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Signets : 
  Realization of the Diagram of Inverter Signaling        Realization of the Diagram of Signaling by means of electronic symbols    Footer  


Example of Practical Application of Karnaugh Tables :


4. - EXAMPLE OF PRACTICAL APPLICATION OF KARNAUGH TABLES

4. 1. - Suppose that when the alarm rings in the morning, we want to know how to dress before even opening the shutters.

For this we have a suitable device placed outside and including a thermometer and a barometer and which gives us the following indications : warm weather, mild weather, cold weather, rainy weather.

We decide that in all cases, we will put pants, shoes and a shirt.

It remains now to determine whether it is necessary to put in a jacket, a coat, a raincoat or take an umbrella.

We will call the input variables :

  warm weather    :  c

  nice weather      :  d

  cold weather      f

  rainy weather     :  p

The output variables will be materialized by lamps and will be called :

  jacket                  :  V

  coat                     :  M

  raincoat               :   I

  umbrella              P

It will be admitted that when it is soft it is a jacket, when it is cold it is put on a jacket and a coat and when it is hot, we put nothing. When it rains, we put a raincoat on the jacket or we take an umbrella when we already have a coat or when it is hot.

From these postulates, let's establish the truth table of the system (Figure 70).

We have reserved a column that corresponds to the impossibilities. Indeed, it can not be hot and cold for example.

Exemple_de_table_de_verite.gif

Let's draw Karnaugh's tables from the truth table for each output variable.

4. 2. - KARNAUGH TABLE RELATING TO THE JACKET   (Figure 71)

Tableau_de_Karnaugh_pour_la_veste.gif

Algebre_de_Boole_pour_la_veste.gif

This confirms Karnaugh's picture.

4. 3. - TABLE OF KARNAUGH RELATING TO THE COAT   (Figure 72)

Tableau_de_Karnaugh_pour_le_manteau.gif

Algebre_de_Boole_pour_le_manteau.gif

Which confirms Karnaugh's picture.

4. 4. - TABLE OF KARNAUGH RELATING TO THE RAINCOAT   (Figure 73)

Tableau_de_Karnaugh_pour_l_impermeable.gif

4. 5. - TABLE OF KARNAUGH RELATING TO UMBRELLA   (Figure 74)

Tableau_de_Karnaugh_pour_le_parapluie.gif

Algebre_de_Boole_pour_le_parapluie.gif

HAUT DE PAGE 4. 6. - REALIZATION OF THE INVERTER SIGNALING DIAGRAM

Inverter contacts will be used, which will be controlled by the appropriate device mentioned above, as shown in Figure 75.

Contacts_inverseurs.gif

In the example of Figure 75, depending on the position of the switch, we have : d = 1 (high), d = 0 (low) is D_BARRE, in such a way that d = doux and D_BARRE = non doux.

      a) - Diagram for the jacket (Figure 76)

Schema_pour_la_veste.gif

In the diagram of Figure 76, the case is shown Formule_de_la_veste.gif, that is to say that the lamp (take your jacket) is lit, since the weather is mild.

It's easy to understand that the lamp (take your jacket) will also be on for position switches Formule_de_la_veste1.gif, that is, mild weather.

      b) - Diagram relating to the mantle (Figure 77)

Using the same conventions, one can, knowing equality Formule_du_manteau.gif, establish the diagram.

Schema_pour_le_manteau.gif

      c) - Diagram relating to the raincoat (Figure 78) : Formule_a_l_impermeable.gif

Schema_pour_l_impermeable.gif

      d) - Diagram relating to the umbrella (Figure 79)

Algebre_de_Boole_pour_le_parapluie1.gif

Schema_pour_le_parapluie.gif

      e) - General scheme (Figure 80)

It represents the case where the lamps V and I are lit.

Schema_general.gif

In the truth table (Figure 70), we have provided a column called "impossibilities" or system failures.

There is, indeed, absolute impossibility that the indications hot and cold, soft and cold, warm and soft, etc ... are present simultaneously.

These cases are impossible or the system is down.

From these cases, a lamp can be lit to indicate that the system is not functioning properly.

      f) - Karnaugh Detection Default Table (Figure 81)

Tableau_de_Karnaugh_pour_la_detection_defaut.gif

From Karnaugh's table, let's establish the equation of D.

Formule_de_detection_de_defaut.gif

This leads us to the diagram of the fault indicator shown in Figure 82.

Schema_indicateur_de_defaut.gif

HAUT DE PAGE 4. 7. - REALIZING THE SIGNALING DIAGRAM USING ELECTRONIC SYMBOLS

      a) - Diagram of the signaling of the jacket (Figure 83)

Signalisation_veste.gif

      b) - Diagram of mantle signaling (Figure 84)

Formule_du_manteau.gif

Signalisation_du_manteau.gif

      c) - Diagram of the signage of the raincoat (Figure 85)

Formule_a_l_impermeable.gif

Signalisation_de_l_impermeable.gif

      d) - Diagram of the umbrella signaling (Figure 86)

Algebre_de_Boole_pour_le_parapluie1.gif

Signalisation_du_parapluie.gif

      e) - General scheme (Figure 87)

Schema_general1.gif

The general scheme will be constituted by the assembly of the other schemes so that the number of logical operators is minimum.

For this purpose, we examine the partial diagrams and knowing for example that d_barre is used in the partial diagrams of V and P can thus use only one inverter to produce the signal d_barre from the variable d. It is thus possible with a little common sense to gather all in a single scheme.

      f) - Diagram of the fault signaling (Figure 88)

Schema_de_la_signalisation_defaut.gif

In the next theory, we will examine on the one hand new methods for transforming Boolean expressions and on the other logical circuits derived from the three fundamental circuits AND, OR, NO, seen in this theory.

The circuits examined so far are the basis of all so-called "combinatorial" assemblies, that is to say the state (1 or 0) of the outputs depends only on the state of the inputs at the moment we examine mounting.

We finish this second lesson.


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