A Chaotic Circuit

Analysis of a Simple Electric Circuit That Displays Chaotic Behavior

Here is a simple circuit containing a sinusoidal voltage source, a resistor, an inductor, and a diode. The presence of the diode introduces nonlinearity and allows for the possibility of complex behavior.

[Graphics:HTMLFiles/circuit_2.gif]

We model the diode as having a piecewise-linear capacitance with a small offset voltage.

The equations for the charge and the current in the diode can be written directly in Mathematica.

eqns = {q^'(t) ==
i(t),
L _ 1 i^'(t) ==  sin(2 π  t) - (({q(t)} (C _ 2 - C _ 1))/(2 C _ 1 C _ 2) + (q(t) (C _ 1 + C _ 2))/(2 C _ 1 C _ 2) + e _ 0) + i(t) (-R _ 1), q(0) == 0, i(0) == 0} ;

This substitutes values for parameters into the equations.

eqns = eqns /. {C
_ 1
-> 0.1 × 10^(-6), C _ 2 -> 400. × 10^(-12), R _ 1 -> 60., L _ 1 -> 100. × 10^(-6),  -> 700000., E _ 0 -> 0.1} ;

It is possible to solve the equations by using the Mathematica function NDSolve.

sol[v_] :=
NDSolve[eqns
/.  -> v, {q, i}, {t, 0, 1.4 × 10^(-4)}, MaxSteps -> 10^4] ;

The function Parametric plots the charge versus the current in the above circuit.

Parametric[voltage2_]
:= Module[{v = ToExpression[voltage2]}, ParametricPlot[Evaluate[{q[t], i[t]} /. sol[v]], {t, 1.2 * 10^-4, 1.4 * 10^-4}, PlotLabel -> "Charge Vs Current (" <> ToString[v] <> " V)"]]

Parametric[0.4]

[Graphics:HTMLFiles/circuit_8.gif]


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