Analyze an Electric Circuit 

Find currents in an electric circuit with two voltage sources and three resistors.

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[\(J\), \(2\)]\)", 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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.}]\);

Find closed circuits.

In[2]:=

X cycles = EdgeCycleMatrix[g];

Kirchhoff’s first law at the junction J₄.

In[3]:=

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

Out[3]=

Kirchhoff’s second law.

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]=

Find currents.

In[7]:=

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

Out[7]=