Modern alternators produce an e.m.f. which is for all practical purposes sinusoidal (i.e., a sine curve), the equation between the e.m.f. and time being
e =E max= sin wt
where, e = Instantaneous voltage ; E= = Maximum voltage ;
wt = Angle through which the armature has turned from neutral.
Taking the frequency as f hertz (cycles per second), the value of ro will be 2rtf, so that the equation reads
e = E max sin (2πf)t.
The graph of the voltage will be as shown in Fig. 5.
1. Cycle. One complete set of positive and negative values of an alternating quantity is known as a cycle. A cycle may also sometimes be specified in terms of angular measure. In that case, one complete cycle is said to spread over 360° or 21t radians.
2. Amplitude. The maximum value, positive or negative, of an alternating quantity, is known as its amplitude.
3. Frequency (f). The number of cycles/second is called the frequency of the alternating quantity.
Its unit is hertz (Hz).
4. Time Period (T). The time taken by an alternating quantity to complete the cycle is called its time period. For example, a 50 hertz (Hz) alternating current has a time period of 1 / 50 second.
Time period is reciprocal of frequency,
T = 1 / f (or f = 1 / T)
5. Root mean square (R.M.S.) value. The r.m.s. (or effective) value of an alternating current is given by that steady (D.C.) current which when flowing through a given circuit for a given time produces the same heat as produced by the alternating current when flowing through the same circuit for the same time.
R.M.S. value is the value which is taken for power purposes of any description. This value is obtained by finding the square root of the mean value of the squared ordinates for a cycle or half cycle (See Fig. 5).
This is the value which is used for all power, lighting and heating purposes, as in these cases the power is proportional to the square of the voltage.
Refer Fig. 5.
The equation of sinusoidal alternating current is given as :
i =I max sin θ
The mean of squares of the instantaneous values of current over half cycle is
Note. While solving problems, the values of given current and voltage should always be taken as the r.m.s. values, unless indicated otherwise.
6. Average or mean value. The average value of an alternating current is expressed by that steady current which transfers across any circuit the same charge as is transferred by that alternating current during the same time.
The mean value is only of use in connection with processes where the results depend on the current only, irrespective of the voltage, such as electroplating or battery charging.
Refer Fig. 6.
The value of instantaneous current is given by
i =I max sin θ
Refer Fig. 6. The value of instantaneous current is given by:
Limits are taken from 0 to π , since only first half cycle is considered. For whole cycle, the average value of sine wave is zero.
= 1 / π . I max [ 1- (-1)[ = 2 / π . I max
I av = 0.637 I max
Note. In case of unsymmetrical altenating current viz. half-wave rectified current the average value must always be taken over the whole cycle.