Period and frequency of the
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Value of the AC current |
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Created it, 05/10/15
Update it, 05/11/25
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AC CURRENT “2nd PART”
2. 1. - CHART OF THE AC CURRENT
Let us try to determine the pace of the AC current for a complete rotation of the flow of induction. We know already that in the time t = 0s and that at time t = 4s is intensity is null, it remains us to find the various intensities taken by the current between these two times.
Let us observe figure 3 on this subject, the current is null when the lines of the flow of induction are horizontal (figures 3-a and 3-e), whereas it circulates in all the other cases. In addition, of what we know about the Lenz's law, we can say that intensity I of the induced current is maximum when the flow embraced by the whorl is null (case of the figures 3-c and 3-g).
From these four known positions, we deduce that intensity I of the induced current is a function of the angle formed between the lines of inductive flow and the horizontal one.
If we symbolize the whole of the lines of induction of flow by a vector (portion of right-hand side directed having an origin and an end and of which the length is a function of the intensity of the force that it represents), this enables us to work out the various cases of figure 4.
The origin of the vector is item 0 located at the center of the whorl. Now let us make turn the vector around item 0.
The nine cases of the figure 4-a determine nine positions of the vector compared to the horizontal one. By lowering the end of the vector on an axis perpendicular to horizontal and passing by item 0, we obtain nine segments of right-hand side d0 to d8 (drawn in red on the figure 4-a) of which the length is a function of the angle formed between the flow of induction and the horizontal one.
These nine segments are deferred only on the 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 segment of right-hand side which it determines is null, this is the case of segments OA (d0), OE (d4) and OI (d8). Appear 4-b, the points A, E and I is confused with the horizontal one.
When the vector is perpendicular to the horizontal one, the segment of right-hand side which it determines is maximum and corresponds to the length of the vector, this is the case of segments OC (d2) and OG (d6).
According to these data, let us determine the pace of the AC current.
The nine points A, B, C,… that we have just determined and who are materialized on the figure 4-b indicate only the value of the intensity of the current taken to each second, but nothing at every moment indicates the value taken by this one.
To obtain the wished result, it is enough to connect between them the nine points known by taking account of the fact that between each one of these points the current does not vary in a linear way. The points thus joined together give the curve represented figure 4-c. A curve of this pace bears the name of sinusoid and we will say that the AC current takes a sinusoidal form.
The sinusoid, being able to be used at any moment to indicate the intensity taken by a AC current, is thus used for the chart of the sinusoidal alternating currents whose example is illustrated figure 5.
The dotted horizontal line of the figure 4-c is replaced by a line in continuous feature where the seconds are deferred. This line constitutes the time scale over which one centimetre corresponds to a one second time. This line is symbolized by the initials t (s) located at its right end. The fact of reading time on a line should not appear strange to you especially when it is known that we read on a circle the hour 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 on a time scale. This line constitutes the scale of the currents on which one centimetre corresponds to an amp. This line is symbolized by the initials I (A) located at its higher end. Let us note that the two straight lines are intercepted at the point O of figure 5. This point constitutes the origin of the time scale, but by that of the scale of the currents bus on this one figures preceded by a sign appear “-”, which should not astonish you considering the AC current circulates in two different directions.
The two perpendicular lines are also called axes and constitute with the sinusoid a Cartesian diagram. Figure 5 are deferred two examples of use of this diagram.
The first example enables us to determine the intensity of the current after 1,5 seconds. For this let us defer on the axis of times the point A1, 1,5 cm after the point O (1 cm = 1s). This A1 point, let us raise the perpendicular, this one intercepts the sinusoid at point A. the distance which separates the A1 points and A represents the intensity that we want to know. This intensity can be read on the axis of the currents by tracing point A horizontal which intercepts the axis of the currents at the A2 point. Knowing that on this axis 1 cm = 1A, it is enough to measure distance OA2 and to convert the result into amp. We thus find in this case 1,9 cm we can say that at the end of 1,5 second the intensity of I is 1,9 A.
The same procedure can be adopted for our second example, in which we wish to know the intensity of the current after 7,8 s. The difference with the preceding case is that the point B, obtained on the sinusoid, determines a B2 point located on the axis of the currents below the point O. Since distance OB2 is 0,7 cm, the intensity of the current is of - 0,7 A ; the sign “-” specifies that this current circulates in contrary direction compared to the current considered in the preceding example.
Now that we know to represent a AC current, analyze its characteristics.
2. 2. -
PERIOD
AND FREQUENCY OF THE AC CURRENT
Until now, we considered that the flow of induction carried out only one rotation generating a current whose pace is represented figure 5.
If the flow of induction continues its rotation and achieves others turns, each one in an identical time, it will generate with each turn a current of pace identical to that of figure 5. A full rotation of flow determines a cycle of the AC current.
The AC current is a succession of equal cycles all between them; having finished a cycle, it begins an identical following and this succession gives the curve represented figure 6-a.
Since all the cycles are identical when we represent a AC current graphically, will not draw we that only one like that figure 5 was made.
To facilitate our preceding explanations, we had put forth the assumption that the flow of induction carried out a complete rotation in 8 seconds but generally, in practice, this one turns much more quickly; to approach reality, we in the figure 6-a chose a time of rotation of 1 second.
To allow a correct analysis of a complete cycle, we should have changed scale and appear 6-a, 1 cm of the time scale corresponds, either to 1 second, but at 0,25 s. On the other hand, we preserved the same scale for the current, that is to say 1 cm for 1 A.
Comparison of figures 5 and 6-a, it arises that to achieve in both cases an identical cycle, the current of the figure 6-a puts 1 second while it spends of them 8 seconds in figure 5.
We have just established here a second characteristic of the AC current, because so that two alternating currents are equal, it is not enough that they take the same values of intensity, but it is necessary that these values are equal at every moment, in other words that they carry out a complete cycle in same time.
One calls period (symbol T) the time put by the AC current to achieve a complete cycle. The unit of the period is thus the second (symbol s).
Appear 6-a, the period of the AC current represented is of 1s ; in this same figure, we note that the period is divided into two called equal parts positive half-wave and negative half-wave. Two alternations owe their name with the values of the currents which they determine on the scale of the current (positive numbers for the positive half-wave and negative numbers for the negative half-wave).
The fact of dividing one period into two alternations does not appear obvious. However, this becomes a need when it is known that the current circulates in a direction during the positive half-wave and in the contrary direction during the negative half-wave and that, as we will see it in the next lessons, certain electronic components react differently according to direction's of the current which crosses them.
Now let us examine the AC current of the figure 6-b of which the period is 0,5 s.
If we compare it with the current of the figure 6-a, we note that it achieves two cycles while that of the figure 6-b achieves one of them. From this, we deduce that for a given time, plus the period is small plus the number of cycles is tall.
A AC current can as characterized by the number of cycles as it carries out in one second. One calls frequency (symbol F) the number of cycles achieved by a AC current in one second.
The frequency F is measured in cycles a second, unit to which it is given the name of hertz (symbol Hz) in homage to the German physicist Heinrich HERTZ (1857-1894) whose experiments reflect in obviousness the electromagnetic wave propagation.
The current of the figure 6-a has a frequency of 1 Hz since it achieves 1 cycle in one second, while that of the figure 6-b has a frequency of 2 Hz considering in the same second it achieves 2 cycles.
The AC current that we use in the dwellings with fine servants has a frequency of 50 Hz, which means that it achieves 50 cycles into 1 second.
In certain particular apparatuses (like the radio operator receivers or the television sets), there are currents which achieve thousands even million cycles a second, from where need for using to quantify the frequency either the hertz but the kilocycle (symbol kHz) which is worth 1 000 Hz or the megahertz (symbol MHz) equal to 1 000 000 Hz. Obviously, from the currents of so high frequency are not obtained any more of the manner described hitherto, i.e. while making turn a flow of induction : it is indeed not possible to make achieve to a primary circuit of the thousands or million turns a second. For the production of these currents known as high-frequency (HF), we will have recourse to particular circuits : electronic oscillators.
As we have just seen it, the period and the frequency are closely dependant between them. This union is sealed by the following relation :
Of this relation, we see that the period and the frequency are two sizes inversely proportional.
Let us apply this formula to calculate the period of the AC current of the sector:
We know now that the AC current of the sector spends 20 ms to achieve a complete cycle. In the same way, if we know the period T of a AC 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 AC
current: it is about its pulsation (symbol
is
read omega) and which is expressed in radians
a second (rd / s). The pulsation is
obtained using the relation :
The pulsation characterizes the number of revolutions of the vector symbolizing inductive flow in the figure 4-a. This size, as you will see it in forthcoming lessons, is mainly necessary for calculations relating to electric circuits supplied with an alternating voltage.
2. 3. - VALUE OF THE AC CURRENT
To represent a AC current, it is necessary to indicate its frequency and the pace of the sinusoid which gives the intensity of the current at every moment.
We observe however that the intensity of the current varies constantly and we do not know which value to choose to characterize with precision this intensity: if we consider the AC current represented figure 7, logic would dictate to choose, for the maximum value reached by the current during one period.
This value is reached twice by the current : first once in the middle of the positive half-wave and second once in the middle of the negative half-wave. These two values are respectively called IM and - IM. This value taken by the current is called maximum value of the AC current.
The Cartesian diagram of figure 7 immediately gives value IM of the current represented which is 3 A. In addition, this same diagram enables us to determine the frequency of this current :
The AC current of figure 7 is characterized by the two following parameters :
However, we can at every moment determine the
value of the current according to the swing angle of inductive flow. Indeed, a
sinusoidal function has as an equation y = ax
in which x = sin
(is read sine phi).
For the case which interests us,
is
the angle described by inductive flow and the horizontal one (figure 4-a) while
(a) is value IM of the current and that (y)
gives the instantaneous value (i) current to
the angle
considered.
We can write : i = IM sin
Using the figures 4-a and 7, let us apply this equation to determine certain values of the current.
= 0°, sine 0° = 0. Current i is thus null, we are at
the beginning of the cycle.
= 90°, flow achieved a quarter of turn, sin 90° = + 1 ------) i = IM = 3 A.
= 180°, flow achieved a half-turn, sin 180° = sin 0° = 0. The current is null.
= 270°, flow achieved three quarters of turn. sin 270° = - 1 ---------) i = -
IM = - 3 A.
= 360°, flow achieved a full rotation. sin 360° = sin 0° = 0. The current is
null.
The value of the sine of an angle is given by a table called trigonometrical table. This one gives the sine of an angle some is its value, and we can thus know the value of the current whatever the position of flow compared to the horizontal one.
Since the maximum value of the current is reached twice by this one, we thus include/understand that this value perhaps is not very adapted to characterize a AC current, would be this only to determine the heating effect produced by such a type of current. This effect of the current is independent of its direction of circulation : indeed, so that there is production of heat, it is enough that a current crosses a resistance and it does not matter that it circulates in a direction or the other. The production of heat is the same one for each of two alternations.
To evaluate the heating effect of the current, we can refer to one whole period. We can apply the formula then w = R x I2 x t in which t = T. Whereas it is easy for us to know R and T, we do not know which value I to choose considering this one changes constantly and consequently the at every moment released value. It is thus necessary to utilize a new characteristic of the AC current which is its effective intensity symbolized by the Ieff initials.
The effective intensity of a AC current is translated in the following way :
It is the intensity of a D.C. current which would produce in same resistance, the same quantity of heat as this AC current.
NOTE :
Taking into account this definition, the effective intensity can, according to the international system of measurement S.I, being symbolized by simple letter I.
There is a relation between the effective intensity and the maximum intensity of a sinusoidal current.
This relation corresponds to the report / ratio :
NOTE :
The demonstration of this relation calling upon mathematical knowledge leaving our program, will thus not speak we. In addition to the maximum value and the effective value of a AC current, there is also its average value (Imoy symbol) which is defined as follows :
The average intensity of a AC current is the intensity of the D.C. current which would transport during the interval of time considered (one period) the same quantity of energy.
In the case of a sinusoidal AC current, the average value of an alternation is given by the relation :
If we calculate the average value for one complete period, we find a value zero.
To explain this briefly, it is enough to think that the electrons, during the positive half-wave, move in a direction, and that during the negative half-wave these same electrons remake the same way in the other direction and return to their starting point; thus the quantity of energy provided by the generator is null (but not that necessary to its operation, which it, is quite real).
The type of current that we chose for our explanations is a AC current of null average value like that delivered by the sector, but as you will see it thereafter there are alternating currents of nonnull average value from where need for having introduced this concept.
Throughout this lesson, we saw how to graphically represent a AC current and how starting from this graph it is possible for us to determine all the characteristics of such a current.
The sizes relating to the sinusoidal AC current are gathered in the table of figure 8.
In the next lesson, we will analyze either the AC current but the alternating voltage and we will see the relations which bind these two sizes according to the type of circuits that they feed.
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