Vectorial
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Created it, 05/10/15
Update it, 06/02/19
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ALTERNATING VOLTAGE “1st PART”
We saw in the preceding lesson how a AC current graphically is represented, and which are the sizes which characterize it ; we now will examine the alternating voltage.
1. - ALTERNATING VOLTAGE
To determine the form of the alternating voltage, we must remember that it is bound by the law of OHM to the current which traverses resistance ; as we know already the form of the AC current, it is easy for us to seek the voltage waveform which determines the passage of this current in resistance.
Let us see, for example, how one can graphically represent the alternating voltage which makes circulate the AC current in a circuit including/understanding a resistance of 5 ohms (figure 1-a).
We observe initially that at the beginning of the period (0 s), with half (0,1 s) and at the end of the period (0,2 s), the current is null : if at these moments, the current does not circulate in the circuit, that means that the power provided by the generator is also null.
When, on the contrary, the current reaches the maximum value (Imax) after 0,05 s, and (- Imax) after 0,15 s, the tension also reaches at these moments the corresponding maximum value Vmax and - Vmax (certain books the electronic ones will note Imax by IM and Vmax by VM).
Since the maximum value of the current is three amps, it is necessary to run this current in a resistance of five ohms, a tension which, according to the law of OHM, is given by the product of the value of resistance by that of the current: the maximum value of the tension is thus of 5 x 3 = 15 V.
In the same way, we can obtain the value of the tension in another unspecified moment, by multiplying the value of resistance by that of the current taken in this moment.
The values thus obtained are deferred on the diagram of the figure 1-b. On the horizontal axis of this diagram, times are indicated same manner as for the current, i.e. each centimetre corresponds to 0,025 s. On the vertical axis of the diagram, the tensions (U) expressed in volts (V) are now indicated ; 1 cm corresponds to 5 V, like specifies it the inscription “1 cm = 5 V” deferred in top and on the right.
As the Imax value of the current is reached at the end of 0,05 s, the Vmax value of the tension corresponds to this same moment ; one proceeds in the same way for the value - Imax, but since this value is under the horizontal axis, the corresponding value of the tension - Vmax is also under this axis.
If the other values of the tension are in the same way deferred, for example those taken at the end of 0,025 s, 0,075 s, 0,125 s and 0,175 s, one obtain a certain number of points which make it possible to trace the line of the figure 1-b by joining them ; this curve indicates the alternative voltage waveform. As one could expect it, it is also a sinusoid, and we can thus say that the alternating voltage has, it also, a sinusoidal form.
(To facilitate the task to you, we defer the same diagram below).
As we already did for the current, we can thus regard the alternating voltage as a succession of identical cycles all, in each one whose the same values are always repeated.
Time that the tension puts to achieve a cycle is called period of the alternating voltage and it is also divided to him into two equal half-periods, one positive and the other negative one; in the first, the values of the tension are indicated by positive numbers, while in the second, the same values are indicated by negative numbers.
This fact indicates to us that the generator reverses its polarities with each half-period ; on the figure 1-b, the values of the power provided by the generator are indicated by positive numbers when this one has its polarities as on the figure 1-c, while the values of the tensions are indicated by negative numbers when the generator inverts its polarities as on the figure 1-d.
Let us note that, after having indicated by A and B the poles of the generator, one considered that the positive current was that which left pole A of the generator, as on the figure 1-c, and which the negative current was on the contrary, that which entered the generator by pole A, as on the figure 1-d. One in the same way considered as the tension was positive or negative when the pole of the generator indicated by A was positive, as on the figure 1-c, or negative, as on the figure 1-d.
This process should be remembered because it will be still used in this lesson.
We observe now that the alternating voltage must have obviously the same period that the current, besides as one checks it immediately by comparing the figures 1-b and 1-a : the current and the tension achieve the same cycle in one second and they thus have the same frequency.
To characterize an alternating voltage, we must thus indicate, not only its period or its frequency, but also its value : as for the current, one indicates the effective value which one obtains while dividing per 1,41 the maximum value of the alternating voltage (valid only for one sinusoidal mode).
When we say, for example, that the tension of the network available in our dwellings has a value of 110 V or of 220 V, we refer to the effective value of this tension ; the tension indicated on the electricals appliance which function with the AC current is the effective value of the tension with which one must supply these apparatuses.
One calls effective value of a sinusoidal alternating voltage at the boundaries of a resistance R, the value of a tension continues of which the current would produce the same calorific effects as the sinusoidal AC current in same resistance R and during same time. In this way, one establishes an equivalence between the alternating voltage and the tension continues, with regard to the calorific effects produced by the currents which they put in circulation.
In practice, that means, for example, that soldering iron envisaged for a tension of 220 V can be fed either by an alternating voltage which has an effective value of 220 V, or by a continuous tension of 220 V. In both cases, the soldering iron produced the same quantity of heat because the alternating voltage makes circulate in its resistance a current whose effective value is equal to that of that which the tension makes circulate continues.
The D.C. current which circulates in the soldering iron can be obtained, according to the law of OHM, by dividing the tension continues by the resistance of the soldering iron; in the same way, if one divides the effective value of the alternating voltage applied to the soldering iron by his resistance, one can obtain the effective value of the current which feeds it.
We thus see that with the effective values, we can carry out calculations relating to the AC current as if it were about a D.C. current, not only with regard to the heat produced by the current, but also for the use of the law of OHM.
But it should well be remembered that equivalence between the AC current and the D.C. current is valid only with regard to the heating effect : that means that we can consider the AC current as a D.C. current only in the circuits formed exclusively of resistive elements which transform the electric power into heat.
In the preceding lessons, we saw other elements, the such condensers and the reels, which do not dissipate the electric power but store it by creating an electric or magnetic field respectively.
For the condensers and the reels, equivalence between the AC current and the D.C. current are not valid any more, since these elements do not give place to the production of heat. However, in the case of the circuits which include/understand condensers or reels, one also indicates the effective value of the tension and the current, but always in reference to the heating effect.
When one says, for example, that a reel is traversed by a AC current of an effective value of 2 A, one indicates a AC current which would produce, if it crossed a resistance, same quantity of heat as that produced by a D.C. current of 2 A.
We must thus recall us that, for the alternative and sinusoidal sizes, one always indicates the effective value, even when it acts of circuits in which no heating effect occurs.
As we saw previously, resistance offered by the resistive element to the AC current is obtained by dividing the effective value of the tension by that of the current, just as in continuous mode. That means that “resistances” have the same behavior in continuous mode that in alternative mode, because they offer in the passing of the two currents a resistance which is equal for the two modes to the report/ratio of the tension by the current.
On the contrary, the condensers and inductances will require to be examined separately according to the mode applied.
2. - VECTORIAL REPRESENTATION
OF THE ALTERNATIVE SIZESThe representation used until now, consisting in drawing the sinusoid which indicates the values of the tension or the current lasting a complete cycle, is not very convenient ; one thus adopted another system of representation which with the advantage of highlighting much properties of the alternative sizes which do not appear clearly on the drawing of the sinusoid.
The sinusoid representing an alternative size is drawn like illustrated figure 2, while making turn a segment length equal to the maximum of the sinusoid and by point by point deferring the ordinates of its end.
Thus to represent a tension or a AC current, it is enough to defer in an orthogonal reference mark projections perpendicular to the y-axis of the vector turning to the selected moments.
This method of layout bears the name of vectorial representation where the sinusoidal alternative sizes are obtained thanks to a vector characterized by :
- a length proportional to the maximum amplitude (or elongation) of the alternative and sinusoidal size ; if for example, one wants to represent a sinusoidal tension of Vmax value = 300 V, one can choose to take a scale with 1 cm for 100 V and to draw a vector length 3 cm corresponding to 300 V.
- a constant speed in anti-clockwise direction whose full rotation represents the period of the sinusoidal size. Thus the vector achieves in one second a number of revolutions equal to the frequency of the size considered.
- a position for which its ordinate represents the value of the size considered at this precise moment.
For better including/understanding this method, we give six examples are deferred figure 3.
The figure 3-a represents a sinusoidal AC current of maximum value Imax = 0,5 A corresponding to 1 cm on the y-axis. The vector representative of 1 cm length is at the origin of times (0s) in horizontal position, meaning that its ordinate is null, just like is the value of the sinusoid. This vector having carried out a full rotation, the values of its ordinate taken at moments precise and deferred in the orthogonal reference mark, will describe the sinusoidal shape of the current of the figure 3-a.
The figure 3-b represents a sinusoidal alternating voltage of maximum value Vmax = 20 V corresponding to 2 cm on the y-axis. The vector 2 cm length representative in advance of 90° compared to the origin of times (0s) is in driving position, meaning that its ordinate is maximum and positive, just like is the value of the sinusoid. This vector having carried out a full rotation, the values of its ordinate taken at moments precise and deferred in the orthogonal reference mark will describe the sinusoidal shape of the current of the figure 3-b.
The figure 3-c represents a sinusoidal AC current of maximum value Imax = 1,5 A where 1 cm corresponds here to (1 A) on the y-axis. The vector representative 1,5 cm length placed at 180° of the origin of times (0s) in horizontal position, is directed towards the left (contrary with that of the figure 3-a), meaning as well as the value of the sinusoid is null. Only this time, the sinusoid will begin with negative values when the vector carries out its full rotation in anti-clockwise direction ; the layout remains the same one as the precedents (figures 3-a and 3-b).
The figure 3-d represents a sinusoidal alternating voltage of maximum value Vmax = 50 V where 1cm corresponds here to 100 V on the y-axis. The vector 0,5 cm positioned length representative with 270° of the origin of times (0s) in driving position, is directed to the bottom (opposite with that of the figure 3-b) because the sinusoid begin with a extremum (maximum value) negative.
One realizes in the examples given that the vector is positioned at the place where the sinusoid starts to indicate its value to this precise moment (t = 0s). It should be known that the angle formed by the vector and the horizontal axis is reached according to a time (t) which will be deferred on the X-coordinate (horizontal axis), to see figure 2.
On the example of the figure 3rd, the vector of the Imax current = 0,5 A of the figure 3-a is represented with an advance of 30° compared to the origin of times. On the figure 3-f, the vector of the Imax current = 1,5 A is represented with an advance of 330° compared to the origin of times.
In general, it is more interesting, at a given moment, to compare the position of the vectors to know their value.
Figure 4 illustrates the vectors representative of the tension and the current in a purely resistive circuit like that of figure 1. Since the two sinusoids begin in the beginning from times, the two vectors horizontal and are superimposed.
The two vectors, turning around item 0 at the same speed, generate two sinusoids (V, I) of the same pace, obtained by the ordinate of the two revolving vectors.
When the values of two sizes reach at the same time the extremums (maximum and minimum) and join on the x-axis to cancel itself, one says these two sinusoids which they are in phase.
Thus when two alternative sizes of the same frequency are in phase, the representative vectors are superimposed and turn at the same speed.
With each turn the vector describes an angle of 360° ; if it achieves a turn in one second, its number of revolutions will be of 360° a second (360° / s), for “n” turns a second will be n x 360° a second.
Generally, the angles are expressed in radian : angles which intercept an arc of circle length equal to the ray. Like 360° correspondent to 2 n radians, one can express the number of revolutions by 2n x n radians a second.
A cycle of an alternative size corresponds to a
full rotation of the representative vector and the number of revolutions “n”
carried out in one second represents the number of cycles a second sinusoid,
still called frequency (F). The speed of the
representative vector will be called pulsation
(symbol 
omega-unit of measurement in rad / s)
and will be expressed by the relation :
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