Single phase alternating
There are different types of variable quantities, but the alternating signal has certain characteristics, it is:
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Bidirectional (oscillating between positive and negative values)
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Periodic (whose variations are reproduced identically to themselves at regular time intervals)
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Sinusoidal (which evolves over time like a sinusoid)
" The period " of a sinusoidal alternating current is the constant duration which separates two consecutive instants, in the same direction, between two identical points.
The period is designated by the capital letter " T " and is measured in " seconds ".
In Europe, the period of the sinusoidal current of the network is: T = 1/50 sec
The period
The frequency
" Frequency " is a periodic phenomenon that represents the number of complete cycles of an oscillation or wave that occur per unit of time (in seconds) .
In the International System of Units (SI) , frequency is expressed in hertz (Hz), a tribute to the German physicist Heinrich Hertz .
When the phenomenon can be described mathematically by a periodic function of time, that is to say a function F(t) such that there exist constants Ti for which, whatever t, F(t+Ti) = F(t), then the smallest of the positive values of these constants Ti is the period "T" of the function.
In this case, and the frequency "f" is the inverse of the period "T".
For one turn, the angle “ φ ” described is 2 π[rad] and therefore: C = 2 π r
If we consider that the circumference "c" of a circle of radius "r" is equal to 1, we deduce that: C = 2 π
As in 1 revolution, the time traveled corresponds to 1 period ( T ), we can write that the angular speed ω:
ω = (1 circumference)/(1 period) = (2π )/T
As: T = 1/f we deduce:
Angular velocity or pulsation
You noticed that our sinusoidal signal represents the sine function by a vector rotating in a circle.
For each angle value, our vector takes on a different value and so we can represent our signal as a vector rotating at angular velocity .
Angular velocity , also called pulsation "ω", defines the number of " radians " traveled per " second " by the radius vector rotating inside the circle.
Before developing the angular velocity, we must define the notion of radian.
Radian concepts
"A radian" is equivalent to the angle which, having its vertex at the center of a circle, intercepts on the circumference of this circle an arc of a length equal to that of the radius of the circle.
The drawn angle represents 1 radian and its radius is 1.
Circumference: C = 2 .π.r = π.d
So: 360° = 2.π [rad] and 180° = π [rad]
Speed :
v= space/time= e/t
Correlation between speed "v" and angular speed "ω".
Angular velocity:
ω= (circumference of the circle)/Period= C/T
Instantaneous magnitudes
" An instantaneous value" of a variable quantity is the value that this quantity can take at any instant.
In our case, the usual electrical quantities are: current intensity, voltage and power.
To represent these " instantaneous " quantities, we will always use lowercase letters i(t), u(t), p(t).
Determine the period of the following oscillograms.
T1 = 5 x 0.5 ms = 2.5 ms
T2 = 8 x 0.5 ms = 4 ms
Instantaneous power
Instantaneous electrical power is expressed by the product of the instantaneous voltage “u” and the instantaneous intensity “i”.
Whatever the electrical regime (continuous, transient, sinusoidal alternating, etc.), the relationship is always true.
Effective values
" The effective value" of a " variable quantity " (voltage, current) over the time of " period (T) ", is equal to the value of a " continuous quantity " (voltage, current) dissipating the same energy through a " resistance (R) " over a period T".
Example :
The effective value of a " variable current " (I), produces the same work as a " direct current ", in the " same load " and during " the same time interval ".
The effective value of this "variable" current will then be the same as that of the direct current.
NB:
The effective value of a current (I) is then expressed as the square root of the mean of the square of the intensity calculated over a period T.
The English term RMS means " root mean square ".
Electrical experiment and logical deduction
We will start by experimenting with a resistance "R", supplied first by a direct current (DC) power source and then supplied by its equivalent in alternating current (AC).
It would be appropriate to establish an equivalence between alternating voltage and direct voltage, from the point of view of their thermal effects.
One may wonder whether a sinusoidal alternating voltage of 200V peak produced the same Joule effect as a direct voltage of 200V across the same resistance.
Calculations of DC dissipated power
Observation:
At all times, with a direct voltage source (DC), the average dissipated power is constant and in our case is 2000 W.
Calculations of dissipated power in Alternating Current (AC)
Observation:
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The power dissipated in a resistor is " variable ".
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" Average power " is half of the maximum power.
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When voltage and current become negative, power remains " positive " because the product of two "negative" numbers is positive.
Conclusion
With the AC voltage source, the average power dissipated in (AC) by a resistor is half of the power dissipated in (DC) with a DC voltage source.
We must conclude that these two voltages, alternating (peak) and continuous, are not equivalent.
Demonstration of effective value
What direct current (DC) voltage would need to be applied to the resistor in order to cause the same thermal effect as the alternating current?
We will start from the principle of calculating energies in order to determine the relationship which exists between maximum value and effective value.
We compare the energy dissipated in continuous mode with the energy in alternating mode by superimposing them.
Area of continuously dissipated energy
Area of energy dissipated in alternating current
Let's compare the two areas with each other.
Since in our case the resistance and the time base (period) are identical, we can remove them from the equation
We remove the square of the effective current by the square root and we obtain:
Appendix: Mathematical developments
Mathematical expression of a sinusoidal effective signal
Continuous efficient energy:
E = R . I² . T
Alternative instant energy:
dE = R . i² (t) . dt
Total energy dissipated over a period "T"
In our case, the two dissipated energies being identical, we can write:
Since the resistance is the same, we can simplify by “R”.
We can transfer the time "T" to the integral:
We remove the square of the effective signal, to find the expression of any effective signal.
Mathematical development of an effective current
We will consider the following equations as a starting point:
We can take the max current squared out of the integral, because it is a constant.
We replace the sine squared by its linearized expression.
We can separate the integral into a sum of 2 integrals
The integral of a cosine over a multiple of its period is "zero".
We perform the integral of 1/2 over the period "T"
We simplify by "T"
We eliminate the square
In our case, we consider the instantaneous value ( i , u ) as the opposite side and the max value ( Î , Û ) as the hypotenuse.
Pythagorean theorem
The vector “ω” rotates at constant speed and the time required to travel 2 π [rad] is a period T.
It is therefore possible to establish a ratio allowing the calculation of the angle traveled "α" during a time difference "∆t" separating the origin "0" from time "t1".
We therefore deduce the formulas for the instantaneous values of the intensity of the electric current and the voltage.
The argument of the sine, " ωt ", is called the phase (or phase angle) of the oscillation.