Eisenkern: Dynamoblech (Dbl)

Approximation der Zeitfunktion H(t) (Messkurve)

Messdaten:

datenz = {{0, 59.6}, {0.12566, 61.7}, {0.25132, 68}, {0.37699, 
            72.3}, {0.50265, 85.1}, {0.62831, 102.1}, {0.75398, 
            131.9}, {0.87964, 170.2}, {1.00531, 225.5}, {1.13097, 
            293.6}, {1.25664, 383}, {1.38230, 446.8}, {1.50796, 
            476.6}, {1.63363, 476.6}, {1.75929, 400}, {1.88495, 
            302.1}, {2.01062, 212.7}, {2.13628, 148.9}, {2.26194, 
            85.1}, {2.38761, 
            34}, {2.51327, -8.5}, {2.63894, -34}, {2.76460, -42.6}, {2.89026, -51.1}, {3.01593, -55.3}, {3.14159, -59.6}, {3.26726, -61.7}, {3.39292, -68}, {3.51858, -72.3}, {3.64424, -85.1}, {3.76991, -102.1}, {3.89557, -131.9}, {4.02124, -170.2}, {4.146902, -225.5}, {4.27257, -293.6}, {4.39823, -383}, {4.523893, -446.8}, {4.64956, -476.6}, {4.77522, -476.6}, {4.90088, -400}, {5.02655, -302.1}, {5.15221, -212.7}, {5.27787, -148.9}, {5.40354, -85.1}, {5.52920, -34}, {5.65487, 8.5}, {5.78053, 34}, {5.90619, 42.6}, {6.03186, 51.1}, {6.15752, 
            55.3}, {6.28318, 59.6}};

b1 = 
        ListPlot[datenz, 
          PlotStyle -> {Hue[0.7], PointSize[0.015]}];

Approximation

Happrozeit9 = 
      Fit[datenz, {Sin[x], Sin[3 x], Sin[5 x], Sin[7 x], Sin[9 x], 
          Cos[x], Cos[3 x], Cos[5 x], Cos[7 x], Cos[9 x]}, x]

b1z = 
        Plot[Happrozeit9, {x, 0, 2 Pi}, 
          PlotStyle -> {Hue[0.0]}];

Show[b1, b1z];

Hk

x = 0;
Happrozeit9
Clear[x]

H(Subscript[μ, max])

x = Pi/2;
Happrozeit9
Clear[x]

Max[Table[Happrozeit9, {x, 0, Pi, Pi/180}]]

Approximation der Zeitfunktion B(t) (Messkurve) \(B\&^\) =1.2 T

Messdaten:

datenb = {{0, 0}, {0.12566, 0.128}, {0.25132, 0.276}, {0.37699, 
            0.447}, {0.50265, 0.564}, {0.62831, 0.691}, {0.75398, 
            0.829}, {0.87964, 0.915}, {1.00531, 1}, {1.13097, 
            1.095}, {1.25664, 1.149}, {1.38230, 1.181}, {1.50796, 
            1.202}, {1.63363, 1.202}, {1.75929, 1.181}, {1.88495, 
            1.149}, {2.01062, 1.095}, {2.13628, 1}, {2.26194, 
            0.915}, {2.38761, 0.829}, {2.51327, 0.691}, {2.63894, 
            0.564}, {2.76460, 0.447}, {2.89026, 0.276}, {3.01593, 
            0.128}, {3.14159, 
            0}, {3.26726, -0.128}, {3.39292, -0.276}, {3.51858, -0.447}, {3.64424, -0.564}, {3.76991, -0.691}, {3.89557, -0.829}, {4.02124, -0.915}, {4.146902, -1}, {4.27257, -1.095}, {4.39823, -1.149}, {4.523893, -1.181}, {4.64956, -1.202}, {4.77522, -1.202}, {4.90088, -1.181}, {5.02655, -1.149}, {5.15221, -1.095}, {5.27787, -1}, {5.40354, -0.915}, {5.52920, -0.829}, {5.65487, -0.691}, {5.78053, -0.564}, {5.90619, -0.447}, {6.03186, -0.276}, {6.15752, -0.128}, {6.28318, 
            0}};

bb1 = 
        ListPlot[datenb, 
          PlotStyle -> {Hue[0.7], PointSize[0.015]}];

Approximation bis 9. Harmonische

Bapprozeit9 = 
      Fit[datenb, {Sin[x], Sin[3 x], Sin[5 x], Sin[7 x], Sin[9 x]}, 
        x]

bb1z = 
        Plot[Bapprozeit9, {x, 0, 2 Pi}, 
          PlotStyle -> {Hue[0]}];

Show[bb1, bb1z];

Ummagnetisierungsschleife für Approxination der Induktion

a = Table[i, {i, -500, 500, 50}];

b = Table[i, {i, -1.2, 1.2, 0.1}] // Chop;

bildbp1 = 
        ParametricPlot[{Happrozeit9, Bapprozeit9}, {x, 0, 2 π}, 
          PlotStyle -> Hue[0], GridLines -> {a, b}];

Aummag12b9h9 = 
      Integrate[D[Bapprozeit9, x] Happrozeit9, {x, 0,  2  π}] // 
        Chop

Für Grundwelle der Induktion

Aummag12b1h9 = 
      Integrate[D[1.2 Sin[x], x] Happrozeit9, {x, 0,  2  π}] // 
        Chop

bildbp5 = 
        ListPlot[
          Transpose[{Table[
                datenz[[i, 2]], {i, 
                  Length[datenz]}], 
Table[
                datenb[[i, 2]], {i, Length[datenb]}]}], 
          PlotStyle -> {Hue[0.3], PointSize[0.012]}];

Show[bildbp1, bildbp5];

Darstellung der approximierten Zeitfunktion bis zur 9. Harmonischen von B über (Subscript[H, ν])

bildbp2a = 
        ParametricPlot[{66.174 Cos[x] - 8.308 Cos[3 x] + 
              4.126 Cos[5 x] - 4.503 Cos[7 x] + 0.931 Cos[9 x], 
            Bapprozeit9}, {x, 0, 2 π}, PlotStyle -> Hue[0.7], 
          GridLines -> Automatic];

Hν9 = 
      Fit[datenz, {Cos[x], Cos[3 x], Cos[5 x], Cos[7 x], Cos[9 x]}, 
        x]

Integrate[D[1.2 Sin[x], x] Hν9, {x, 0,  2  π}] // 
      Chop

Magnetisierungskennlinie

Hμ9zeit = 
      Fit[datenz, {Sin[x], Sin[3 x], Sin[5 x], Sin[7 x], Sin[9 x]}, 
        x]

bildmag9 = 
        ParametricPlot[{Hμ9zeit, Bapprozeit9}, {x, 0, 2 π}, 
          PlotStyle -> Hue[0.7]];

daten1 = {{0.05, 10}, {0.125, 20}, {0.2, 24}, {0.25, 27}, {0.3, 
            30}, {0.4, 39}, {0.5, 49}, {0.55, 55}, {0.6, 62.5}, {0.65, 
            73}, {0.7, 87}, {0.75, 100}, {0.8, 120}, {0.85, 140}, {0.9, 
            160}, {0.95, 190}, {1. , 228}, {1.05, 275}, {1.1, 335}, {1.15, 
            410}, {1.2, 500}, {1.25, 615}};

daten1ge = 
        Table[{daten1[[i, 2]], daten1[[i, 1]]}, {i, 
            Length[daten1]}];

bilddge = 
        ListPlot[daten1ge, 
          PlotStyle -> {PointSize[0.01], Hue[0]}];

Show[bildmag9, bilddge];

Show[bildbp1, bildbp5, bildmag9, bilddge];

Nur für Grundwelle der Induktion (1.2 sin(ω t ))

bb1zg = 
        Plot[1.2 Sin[x], {x, 0, 2 Pi}, 
          PlotStyle -> {Hue[0.0]}];

Show[bb1, bb1zg];

Ummagnetisierungsschleife für Grundwelle der Induktion

bildbp12 = 
        ParametricPlot[{Happrozeit9, 1.2 Sin[x]}, {x, 0, 2 π}, 
          PlotStyle -> Hue[0], GridLines -> {a, b}];

Aummag12b1h9 = 
      Integrate[D[1.2 Sin[x], x] Happrozeit9, {x, 0,  2  π}] // 
        Chop

Show[bildbp12, bildbp5];

Darstellung der  Grundwelle von B  über der approximierten Zeitfunktion von (Subscript[H, ν])bis zur 9. Harmonischen (Subscript[H, ν]) 9 (B)

bildbp2ag = 
        ParametricPlot[{66.174 Cos[x] - 8.308 Cos[3 x] + 
              4.126 Cos[5 x] - 4.503 Cos[7 x] + 0.931 Cos[9 x], 
            1.2 Sin[x]}, {x, 0, 2 π}, PlotStyle -> Hue[0.7], 
          GridLines -> Automatic];

Magnetisierungskennlinie

Hμ9zeit = 
      Fit[datenz, {Sin[x], Sin[3 x], Sin[5 x], Sin[7 x], Sin[9 x]}, 
        x]

bildmag12 = 
        ParametricPlot[{Hμ9zeit, 1.2 Sin[x]}, {x, 0, 2 π}, 
          PlotStyle -> Hue[0]];

Show[bildmag12, bilddge];

Darstellung von Hν9zeit über Bmax cos x ⟹ aus der Verlustdichteschleife wird eine eindeutige Kennlinie

bildhν9b1 = 
        ParametricPlot[{1.2 Cos[x], Hν9}, {x, 0, 2  π}, 
          PlotStyle -> Hue[0]];

Fragestellung:

Ist Hν9(B) identisch mit der Kennlinie der Koerzitivfeldstärke \(TraditionalForm\`\(\(H\_k\)\&^\)(\(B\&^\))\)?

Dazu wird Datensatz daten2 eingegeben (\(TraditionalForm\`\(\(H\_k\)\&^\)\)-Werte sind den zugehörigen Ausdrucken der Ummagnetisierungsschleifen entnommen.

daten2 = {{0, 0}, {0.125, 9.4}, {0.25, 16.6}, {0.5, 27.1}, {0.75, 
            36.2}, {1. , 47.9}, {1.2, 60.6}};

b1d2 = 
        ListPlot[daten2, 
          PlotStyle -> {Hue[0.7], PointSize[0.015]}];

Hk9appro[x_] = Fit[daten2, {x, (x^3), (x^5), (x^7), (x^9)}, x]

b1appro = 
        Plot[Hk9appro[x], {x, 0, 1.2}, 
          PlotStyle -> {Hue[0.7]}];

Show[b1d2, b1appro, bildhν9b1];

Hk9 = TrigReduce[Hk9appro[1.2 Cos[x]]] // Simplify // 
        Chop

bildb2verl = 
        ParametricPlot[{Hk9, 1.2 Sin[x]}, {x, 0, 2 π}];

Integrate[D[1.2 Sin[x], x] Hk9, {x, 0,  2  π}] // Chop

Show[bildb2verl, bildbp2ag];

bildbpappro = 
        ParametricPlot[{Hμ9zeit + Hk9, 1.2 Sin[x]}, {x, 0, 2 π}, 
          PlotStyle -> Hue[0.7], GridLines -> {a, b}];

Show[bildbp12, bildbp5, bildbpappro];