The rms value of alternating current  i=2sin100πt+2cos100πt+300 is (in ampere)

# The rms value of alternating current  $i=2\mathrm{sin}100\text{π}t+2\mathrm{cos}\left(100\text{π}t+{30}^{0}\right)$ is (in ampere)

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### Solution:

1. Given  $i=2\mathrm{sin}100\text{π}t+2\mathrm{cos}\left(100\text{π}t+{30}^{0}\right)$
The equation can be written as  $i=2\mathrm{sin}100\text{π}t+2\mathrm{sin}\left(100\text{π}t+{120}^{0}\right)$
So phase difference  $\varphi ={120}^{0}$
${\left({I}_{m}\right)}_{res}=\sqrt{{A}_{1}^{2}+{A}_{2}^{2}+2{A}_{1}{A}_{2}\mathrm{cos}\varphi }$
${A}_{1}=2;\text{\hspace{0.17em}}{A}_{2}=2$ .
${I}_{rms}=\text{\hspace{0.17em}}\frac{{\left({I}_{m}\right)}_{res}}{\sqrt{2}}=\frac{\sqrt{4+4+2×2×2\left(-\frac{1}{2}\right)}}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$
So effective value or rms value  $2/\sqrt{2}=\sqrt{2}A$
Therefore, the correct answer is 1.41

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