Transcript Poster: Chaos in a diode - Department of Physics and Astronomy

CHAOS IN A DIODE: PERIOD DOUBLING AND CHAOTIC
BEHAVIOR IN A DIODE DRIVEN CIRCUIT
Maxwell Mikel-Stites
University of Rochester, Rochester, NY, 14627
Diode Voltage vs Driving Voltage
3.5
•In order to find the resonant frequency of the circuit, fix the amplitude to a low value
(~20mV) and fix a low frequency (~30khz) for some chosen inductance. Next, increase the
frequency until the peak to peak voltage stops increasing and begins to decrease. This
point is the resonant frequency for the circuit.
3
Diode Voltage (V)
The purpose of this experiment was to examine period
doubling and chaotic behaviour in a diode-driven
circuit. This allows us to learn more about the physics
behind the diode and its interaction with the circuit, as
well as the manifestation of chaos in a system as a
result of increasing bifurcation in the diode voltages. It
was possible to observe this type of behaviour clearly,
even with varying data quality; even in the worst case,
the bifurcations leading into chaos were clearly
defined and were relatively easy to examine.
Procedure
4
2.5
2
1.5
1
0.5
•In order to examine chaos, increase the amplitude in small increments (0.20V or less) at
this resonant frequency and measure the peak to peak voltages displayed on the
oscilloscope.
•Bifurcations should appear at successively closer voltages, at varying voltages depending
on the inductance chosen.
•With the experimental setup provided, it was only possible to observer the first three
bifurcations before the system developed into chaos in observed cases.
0
0
5
10
15
20
25
Driving Voltage (V)
•Resonant frequency=73.7khz
•Inductance=10 ohms
•Bifurcations at 3.6, 10.6, 15.1V
•Chaos at ~18V
•Fiegenbaum’s constant=1.56; too few bifurcations
to measure accurately.
Diode Voltage vs Driving Voltage
3
2.5
•Constructed out of a combination of a p and n type
semiconductors
•Diode not perfect; causes finite time for current
reversal.
•Causes forward and reverse bias to alternate
•Forward; diode acts as a resistor
•Reverse; causes diode to act as a capacitor
•The interactions between these parameters with
increasing voltage causes the signal to bifurcate as
it is read from the diode.
Diode voltage (V)
The Diode Itself
0
The above pictures, from left to right, show the oscilloscope graph of the peak to peak diode
voltage v.s. time for the initial voltage, first voltage bifurcation and the Lissajous graph of
driving voltage v.s. diode voltage. For the first bifurcation, this was found at approximately
3.20 volts for one run and approximately 3.6 volts for another.
Above are the graphs of the second bifurcation, found at approximately 8.20
volts and 10.6 volts for the two runs.
Diode Circuit
1) Diode
To the left is
the Lissajous
graph of the
third
bifurcation,
since the
diode
voltage
graph is
nearly
indistinguish
able from the
second
bifurcation in
many cases.
2)2400 ohms
3
2
3)185 ohms
4
1
4)590 ohms
From Signal Generator
Shortly after the third bifurcation, the graph evolves into chaos, with the leftmost picture
the peak to peak diode voltage, the center the Lissajous graph of chaos and the
rightmost a closeup of the chaos visible in each peak of the graph, as the increasing
number of bifurcations overlap each other.
Circuit Diagram
Signal Generator
2
3
Ground
4
Ground
Ground
2
4
6
8
10
12
Driving Voltage (V)
Oscilloscope
Inductor
1
0
inductor
To Oscilloscope
1.5
0.5
The below photos detail the setup for the entire
lab and the provided equipment. For the
experiment, the inductor was set to
approximately 10mH, since it was the lowest
inductance value possible with the given
inductor.
Signal
Generator
2
•Resonant
frequency= 73.7khz
•Inductance=~10
ohms. There is
some uncertainty
due to problems
with the inductor.
•This caused earlier
bifurcations.
•Change in bifurcations
consistent with higher
inductance; system likely not
at resonance.
•Feigenbaum’s constant
evaluated to be approximately
4.167.
•Probably due to error in the
equipment, since system not
at resonance for a higher
inductance.
The equation used to produce the below graph
was xn+1 = r xn (1 – xn), using the initial values of xo
= 0.7, r = {2.5 : 0.015 : 4.0}
This model demonstrates what would
happen if we were able to observe
bifurcations beyond third order with our
diode/oscilloscope setup. As the bifurcations
increase, in addition to the readily
observable chaos, there are island of
stability appearing at systematic intervals. In
order to observe this behaviour
experimentally, one would have to continue
to increase the voltage extremely carefully in
order to observe one, as the peaks for
stability are quite narrow.
Ground
Measurable Quantities
•Feigenbaum’s Constant:The ratio of the difference
between the bifurcations; as the number of
bifurcations goes to infinity, it approaches 4.6669.
•Driving voltage
•Peak to peak diode voltage
•Frequency
•Inductance
Overall, the data gathered accurately demonstrates only the progress of increasing bifurcations leading to chaos, and also showed that the
inductance of the system also greatly affects the voltages at which the bifurcations are observed. In this way, by increasing the inductance
value, one could cause the bifurcation pattern to emerge earlier and earlier. Similarly, one could decrease the inductance to cause bifurcations
to appear at later intervals. In order to improve the lab, it would be beneficial to obtain better equipment, such as more reliable inductors and a
digital oscilloscope.
Chris Osborn, for the code involved in producing the chaos model and
Dan Richman and Chris Osborn for their assistance in gathering data.