chaos in a diode observing period-doubling and chaotic behavior in a driven diode circuit chris...
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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-typeHoles 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-typeReinforces 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
• Function generator output: sine– Frequency is circuit’s resonance frequency (73.7 kHz)– Amplitude adjustable from millivolts to tens of volts
• Inductor set to ten mH• Oscilloscope reads generator voltage and diode
voltage
1. Fix the driving frequency2. Start at a low voltage amplitude3. Increase amplitude in steps of tenths4. Observe multiplying curves on diode voltage
display
Observing the CircuitWhat 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
Original computation in Matlab for the longtime convergence of xn
• produces a sequence of terms whose long-time convergence is the bifurcation diagram
• xo = 0.7, r = {2.5 : 0.015 : 4.0}
xn+1 = r xn (1 – xn)
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