(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 132714, 3213]*) (*NotebookOutlinePosition[ 133429, 3238]*) (* CellTagsIndexPosition[ 133385, 3234]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Mean Power is not RMS Power for a sinusoidal voltage.\n", StyleBox["Dr. David Kirkby, G8WRB. 26/3/2005", FontSize->18] }], "Title", FontSize->24], Cell[CellGroupData[{ Cell[TextData[{ "\n", StyleBox["This was done in an attempt to settle an argument about RMS \ power, that appeared on the 'amps' mailing list in March 2005. Not that \ arguments every seem to get settled there (especially the parasitic \ oscillations), but this might be a bit more scientific. ", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell["Definition of RMS", "Subsection"], Cell[BoxData[{\(This\ is\ the\ definition\ of\ the\ RMS\ value\ of\ a\ \ repetitive\ waveform . \ You\ Integrate\ over\ the\ time\ period\ T\), "\[IndentingNewLine]", RowBox[{\(rms = \@\(\(\[NegativeThinSpace]\)\(\(1\/T\) \ \(\[Integral]\_0\%T\ \ \(\(f\) \(\((t)\)\^2\)\(\ \ \)\(dt\)\(\ \[NegativeThinSpace]\)\)\)\)\)\%2\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ "where", " ", "T", " ", "is", " ", "the", " ", "period", " ", "of", " ", "the", " ", \(waveform . \ This\), " ", "is", " ", "valid", " ", "for", " ", StyleBox["any", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "repetitive", " ", \(\(waveform\)\(.\)\(\ \)\)}]}], "Text"], Cell[TextData[{ "Definition of mean value of a continuous repetitive waveform\n", Cell[BoxData[ \(mean = \(1\/T\) \(\[Integral]\_0\%T\ \ \(\(f\) \ \((t)\)\(\ \ \)\(dt\)\(\[NegativeThinSpace]\)\)\)\)]], "\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the RMS value of a sine wave. \ \>", "Subsection"], Cell["\<\ First find the RMS value of a sine wave, with a peak of 1. 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We define the voltage as sin(t). \ \ \>", "Text"], Cell[BoxData[ \(\(\(voltage[t_] := Sin[t]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(rms = Sqrt[1/\((2\ Pi)\)\ * Integrate[voltage[t]^2, {t, 0, 2\ Pi}]\ ]\)], "Input"], Cell[BoxData[ \(1\/\@2\)], "Output"] }, Open ]], Cell["\<\ Convert the above to a floating point number. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input"], Cell[BoxData[ \(0.7071067811865475`\)], "Output"] }, Open ]], Cell["\<\ So no great surprise there. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Finding the mean of a sine wave \ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(1/\((2\ \ Pi)\)\)\(\ \)\(*\)\(Integrate[ voltage[t], {t, 0, 2\ Pi}]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ So the mean value of voltage on a sine wave is 0, the rms is 0.707 \ times the peak. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Instantaneous power. 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", "Subsection"], Cell[TextData[{ "To find the rms value of the power, we note that power is the product of \ the voltage and current. and go back to our definition of RMS\n", Cell[BoxData[ \(rms = \@\(\(\[NegativeThinSpace]\)\(\(1\/T\) \(\[Integral]\_0\%T\ \ \ \(\(f\) \(\((t)\)\^2\)\(\ \ \)\(dt\)\(\[NegativeThinSpace]\)\)\)\)\)\%2\)]], "\n " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(rmsPower = Sqrt[\ \((1/\((2\ Pi)\))\)* Integrate[\ \((voltage[t]\ *\ current[t])\)^2, {t, 0, 2\ Pi}]]\)], "Input"], Cell[BoxData[ \(\@\(3\/2\)\/2\)], "Output"] }, Open ]], Cell["\<\ Now to find the numerical value of that. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(N[%]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\) \)\)\)], "Input"], Cell[BoxData[ \(0.6123724356957945`\)], "Output"] }, Open ]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell["\<\ Finally, here is a graph showing the voltage (green), the power \ (red) and that of a rectified sine 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