Two waveforms are represented by the following equations:
$$i_1=10cos(omegat)-7cos(3omegat)-3sin(5omegat)$$
$$i_2=10sin(omegat)+3cos(3omegat)+7cos(5omegat)$$
How do their RMS values compare?
A. RMS values of $$i_1(t)$$ and $$i_2(t)$$ are nonzero and equal.
B. RMS value of $$i_1(t)$$ is larger than that of $$i_2(t)$$.
C. RMS value of $$i_1(t)$$ is smaller than that of $$i_2(t)$$.
D. RMS values of $$i_1(t)$$ and $$i_2(t)$$ are each zero.

1 Answer

James Dowd

Updated on December 30th, 2020

A waveform which consists of a sum of sinusoids of different frequencies, as both $i_{1}$ and $i_{2}$ , has an RMS value of the square root of the sum of the squares of the RMS values of each sinusoid. This is shown below:

$I_{R M S} = \sqrt{{I_{0}}^{2}_{R M S} + {I_{1}}^{2}_{R M S} + . . . + {I_{n}}^{2}_{R M S}}$

The equation for each individual RMS voltage can be found on page 359 of the FE reference handbook.

$X_{R M S} = \frac{X_{m a x}}{\sqrt{2}}$

Since a cos or sin function can only be a maximum of 1, the max of each sinusoid function corresponds to its coefficient. Therefore, for $i_{1}$ :

${I_{1}}_{R M S} = \sqrt{{I_{0}}^{2}_{R M S} + {I_{1}}^{2}_{R M S} + . . . + {I_{n}}^{2}_{R M S}} {I_{1}}_{R M S} = \sqrt{({\frac{10}{\sqrt{2}}}^{2}) + ({\frac{- 7}{\sqrt{2}}}^{2}) + ({\frac{- 3}{\sqrt{2}}}^{2})} {I_{1}}_{R M S} = \sqrt{(\frac{100}{2}) + (\frac{49}{2}) + (\frac{9}{2})} = \sqrt{79} {I_{1}}_{R M S} = 8.89$

And for $i_{2}$ :

${I_{2}}_{R M S} = \sqrt{{I_{0}}^{2}_{R M S} + {I_{1}}^{2}_{R M S} + . . . + {I_{n}}^{2}_{R M S}} {I_{2}}_{R M S} = \sqrt{({\frac{10}{\sqrt{2}}}^{2}) + ({\frac{3}{\sqrt{2}}}^{2}) + ({\frac{7}{\sqrt{2}}}^{2})} {I_{2}}_{R M S} = \sqrt{(\frac{100}{2}) + (\frac{9}{2}) + (\frac{49}{2})} = \sqrt{79} {I_{2}}_{R M S} = 8.89$

Therefore, the RMS values of both are clearly nonzero and equal to each other. Our answer is A.

1 Answer

James Dowd

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