Question

Show that average heat produced during a cycle of AC is same as produced by DC with i=i_("rms").

Answer 1

Instantaneous value

The instantaneous value is “the value of an alternating quantity (it may ac voltage or ac current or ac power) at a particular instant of time in the cycle”. There are uncountable number of instantaneous values that exist in a cycle.

Average value:

The average value is defined as “the average of all instantaneous values during one alternation”. That is, the ratio of the sum of all considered instantaneous values to the number of instantaneous values in one alternation period.

Whereas the average value for the entire cycle of alternating quantity is zero. Because the average value obtained for one alteration is a positive value and for another alternation is a negative value. The average values of these two alternations (for entire cycle) cancel each other and the resultant average value is zero.

Answer 2

Now , we known Instantaneous Heat produced in time 'dt' across resistance 'R' is

•) dH = I^2*R*dt

Where , in AC, I = i * sinwt and i is the peak current

=> dH = i^2 * sinwt^2*Rdt

•) Now , let H' be the average heat produced ,

=> H' = § dH / § dt

=> H' = § (i^2*R*sinwt^2)dt / §dt

•) Now , here limits of integrals ranged from 0 to T , where T = 2π/w

And T is the time period and w is the angular frequency .

•) Now ,

H' = i^2*R*[ t - sin2wt/2w] / 2 [t]

H' = i^2*R/2 = (Irms)^2*R

Where Irms is root mean square current and Irms = i /√2.