Question

An alternating current is given by i = i1 sin ωt + i2 cos ωt. The r.m.s. current is given by

• A. √ (i12 + i22)/ 2
• B. (i1 + i2)/ √2
• C. √ (i12 + i22)/ √2
• D. (i1 - i2)/ √2

Answer 1

Answer: (A)

√ (i12 + i22)/ 2

Answer 2

Given: i = i1 sin ωt + i2 cos ωt
Step 1: Square the expression
i2 = (i1 sin ωt + i2 cos ωt)2
= i12 sin2 ωt + i22 cos2 ωt + 2i1i2 sin ωt cos ωt
Step 2: Average over one time period
The average of sin2 ωt and cos2 ωt over one time period is 1/2.
The average of sin ωt cos ωt over one time period is 0.
Therefore, the average of i2 over one time period is:
(i12 sin2 ωt + i22 cos2 ωt + 2i1i2 sin ωt cos ωt)/2
= (i12 + i22)/2
Step 3: Take the square root
r.m.s. current = √((i12 + i22)/2)
Comparing this result to the given options:
(1) √ (i12 + i22)/ 2 -> This matches our calculated expression.
(2) (i1 + i2)/ √2 -> This is not equivalent to our expression.
(3) √ (i12 + i22)/ √2 -> This is not equivalent to our expression.
(4) (i1 - i2)/ √2 -> This is not equivalent to our expression.
Therefore, the correct answer is (A) √ (i12 + i22)/ 2.