(************** Content-type: application/mathematica **************
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Notebook[{

Cell[CellGroupData[{
Cell["\<\
Series-Parallel Circuit Construction:
a Narrative with a Surprise.\
\>", "Title",
  TextAlignment->Center,
  TextJustification->0],

Cell["Barrie Stokes", "Author"],

Cell[TextData[{
  "Discipline of Clinical Pharmacology\nSchool of Medicine and Public Health\n\
Faculty of Health\nUniversity of Newcastle\nNSW Australia\nph  (+61) 02 4921 \
1832\nfax (+61) 02 4960 2088\n",
  ButtonBox["Barrie.Stokes@newcastle.edu.au",
    ButtonData:>{"Barrie.Stokes@newcastle.edu.au", None},
    ButtonStyle->"Hyperlink"]
}], "TextAboutAuthor"],

Cell[CellGroupData[{

Cell["About the Author", "SectionAboutAuthor"],

Cell[TextData[{
  "My professional work is now as an applied Bayesian statistician and \
modeler in medical research. I have worked in operations research and as a \
consultant statistician in a very wide range of research areas, and have been \
a devoted ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " user since 1992. Classical music and hi-fi have been enthusiasms since a \
much earlier date. This is my first really serious foray into more symbolic \
programming, as opposed to the numeric and graphic analysis which is my usual \
fare."
}], "TextAboutAuthor"],

Cell[TextData[{
  "The ohmic impedance (resistance) of the load of the LP gramophone \
cartridge has a non-trivial effect on the ultimate fidelity of the sound \
produced by the complete audio reproduction system. Several high fidelity \
equipment manufacturers supply a selection of high-precision resistors \
intended to be used by the purchaser to form a series-parallel network having \
a total resistance giving optimal sound performance.\n\nAfter purchasing such \
a piece of equipment, the author became curious  as to the possibility of \
using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to obtain an exhaustive list of all the possible circuits and resulting \
resistance values, given any particular set of resistors.\n\nA method of \
generating all possible networks based on a \"syntactic\" representation of \
such networks is presented, taking advantage of the great flexibility offered \
by ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " in representing data structures. Some issues connected with the \
variations in the values of manufactured resistors and the robustness of \
series-parallel circuits will also be considered.\n\nThe main interest here \
may well be that, although the method is successful in practical terms, it is \
not quite completely successful in mathematical terms, and at the time of \
writing the author does not understand why."
}], "Abstract"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Initialisation.", "Section"],

Cell[BoxData[
    \(\($Line = 0;\)\)], "Input",
  CellOpen->False],

Cell[BoxData[{
    \(Needs["\<DiscreteMath`Combinatorica`\>"]\), "\n", 
    \(Needs["\<Statistics`\>"]\), "\n", 
    \(Needs["\<Graphics`\>"]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Functions.", "Section"],

Cell["\<\
This function is just a slightly modified version of Flatten[list, n], where \
the depth to which Flattening proceeds is specified by how close to the \
bottom of the nesting structure of \"list\" we wish to go. The idea will be \
to Flatten some lists which are complex Lists-of-Lists-of-Lists, etc., just \
to the required degree.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ separateElements\ ]\), "\[IndentingNewLine]", 
    \(separateElements[\ list_List, \ d_\ ]\  := \ 
      Flatten[\ list, \ Length[\ Dimensions[\ list\ ]\ ] - d\ ]\)}], "Input"],

Cell[BoxData[{
    \(Clear[\ series, \ paral\ ]\), "\n", 
    \(series[\ x_, \ y_\ ]\  := \ x + y\), "\[IndentingNewLine]", 
    \(paral[\ x_, \ y_\ ]\  := \ 1\/\(\(1\/x + 1\/y\)\(\ \)\)\)}], "Input"],

Cell["\<\
We need Boolean tests to say whether a character is or is not a left or a \
right parenthesis:\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ lbrQ\ ]\), "\[IndentingNewLine]", 
    \(lbrQ[\ n_\ ]\  := \ n \[Equal] 40\)}], "Input"],

Cell[BoxData[{
    \(Clear[\ nonLbrQ\ ]\), "\[IndentingNewLine]", 
    \(nonLbrQ[\ n_\ ]\  := \ n\  != 40\)}], "Input"],

Cell[BoxData[{
    \(Clear[\ rbrQ\ ]\), "\[IndentingNewLine]", 
    \(rbrQ[\ n_\ ]\  := \ n \[Equal] 41\)}], "Input"],

Cell[BoxData[{
    \(Clear[\ nonRbrQ\ ]\), "\[IndentingNewLine]", 
    \(nonRbrQ[\ n_\ ]\  := \ n\  != 41\)}], "Input"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " already has a built-in Boolean test for a list: "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ListQ\  /@ \ {\ \ aaa, \ \ \ \ \ {aaa}, \ \ \ \ {{aaa}}\ \ \ \ \ \ \
}\)], "Input"],

Cell[BoxData[
    \({False, True, True}\)], "Output"]
}, Open  ]],

Cell["\<\
We need to define the complementary Boolean test for a non-list, to be used \
in the following function for \"denesting\" expressions like {{{a,b,c}}}:\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ nonListQ\ ]\), "\[IndentingNewLine]", 
    \(nonListQ[\ x_\ ]\  := \ \(! ListQ[\ x\ ]\)\)}], "Input"],

Cell["Here's what they do:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(ListQ\  /@ \ {\ \ aaa, \ \ \ \ \ {aaa}, \ \ \ \ {{aaa}}\ \ \ \ \ \ }\), \
"\n", 
    \(nonListQ\  /@ \ {\ \ aaa, \ \ \ \ \ {aaa}, \ \ \ \ {{aaa}}\ \ \ \ \ \ }\
\)}], "Input"],

Cell[BoxData[
    \({False, True, True}\)], "Output"],

Cell[BoxData[
    \({True, False, False}\)], "Output"]
}, Open  ]],

Cell["\<\
This function uses nonListQ to strip off repeated enclosing \"{\" and \
matching \"}\" braces from terms like {{{a,b,c}}}:\
\>", "Text"],

Cell[BoxData[{\(Clear[\ denestTerm\ ]\), "\n", 
    RowBox[{\(denestTerm[\ {a_?ListQ}\ ]\  := \ \ denestTerm[\ a\ ]\), " ", 
      StyleBox[\( (*\ 
          remove\ extra\ sets\ of\ enclosing\ "\<{\>"\ and\ "\<}\>"\ until\ \
none\ remain\ *) \),
        FontColor->GrayLevel[0.500008]]}], "\n", 
    RowBox[{\(denestTerm[\ a_\ ]\  := \ \ a\), " ", 
      StyleBox[\( (*\ leave\ a\ non - nested\ list\ unaltered\ *) \),
        FontColor->GrayLevel[0.500008]]}], "\n", 
    RowBox[{\(denestTerm[\ a_?nonListQ\ ]\), " ", ":=", "  ", "a", " ", 
      StyleBox[\( (*\ leave\ isolated\ non - lists\ altered\ *) \),
        FontColor->GrayLevel[0.500008]]}]}], "Input"],

Cell[TextData[{
  "These next two functions convert a nested representation to a circuit \
resistance expression via the T1 transformation rules in which \",\" means \
series and \"},{\" means parallel, and ",
  StyleBox["vice versa ",
    FontSlant->"Italic"],
  "for the T2 rules.\n\nNote: Although this code looks fussy, using \
replacement rules on lists of character codes, the average timing for one \
conversion of a 4-element nested expression is ",
  Cell[BoxData[
      \(0.0008387500000000004`\)], "Output"],
  " seconds, or around 1190 conversions/sec on a Toshiba TE2000 1.7 MHz \
Pentium III laptop. On the development machine, with a 7-element nested \
expression \[Ellipsis]"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(nestStructureToResistanceT1[\ {{t}, {u, {{v, {{w}, {x}}, 
              y}, {z}}}}\ ]\), "\n", 
    \(\(Do[\ 
          nestStructureToResistanceT1[\ {{t}, {u, {{v, {{w}, {x}}, 
                    y}, {z}}}}\ ], \ {10000}\ ]\  // Timing;\)\), "\n", 
    \(%\[LeftDoubleBracket]1\[RightDoubleBracket]\ /\ 10000\), "\n", 
    \(1\/%\)}], "Input"],

Cell[BoxData[
    \(1\/\(1\/t + 1\/\(u + 1\/\(1\/\(v + 1\/\(1\/w + 1\/x\) + y\) + \
1\/z\)\)\)\)], "Output"],

Cell[BoxData[
    \(0.00038440000000000013`\ Second\)], "Output"],

Cell[BoxData[
    \(2601.4568158168568`\/Second\)], "Output"]
}, Open  ]],

Cell[BoxData[{\(Clear[\ 
      nestStructureToResistanceT1\ ]\), "\[IndentingNewLine]", 
    RowBox[{\(nestStructureToResistanceT1[\ originalTestExpression_\ ]\), " ",
       ":=", "\[IndentingNewLine]", 
      RowBox[{"Module", "[", " ", 
        RowBox[{\({testExpression, \ codeExpression, \ series, \ paral, \ 
            codeValue}\), ",", "\[IndentingNewLine]", "\[IndentingNewLine]", 
          
          StyleBox[\( (*\ 
              convert\ originalTestExpression\ to\ a\ list\ of\ character\ \
codes\ *) \),
            FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
          
          RowBox[{\(testExpression\  = \ 
              ToCharacterCode[\ ToString[\ originalTestExpression\ ]\ ]\), 
            ";", "\[IndentingNewLine]", 
            StyleBox[\( (*\ remove\ any\ spaces\ *) \),
              FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
            RowBox[{"testExpression", "=", 
              RowBox[{"testExpression", "//.", 
                RowBox[{\({x___, 32, y___}\), "\[Rule]", 
                  RowBox[{"{", \(x, y\), 
                    StyleBox["}",
                      FontColor->GrayLevel[0]]}]}]}]}], 
            StyleBox[" ",
              FontColor->GrayLevel[0]], ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\(\(\(\ \)\(convert\ all\ {\ 
                          to\ \(\((\)\(\(,\)\(\ \)\(and\ all\)\)\(\ \)\)\)\(}\
\)\)\ to\)\(\ \)\()\)\ *) \),
              FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
            RowBox[{"testExpression", " ", "=", " ", 
              RowBox[{"testExpression", " ", "//.", " ", 
                RowBox[{\({x___, 123, y___}\), "\[Rule]", 
                  RowBox[{"{", 
                    RowBox[{"x", ",", 
                      StyleBox["40",
                        FontColor->GrayLevel[0]], 
                      StyleBox[",",
                        FontColor->GrayLevel[0]], 
                      StyleBox[" ",
                        FontColor->GrayLevel[0]], 
                      StyleBox["y",
                        FontColor->GrayLevel[0]]}], 
                    StyleBox["}",
                      FontColor->GrayLevel[0]]}]}]}]}], 
            StyleBox[";",
              FontColor->GrayLevel[0]], 
            
            StyleBox[\(testExpression\  = \ 
                testExpression\  //. \ {x___, 125, y___} \[Rule] {x, 41, y}\),
              
              FontColor->GrayLevel[0]], 
            StyleBox[";",
              FontColor->GrayLevel[0]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0]], 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<!),!(\>"\ with\ "\<~series~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontWeight->"Plain",
              FontSlant->"Italic",
              FontColor->GrayLevel[0]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?nonRbrQ, 44, x_?nonLbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 115, 101, 114, 105, 
                    101, 115, 126, x, \ y, z}\), " ", ";", 
            "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<),!(\>"\ with\ "\<~series~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            "\[IndentingNewLine]", \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?rbrQ, 44, x_?nonLbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 115, 101, 114, 105, 
                    101, 115, 126, x, \ y, z}\), " ", ";", 
            "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<!),(\>"\ with\ "\<~series~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0.500008]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?nonRbrQ, 44, x_?lbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 115, 101, 114, 105, 
                    101, 115, 126, x, \ y, z}\), " ", ";", 
            "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<),(\>"\ with\ "\<~paral~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0.500008]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?rbrQ, 44, x_?lbrQ, y_, 
                    z___} \[Rule] {u, v, w, \ 126, 112, 97, 114, 97, 108, 
                    126, x, \ y, z}\), " ", ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                convert\ testExpression\ from\ list\ of\ character\ codes\ to\
\ a\ string, \ and\ then\ to\ code\ expression\ *) \),
              FontColor->GrayLevel[0.500008]], 
            "\n", \(codeExpression\  = \ 
              ToExpression[\ FromCharacterCode[\ testExpression\ ]\ ]\)}]}], 
        " ", "\[IndentingNewLine]", "]"}]}]}], "Input"],

Cell[BoxData[{\(Clear[\ 
      nestStructureToResistanceT2\ ]\), "\[IndentingNewLine]", 
    RowBox[{\(nestStructureToResistanceT2[\ originalTestExpression_\ ]\), " ",
       ":=", "\[IndentingNewLine]", 
      RowBox[{"Module", "[", " ", 
        RowBox[{\({testExpression, \ codeExpression, \ series, \ paral, \ 
            codeValue}\), ",", "\[IndentingNewLine]", 
          
          StyleBox[\( (*\ 
              convert\ originalTestExpression\ to\ a\ list\ of\ character\ \
codes\ *) \),
            FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
          
          RowBox[{\(testExpression\  = \ 
              ToCharacterCode[\ ToString[\ originalTestExpression\ ]\ ]\), 
            ";", "\[IndentingNewLine]", 
            StyleBox[\( (*\ remove\ any\ spaces\ *) \),
              FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
            RowBox[{"testExpression", "=", 
              RowBox[{"testExpression", "//.", 
                RowBox[{\({x___, 32, y___}\), "\[Rule]", 
                  RowBox[{"{", \(x, y\), 
                    StyleBox["}",
                      FontColor->GrayLevel[0]]}]}]}]}], 
            StyleBox[" ",
              FontColor->GrayLevel[0]], ";", "\[IndentingNewLine]", 
            StyleBox[\( (*\ remove\ enclosing\ {\ and\ }\ *) \),
              FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
            RowBox[{"(*", 
              RowBox[{
                RowBox[{\(\(**\)\(\ \)\(testExpression\)\), " ", "=", " ", 
                  RowBox[{"Drop", "[", " ", 
                    RowBox[{\(Rest[\ testExpression\ ]\), ",", 
                      StyleBox[" ",
                        FontColor->GrayLevel[0]], 
                      StyleBox[\(-1\),
                        FontColor->GrayLevel[0]]}], 
                    StyleBox[" ",
                      FontColor->GrayLevel[0]], 
                    StyleBox["]",
                      FontColor->GrayLevel[0]]}]}], ";"}], " ", "***)"}], 
            "\[IndentingNewLine]", 
            
            StyleBox[\( (*\(\(\(\ \)\(convert\ all\ {\ 
                          to\ \(\((\)\(\(,\)\(\ \)\(and\ all\)\)\(\ \)\)\)\(}\
\)\)\ to\)\(\ \)\()\)\ *) \),
              FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
            RowBox[{"testExpression", " ", "=", " ", 
              RowBox[{"testExpression", " ", "//.", " ", 
                RowBox[{\({x___, 123, y___}\), "\[Rule]", 
                  RowBox[{"{", 
                    RowBox[{"x", ",", 
                      StyleBox["40",
                        FontColor->GrayLevel[0]], 
                      StyleBox[",",
                        FontColor->GrayLevel[0]], 
                      StyleBox[" ",
                        FontColor->GrayLevel[0]], 
                      StyleBox["y",
                        FontColor->GrayLevel[0]]}], 
                    StyleBox["}",
                      FontColor->GrayLevel[0]]}]}]}]}], 
            StyleBox[";",
              FontColor->GrayLevel[0]], 
            
            StyleBox[\(testExpression\  = \ 
                testExpression\  //. \ {x___, 125, y___} \[Rule] {x, 41, y}\),
              
              FontColor->GrayLevel[0]], 
            StyleBox[";",
              FontColor->GrayLevel[0]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0]], 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<!),!(\>"\ with\ "\<~paral~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontWeight->"Plain",
              FontSlant->"Italic",
              FontColor->GrayLevel[0]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?nonRbrQ, 44, x_?nonLbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 112, 97, 114, 97, 108, 
                    126, x, \ y, z}\), "  ", ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<),!(\>"\ with\ "\<~paral~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            "\[IndentingNewLine]", \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?rbrQ, 44, x_?nonLbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 112, 97, 114, 97, 108, 
                    126, x, \ y, z}\), " ", ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<!),(\>"\ with\ "\<~paral~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0.500008]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?nonRbrQ, 44, x_?lbrQ, 
                    y_, z___} \[Rule] {u, v, w, \ 126, 112, 97, 114, 97, 108, 
                    126, x, \ y, z}\), "  ", ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                replace\ all\ "\<),(\>"\ with\ "\<~series~\>"\ *) \),
              FontColor->GrayLevel[0.500008]], 
            StyleBox["\[IndentingNewLine]",
              FontColor->GrayLevel[0.500008]], \(testExpression\  = \ 
              testExpression\  //. \ {u___, v_, \ w_?rbrQ, 44, x_?lbrQ, y_, 
                    z___} \[Rule] {u, v, w, 126, 115, 101, 114, 105, 101, 
                    115, 126, x, \ y, z}\), " ", ";", "\[IndentingNewLine]", 
            
            StyleBox[\( (*\ 
                convert\ testExpression\ from\ list\ of\ character\ codes\ to\
\ a\ string, \ and\ then\ to\ code\ expression\ *) \),
              FontColor->GrayLevel[0.500008]], 
            "\n", \( (*\(\(\(**\)\(\ \)\(Print[\ 
                    originalTestExpression\ \ \ ]\)\);\)\ ***) \), 
            "\[IndentingNewLine]", \( (*\(\(\(**\)\(\ \)\(Print[\ 
                    FromCharacterCode[\ 
                      testExpression\ ]\ \ ]\)\);\)\ ***) \), 
            "\n", \(codeExpression\  = \ 
              ToExpression[\ FromCharacterCode[\ testExpression\ ]\ ]\)}]}], 
        " ", "\[IndentingNewLine]", "]"}]}]}], "Input"],

Cell["\<\
For use in following defined functions we need a couple of custom Boolean \
functions. The first is \[Ellipsis]\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ onlyVariableQ\ ]\), "\[IndentingNewLine]", 
    \(onlyVariableQ[\ 
        x__\ ]\  := \ \(! \((\((x\  === Plus\ )\)\  || \ \((x === \ 
                Power)\) || \ \((x === \ \(-1\))\) || \ \((Length[x] > 
                0)\))\)\)\)}], "Input"],

Cell["\<\
We also define the logically complementary function, for explanatory reasons.\
\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ notOnlyVariableQ\ ]\), "\[IndentingNewLine]", 
    \(notOnlyVariableQ[\ 
        x__\ ]\  := \ \((x\  === Plus\ )\)\  || \ \((x === \ 
            Power)\) || \ \((x === \ \(-1\))\)\)}], "Input"],

Cell["Here's how they work:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(onlyVariableQ\  /@ \ \ \ \ \ \ \ \ {x1, \ x12, \ Power, \ 
        Plus, \ \(-1\), \ 1\/x1}\), "\n", 
    \(notOnlyVariableQ\  /@ \ {x1, \ x12, \ Power, \ Plus, \ \(-1\), \ 
        1\/x1}\)}], "Input"],

Cell[BoxData[
    \({True, True, False, False, False, False}\)], "Output"],

Cell[BoxData[
    \({False, False, True, True, True, False}\)], "Output"]
}, Open  ]],

Cell["\<\
The following applications of notOnlyVariableQ[ ] shows that in the structure \
expressions of two topologically identical, i.e., equivalent circuits, the \
Heads Power and Plus, and the -1 involved in a reciprocal,  can be in \
different places, so this turns out to be no basis for an equivalence test.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Position[\ 
      1\/\(1\/x1 + 1\/\(1\/\(1\/x2 + 1\/x3\) + x4\)\), _?
        notOnlyVariableQ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({{0}, {1, 0}, {1, 1, 0}, {1, 1, 2}, {1, 2, 0}, {1, 2, 1, 0}, {1, 2, 1, 
        1, 0}, {1, 2, 1, 1, 1, 0}, {1, 2, 1, 1, 1, 1, 0}, {1, 2, 1, 1, 1, 1, 
        2}, {1, 2, 1, 1, 1, 2, 0}, {1, 2, 1, 1, 1, 2, 2}, {1, 2, 1, 1, 
        2}, {1, 2, 2}, {2}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Position[\ 
      1\/\(1\/x1 + 1\/\(x2 + 1\/\(1\/x3 + 1\/x4\)\)\), _?
        notOnlyVariableQ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({{0}, {1, 0}, {1, 1, 0}, {1, 1, 2}, {1, 2, 0}, {1, 2, 1, 0}, {1, 2, 1, 
        2, 0}, {1, 2, 1, 2, 1, 0}, {1, 2, 1, 2, 1, 1, 0}, {1, 2, 1, 2, 1, 1, 
        2}, {1, 2, 1, 2, 1, 2, 0}, {1, 2, 1, 2, 1, 2, 2}, {1, 2, 1, 2, 
        2}, {1, 2, 2}, {2}}\)], "Output"]
}, Open  ]],

Cell["\<\
The complementary Boolean test onlyVariableQ[ ] will turn out to be useful \
\[Ellipsis]\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Position[\ 
      1\/\(1\/x1 + 1\/\(1\/\(1\/x2 + 1\/x3\) + x4\)\), _?
        onlyVariableQ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({{1, 1, 1}, {1, 2, 1, 1, 1, 1, 1}, {1, 2, 1, 1, 1, 2, 1}, {1, 2, 1, 
        2}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Position[\ 
      1\/\(1\/x1 + 1\/\(x2 + 1\/\(1\/x3 + 1\/x4\)\)\), _?
        onlyVariableQ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({{1, 1, 1}, {1, 2, 1, 1}, {1, 2, 1, 2, 1, 1, 1}, {1, 2, 1, 2, 1, 2, 
        1}}\)], "Output"]
}, Open  ]],

Cell["\<\
\[Ellipsis] because it has clearly found the positions of the different \
symbols x1, x2, etc., and in combination with the built-in function Extract[ \
], we can reproduces a list of all the symbols in a structure expression for \
a circuit.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ extractElements\ ]\), "\[IndentingNewLine]", 
    \(extractElements[\ struct_\ ]\  := \ 
      Extract[\ struct, \ \ Position[\ 
          struct, _?onlyVariableQ]\ ]\)}], "Input"],

Cell["A test \[Ellipsis]", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(extractElements[\ 
      1\/\(1\/x3 + 1\/\(x2 + 1\/\(1\/x1 + 1\/\(x4 + x5 + 1\/x6\)\)\)\)\ ]\)\(\
\ \)\)\)], "Input"],

Cell[BoxData[
    \({x3, x2, x1, x4, x5, x6}\)], "Output"]
}, Open  ]],

Cell["\<\
extractElements[ ] is used inside the following function, since we need a \
list of the symbols in each of two structures we are testing for equivalence \
in order to construct the rules for substituting different highly precise \
numerical values into the two structures in all possible ways.\
\>", "Text"],

Cell[BoxData[
    RowBox[{\(equivalentStructuresQ[\ struct1_, \ struct2_\ ]\), " ", ":=", 
      "\[IndentingNewLine]", 
      RowBox[{"Module", "[", " ", "\[IndentingNewLine]", 
        RowBox[{\({perms, \ nPerms, \ elementSymbols1, elementSymbols2, \ 
            nElements, \ allRules, allCircuitValues1, \ allCircuitValues2}\), 
          ",", "\[IndentingNewLine]", "\[IndentingNewLine]", 
          
          RowBox[{\(elementSymbols1\  = \ 
              Sort[\ extractElements[\ struct1\ ]\ ]\), ";", 
            "\[IndentingNewLine]", \(elementSymbols2\  = \ 
              Sort[\ extractElements[\ struct2\ ]\ ]\), ";", 
            "\[IndentingNewLine]", "\[IndentingNewLine]", 
            RowBox[{"If", "[", " ", 
              
              RowBox[{\(Length[\ elementSymbols1\ ] \[Equal] 
                  Length[\ elementSymbols2\ ]\), ",", " ", 
                
                StyleBox[\( (*\ 
                    circuits\ must\ at\ least\ have\ same\ \ number\ of\ \
symbols\ *) \),
                  FontColor->GrayLevel[0.500008]], 
                StyleBox["\[IndentingNewLine]",
                  FontColor->GrayLevel[0.500008]], 
                
                RowBox[{\(nElements\  = \ Length[\ elementSymbols1\ ]\), ";", 
                  "\[IndentingNewLine]", 
                  RowBox[{"(", 
                    
                    RowBox[{\(perms\  = \ 
                        Permutations[\ 
                          Table[\ Random[], \ {nElements}\ ]\ ]\), ";", 
                      "\n", \(nPerms\  = \ Length[\ perms\ ]\), ";", 
                      "\n", \(allRules1\  = \ {}\), ";", 
                      "\n", \(Do[\ 
                        AppendTo[\ allRules1, \ 
                          Thread[\ 
                            Rule[\ elementSymbols1, 
                              perms\[LeftDoubleBracket]
                                i\[RightDoubleBracket]\ ]\ ]\ ], \ {i, 1, 
                          nPerms}\ ]\), ";", 
                      "\[IndentingNewLine]", \(allRules2\  = \ {}\), ";", 
                      "\n", \(Do[\ 
                        AppendTo[\ allRules2, \ 
                          Thread[\ 
                            Rule[\ elementSymbols2, 
                              perms\[LeftDoubleBracket]
                                i\[RightDoubleBracket]\ ]\ ]\ ], \ {i, 1, 
                          nPerms}\ ]\), ";", 
                      "\[IndentingNewLine]", \(allCircuitValues1\  = 
                        Release[\ Hold[\ \ struct1\ ]\  /. \ allRules1\ ]\), 
                      ";", " ", 
                      
                      StyleBox[\( (*\ 
                          all\ possible\ substitutions\ in\ structure\ 1\ *) \
\),
                        FontColor->GrayLevel[0.500008]], 
                      "\[IndentingNewLine]", \(allCircuitValues2\  = 
                        Release[\ Hold[\ \ struct2\ ]\  /. \ allRules2\ ]\), 
                      ";", " ", 
                      
                      StyleBox[\( (*\ 
                          all\ possible\ substitutions\ in\ structure\ 2\ *) \
\),
                        FontColor->GrayLevel[0.500008]], 
                      "\[IndentingNewLine]", \(Length[\ 
                          Intersection[\ allCircuitValues1, \ 
                            allCircuitValues2\ ]\ \ ] != 0\)}], ")"}]}], ",", 
                " ", 
                StyleBox[\( (*\ any\ common\ \(\(values\)\(?\)\)\ *) \),
                  FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", 
                "False"}], " ", 
              StyleBox[\( (*\ structures\ don' t\ have\ same\ symbols\ *) \),
                FontColor->GrayLevel[0.500008]], "\[IndentingNewLine]", " ", 
              "]"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input"],

Cell["\<\
Topologically identical, i.e., equivalent, circuits with same symbols:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 1\/\(1\/\(x1 + x2\) + 1\/x3 + 1\/x4\), \ 
      1\/\(1\/x1 + 1\/\(x2 + x3\) + 1\/x4\)\ ]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell["\<\
Topologically identical, i.e., equivalent, circuits with different symbols:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 1\/\(1\/\(aaa + bbb\) + 1\/ccc + 1\/ddd\), \ 
      1\/\(1\/x1 + 1\/\(x2 + x3\) + 1\/x4\)\ ]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell["\<\
Topologically non-identical, i.e., non-equivalent, circuits with same \
symbols:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 1\/\(1\/\(x1 + x2\) + 1\/x3 + 1\/x4\), \ 
      x1 + 1\/\(1\/x2 + 1\/x3 + 1\/x4\)\ ]\)], "Input"],

Cell[BoxData[
    \(False\)], "Output"]
}, Open  ]],

Cell["\<\
Topologically non-identical, i.e., non-equivalent, circuits with same \
symbols:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 
      1\/\(1\/x1 + 1\/\(x2 + 1\/\(1\/x3 + 1\/x4\)\)\), \ 
      x1 + 1\/\(1\/x2 + 1\/x3 + 1\/x4\)\ ]\)], "Input"],

Cell[BoxData[
    \(False\)], "Output"]
}, Open  ]],

Cell["\<\
Topologically identical, i.e., equivalent, circuits with same symbols:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 
      1\/\(1\/x1 + 1\/\(x2 + 1\/\(1\/x3 + 1\/x4\)\)\), \ 
      1\/\(1\/\(x1 + 1\/\(1\/x2 + 1\/x3\)\) + 1\/x4\)\ ]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell["\<\
Topologically non-identical, i.e., nonequivalent, circuits because of \
differing numbers of elements:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(equivalentStructuresQ[\ 
      1\/\(1\/x1 + 1\/\(x2 + 1\/\(1\/x3 + 1\/x4\)\)\), \ 
      1\/\(1\/\(x1 + 1\/\(1\/x2 + 1\/x3\)\)\)\ ]\)], "Input"],

Cell[BoxData[
    \(False\)], "Output"]
}, Open  ]],

Cell["\<\
This function \"repairs\" the effect of Sort[ ] applied to individual circuit \
expressions when they represent an essentially parallel circuit.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[\ repairFunction\ ]\), "\[IndentingNewLine]", 
    \(repairFunction[\ x__\ ]\  := \ 
      If[\ Head[x] === Plus, \ x, \ 
        1\/x\[LeftDoubleBracket]2\[RightDoubleBracket]]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Section 1 - Introduction.", "Section"],

Cell["\<\
The modern digital systems\[LongDash]CD, HDCD, DVD-A, SACD\[LongDash]for \
sound reproduction were preceded by an electromechanical system which was \
initially developed before 1900, and continues to be refined in the present \
day. The flat black gramophone record, or LP (for Long Playing) was \
introduced in the mid 1950s, and audiophiles regard the period from then \
until around 1970 as the \"Golden Age of the LP\". It is possible today to \
purchase extremely expensive turntables, arms, and cartridges to play both \
old and newly pressed LPs, and many audiophiles in what is termed the high \
end of the market seriously claim that such equipment produces music of even \
greater fidelity than the best of modern digital systems.

The basic phonographic mechanism for storing and recovering the oscillating \
voltage signal originally produced by the microphone in the original \
recording venue is illustrated in Figures 1 and 2.\
\>", "Text"],

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CBCBCCCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCBCCCDCECDCCCAC9CACCCED0D1D1D0D0
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CDCDCDCECFCFCFD0D0D1D1D2D1D0CFCDCDCDCECFCECECDCDCCCBCBCACDCCCBC9C7C5C4C3
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CBCDCAC2BFBEB6AAAAA9A8A7A8A9AAABACADAEB1B5B9BDC0C5C5C5C7C9CDD0D2D1D1D1D1
D1D1D0D0D3D3D3D3D3D3D3D3CFCECDCCCDCECFD0CFCFCECECDCCCCCBCBCBCAC9C7C6C5C5
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CBCACAC9CDCDCED0D1D2D3D4CFCECECDCCCCCBCBC8C9CACBCCCDCECFD4D6D9DCDDDCDAD8
DCDCDBDAD9D8D7D6D5D4D2D1D0D1D2D3D0D2D3D5D6D6D5D4DCDBD9D7D6D7D8D8D9D9D8D8
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ECEDEEEEECEBEAE9E7E6E5E4E4E4E4E4E4E4E4E4E4E4E4E4E4E4E4E4
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C8C6C4C2C2C4C6C8C6C6C6C6C6C6C6C6C5C5C6C6C6C4C2C1C1C1C1C1BFBCBAB8B6B5B2AE
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D4D4D4D4D0D0D1D1D2D3D3D4D6D6D5D4D4D3D2D2D0D0D1D1D2D3D3D4D8D8D8D8D8D8D8D8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ADABACADA6A1ACBCC1C6CACBC9C8CACCD0D0CFCCC7C0BAB6B2B1B0AFAFB0B2B3B2B6BDC5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ADABAEB1ABA4AAB7C0C4C9CBC9C9CBCECFD1D3D3CFC7BFBAB4B3B2B1B1B2B3B4B2B6BBC1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A8A7ACB3B0AAADB7BEC3C8CAC9C9CBCECDD1D6D8D5CDC4BEB8B7B5B3B3B3B4B5B3B6BABF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A2A1A8B3B5B1B3BCBDC1C6C7C6C5C7CACBCFD5D8D7D0C8C3BBBAB8B6B5B5B6B6B6B7BABD
C0C1C2C3C5C6C7C8C9C8C7C6C3C3C3C3C3C3C3C3C7C6C6C5C4C4C3C3BDC1C4C5C3C4C7CB
C9C5C2C3C7CACAC8C0C1C2C4C6C7C9C9C6C6C7C8C8C9CACACCCCCBCBCACAC9C9BDBCBAB8
B8B8B9BAC0BDBAB7B8BBC0C3C9C7C6C7CACED1D3D5CBBDB1A9A1989195928F8C8982776F
73767A7B78736E6C76787A7C7E7E7E7E7676777A7E8286888F919394969AA0A5BAC3CED3
CFC9C4C3BBB6AFACABA9A6A2A8A0978F8E91969A9B9A98989A9EA3A7A5A8ACB1B5B8B9BA
BCBDBFC1C3C5C7C8C9CED4D6D4D2D2D3C7C8CACDCFD2D4D5D5D4D4D3D2D1D0D0D2D2D2D2
D2D2D2D2D3D3D4D5D5D6D7D7D6D6D5D4D4D3D2D2D3D3D4D5D5D6D7D7D8D8D8D8D8D8D8D8
D8D9D9D8D6D4D1D0C8C8CACCCFD2D5D6DAD8D6D4D2D0CFCFCFD1D3D3D2D1D0D0C9CED6DD
E1E1DFDED8D6D4D2D1D1D2D2D0D1D3D5D6D7D8D8D7D7D7D7D8D9DADBD6D6D6D6D8DADDDF
DEDDDCDBDADADADAD7D7D6D5D5D4D3D3D1CECAC7C7CCD3D8D8D9DBDCDBD9D7D5D8D8D9DA
DADBDCDCDEDFDFE0E1E1E2E2E1E0DFDEDDDDDDDDDAD7D5D7DBDCDBD8DADADADADADADADA
DCDCDCDCDCDCDCDCD8D9DBDCDCDBD9D8D9DADADADADADADBDADADADADADADADADCDCDCDC
DCDCDCDCDADADADADADADADADCDCDCDCDCDCDCDCDCDCDBDADAD9D8D8D3D0CCCBCCCBC7C3
C5C4C2C1C1C1C2C3C7C7C6C7C8CBCDCFD0D1D3D5D8DADCDDDFE0E2E3E4E3E2E1DCDAD8D6
D5D4D5D5D5D5D5D5D6D7D8D9D9DADADBDBDBDBDADBDBDBDCDCDDDDDDDDDEE0E1E1E0DEDD
E0E0DFDDDCDAD9D9D9D9DADADBDBDCDCDEDFE1E4E7EAECEDEEEEEFF0F0F1F2F2F0F0F0F0
F0F0F0F0EFEFF0F0F0EEECEBEBEAEAE9E8E8E7E7EAEAEAEAEAEAEAEA
A19FA5B1B7B5B8C0BCC0C4C4C2C0C1C4C8CBD0D4D3CFCAC6BFBDBBB9B7B7B7B8BABABBBD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A7A1A4AFB6B5B8C0BCBFC2C1BDBBBBBDC5C7CACDCECCCAC8C1C0BDBBB9B8B8B9BEBDBDBE
BFC0C2C3C1C2C4C6C7C7C6C5C8C8C8C8C8C8C8C8C8C8C7C7C6C5C5C4C3C6C8C6C2C0C2C5
C7C3C0C1C5C8C7C6C5C5C5C5C6C6C6C6C6C6C7C8C8C9CACAC7C7C7C8C8C8C8C8CFCCC9C5
C2C1C0C0C0C0C1C2C3C4C5C6C8C6C7CCD1D1CCC5BCAE9C9392918B85847F797677777572
6D71757471707377767A7F848584807D7B7B7B7C7F8285878888898D92969999A8B9CDD7
D4CECCCDC1BCB5B1AFADA8A5ADA1928A8C91959699999A9B9DA0A3A4A1A4A8AEB2B6B8B9
BCBDBFC1C3C5C7C7C7CDD3D7D6D6D7D9D3D3D4D5D6D7D8D8DBDAD9D7D6D4D3D2D7D7D7D7
D7D7D7D7D5D6D6D7D8D8D9D9D6D6D5D4D4D3D2D2D5D6D6D7D8D8D9D9D8D8D8D8D8D8D8D8
D9D9D8D7D5D1CECCCDCED0D3D5D6D7D7D8D6D3D0CFCFD1D2CCD1D4D3CDCACBCECDCFD3D6
D7D7D6D5D1D0CFCFCFD0D2D3D1D3D6D9DBDBDAD9D9D7D6D5D6D8DBDDD2D2D3D4D6DADDDF
E2E0DDDBDADBDDDED9D9D8D8D7D6D6D5D1D3D5D2CDCCD0D5D6D9DDE0E0DDDAD7D8D8D9DA
DADBDCDCE0E0E1E1E2E3E3E4E4E2E0DDDDDEDFE1DEDBD9DADDDEDCD9DADADADADADADADA
DEDEDEDEDEDEDEDED8D9DBDCDCDBD9D8D8D8D9DADADBDCDCDCDCDCDCDCDCDCDCDEDEDEDE
DEDEDEDEDFDFDFDFDFDFDFDFDCDCDCDCDCDCDCDCDCDCDBDADAD9D8D8D5D1CECDCECDC9C5
C5C4C3C2C2C3C5C6CBCBCACBCCCFD1D3D4D6D8DBDEE1E3E5E3E4E4E4E3E1DFDDDAD9D8D6
D5D6D7D8D7D5D4D3D3D6D8DADBDDE0E3E4E3E1E0E1E1E1E1E1E0E0E0E0E1E2E3E3E2E1E0
E2E1E0DFDDDCDBDADCDCDCDCDBDBDBDBDDDEE0E3E6E9EBECECECEEEFF1F2F4F4F2F2F2F2
F2F2F2F2F0F1F2F2F1EFEDECEBEBEBEAE9E8E8E8EAEAEAEAEAEAEAEA
ADA5A5AEB4B4B7BDBBBFC1BFBBB7B7B8C3C4C6C8CACAC9C9C3C1BEBCBAB9B9B9C0BFBFBE
BFC1C2C4C1C3C5C7C8C8C8C7CACACACACACACACACFCECECDCCCCCBCBC6C8C9C6C2BFC0C2
C8C4C1C2C6C9C9C7C9C9C9C8C8C7C7C7C6C6C7C8C8C9CACAC9C9CACACBCBCBCBCBC9C5C0
BCBAB8B8BBBEC1C5C8C9CACACAC8C9CED3D0C6BDB9AA999192938F8987807976787A7B7B
72767A7875767D8375797F8485837E7B81807F7E7F80828389898A8F959A9B9BA6B8CFDA
D7D1D0D1C5BFB8B3B2AFABA7A99F938E9093929098999B9DA0A1A3A4A0A3A7ADB2B6B8B9
BEBFC1C3C5C7C9CAC7CDD4D8D7D7D9DBDADADADADBDBDBDBDFDEDDDBD9D8D6D6D8D8D8D8
D8D8D8D8D8D9D9DADBDBDCDCD6D6D5D4D4D3D2D2D8D9D9DADBDBDCDCD8D8D8D8D8D8D8D8
D9D9D8D7D4D1CECCCFD1D3D5D7D7D7D7D7D5D2CFCED0D2D3CBD1D5D3CCC8CACDD0D1D3D5
D5D5D4D4D0CFCECECFD0D2D3D1D4D7DBDCDCDAD9D9D8D6D4D5D8DBDDD1D1D1D3D5D9DCDE
E5E3E0DDDDDEE0E1DCDCDBDBDAD9D9D8D0D5D8D5CFCDD0D4D8DBE0E3E3E1DDDAD8D8D9DA
DADBDCDCE0E0E1E2E2E3E4E4E5E3E0DDDDDEE0E1E0DCDADBDEDEDCD9DADADADADADADADA
DEDEDEDEDEDEDEDED8D9DBDCDCDBD9D8D7D8D8D9DBDCDCDDDCDCDCDCDCDCDCDCDEDEDEDE
DEDEDEDEE0E0E0E0E0E0E0E0DCDCDCDCDCDCDCDCDCDCDBDADAD9D8D8D6D3CFCECFCECAC6
C5C4C3C2C3C4C6C7CCCCCBCCCDD0D2D4D8D9DBDFE2E5E8E9E6E7E7E6E5E2E0DEDBDAD8D7
D6D7D8D9D9D8D6D4D5D8DBDDDCDFE2E5E6E5E4E2E3E3E3E3E2E2E2E1E2E3E5E6E6E5E3E2
E2E1E0DFDDDCDBDADFDFDFDEDEDDDDDDDCDDE0E2E5E8EAECEBECEDEFF1F3F4F5F2F2F2F2
F2F2F2F2F0F1F2F2F1F0EEEDEEEEEDECECEBEAEAEAEAEAEAEAEAEAEA
A9AAADB1B4B8BBBCC0C0C0BFBDB9B6B5BDBBB9BCC2C8CCCDC7C4C1C1C1BFBAB5BCBCBDBE
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  "The tip of the stylus rests in the wiggly groove, and the rotation of the \
record means that the stylus is made to oscillate both laterally and \
vertically as it \"tracks\" the groove. Typical groove width is 75 microns \
(",
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elliptical cross-section, are 5-6 microns by 35-120 microns. The wavelength \
of a 20 kHz signal near the inner grooves of a 30 cm LP is around 18 microns, \
while a medium level 1 kHz signal has a sideways amplitude of  about 11 \
microns. The same signal at a still-audible 40 dB lower level has an \
amplitude of around 110 ",
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signal requires an amplitude of ",
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  ". Modern cartridges can trace such a waveform, and the signal when \
simplified can easily be heard as part of a musical presentation. The power \
output of  a moving coil cartridge at 1 kHz is around 15 ",
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  "The stylus is attached to a light but rigid cantilever, a small thin metal \
tube, often beryllium or boron/aluminium, which is suspended in a suitable \
elastomeric\[LongDash]butyl rubber or neoprene\[LongDash]mounting (the orange \
block in Figure 2). This arrangement allows the oscillations of the stylus to \
be faithfully transmitted to the end of the cantilever carrying either a \
magnet or a coil. Figure 2 illustrates schematically a so-called moving ",
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  " (MM) design, in which Faraday's law of induction operates. Faraday's law \
says that if an electrically conductive loop\[LongDash]here the turns of the \
coil\[LongDash] is placed in a time-varying magnetic field, the varying \
magnetic flux through the loop induces an emf (\"electro-motive force\") in \
the loop which sets electrons in motion, i.e., produces a current flow in the \
loop. As a physicist would say,"
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  " represents the rate of change of the magnetic flux through the coil. \
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is attached to the cantilever, and fixed magnets are positioned along the \
axis of the coil, the moving ",
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