An alternating current is given by the equation `i = (i_1 cos omegat + i_2 sin omegat)`. The rms current is given by ……..

Options

A. `1/sqrt2 (i_1+i_2)`
B. `1/sqrt2(i_1+i_2)^2`
C. `1/sqrt2(i_1^2 + i_2^2)^(1/2)`
D. `1/2(i_1^2 + i_2^2)^(1/2)`

Answer

C

Explanation

`i=i_1 cos omegat + i_2 sin omegat`
`lt i^2 gt = lt (i_1 cos omegat + i_2 sin omegat)^2gt`
` lt i^2 gt = lt i_1^2 cos^2 omegat gt + lt i_2^2 sin^2 omegat gt + lt 2i, i_2 sin omegat cos omegat gt` ….(1)
But `lt i_1^2 cos^2 omegat gt = i_1^2 lt (1+cos 2 omegat)/2 gt`
`=i_1^2 (:1/2:)+(:(cos2omegat)/2:)]`
Average of `cos2omegat` one period =Zero
`therefore lt i_1^2 cos^2 omegat gt =i_1^2/2`....(2)
Similarly `lt i_2^2 sin^2 omegat gt = i_2^2/2`...(3)
And value of `lt2sin omegat cos omegatgt` on one period =0 ...(4)
`therefore` From equation (1),(2) , (3) and (4)
`lti^2gt=i_1^2/2+i_2^2/2+0`
`=(i_1^2+i_2^2)/2`
`therefore i_(rms)=sqrt(lti^2gt)=sqrt((i_1^2+i_2^2)/2)=1/sqrt2(i_2^2+i_2^2)^(1/2)`