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Created it, 06/09/09
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2. - RESEARCH OF THE DIAGRAMS IN SEQUENTIAL LOGIC
This chapter presents a certain number of information which, if they are very important in automatism and somewhat exceed the framework of this lesson, is not essential to the comprehension of the numerical electronic systems which will follow.
2. 1. - SEARCH FOR A DIAGRAM BY THE METHOD OF THE PHASES
Let us suppose that one lays out, to order the lighting of a lamp L, of two push-buttons, one called “m” or “walk” and other “a” or “stop”.
Figure 5 represents the table of operation of the assembly.
We can see by examining figure 5 which there exists for two identical combinations of the variables of entries “a” and “m” two logical states different for L (combinations ringed in red).
This is new. We cannot thus solve the problem by the traditional combinative method. Indeed, the table of Karnaugh does not admit that a value 1 or 0 per box or combination of the variables of entries.
This is why it is necessary to introduce a secondary variable still called memory which makes it possible to preserve the memory of the former events and to establish a chronology. In electric technology, this memory is materialized by a relay X which is a variable of exit and its contact x which is a new variable of entry.
The former state of the exit (taking into account the introduced delay) is thus taken into account like an additional variable of called entry secondary variable. It is called internal variable, in electronics.
By convenience, we will take X = L, which is not obligatory; indeed, one could also imagine X = f (L).
We can rewrite the table of operation in the way indicated figure 6.
In theory 2, we saw the use of the tables of Karnaugh in whom the variables of entries were independent of the exits. However here it is not any more the case bus L = X = x and a variation of L involves a variation of x.
We have to make here, into sequential, with circuits rebouclés on themselves for which one cannot use the method of Karnaugh.
From the new table of operation where each combination of the entries determines a stable state of the exits, we can represent by means of a graph called chronogram the evolution of m, a, x, X and L according to time (figure 7).
The chronogram of figure 7 takes account of the delay with the rise of relay X, indeed, it exists a time q1 during which the reel of relay X is traversed by a current whereas its normally-open contact is not closed yet. There is also a delay q2 when the contact falls down. This does not appear in the table of operation of figure 6.
One will call, transients, times q1 and q2 of joining and takeoff of contact x of relay X.
We see that L
will be lit during the phases : transient 1,
OR stable state 
,
OR stable state 
.
We can thus write by looking at the
chronogram for 1 ,
that L = X will be equal to :
However, as we saw
in a preceding theory : a + 
b
= a + b donc X =
(m + x). (Return
continuation)
The diagram of figure 8 materializes the found equation.
The logigramme is represented by figure 9.
By examining figure 9, we can notice that the delay between X and x does not occur ; however, there is a delay due to the time of transit through the door OR and carries it AND which conditions the loop of reaction figure 9.
These concepts of time of transit are specified in theory 5 like in technology 4 entitled “Summary Digital and Fundamental Technology”.
Knowing that according to the theorem of MORGAN one can replace one AND by NOR at the complémentées entries and that in addition one OR can be replaced by an NOR follow-up of a reverser, the diagram becomes that of figure 10.
The simplified diagram is then that of figure 11.
The circuit obtained is an on/off bistable rocker.
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A representation much more widespread of this same circuit is given figure 12. This assembly is still called FLIP-FLOP : We will speak again about it rather lengthily thereafter.
2. 2. - METHOD OF THE PHASES GENERALIZED
A method that we will not show here because it largely exceeds the framework of this lesson makes it possible to replace algebraic simplification by tables of Karnaugh.
This wide method of the phases can apply to all the sequential devices.
One calls phase the time during which proceeds a transitory or stable state.
2. 2. 1. - DIAGRAM OF THE PHASES
The diagram of the phases is represented by a squaring (figure 13).
a) To the higher part of each column, one indicates the number of the phase. Whatever the duration of the phases (a few nanoseconds or a few seconds) the columns will be identical.
b) One indicates, figure 14, opposite each line, successively in the order of their implementation :
Variables of entries in small letters.
Bodies of excitation located by capital letters.
Secondary variables of transfer in the order of their use represented by the
tiny ones.
Exits in capital letters.
For each variable of entry and each secondary variable of transfer, one defines a weighting p which is a number such as for the first variable of entry p = 21 - 1 = 1, for the second variable p = 22 - 1 = 2…, for Nième variable p = 2n - 1.
In the example already solved previously, the table of the phases is represented figure 14.
2. 2. 2. - LAYOUT
OF THE DIAGRAM
Let us examine the table of figure 14.
Phase 0 represents by convention the at-rest state of the system.
The fatty features traced on the horizontal lines of the table corresponding to the variables, represent these variables or contacts with state 1 during each phase.
The horizontal discontinuous fatty features represent state 1 of the excitations in each phase.
With each time the modification of the state of a variable of entry (including the secondary variables) involves a change of state of an exit (including the excitations) this is materialized by a vertical arrow (in fat).
However, the dependences between excitations and variable secondary of transfer will be marked by oblique arrows to take account of the appreciable delay (time q = of transfer) introduced by the retro-coupling.
Weighting by phase Ph appearing in the table is the sum of weightings p of each variable to state 1 (entries and transfers), carried out vertically inside a given phase.
Rules of establishment of the diagram :
Only one variable must change
with each phase.
The secondary variables are always late of a phase compared to the state of
excitation.
An identical weighting in two phases but producing states different from the
excitations or exits introduces obligatorily a new secondary variable : it is
necessary to add an additional excitation and its transfer.
We will see now more in detail how we arrived at the development of the table of figure 14.
2. 2. 3. - RESEARCH OF THE
DIAGRAM OF THE PARAGRAPH 2. 1. BY
THE METHOD OF THE PHASES GENERALIZED
Figure 15 gives the table of operation relating to the case which interests us (figure 8).
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b) Construction of the diagram.
From the table of preceding operation, let us start to draw the picture of the phases of the system; this table represented figure 16 comprises three horizontal lines since there are two variables of entries a and m and an exit L.
The table can break up vertically in the following way :
Phase 0 :
Position rest a = 0, m = 0, L = 0
No body is actuated, it is not necessary to trace of feature in fat.
Weighting Ph of this phase is 0.
Phase 1 :
a = 0, m = 1, L = 1
The action to support on m lights the lamp. One traces a fatty feature for m and L. Weighting Ph is 2. The vertical arrow indicates that m = 1 involves L = 1.
Phase 2 :
a = 0, m = 0, L = 1
One prolongs the fatty feature of phase 1 for L. weighting Ph of this phase is 0.
CAUTION :
For phases 0 and 2, weighting is 0, but the variables of entries are in both cases with 0. As we announced previously, it is thus necessary to introduce a secondary variable to differentiate these two cases.
A complementary relay X and its secondary variable of transfer x allows this discrimination by change of the weighting of phase 2.
Figure 17 shows the new table of the phases obtained with these new data.
Phase 1 :
a = 0, m = 1, L = 1
The action of m causes the excitation of X (vertical arrow directed of m towards X).
The action of m also causes the lighting of L from where prolongation of the arrow vertically towards L.
The contact of transfer x ordered by X is late of a phase compared to the excitation (one traces an oblique arrow directed of X towards x).
If one slackens m during phase 1, one will let L die out. In order to mitigate this risk, x not being yet with 1, it is appropriate to prolong the action of m during the following phase to make sure that the impulse on m was memorized.
One will proceed always thus when a variable excites a new relay.
The weighting of phase 1 will be 21 = 2.
Phase 2 :
a = 0, m = 1, L = 1
Only one variable changes state compared to the phase, indeed, only x passes to 1 whereas m, as we decided in phase 1 was maintained to 1, X remaining stuck.
It is necessary to prolong the fatty feature of m, the dotted feature of X, the fatty feature of L. weighting Ph passes to 6.
Phase 3 :
a = 0, m = 0, L = 1
One slackens m (the fatty feature stops), excitation X is memorized by x (one prolongs the dotted lines on X and the features fatty on x and L). Between phases 2 and 3, only one variable changes state : m which is slackened.
Weighting Ph of this phase is 4.
Phase 4 :
a = 1, m = 0, L = 0
One supports on “a” what involves that X passes to 0 and L also. One thus traces an arrow vertical feature towards X and L.
x will follow X with a delay of a phase.
One traces an oblique feature of X towards x.
One realizes that if one slackens “a” during phase 4 whereas “x” did not change a state, a risk of operation can appear ; L being re-ignited owing to the fact that the action on “a” was not memorized by “x”, secondary variable of entry.
It is thus advisable to prolong the influence of “a” in the following phase.
Weighting Ph of this phase is 5.
Phase 5 :
a = 1, m = 0, L = 0
The action on “a” is prolonged, one traces the fatty feature. There is no more excitation of transfer. Weighting Ph of this phase is 1.
Phase 6 :
has = 1, m = 1, L = 0
This phase makes it possible to highlight the case where the action on “a” is prolonged and which one supports on “m”.
Two full features are traced. The priority is given to the stop, the action on “m” thus does not have an effect it is not necessary to trace other features.
Phase 7 :
To return to phase 0 (“a” and “m” slackened), it is necessary to pass by phase 7 so that only a variable commutates. One prolongs the action of “a”.
Phases 5 and 7 have same weighting and are identical, the action on “a” not involving any change.
By maintaining “m” and while slackening one “a” would then have fallen down on phase 1.
All the cases were considered, the diagram is thus complete.
c) Deduction of the equation of the exit (lamp L).
We saw in the chapter 2. 1. which it is possible to draw the equation directly from the diagram. But if several secondary variables are necessary (and it a number is unknown with the reading of the statement of the problem) this is practically impossible and the risks are more difficult to detect.
Let us use the table of Karnaugh.
The table is composed of 2n boxes (n representing the total number of predicated variables and secondary).
In order to avoid in electronics and electricity of the risks of operation, it is obligatory to practice groupings which are recut.
Let us number into decimal the boxes of the table of Karnaugh (figure 18)
according to the binary weight of the entries for example for the box :
Let us defer value 1 in the boxes whose
decimal number corresponds to weighting Ph
by phase for which X = 1.
The table of Karnaugh is drawn up for each excitation, here only one : X, therefore only one table. The table obtained is that of figure 18.
One draws from the table the following equation :
X = m
+ x
X = (m
+ x)
The found equation quite identical to that is obtained chapter 2. 1.
It is enough to make L = X and to draw the logigramme that you can find figure 9
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