Einen Stromkreis analysieren 

Ermitteln Sie die Stromstärken in einem elektrischen Stromkreis mit zwei Spannungsquellen und drei Widerständen.

In[1]:=

X g = \!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{"\!\(\*SubscriptBox[\(J\), \(1\)]\)", "\!\(\*SubscriptBox[\(J\), \(2\)]\)", "\!\(\*SubscriptBox[\(J\), \(3\)]\)", "\!\(\*SubscriptBox[\(J\), \(4\)]\)"}, {{{1, 4}, {4, 2}, {4, 3}, {2, 1}, {3, 1}}, Null}, { AspectRatio -> Full, EdgeLabels -> { DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(4\)]\)", "\!\(\*SubscriptBox[\(J\), \(2\)]\)"] -> Placed[ Labeled[ Graphics[ GeometricTransformation[ GeometricTransformation[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], Line[{{-90., 0.}, {90., 0.}}], Rectangle[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], Rectangle[{90, -10}, {110, 10}], Rectangle[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], { 0, 0}], ImageSize -> 65], Style["\!\(\*SubscriptBox[\(R\), \(2\)]\)", 12], Top], {0.5, { Rational[1, 2], 0.39}}], DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(4\)]\)", "\!\(\*SubscriptBox[\(J\), \(3\)]\)"] -> Placed[ Labeled[ Graphics[ GeometricTransformation[ GeometricTransformation[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], Line[{{-90., 0.}, {90., 0.}}], Rectangle[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], Rectangle[{90, -10}, {110, 10}], Rectangle[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], { 0, 0}], ImageSize -> 65], Style["\!\(\*SubscriptBox[\(R\), \(3\)]\)", 12], Top], {0.66, { Rational[1, 2], 0.39}}], DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(1\)]\)", "\!\(\*SubscriptBox[\(J\), \(4\)]\)"] -> Placed[ Labeled[ Graphics[ GeometricTransformation[ GeometricTransformation[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], Line[{{-90., 0.}, {90., 0.}}], Rectangle[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], Rectangle[{90, -10}, {110, 10}], Rectangle[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], { 0, 0}], ImageSize -> 65], Style["\!\(\*SubscriptBox[\(R\), \(1\)]\)", 12], Top], {0.6, { Rational[1, 2], 0.38}}], DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(3\)]\)", "\!\(\*SubscriptBox[\(J\), \(1\)]\)"] -> Placed[ Labeled[ Graphics[{{ GrayLevel[1], Rectangle[{0, -1}, {0.3, 1}]}, { Thickness[0.3], Line[{{0, -0.5}, {0, 0.5}}]}, { Thickness[0.5], Line[{{0.5, -1}, {0.5, 1}}]}}, ImageSize -> 8], Style["\!\(\*SubscriptBox[\(V\), \(2\)]\)", 12], Top], {0.25, { Rational[1, 2], 0.39}}], DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(2\)]\)", "\!\(\*SubscriptBox[\(J\), \(1\)]\)"] -> Placed[ Labeled[ Graphics[{{ GrayLevel[1], Rectangle[{0, -1}, {0.3, 1}]}, { Thickness[0.5], Line[{{0, -1}, {0, 1}}]}, { Thickness[0.3], Line[{{0.5, -0.5}, {0.5, 0.5}}]}}, ImageSize -> 8], Style["\!\(\*SubscriptBox[\(V\), \(1\)]\)", 12], Top], {0.5, { Rational[1, 2], 0.39}}]}, EdgeLabelStyle -> {5}, EdgeShapeFunction -> { DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(2\)]\)", "\!\(\*SubscriptBox[\(J\), \(1\)]\)"] -> ({ Arrowheads[{{0.05, 0.85}}], Arrow[#]}& ), DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(3\)]\)", "\!\(\*SubscriptBox[\(J\), \(1\)]\)"] -> ({ Arrowheads[{{0.05, 0.8}}], Arrow[{ Part[#, 1], { Part[#, 2, 1], Part[#, 1, 2]}, Part[#, -1]}]}& ), DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(4\)]\)", "\!\(\*SubscriptBox[\(J\), \(3\)]\)"] -> ({ Arrowheads[{{0.05, 0.2}}], Arrow[{ Part[#, 1], { Part[#, 1, 1], Part[#, 2, 2]}, Part[#, -1]}]}& ), DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(1\)]\)", "\!\(\*SubscriptBox[\(J\), \(4\)]\)"] -> ({ Arrowheads[{{0.05, 0.92}}], Arrow[{ Part[#, 1], { Part[#, 1, 1], 1}, { Part[#, -1, 1], 1}, Part[#, -1]}]}& ), DirectedEdge[ "\!\(\*SubscriptBox[\(J\), \(4\)]\)", "\!\(\*SubscriptBox[\(J\), \(2\)]\)"] -> ({ Arrowheads[{{0.05, 0.2}}], Arrow[#]}& )}, EdgeStyle -> { GrayLevel[0]}, EdgeWeight -> {-100, -200, -300, 3, -4}, ImagePadding -> {{10, 15}, {10, 25}}, ImageSize -> {236., 149.}, PerformanceGoal -> "Quality", VertexCoordinates -> {{-1.5, 0}, {-0.5, 0}, {-0.5, -1}, { 1.5, 0}}, VertexLabels -> {"Name"}, VertexStyle -> { GrayLevel[0]}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.028937842778793414`], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], GrayLevel[0], { TagBox[{ Arrowheads[{{0.05, 0.92}}], ArrowBox[{{-1.5, 0.}, {-1.5, 1}, {1.5, 1}, {1.5, 0.}}]}, "DynamicName", BoxID -> "EdgeLabelID$1"], InsetBox[ FormBox[ StyleBox[ TemplateBox[{ GraphicsBox[ GeometricTransformationBox[ GeometricTransformationBox[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], LineBox[{{-90., 0.}, {90., 0.}}], RectangleBox[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], RectangleBox[{90, -10}, {110, 10}], RectangleBox[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], {0, 0}], ImageSize -> 65], StyleBox["\"\\!\\(\\*SubscriptBox[\\(R\\), \\(1\\)]\\)\"", 12, StripOnInput -> False]}, "Labeled", DisplayFunction -> (FormBox[ GridBox[{{ ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}, { TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"]}}, GridBoxAlignment -> { "Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {2, 1}], TraditionalForm]& ), InterpretationFunction -> (RowBox[{"Labeled", "[", RowBox[{#, ",", #2, ",", "Top"}], "]"}]& )], 5, StripOnInput -> False], TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$1", Automatic, Scaled[0.6]]], ImageScaled[{ Rational[1, 2], 0.38}], BaseStyle -> "Graphics"]}, { TagBox[{ Arrowheads[{{0.05, 0.85}}], ArrowBox[{{-0.5, 0.}, {-1.5, 0.}}]}, "DynamicName", BoxID -> "EdgeLabelID$2"], InsetBox[ FormBox[ StyleBox[ TemplateBox[{ GraphicsBox[{{ GrayLevel[1], RectangleBox[{0, -1}, {0.3, 1}]}, { Thickness[0.5], LineBox[{{0, -1}, {0, 1}}]}, { Thickness[0.3], LineBox[{{0.5, -0.5}, {0.5, 0.5}}]}}, ImageSize -> 8], StyleBox["\"\\!\\(\\*SubscriptBox[\\(V\\), \\(1\\)]\\)\"", 12, StripOnInput -> False]}, "Labeled", DisplayFunction -> (FormBox[ GridBox[{{ ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}, { TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"]}}, GridBoxAlignment -> { "Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {2, 1}], TraditionalForm]& ), InterpretationFunction -> (RowBox[{"Labeled", "[", RowBox[{#, ",", #2, ",", "Top"}], "]"}]& )], 5, StripOnInput -> False], TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$2", Automatic, Scaled[0.5]]], ImageScaled[{ Rational[1, 2], 0.39}], BaseStyle -> "Graphics"]}, { TagBox[{ Arrowheads[{{0.05, 0.8}}], ArrowBox[{{-0.5, -1.}, {-1.5, -1.}, {-1.5, 0.}}]}, "DynamicName", BoxID -> "EdgeLabelID$3"], InsetBox[ FormBox[ StyleBox[ TemplateBox[{ GraphicsBox[{{ GrayLevel[1], RectangleBox[{0, -1}, {0.3, 1}]}, { Thickness[0.3], LineBox[{{0, -0.5}, {0, 0.5}}]}, { Thickness[0.5], LineBox[{{0.5, -1}, {0.5, 1}}]}}, ImageSize -> 8], StyleBox["\"\\!\\(\\*SubscriptBox[\\(V\\), \\(2\\)]\\)\"", 12, StripOnInput -> False]}, "Labeled", DisplayFunction -> (FormBox[ GridBox[{{ ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}, { TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"]}}, GridBoxAlignment -> { "Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {2, 1}], TraditionalForm]& ), InterpretationFunction -> (RowBox[{"Labeled", "[", RowBox[{#, ",", #2, ",", "Top"}], "]"}]& )], 5, StripOnInput -> False], TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$3", Automatic, Scaled[0.25]]], ImageScaled[{ Rational[1, 2], 0.39}], BaseStyle -> "Graphics"]}, { TagBox[{ Arrowheads[{{0.05, 0.2}}], ArrowBox[{{1.5, 0.}, {-0.5, 0.}}]}, "DynamicName", BoxID -> "EdgeLabelID$4"], InsetBox[ FormBox[ StyleBox[ TemplateBox[{ GraphicsBox[ GeometricTransformationBox[ GeometricTransformationBox[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], LineBox[{{-90., 0.}, {90., 0.}}], RectangleBox[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], RectangleBox[{90, -10}, {110, 10}], RectangleBox[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], {0, 0}], ImageSize -> 65], StyleBox["\"\\!\\(\\*SubscriptBox[\\(R\\), \\(2\\)]\\)\"", 12, StripOnInput -> False]}, "Labeled", DisplayFunction -> (FormBox[ GridBox[{{ ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}, { TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"]}}, GridBoxAlignment -> { "Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {2, 1}], TraditionalForm]& ), InterpretationFunction -> (RowBox[{"Labeled", "[", RowBox[{#, ",", #2, ",", "Top"}], "]"}]& )], 5, StripOnInput -> False], TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$4", Automatic, Scaled[0.5]]], ImageScaled[{ Rational[1, 2], 0.39}], BaseStyle -> "Graphics"]}, { TagBox[{ Arrowheads[{{0.05, 0.2}}], ArrowBox[{{1.5, 0.}, {1.5, -1.}, {-0.5, -1.}}]}, "DynamicName", BoxID -> "EdgeLabelID$5"], InsetBox[ FormBox[ StyleBox[ TemplateBox[{ GraphicsBox[ GeometricTransformationBox[ GeometricTransformationBox[{ AbsoluteThickness[1], Opacity[1], Dashing[{}], LineBox[{{-90., 0.}, {90., 0.}}], RectangleBox[{-110, -10}, {-90, 10}], { EdgeForm[{ Opacity[1], AbsoluteThickness[1], Dashing[{}]}], FaceForm[ RGBColor[1, 1, 1]], RectangleBox[{90, -10}, {110, 10}], RectangleBox[{-70, -30}, {70, 30}]}}, {{{0.1, 0}, {0, 0.1}}, {0, 0}}], {0, 0}], ImageSize -> 65], StyleBox["\"\\!\\(\\*SubscriptBox[\\(R\\), \\(3\\)]\\)\"", 12, StripOnInput -> False]}, "Labeled", DisplayFunction -> (FormBox[ GridBox[{{ ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}, { TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"]}}, GridBoxAlignment -> { "Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {2, 1}], TraditionalForm]& ), InterpretationFunction -> (RowBox[{"Labeled", "[", RowBox[{#, ",", #2, ",", "Top"}], "]"}]& )], 5, StripOnInput -> False], TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$5", Automatic, Scaled[0.66]]], ImageScaled[{ Rational[1, 2], 0.39}], BaseStyle -> "Graphics"]}}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], GrayLevel[0], TagBox[{ TagBox[ DiskBox[{-1.5, 0.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$1"], InsetBox[ FormBox["\"\\!\\(\\*SubscriptBox[\\(J\\), \\(1\\)]\\)\"", TraditionalForm], Offset[{2, 2}, DynamicLocation["VertexID$1", Automatic, {Right, Top}]], ImageScaled[{0, 0}], BaseStyle -> "Graphics"]}, "DynamicName", BoxID -> "VertexLabelID$1"], TagBox[{ TagBox[ DiskBox[{-0.5, 0.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$2"], InsetBox[ FormBox["\"\\!\\(\\*SubscriptBox[\\(J\\), \\(2\\)]\\)\"", TraditionalForm], Offset[{2, 2}, DynamicLocation["VertexID$2", Automatic, {Right, Top}]], ImageScaled[{0, 0}], BaseStyle -> "Graphics"]}, "DynamicName", BoxID -> "VertexLabelID$2"], TagBox[{ TagBox[ DiskBox[{-0.5, -1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$3"], InsetBox[ FormBox["\"\\!\\(\\*SubscriptBox[\\(J\\), \\(3\\)]\\)\"", TraditionalForm], Offset[{2, 2}, DynamicLocation["VertexID$3", Automatic, {Right, Top}]], ImageScaled[{0, 0}], BaseStyle -> "Graphics"]}, "DynamicName", BoxID -> "VertexLabelID$3"], TagBox[{ TagBox[ DiskBox[{1.5, 0.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$4"], InsetBox[ FormBox["\"\\!\\(\\*SubscriptBox[\\(J\\), \\(4\\)]\\)\"", TraditionalForm], Offset[{2, 2}, DynamicLocation["VertexID$4", Automatic, {Right, Top}]], ImageScaled[{0, 0}], BaseStyle -> "Graphics"]}, "DynamicName", BoxID -> "VertexLabelID$4"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], { CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{10.980693309010109`, 233.14020984366067`}, {-79.55937532285577, 76.7513936293538}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], AspectRatio->Full, DefaultBaseStyle->{ "NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->{{10, 15}, {10, 25}}, ImageSize->{236., 149.}]\);

Ermitteln Sie die geschlossenen Stromkreise.

In[2]:=

X cycles = EdgeCycleMatrix[g];

Das 1. Kirchhoffsche Gesetz bei Verzweigung J₄.

In[3]:=

X c1 = Total[{EdgeList[g, "\!\(\*SubscriptBox[\(J\), \(4\)]\)" \[DirectedEdge] _], \ -EdgeList[ g, _ \[DirectedEdge] "\!\(\*SubscriptBox[\(J\), \(4\)]\)"]}, -1] == 0

Out[3]=

Das 2. Kirchhoffsche Gesetz.

In[4]:=

X weights = PropertyValue[g, EdgeWeight]; elist = EdgeList[g];

In[5]:=

X c2 = Total[elist*weights*cycles[[1]]] == 0

Out[5]=

In[6]:=

X c3 = Total[elist*weights*cycles[[2]]] == 0

Out[6]=

Ermitteln Sie die Stromstärke.

In[7]:=

X Solve[{c1, c2, c3} /. Thread[EdgeList[g] -> {i1, i2, i3, 1, 1}], {i1, i2, i3}]

Out[7]=