Physics >> Electrostatics >> A.C. Mean Value
A.C. Mean Value
Since the alternating current is positive during the first half cycle (0 to T/2) and negative during the other half cycle (T/2 to T), therefore the mean or average value of over one complete cycle is zero. We can however find the mean or average value of A.C. over any half cycle.
The mean or average value of A.C. over any half cycle is defined as that value of steady current which would send the same amount of charge through a circuit in the time of half cycle as is sent by the A.C. through the same circuit in the same time.
To calculate its value let an alternating current be represented by
I= I0 sin w t
If the strength of current is assumed to remain constant of a small time, di then small amount of charge sent in a small time dt is
dq = I dt
Let q be the total charge sent by A.C. in the first half cycle
∴ q= ∫I dt
Using we get
Q = 0∫T/2 Io sin w t. dt = I0 [- cos w t /w] T/2
= - I0 /w [(cos w T/2) – cos θ]
=- I0 / w [cos π - cos θ] (∴ wT = 2π)
Q = - I0 / w [-1-1] = 2I0/w
If Im represents the mean or average value of A.C. over the 1st half cycle then
Q = Im × (T/2)
From and we get
Im X (T/2) = 2 I0/w = 2I0 . (T/2π)
Or Im = 2/π I0 = 0.637 I0
Hence mean or average value of over positive half cycle is 0.637 times the peak value of A.C. 63.7% of the peak value.
Similarly the mean or average value of A.C. over the negative half cycle is obtained by integrating (15) within the limits T/2 to T. it comes out to be -0.637 I0 hence the mean or average value of A.C. over one complete cycle is 0.637 Io – 0.637 I0 = zero.
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