2. 1. -GRAPHIC REPRESENTATION OF ALTERNATIVE CURRENT
Let's try to determine the shape of the alternating current for a complete rotation of the induction flux. We already know that at time t = 0s and that at time t = 4s are intensity is zero, we have to find the different intensities taken by the current between these two times.
Note in this regard Figure 3, the current is zero when the lines of the induction flow are horizontal (Figures 3-a and 3-e), while it circulates in all other cases. On the other hand, from what we know about the LENZ law, we can say that the intensity I of the induced current is maximum when the flux embraced by the turn is zero (case of Figures 3-c and 3-g).
From these four known positions, we deduce that the intensity I of the induced current is a function of the angle formed between the lines of the inducing flux and the horizontal.
If we symbolize the set of induction lines of the flux by a vector (oriented straight portion having an origin and an end and whose length is a function of the intensity of the force that it represents), this allows us to elaborate the different cases of figure 4.
The origin of the vector is the 0 point in the center of the turn. Let's now rotate the vector around point 0.
The nine cases of Figure 4-a determine nine positions of the vector in relation to the horizontal. By lowering the end of the vector on an axis perpendicular to the horizontal and passing through the point 0, we obtain nine line segments d0 to d8 (drawn in red in Figure 4-a) whose length is a function of the angle formed between the induction flux and the horizontal.
These nine segments are reported alone in Figure 4-b and their respective ends are indicated by the points A, B, C, D, E, F, G, H and I.
When the vector is horizontal, the line segment that it determines is zero, this is the case of segments OA (d0), OE (d4) and OI (d8). Figure 4-b, points A, E and I coincide with the horizontal.
When the vector is perpendicular to the horizontal, the line segment that it determines is maximal and corresponds to the length of the vector, this is the case of the segments OC (d2) and OG (d6).
Based on these data, determine the pace of the alternating current.
At time t = 0s, point A coincides with point 0 (d0) : the intensity of the induced current is zero. from t = 0s to t = 2s, the induced current increases and its intensity is maximum at t = 2s : the line segment OC (d2) is also maximal. Between these two positions, the vector determines at time t = 1s the line segment OB (d1). from t = 2s to t = 4s, the induced current decreases and its intensity is zero at t = 4s : the point E coincides with the point O (d4).
Between these two positions, place at time t = 3s the point D corresponding to the segment OD (d3) at t = 4s, the current reverses its direction of circulation from t = 4s to t = 6s, the induced current increases again and its intensity reaches a second maximum at t = 6s. The right segment OG (d6) is maximal. Between these two positions, place at time t = 5s the point F, determined by the segment OF (d5). from t = 6s to t = 8s, the current decreases and its intensity vanishes at t = 8s : the point I coincides with the point O (d8). at t = 8s, the direction of the current is reversed again and a second cycle begins.
The nine points A, B, C, ... that we have just determined and which are shown in Figure 4-b only show the value of the intensity of the current taken every second, but nothing indicates the value taken by it at every moment.
To obtain the desired result, it is sufficient to connect the nine known points together, taking into account that between each of these points the current does not vary linearly. The points together give the curve shown in Figure 4-c. A curve of this pace is called sinusoid and we will say that the alternating current has a sinusoidal pace.
The sinusoid, which can be used to indicate at any time the intensity taken by an alternating current, is therefore used for the graphical representation of the sinusoidal alternating currents, an example of which is illustrated in Figure 5.
The dashed horizontal line of Figure 4-c is replaced by a straight line where the seconds are reported. This line constitutes the time scale on which a centimeter corresponds to a time of one second. This line is symbolized by the symbol t (s) located at its right end. Reading time on the right should not seem strange especially when you know that we read on a circle the time indicated by a clock or a watch.
Since the intensity of the current is materialized by the distance between each point of the sinusoid and the horizontal line, it is very useful to note the values of this intensity on a vertical line perpendicular to the time scale. This line is the scale of the currents on which a centimeter corresponds to an ampere. This line is symbolized by the symbol I (A) at its upper end. Note that the two lines intersect at point O in Figure 5. This point constitutes the origin of the time scale, but by that of the scale of the currents because on it appear figures preceded by a sign "-", which should not surprise you since the alternating current flows in two different directions.
The two perpendicular lines are also called axes and constitute with the sinusoid a Cartesian diagram. Figure 5 are reported two examples of use of this diagram.
The first example allows us to determine the intensity of the current after 1.5 seconds. For this we report on the time axis the point A1, 1.5 cm after the point O (1 cm = 1s). From this point A1, let us raise the perpendicular, this one intercept the sinusoid at the point A. The distance which separates the points A1 and A represents the intensity which we want to know. This intensity can be read on the axis of currents by drawing from point A a horizontal intercepting the axis of currents at point A2. Knowing that on this axis 1 cm = 1A, it is enough to measure the distance OA2 and to convert the result into ampere. We find in this case 1.9 cm so we can say that after 1.5 seconds the intensity of I is 1.9 A.
The same procedure can be adopted for our second example, in which we want to know the intensity of the current after 7.8 s. The difference with the previous case is that the point B, obtained on the sinusoid, determines a point B2 situated on the axis of currents below the point O. Since the distance OB2 is 0.7 cm, the intensity of the current is - 0.7 A ; the sign "-" indicates that this current flows in the opposite direction relative to the current considered in the previous example.
Now that we know how to represent an alternating current, let's analyze its characteristics below, (paragraph 2. 2.).
2. 2. -PERIOD AND FREQUENCY OF ALTERNATIVE CURRENT
Until now, we have considered that the induction flux only performs a single rotation generating a current whose rate is represented in Figure 5.
If the induction flow continues its rotation and completes other turns, each in an identical time, it will generate at each turn a current of identical appearance to that of Figure 5. A complete turn of the flow determines a current cycle alternative.
The alternating current is a series of cycles all equal to each other ; having completed a cycle, it begins an identical following and this succession gives the curve represented figure 6-a.
As long as all the cycles are identical when we graphically represent an alternating current, we will draw only one as it was done figure 5
To facilitate our previous explanations, we hypothesized that the induction flow performed a complete rotation in 8 seconds but generally, in practice, it turns much faster ; to get closer to reality, we have in figure 6-a chosen a rotation time of 1 second.
To allow a correct analysis of a complete cycle, we had to change scale and figure 6-a, 1 cm of the time scale corresponds, no longer to 1 second, but to 0.25 s. On the other hand, we kept the same scale for the current, that is 1 cm for 1 A.
From the comparison of Figures 5 and 6-a, it appears that to achieve in both cases an identical cycle, the current of Figure 6-a puts 1 second while it takes 8 seconds in Figure 5.
We have just established here a second characteristic of the alternating current, because for two alternating currents to be equal, it is not enough that they take the same intensity values, but these values ??must be equal at every moment, otherwise says they perform a complete cycle at the same time.
The time taken by the alternating current to complete a complete cycle is called the period (symbol T). The unit of the period is therefore the second (symbol s).
Figure 6-a, the period of alternating current shown is 1s ; in this same figure, we find that the period is divided into two equal parts called positive alternation and negative alternation. The two alternations owe their name to the values of the currents they determine on the scale of the current (positive numbers for the positive alternation and negative numbers for the negative alternation).
To divide a period into two alternations does not seem obvious. However, this becomes a necessity when one knows that the current flows in one direction during the positive and the opposite halfway during the negative alternation and that, as we will see in the next lessons, some electronic components react differently. according to the direction of the current passing through them.
Now let's look at the alternating current in Figure 6-b whose period is 0.5 s.
If we compare it with Figure 6-a, we note that it performs two cycles while that of Figure 6-b accomplishes only one. From this, we deduce that for a given time, the shorter the period, the greater the number of cycles.
An alternating current can also be characterized by the number of cycles that it performs in one second. Frequency (symbol F) is the number of cycles performed by an alternating current in one second.
The frequency F is measured in cycles per second, to which is given the name of hertz (symbol Hz) in tribute to the German physicist Heinrich HERTZ (1857-1894) whose experiments showed the propagation of electromagnetic waves.
The current of figure 6-a has a frequency of 1 Hz since it completes 1 cycle in one second, while that of figure 6-b has a frequency of 2 Hz since in the same second it carries out 2 cycles.
The alternating current that we use in homes for domestic purposes has a frequency of 50 Hz, which means that it completes 50 cycles in 1 second.
In some particular devices (such as radio receivers or televisions), currents exist which can be thousands or even millions of cycles per second, hence the need to use frequency quantization not only for hertz but for kilohertz (symbol kHz) which is equal to 1 000 Hz or the megahertz (MHz symbol) equal to 1 000 000 Hz. Of course, currents of such high frequency are no longer obtained in the manner previously described, that is to say say by rotating an induction flow : it is indeed not possible to make an inductive circuit perform thousands or millions of revolutions per second. For the production of these so-called high-frequency (HF) currents, we will resort to particular circuits : the electronic oscillators.
As we have just seen, the period and frequency are intimately linked. This union is sealed by the following relation :
From this relation, we see that the period and the frequency are two inversely proportional quantities.
Let's apply this formula to calculate the Alternating Current period of the sector :
T = 1 / F = 1 / 50 Hz = 0,02 s = 20 ms (milliseconds).
We now know that the Alternating Current of the sector puts 20 ms to complete a complete cycle. Similarly, if we know the period T of an alternating current, for example 10 µs, by applying the relation in its form F = 1 / T we deduce that this current has a frequency of 100 kHz.
There is a third parameter characterizing the alternating current : it is about its pulsation (symbole reads oméga) and which is expressed in radians per second (rd / s). The pulsation is obtained using the relation :
The pulsation characterizes the speed of rotation of the vector symbolizing the inducing flux in Figure 4-a. This magnitude, as you will see in future lessons, is mainly needed for calculations relating to electrical circuits powered by an alternating voltage.
To represent an alternating current, it is necessary to indicate its frequency and the pace of the sinusoid which gives at each instant the intensity of the current.
We observe, however, that the intensity of the current varies constantly and we do not know what value to choose precisely to characterize this intensity : if we consider the alternating current shown in Figure 7, the logic would dictate to opt for the maximum value reached by the current during a period.
This value is reached twice by the current : a first time in the middle of the positive half cycle and a second time in the middle of the negative half cycle. These two values are respectively called IM and - IM. This value taken by the current is called the maximum value of the alternating current.
The Cartesian diagram in Figure 7 immediately gives the IM value of the current represented, which is 3 A. On the other hand, this same diagram enables us to determine the frequency of this current :
Indeed, his period T = 0,2 s --------------) F = 1 / 0,2 = 5 Hz
The alternating current of Figure 7 is characterized by the following two parameters :
IM = 3A
F = 5 Hz
However, we can determine at any moment the value of the current as a function of the angle of rotation of the inducing flux. Indeed, a sinusoidal function has for equation y = ax in which (reads sinus phi)
For the case that interests us,
is the angle described by the inductor flux and the horizontal (figure 4-a) while (a) is the IM value of the current and that (y) gives the instantaneous value (i) of the current at theangle
considered.
We can write :
i = IM sin
Using Figures 4-a and 7, apply this equation to determine some current values.
= 0°, sinus 0° = 0. The current i is therefore zero, we are at the beginning of the cycle.
= 90°, the flow has completed a quarter turn, sin 90° = + 1 ------) i = IM
= 3 A.
= 180°, the flow completed a half-turn, sin 180° = sin 0° = 0. The current is zero.
= 270°, the stream has completed three quarters of a turn. sin 270° = - 1 ---------) i = - IM = - 3 A.
= 360°, the flow has completed a complete turn. sin 360° = sin 0° = 0. The current is zero.
The value of the sine of an angle is given by a table called a trigonometric table. This gives the sine of an angle whatever its value, and we can know the value of the current regardless of the position of the flow relative to the horizontal.
This gives the sine of an angle whatever its value, and we can know the value of the current regardless of the position of the flow relative to the horizontal.
Since the maximum value of the current is reached twice by it, we understand that this value may not be very suitable for characterizing an alternating current, if only to determine the thermal effect produced by such a type of current. This effect of the current is independent of its direction of circulation : indeed, for there to be heat production, it is enough that a current crosses a resistance and it does not matter that it circulates in one direction or the other. The heat production is the same for each of the two alternations.
To evaluate the thermal effect of the current, we can refer to an entire period. We can then apply the formula w = R x I2 xt in which t = T. While it is easy for us to know R and T, we do not know what value to choose because it changes constantly and consequently value released at every moment. We must therefore introduce a new characteristic of the alternating current which is its effective intensity symbolized by the acronym Ieff.
The effective intensity of an alternating current is expressed as follows :
It is the intensity of a direct current which would produce in the same resistance the same quantity of heat as this alternating current.
NOTE :
In view of this definition, the effective intensity can, according to the international system of measurement S.I., be symbolized by the simple letter I.
There is a relation between the effective intensity and the maximum intensity of a sinusoidal current.
This relationship corresponds to the relationship :
NOTE :
The demonstration of this relationship using mathematical knowledge coming out of our program, we will not talk about it. In addition to the maximum value and the rms value of an alternating current, there is also its average value (Imoy symbol) which is defined as follows :
The average intensity of an alternating current is the intensity of the direct current which would carry during the period of time considered (one period) the same quantity of energy.
In the case of a sinusoidal alternating current, the average value of an alternation is given by the relation :
If we calculate the average value for a complete period, we find a value of zero.
To explain this succinctly, it is enough to think that the electrons, during the positive alternation, move in one direction, and that during the negative alternation these same electrons make the same way in the other direction and come back to their starting point ; therefore the amount of energy supplied by the generator is zero (but not that necessary for its operation, which it is very real).
The type of current that we have chosen for our explanations is an alternative current of zero average value like that delivered by the sector, but as you will see thereafter there are alternative currents of average value non zero hence the need for have introduced this notion.
Throughout this lesson, we have seen how to graphically represent an alternating current and how from this graph it is possible for us to determine all the characteristics of such a current.
The quantities relating to the sinusoidal alternating current are grouped in the table of Figure 8
In the next lesson, we will not analyze the alternating current but the alternating voltage and we will see the relations which link these two quantities according to the type of circuits which they feed.
Period and frequency of the alternating current
reads oméga) and which is expressed in radians per second (rd / s). The pulsation is obtained using the relation :
(reads sinus phi)
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