Download Chaos in a diode

Chaos in a Diode
observing period-doubling and chaotic behavior
in a driven diode circuit
Chris Osborn
Daniel Richman
The Diode
• Passive (not self-powered) circuit component
• Restricts current flow to one direction
• Modern diodes are based on semiconductor p-n junctions
– Situation when diode is not connected to a circuit (a voltage):
Electron and hole recombination in vicinity of junction, depletion of mobile charges;
remaining negative charges in p-type and positive charges in n-type cause potential
difference across junction (field points from n-type to p-type)
– Connection methods:
• Forward bias—positive emf terminal to p-type, negative to n-type
Holes and electrons are pushed toward non-conducting center and oppose built-in
potential barrier; current tunnels through the barrier
• Reverse bias: negative emf terminal to p-type, positive to n-type
Reinforces built-in potential barrier; current is blocked
The Circuit
Characteristics (why nonlinearity)
• Diode has threshold voltage required for conduction
– Forward voltage drop must reach this value for conduction to occur
• Diode has residual conduction during reverse-bias part of cycle
(forward bias means diode “on”, reverse bias means diode “off”)
– Exponential decay of current—recovery time
– Frequency of cycle is faster than recovery time
 Affects starting condition for next forward-bias part of cycle
Experimental Procedures
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Function generator output: sine
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Frequency is circuit’s resonance frequency (73.7 kHz)
Amplitude adjustable from millivolts to tens of volts
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Inductor set to ten mH
Oscilloscope reads generator voltage and diode voltage
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Fix the driving frequency
Start at a low voltage amplitude
Increase amplitude in steps of tenths
Observe multiplying curves on diode voltage display
Observing the Circuit
What are we looking at?
Diode voltage versus time
(# of periods is # of distinct peaks)
Diode voltage versus driving voltage
(# of periods is # of loops)
Chaos
Experimental Data
[Diode voltage versus driving voltage bifurcation graph]
brief chaos between red and blue
chaos between purple and gold
Simulation
xn+1 = r xn (1 – xn)
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produces a sequence of terms whose long-time convergence is the bifurcation diagram
• xo = 0.7, r = {2.5 : 0.015 : 4.0}
Original computation in Matlab for the longtime convergence of xn