An alternating current is given by i = i1 sin(ωt) + i2 cos(ωt). Find the rms value of this current.

(1/sqrt2) (i1² + i2²)^(1/2)
(1/sqrt2) (i1 + i2)²
(1/2) (i1² + i2²)^(1/2)
(1/sqrt2) (i1 + i2)

Correct answer: (1/sqrt2) (i1² + i2²)^(1/2)

Solution

Since sin(ωt) and cos(ωt) are 90 deg out of phase, i = i1 sin + i2 cos can be written as I0 sin(ωt + φ) with I0 = sqrt(i1² + i2²). The rms value of a sinusoid is amplitude/sqrt2, so i_rms = sqrt(i1² + i2²)/sqrt2.

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