Variation of voltage with time is shown in fig

(a) The rms voltage is found to be `Nsqrt((2)/(3))V`. Find N.

Answer

(i) 4, (ii) 1

Explanation

In such cases it is difficult to develop a single equation. Hence, it is usual to cosider tow equations one applicable from 0 to 1 ms and another from 1 to 2 ms.
for t lying between 0 and 1 ms `(V_1) = 4`. For t lying between 1 and 2 ms `V_(2) =-4t+4`
(i) `V_(rms) = sqrt(1/2(int_(0)^(1) V_(1)^(2) dt +int_(1)^(2) V_(2)^(2)dt))`
`V_(rms)^(2) = 1/2 [ int_(0)^(1) 4^(2)dt + int_(1)^(2) (-4t+4)^(2) dt ]`
`=1/2 [|16 t|_(0)^(1) +|(16 t^(3))/(3)|_(1)^(2) +|16t|_(1)^(2)-|(32t^(2))/(2)|_(1)^(2)]=32/3`
`V_(rms) = sqrt((32)/(3)) 4 sqrt((2)/(3)) V, Hence N=4`.
(ii) `V_(av) = 1/2 [int_(0)^(1)V_(1)dt+int_(1)^(2)V_(2)dt] = 1/2[int_(0)^(1) 4 dt + int_(1)^(2) (-4t+4)dt]`
`1/2[|4t|_(0)^(1)+|-(4t^2)/(2)+4t|_(1)^(2)]=1V`.