertices":[{"x":90,"y":335}]},{"from":{"id":"x7","port":"out"},"to":{"id":"or5","port":"in2"}},{"from":{"id":"x4","port":"out"},"to":{"id":"or7","port":"in1"},"vertices":[{"x":95,"y":290}]},{"from":{"id":"x5","port":"out"},"to":{"id":"or7","port":"in2"}},{"from":{"id":"or1","port":"out"},"to":{"id":"or3","port":"in1"}},{"from":{"id":"or2","port":"out"},"to":{"id":"or3","port":"in2"}},{"from":{"id":"or4","port":"out"},"to":{"id":"or6","port":"in1"}},{"from":{"id":"or5","port":"out"},"to":{"id":"or6","port":"in2"}},{"from":{"id":"or5","port":"out"},"to":{"id":"or8","port":"in1"}},{"from":{"id":"or7","port":"out"},"to":{"id":"or8","port":"in2"}},{"from":{"id":"or3","port":"out"},"to":{"id":"y0","port":"in"}},{"from":{"id":"or6","port":"out"},"to":{"id":"y1","port":"in"}},{"from":{"id":"or8","port":"out"},"to":{"id":"y2","port":"in"}}],"subcircuits":{}}

Prioritätsencoder

Abhilfe schafft der Prioritätsencoder. Hier wird eine explizite Auswahl für verschiedene Eingangskonfigurationen getroffen.

$x_3$ $x_2$ $x_1$ $x_0$ $y_1$ $y_0$ Bemerkung
0 0 0 0 0 0 ungültig
0 0 0 1 0 0
0 0 1 @gray(X) 0 1
0 1 @gray(X) @gray(X) 1 0
1 @gray(X) @gray(X) @gray(X) 1 1

$$ \begin{aligned} y_0 &= \overline{x}_3 \overline{x}_2 x_1 + x_3 \\ y_1 &= \overline{x}_3 x_2 \\ \end{aligned} $$

Und im echten Leben? Kommen noch einige Spezialeingänge / -ausgänge dazu Link.

BCD Dekoder

Aufgabe: Vereinfachen Sie die Funktionen für $Q_2$!

$$ \begin{aligned} Q_2 &= D_4 + D_5 + D_6 + D_7 \\ \end{aligned} $$

Analog Digitalwandler

Wir werden dem Encoder bei der Diskussion der Peripherie eines Mikrocontrollers sehr häufig wiederbegegnen. Das Video zeigt einen Anwendungsfall - das Mapping der Ergebnisse eines Analog-Digital-Wandlers auf eine binäre Ausgabe.

BCD Dekoder 1

!?Erläuterung Flashwandler

Multiplexer / Demultiplexer

Eine Mulitplexerschaltung bildet analoge oder digitale Eingangssignale auf einen Kommunikationskanal ab, der Demultiplexer übernimmt die Abbildung auf n Ausgangsleitungen.


    Multiplexer             Demultiplexer
        +                        +
------> |\                      /|------>
        | \                    / |
------> |  +                  +  |------>
        |  | ................>|  |
------> |  +                  +  |------>
        | /                    \ |
------> |/|                    |\|------>
        +||                    ||+
         ||                    ||
       "$a_0$ $a_1$"         "$a_0$ $a_1$"                                       .
Multiplexer Dekoder
mehrere Eingänge, ein Ausgang mehrere Eingänge, mehrere Ausgänge
Steuerleitungen konfiguriereren die Weiterleitung das Mapping wird durch die interne Logik bestimmt
bildet den Pegel einer Eingangsleitung auf die Ausgangsleitung ab wandelt den binären Code in einen unären Code um == aktiviert eine Ausgangsleitung

Multiplexer

Generelle Konfiguration eines 1-aus-k Multiplexer:

  • $n$ Steuerleitungen: $s_{n-1}, ... , s_1, s_0$
  • $k = 2^n$ Eingänge: $x_0, x_1, ... , x_{k-1}$
  • ein Ausgang: y
  • $y = x_i$ für $(s_{n-1}, ... , s_1 , s_0)_2 = i$

Beispiel: 1-zu-4 Multiplexer

Eine Wahrheitstabelle mit 4 Eingangsvariablen und 2 Steuersignalen würde eine entsprechende Größe aufweisen. Allerdings kann die Funktion auch sehr einfach hergeleitet werden. Ein Inputsignal wird nur dann durchgeleitet, wenn die zugehörige Kombination von Steuersignalen anliegt.

$a_1$ $a_2$ Signal
0 0 $x_0$
0 1 $x_1$
1 0 $x_2$
1 1 $x_3$

$y = \overline{a_1} \cdot \overline {a_0} \cdot x_0 + \overline{a_1} \cdot a_0 \cdot x_1 + a_1 \cdot \overline {a_0} \cdot x_2 + a_1 \cdot a_0 \cdot x_3$

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Anwendung in Microcontrollern

BCD Dekoder

Multiplexer als universelle boolesche Funktionsrepräsentation

$x_2$ $x_1$ $x_0$ $y$
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1

BCD Dekoder

Demultiplexer

Generelle Konfiguration eines 1-aus-k Multiplexer:

  • $n$ Steuerleitungen: $s_{n-1}, ... , s_1, s_0$
  • $k = 2^n$ Ausgänge: $y_0, y_1, ... , y_{k-1}$
  • ein Eingang: x
  • $y = x_i$ für $(s_{n-1}, ... , s_1 , s_0)_2 = i$

Beispiel: 2 Bit Adresse -> 4 Ausgänge

$a_0$ $a_1$ $x$ $y_0$ $y_1$ $y_2$ $y_3$
0 0 0
0 0 1 1
0 1 0
0 1 1 1
1 0 0
1 0 1 1
1 1 0
1 1 1 1

$y_0 = x \cdot \overline{a_1} \cdot \overline{a_0}$

$y_1 = x \cdot \overline{a_1} \cdot a_0$

$y_2 = x \cdot a_1 \cdot \overline{a_0}$

$y_3 = x \cdot a_1 \cdot a_0$

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Komperatoren

... siehe Hausaufgabe

Ausblick

Nunmehr können boolesche Funktionen als Schaltnetze abbilden? Was fehlt für den Rechner? Zwei Dinge ... Speicher und arithmetische Operationen.

Hausaufgaben

  1. Entwerfen Sie einen Komperator, der zwei zweistellige Zahlen vergleicht. Definieren Sie dazu zunächst einen Ein-Bit Komperator und nutzen sie diesen als Grundlage für die Zwei-Bit-Variante

  2. Entwickeln Sie ein Schaltnetz, dass die Teilbarkeit durch drei von einer 4-stelligen binären Zahl prüft. Stellen Sie dazu eine Wahrheitstafel auf, minimieren Sie den Ausdruck soweit wie möglich und skizzieren Sie die Verdrahtung der Gatter.

Wer möchte sich mit dem NE555 beschäftigen?

Footnotes

  1. Autor: Always Be Positive, Youtube Video - Flash or Parallel ADC (Analog to Digital Converter) ↩