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Question about rms value of a sine wave 
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#1
May714, 06:15 PM

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Sorry if this sounds like a dumb question, but why is the effective value of a sine wave 0.707, as opposed to 0.637 which is the value generated by finding the definite integral over the domain [0,∏] divided length of the domain?



#2
May714, 06:36 PM

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RMS means "root mean SQUARED", so we have to square the sine wave before integrating. Regarding why we use this measurement, it is essentially a generalization of Euclidean distance: ##\sqrt{\int x(t)^2 dt}## is a limiting form of ##\sqrt{x(t_1)^2 + x(t_2)^2 + \ldots + x(t_n)^2}##, which is the distance between the point ##(x(t_1), x(t_2), \ldots, x(t_n))## and the origin. There are many other reasons to prefer the RMS as well: it plays nicely with how we measure the energy in a random quantity (variable or process), namely the standard deviation. Also, the RMS of a function/signal is preserved when we transform to the frequency domain via the Fourier transform.
Mathematically, "RMS" is also a common way to measure the norm ("size") of a function: we call it the ##L^2## norm. Working in the ##L^2## space is very nice because it is a Hilbert space, unlike the other ##L^p## spaces, and because the Fourier transform is an isometry on the ##L^2## space. Don't worry if these terms are unfamiliar  you may see them eventually if you study advanced mathematics or physics, but otherwise you can probably live a perfectly happy life if you never hear about them again. 


#3
May714, 07:19 PM

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By the way, the calculation you performed is also a common way of measuring the size of a function/signal. In mathematics we call it the ##L^1## norm: ##\int x(t) dt##. It is a limiting form of ##x(t_1) + x(t_2) + \ldots + x(t_n)##, which is another way of measuring the distance between a point and the origin, assuming you are constrained to travel along an orthogonal "grid" to get to the point.



#4
May714, 07:23 PM

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Question about rms value of a sine wave
Alright, thanks for the help!



#5
May714, 10:14 PM

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The 0.637 figure also has its uses, but to situations where we are concerned with average. For example, an electromagnet is roughly linear, so if you applied a rectified sinewave of amplitude A_{v} to the windings of an electromagnet, the field strength produced will have an average value equal to that produced by applying DC of magnitude 0.637A_{v} to the windings. 


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