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Sven Åge Eriksen
http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karnaugh.html#introductionReferanse:
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KARNAUGH DIAGRAMHensikten med Karnaughdiagrammet er å forenkle funksjonsuttrykk ved å gruppere sammen celler som ligger ved siden av hverandre som inneholder 1.
Karnaughdiagrammet er en grafisk metode for forenkling av Boolske uttrykk.
Grunnen til at vi ønsker å forenkle funksjonsutrykk er flere:
Vi ønsker færrest mulig kretser hvis funksjonen skal lages med IC-kretser;det blir billigere, mer oversiktlig, tar mindre plass og færre deler kan gå i stykker.
Hvis funksjonen skal programmeres i en PLS er det også viktig å forenkle funksjonsuttrykk for å ha programmet så raskt så mulig og også så oversiktlig som mulig.
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KARNAUGH DIAGRAMSo far we can see that applying Boolean algebra can be awkward in order to simplify expressions. Apart from being laborious (and requiring the remembering all the laws) the method can lead to solutions which, though they appear minimal, are not. The Karnaugh map provides a simple and straight-forward method of minimising boolean expressions. With the Karnaugh map Boolean expressions having up to four and even six variables can be simplified.
So what is a Karnaugh map?
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KARNAUGH DIAGRAMA Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The Karnaugh map can also be described as a special arrangement of a truth table. The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem.
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KARNAUGH DIAGRAMIntroduksjon: Fra sannhetstabell til Karnaughdiagram
SANNHETSTABELL:
KARNAUGH DIAGRAM:
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KARNAUGH DIAGRAMIntroduksjon: Fra sannhetstabell til Karnaughdiagram
SANNHETSTABELL:
KARNAUGH DIAGRAM:
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KARNAUGH DIAGRAMIntroduksjon: Fra sannhetstabell til Karnaughdiagram
SANNHETSTABELL:
KARNAUGH DIAGRAM:
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KARNAUGH DIAGRAMThe values inside the squares are copied from the output column of the truth table, therefore there is one square in the map for every row in the truth table. Around the edge of the Karnaugh map are the values of the two input variable. A is along the top and B is down the left hand side. The diagram below explains this:
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KARNAUGH DIAGRAMThe values around the edge of the map can be thought of as coordinates. So as an example, the square on the top right hand corner of the map in the above diagram has coordinates A=1 and B=0. This square corresponds to the row in the truth table where A=1 and B=0 and F=1. Note that the value in the F column represents a particular function to which the Karnaugh map corresponds.
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Utlesnings-regler for 2 variable
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EKSEMPEL 1
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KARNAUGH DIAGRAM
EKSEMPEL 1
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KARNAUGH DIAGRAM
EKSEMPEL 1
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KARNAUGH DIAGRAM
•Note that values of the input variables form the rows and columns. That is the logic values of the variables A and B (with one denoting true form and zero denoting false form) form the head of the rows and columns respectively. •Bear in mind that the above map is a one dimensional type which can be used to simplify an expression in two variables. •There is a two-dimensional map that can be used for up to four variables, and a three-dimensional map for up to six variables.
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KARNAUGH DIAGRAM
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KARNAUGH DIAGRAMReferring to the map above, the two adjacent 1's are grouped together. Through inspection it can be seen that variable B has its true and false form within the group. This eliminates variable B leaving only variable A which only has its true form. The minimised answer therefore is Z = A.
The minimised answer therefore is Z = A
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KARNAUGH DIAGRAMVed å bruke Boolsk algebra til å forenkle, får vi:
Z = A + AB Z = A( + B) Z = A
Variable B blirredundant i henhold tilBoolean Theorem T9a.
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EKSEMPEL 2
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KARNAUGH DIAGRAM.
EKSEMPEL 2
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KARNAUGH DIAGRAM.
EKSEMPEL 2
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KARNAUGH DIAGRAMPar med 1‘ere er gruppert som vist under og den forenklede funksjonen blir oppnådd ved å bruke følgende trinn:
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KARNAUGH DIAGRAMPar med 1‘ere er gruppert som vist under og den forenklede funksjonen blir oppnådd ved å bruke følgende trinn:
Note that two groups can be formed for the example given above, bearing in mind that the largest rectangular clusters that can be made consist of two 1s. Notice that a 1 can belong to more than one group.
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KARNAUGH DIAGRAMPar med 1‘ere er gruppert som vist under og den forenklede funksjonen blir oppnådd ved å bruke følgende trinn:
The first group labelled I, consists of two 1s which correspond to A = 0, B = 0 and A = 1, B = 0.
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KARNAUGH DIAGRAMPar med 1‘ere er gruppert som vist under og den forenklede funksjonen blir oppnådd ved å bruke følgende trinn:
Put in another way, all squares in this example that correspond to the area of the map where B = 0 contains 1s, independent of the value of A. So when B = 0 the output is 1. The expression of the output will contain the term:
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KARNAUGH DIAGRAMPar med 1‘ere er gruppert som vist under og den forenklede funksjonen blir oppnådd ved å bruke følgende trinn: For group labelled II corresponds to the area of the map where A = 0. The group can therefore be defined as . This implies that when A = 0 the output is 1. The output is therefore 1 whenever B = 0 and A = 0 Hence the simplified answer is Z = +
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KARNAUGH DIAGRAMFORENKLINGSREGLER
KARNAUGH DIAGRAM
FORENKLINGSREGLER
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KARNAUGH DIAGRAMKarnaugh diagrammet bruker følgende regler til å forenkle funksjonsuttrykk ved å gruppere sammen celler som ligger ved siden av hverandre som inneholder 1:
Grupper kan ikke inneholde celler som inneholder en nullGrupper kan være horisentale eller vertikale, men ikke diagonale:
1:
2:
3:
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KARNAUGH DIAGRAMOversikt over de 8 forenklingsreglene:
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KARNAUGH DIAGRAMThe Karnaugh diagrammet bruker følgende regler til å forenkle funksjonsuttrykk ved å gruppere sammen celler som ligger ved siden av hverandre som inneholder 1:
1:
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KARNAUGH DIAGRAM
2:
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KARNAUGH DIAGRAMRIKTIG
3:
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KARNAUGH DIAGRAM
4:Merk at ingen Boolske lover er brutt, men uttrykket blir ikke minimalisert.
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KARNAUGH DIAGRAM
5:
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KARNAUGH DIAGRAM
6:
Grupper skal overlappe hvis mulig:
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KARNAUGH DIAGRAM
7:
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KARNAUGH DIAGRAM
8:
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OPPGAVER!
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KARNAUGH DIAGRAM
OPPGAVE 1:
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KARNAUGH DIAGRAM.
OPPGAVE 1: LØSNING
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KARNAUGH DIAGRAM.
OPPGAVE 1: LØSNING
By using the rules of simplification and ringing of adjacent cells in order to make as many variables redundant, the minimised result obtained is
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KARNAUGH DIAGRAM.
OPPGAVE 1: LØSNING
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KARNAUGH DIAGRAM
OPPGAVE 2:
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KARNAUGH DIAGRAM.
OPPGAVE 2: LØSNING
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KARNAUGH DIAGRAM.
OPPGAVE 2: LØSNINGBy using the rules of simplification and ringing of adjacent cells in order to make as many variables redundant, the minimised result obtained is
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KARNAUGH DIAGRAM.
OPPGAVE 2: LØSNING
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KARNAUGH-DIAGRAM !
Kompendium side 20v/ Espen Aamodt
Kompendium side 20
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Kompendium side 20v/ Espen Aamodt
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Kompendium side 20v/ Espen Aamodt
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Kompendium side 20v/ Espen Aamodt
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Kompendium side 20v/ Espen Aamodt
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Kompendium side 20v/ Espen Aamodt
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Kompendium side 20v/ Espen Aamodt
![Page 57: 2016.10.30 karnaugh diagram - sae v.12 Sven Åge Eriksen Fagskolen Telemark](https://reader034.vdocuments.mx/reader034/viewer/2022042723/587490191a28abc62f8b9099/html5/thumbnails/57.jpg)
Kompendium side 20v/ Espen Aamodt
![Page 58: 2016.10.30 karnaugh diagram - sae v.12 Sven Åge Eriksen Fagskolen Telemark](https://reader034.vdocuments.mx/reader034/viewer/2022042723/587490191a28abc62f8b9099/html5/thumbnails/58.jpg)
Kompendium side 20v/ Espen Aamodt
![Page 59: 2016.10.30 karnaugh diagram - sae v.12 Sven Åge Eriksen Fagskolen Telemark](https://reader034.vdocuments.mx/reader034/viewer/2022042723/587490191a28abc62f8b9099/html5/thumbnails/59.jpg)
THE END !
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KARNAUGH DIAGRAMKarnaugh diagrammet bruker følgende regler til å forenkle funksjonsuttrykk ved å gruppere sammen celler som ligger ved siden av hverandre som inneholder 1:
Grupper kan ikke inneholde celler som inneholder en nullGruppene må inneholde 2, 4, 8, 16 osv antall enere (2 )RIKTIG RIKTIG
Merk at ingen Boolske lover er brutt, men uttrykket blir ikke minimalisert.
![Page 61: 2016.10.30 karnaugh diagram - sae v.12 Sven Åge Eriksen Fagskolen Telemark](https://reader034.vdocuments.mx/reader034/viewer/2022042723/587490191a28abc62f8b9099/html5/thumbnails/61.jpg)
![Page 62: 2016.10.30 karnaugh diagram - sae v.12 Sven Åge Eriksen Fagskolen Telemark](https://reader034.vdocuments.mx/reader034/viewer/2022042723/587490191a28abc62f8b9099/html5/thumbnails/62.jpg)
KARNAUGH DIAGRAMForenkling av funksjonsuttrykk vha KARNAUGH DIAGRAM
http://www.talkingelectronics.com/te_interactive_index.html