|NOR function||Function OR EXCLUSIVE||Functional check of one OR EXCLUSIVE|
|NOR function EXCLUSIVE||Logical function identity|
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In this lesson, we will finish the examination of the switching functions basic such as functions NAND, NOR, OR EXCLUSIVE… and will solve some problems using the theorems of MORGAN and method of QUINE MAC CLUSKEY.
1. - FUNCTIONS DERIVED FROM THE FUNDAMENTAL FUNCTIONS
1. 1. - FUNCTION NAND (NOT - AND)
Circuit NAND although derived from the circuit AND (AND) is used more and more current that this one. Indeed in the beginning, it was technologically easier to realize and less expensive, which explains why it was the circuit most frequently used. However, this is not completely any more true bus progress technologies saw the prices and the performances of the whole of the circuits to evolve/move very quickly.
1. 1. 1. - CIRCUIT NAND
Its symbol is that of figure 1.
A circuit NAND is obtained by putting in series a door AND and a reverser as represented figure 2.
Let us study the relation existing between a, b and S ; for that let us start from a circuit AND follow-up of a NOT-circuit.
One obtains the truth table of circuit NAND by writing the equation initially c = a . b then S =.
The truth table of the circuit AND is represented figure 3 :
The truth table of the NOT-circuit is reproduced figure 4 :
One can easily deduce the truth table from it from the circuit NAND which is represented figure 5 :
The exit of a circuit NAND is in a logical state 0 only when the two entries are with state 1. It is enough that only one of the entries is to 1 so that the exit becomes 1.
Switching function NAND can be summarized by the Boolean equation according to the number of the entries.
The sign is found . symbolizing the product AND and bars it indicating the complementation.
1. 1. 2. - REPRESENTATION OF EULER (figure 6)
The function AND such as S = a . b is the intersection of A and B (blue hatched surface) whereas the reverse of a . b either is the whole of hatched surfaces red.
1. 1. 3. - ELECTRIC CIRCUIT (Return to the theory N°3 TS)
Figure 7 represents the electric circuit used to fulfill a function NAND with two variables of entries a and b, S being the exit.
The contacts used a and b are normally-open contacts, i.e. opened at rest. If we refer to theory 2, we see that the unit represented in red one AND is carried out with contacts so that a . b = c.
However, C inside the unit woven in green, is a relay whose reel is materialized by the sign . It orders the contact which is a normally-closed contact so that one can write the table of operation represented figure 8.
With positive logical convention, one can write :
Contact closed = 1 ; open contact = 0
Lit lamp = 1 ; extinct lamp = 0
Relay fed = 1 ; relay not fed = 0
From where the truth table represented figure 9 :
We see in this truth table that the exit is always to 1 except for the two entries with 1 or S = 0.
1. 2. - NOR FUNCTION (Return to the theory N°3 TS)
As for the door AND which becomes after addition of a reverser a door NAND, with a circuit OR follow-up of a reverser at its exit one obtains a NOR-circuit whose graphic symbol is that of the figure 10-a :
The NOR-circuit is equivalent to a circuit OR follow-up of a reverser as represented figure 10-b :
Let us study the existing relation between a, b and S, for that let us start from a circuit OR follow-up of a not-circuit.
One obtains the truth table of the NOR-circuit by writing the equation initially C = a + b then S = .
The truth table of the circuit OR is represented figure 11 :
The truth table of the NOT-circuit is represented figure 12 :
The truth table of the NOR-circuit can then be easily deduced (figure 13) :
The exit of a NOR - circuit is in a logical state 1 only when the two entries are to 0.
The NOR switching function can be summarized by the Boolean equation according to the number of the entries.
One finds the sign + symbolizing the logical sum OR and bars it indicating the complementation.
1. 2. 1. - REPRESENTATION OF EULER (figure 14)
The function OR such as S = a + b is the union of A and B (blue hatched surface) whereas the reverse of a + b either is the whole of hatched surfaces red.
1. 2. 2. - ELECTRIC CIRCUIT
Figure 15 represents the electric circuit used to fulfill an NOR function.
The contacts used a and b are normally-open contacts, i.e. opened at rest. If we refer to theory 2, we see that the unit represented in red one OR is carried out with contacts so that a + b = C.
However, C inside the green hatched unit is a relay whose reel is materialized by the sign. It orders the contact which is a normally-closed contact so that one can write the table of following operation (figure 16) :
Taking into account positive logical convention, one can deduce the truth table from figure 17 :
We see in this truth table that the exit is always to 0 except for the two entries with 0 or S = 1.
1. 3. - FUNCTION OR EXCLUSIVE (EXCLUSIVE OR)
1. 3. 1. - TRUTH TABLE
The function OR exclusive is more complex than the whole of the functions than we have just analyzed.
We point out the truth table of the function OR inclusive (figure 18) :
We see that the exit S of the operator OR was to 1 when a OR b or both were to 1.
In the case of it OR exclusive, it will not be the same. Indeed, for S = 1, it will be necessary that a OR b is to 1 exclusively, i.e. S will not be to 1 when a and b are simultaneously to 1. The OR exclusive one as its name indicates it excludes this possibility.
Figure 19 shows the truth table of OR exclusive :
It is written whereas S = a b that one states S equal a OR exclusive b.
The sign is the symbol of OR exclusive in the logical equations.
In the diagrams, one uses the graphic symbol represented figure 20.
1. 3. 2. - REPRESENTATION OF EULER (Figure 21)
The function OR exclusive such as S = a B is the surface hatched such as S = 1 for a = 1 or b = 1, i.e. the meeting of sets A and B other than surface common to A and B.
1. 3. 3. - DIAGRAM CARRIED OUT WITH SIMPLE LOGICAL OPERATORS
As we said to the beginning of this chapter, the function OR exclusive is more complex than functions NAND or NOR.
Let us try by the graphic reasoning to find an equation of S such that the sign disappears to bring back the function OR exclusive to functions AND and OR traditional or functions reversed.
We see that S is made of two distinct surfaces :
- S1 surface included inside A
- S2 surface included inside B.
Let us represent figure 22 S1 surface.
The unit hatched in red is the complement of B compared to Â.
We see that S1 surface, represented by the unit hatched vertically in black, is the intersection of and A.
We can thus write S1 = a bus if A and B are the sets for which a and b are respectively to 1, S1 will be the unit for which a = 1 and b = 0.
We can easily while following the same reasoning to see on figure 23 as S2 = b since S2 is the intersection of and B.
From the truth table of the function OR exclusive (figure 19), we can draw up the table of Karnaugh of this function represented figure 24.
In this respect, it is absolutely fundamental to remember how one carries out a table of Karnaugh starting from the truth table.
The values of the exit S for a combination of a and b given are deferred in the box being to the intersection of the values of a and b considered, related to the sides of the table. We advise you to possibly return to theory 2 in order to re-examine if necessary the tables of Karnaugh.
One deduces some :
red grouping S = a,
green grouping S = b,
From where one can write :
S = b + a
This confirms the result obtained by the graphic decomposition carried out thanks to the representation of Euler.
We can thus now build the diagram of figure 25 which represents a function OR exclusive fulfilled starting from functions AND, OR and NOT.
1. 3. 4. - FROM THE LOGIC DIAGRAM AND RECONSTITUTION FUNCTIONAL CHECK OF THE TRUTH TABLE
On figures 26 a, b, c and d are deferred the four combinations being able to be taken by two entries a and b.
On each one of these figures the various logical levels in entry and exit of each door are deferred.
We can recapitulate the four cases of figure 26 in the table of Karnaugh of figure 27. The table obtained corresponds well in all points to the table of the function OR exclusive.
This last exercise could appear useless to you, actually it of it is not nothing because it is necessary to be smelled perfectly at ease in the tables of Karnaugh, the truth tables and the diagrams in order to be able to make the transformation in a direction or the other without error and possibly during the development of a circuit complexes to be checked by using each time a different method.
1. 4. - NOR FUNCTION EXCLUSIVE (EXCLUSIVE NOR)
The exclusive NOR - circuit whose symbol is represented figure 28 obtains in a way identical to the NAND and NOR but by using an exclusive-OR gate follow-up of a reverser (figure 29).
The logical equation of NOR exclusive at two entries is . This equation indicates well that the NOR exclusive one carries out the operation OR exclusive and complémente the result (bars above).
You can use yourself the process followed for doors NAND and NOR in order to find the truth table. We will restrict ourselves to indicate this table, figure 30, for NOR exclusive at two entries.
A NOR-circuit exclusive, like an exclusive-OR gate, is used to detect the presence of a single signal either a, or b (it is said that the signal a or b is present when it has a logical level 1).
In the case of the exclusive NOR-circuit, one can also check the equality between two signals has and B. Indeed, S is to 1 for a and b simultaneously with 1, but also for a and b simultaneously with 0.
1. 5. - LOGICAL FUNCTION IDENTITY
The logical function identity does not exist as such in the form of integrated circuits. However, it can render services in certain automatism to create safety measures.
Let us suppose, for example, that on the one hand a variable a indicates the direction of rotation of an engine (a = 1 “front walk”, a = 0 “reverse gear”) and, that on the other hand for reasons of safety one must check that a tool is well positioned on the machine according to the selected direction of rotation (b = 1 “front size”, b = 0 “back size”).
It is seen immediately that by reason of safety, it is necessary that the startup is authorized only for a = b = 1 OR for a = b = 0. Matérialisation of the authorization of startup will be carried out for example by means of a lamp.
The equation of such a logical identity is :
S = ab +
That one writes :
S = a b
Figure 31 gives the electric diagram of this logical identity carried out with contacts.
You know now that one AND materializes by contacts in series and one OR by contacts in parallel.
The line of normally-open contacts a, b thus materializes ab whereas the line of normally-closed contacts materializes . The two lines of contacts being in parallel, one thus has :
S = ab +
Figure 32 gives the electric diagram of the same identity logical but carried out this time using electronic doors.
The realization of such a diagram is even simpler, one establishes ab and by means of two doors AND (respectively doors n°1 and n°2), and being obtained before by two reversers.
The circuit OR (n°3 carries) carries out as for him the final operation to obtain :
S = ab +
We will now study negative logic and the theorem of MORGAN on another page in order not to encumber this one.
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