(************** Content-type: application/mathematica ************** CreatedBy='%s' 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[ 119118, 2028]*) (*NotebookOutlinePosition[ 119969, 2059]*) (* CellTagsIndexPosition[ 119925, 2055]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell["\<\ Field Patterns of a Linear Dipole Antenna Array\ \>", "Title", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Description", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This Notebook shows how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used in the analysis of engineering problems. We examine the \ specific example of a linear array of dipole antennas." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ John M. Novak Wolfram Research, Inc.\ \>", "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Basics", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ In this analysis, we are only interested in the field or radiation \ pattern of antenna arrays, which is solely determined by the phases of the \ array elements. The phase of the field of a single dipole element is \ determined by the exponential term. This can be expressed as\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Exp[-I (2 Pi f r)/c]", FontFamily->"Courier", FontWeight->"Bold"], "" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "where ", StyleBox["f", FontWeight->"Bold"], " is the frequency of the transmission, ", StyleBox["r", FontWeight->"Bold"], " is the distance from the element, and ", StyleBox["c", FontWeight->"Bold"], " is the speed of light." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "We can analyze the array by superposing the individual elements. This \ leads us to the following description of the field pattern, assuming", StyleBox[" elems ", FontWeight->"Bold"], "is the number of elements:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["(1/elems) Sum[Exp[-I (2 Pi f r[n])/c + phase[n]], \ {n,1,elems}]", FontFamily->"Courier", FontWeight->"Bold"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This assumes that all the elements are transmitting at the same frequency. \ Note that ", StyleBox["r[n]", FontWeight->"Bold"], " is the distance from the ", StyleBox["n", FontWeight->"Bold"], StyleBox["th", FontVariations->{"CompatibilityType"->"Superscript"}], " element and ", StyleBox["phase[n] ", FontWeight->"Bold"], "is a phase shift. We can further refine this by assuming the elements are \ evenly spaced, by a distance d, along the x axis (a linear array), and we \ determine the distance from each element in terms of the distance from the \ first element (centered at 0,0) and the angle from the x axis. From this, we \ can make a ", StyleBox["Mathematica", FontSlant->"Italic"], " function that, given the various parameters, returns the field pattern at \ a point. Let's call our function ", StyleBox["AF", FontWeight->"Bold"], ", for array factor." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(AF[r_, phi_, f_, d_, shift_, elems_] := \(\[Sum]\+\(n = 1\)\%elems \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \((\(2\ \[Pi]\ f\ \((r - \@\(r\^2 + \((\((n - 1)\)\ d)\)\^2 \ - 2\ r\ \((\((n - 1)\)\ d)\)\ Cos[phi]\))\)\)\/c + \((n - 1)\)\ \ shift)\)\)\)\/elems\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "We also need to set a value for", StyleBox[" c", FontWeight->"Bold"], ". We will assume all measurements are in meters and radians:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(c = 3. \ 10\^8; \)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Far-Field Approximation", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "It is customary to examine the far-field case by allowing", StyleBox[" r ", FontWeight->"Bold"], "to go to a large value. We note that it might be useful to look at an \ approximation to the distance of each element in this case." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell[BoxData[ \(Series[ r - \@\(r\^2 + \((\((n - 1)\)\ d)\)\^2 - 2\ r\ \((\((n - 1)\)\ d)\)\ \ Cos[phi]\), {r, \[Infinity], 1}]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ InterpretationBox[ RowBox[{\(d\ \((\(-1\) + n)\)\ Cos[phi]\), "-", \(\(d\^2\ \((\(-1\) + n)\)\^2 - d\^2\ \((\(-1\) + n)\)\^2\ Cos[phi]\^2\)\/\(2\ r\)\), "+", InterpretationBox[\(O[1\/r]\^2\), SeriesData[ FE`Conversion`Private`r, DirectedInfinity[ 1], {}, 0, 2, 1]]}], SeriesData[ FE`Conversion`Private`r, DirectedInfinity[ 1], { Times[ FE`Conversion`Private`d, Plus[ -1, FE`Conversion`Private`n], Cos[ FE`Conversion`Private`phi]], Times[ -1, Plus[ Times[ Power[ FE`Conversion`Private`d, 2], Power[ Plus[ -1, FE`Conversion`Private`n], 2]], Times[ -1, Power[ FE`Conversion`Private`d, 2], Power[ Plus[ -1, FE`Conversion`Private`n], 2], Power[ Cos[ FE`Conversion`Private`phi], 2]]], Power[ 2, -1]]}, 0, 2, 1]]], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ From this, we can create a simpler function, depending on phi, that \ gives the far-field expression:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(FarAF[phi_, f_, d_, shift_, elems_] := \(\[Sum]\+\(n = 1\)\%elems \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \((\(2\ \[Pi]\ f\ \((n - 1)\)\ d\ Cos[phi]\)\/c + \((n - \ 1)\)\ shift)\)\)\)\/elems\)], "Input", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Examples", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Plots can be generated from these functions. For instance, we \ might wish to examine the conventional far-field representation. To do \ this, let us first load a package that defines polar plots:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Needs["\"]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Standard Far-Field Pattern", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Let us examine the far field for a linear array of six elements \ transmitting at 3 MHz, with the antennas spaced a half wavelength apart, with \ a progressive phase shift of 105 degrees between each element. (The", StyleBox[" PlotRange", FontWeight->"Bold"], " option is to make sure that no outlying part of the graph gets cut off, \ as we are interested in all parts of this graph, not only the densest.)" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell[BoxData[ \(PolarPlot[ Abs[FarAF[phi, 3\ 10\^6, 50, 105\ \[Degree], 6]], {phi, \(-\[Pi]\), \[Pi]}, PlotRange \[Rule] All]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PICT", "\<\ :aX0100[0Hl0m00A0_l<0?ooool0:`0000@0003d0001S`000000000N00402P04 02/1S`3d09R0bP000001R`39000000000000B00004P0000000P00@080000002J i`@000000004c`000?l00?ooooooo`01ooooolc<00;oo?olZZP00ooooomVIP04 oooooca01eEE00000007ZZX0000000Ooo`00000023oo`00VIT08Ooo001VIP0R ool003DKTA^A6i00j QNN5ihGW03_4jlC[a>/0?000002ZZ00mEE@00:ZX03jZZ000ZZP0?ool002ZZ010 c<`cIVLc<0000E9VIVIWoo`1EVIVIVLc<05JIVIVIVIT0EiVIVIUVIP1H 3l0?`0o005U>a4k4C/@0FXg8SLR=b01Kca4k40JE:dD[AB]01Z TKRA^96h06_@_=2ld;`0K000EEAEE01]EEAEE5ED06jZZ5EDEE@0KoolEEAEE01` IVKooc 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Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics3D-\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Onset of Far Field", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ It is also interesting to examine the more general expression for \ the field pattern, dependent on distance. Although it is not strictly \ accurate, as it ignores secondary terms such as coupling between the \ antennas, it gives us a view of the onset of the far-field terms.\ \>", "Text",\ ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "To do this, we need to convert from Cartesian to polar coordinates. It is \ useful to define a function that determines the arctangent of one number over \ another properly (the built-in function ", StyleBox["ArcTan", FontWeight->"Bold"], " only accepts a single argument, so it will not necessarily give the \ required answer for each quadrant)." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(arctan[x_, y_] := Which[x == 0 && y \[GreaterEqual] 0, \[Pi]\/2, x == 0 && y < 0, \(-\(\[Pi]\/2\)\), x < 0 && y \[GreaterEqual] 0, ArcTan[y\/x] + \[Pi], x < 0 && y < 0, ArcTan[y\/x] - \[Pi], True, ArcTan[y\/x]]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A plot can now be generated from this. 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