R.Measures wrote:
>
>>
>>You can not find RMS power by multiplying RMS voltage by RMS current.
>>
>>
>
>I disagree. .
>
Rich, I think you are wrong. Here's my reasoning - I may well be
wrong, as maths never was a good point of mine.
You need to understand calculus to properly work out the RMS of an
arbitrary waveform, as you probably know.
Hence this explanation will really only be of use to someone who knows
integration. But I can't see a simple way of computing the RMS value of
the power, without it. People will just argue, but if you go back to
first principles, you should get the right answer.
This is the definition of RMS
rms = Sqrt [ 1/T * integral_of_waveform^2 ]
where the period you integrate over must be a complete cycle (2 Pi
radians, or an integer multiple of it).
Here I run Mathematica (a maths program from http://www.wolfram.com/) .
This for a simple sine wave, with a peak value of 1.
I integrate sin(t)^2 over 2 Pi radians, then take the square root, which
is what the definition of rms is.
Mathematica 5.1 for Sun Solaris (UltraSPARC)
Copyright 1988-2004 Wolfram Research, Inc.
-- Motif graphics initialized --
In[1]:= Sqrt[1/(2 Pi) Integrate[Sin[t]^2, {t, 0, 2 Pi}]]
1
Out[1]= -------
Sqrt[2]
It gives an answer of 1/sqrt(2), the approximate value of which is
In[2]:= N[%1]
Out[2]= 0.707107
So, the rms value of a sine wave is 0.707 times the peak.
Now assume for simplicity the resistor is 1 Ohm. Hence the current will
follow the voltage exactly and will be sin(t) too. Hence the power is
varies as a sin(t)^2. I shows this below as sin(t) * sin(t), but take
the square of that (effectively sin(t)^4), since the definition of RMS
requires you square up the waveform.
In[6]:= Sqrt[1/(2 Pi) Integrate[ (Sin[t] Sin[t])^2, {t,0,2 Pi}] ]
3
Sqrt[-]
2
Out[6]= -------
2
I doubt that is formatted too well, but it is saying the answer is
sqrt(3/2) / 2. The numerical value of which is:
In[7]:= N[%6]
Out[7]= 0.612372
So for a peak voltage of one volt, the rms value of the voltage is 0.707
V, but the RMS value of the power is 0.612 W.
So we have
1) RMS value of a sinusoidal voltage of 1 V peak = 0.707 V.
2) RMS value of a sinusoidal current of 1 V peak = 0.707 A
2) RMS value of the power (which will not be sinusoidal) is 0.613 W.
Note the instantaneous power will not be sinusoidal, but sin(t)^2. That
is sort of like a rectified sine wave, but it is not *exactly* the same
shape.
--
Dr. David Kirkby,
G8WRB
Please check out http://www.g8wrb.org/
of if you live in Essex http://www.southminster-branch-line.org.uk/
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