Download CHAOTIC CIRCUITS - Green River Community College

CHAOTIC CIRCUITS
The way the electron bounces
1
TOPICS

Linear Circuits
– Inductor
– Diode

Non-Linear Circuits
– How can we tell if a circuit is behaving non-linearly ?
– Diode and inductor
– Transistor




Bifurcation Diagrams
The Feigenbaum number
Why are these systems chaotic ?
Attractors
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MAY POPULATION MODEL
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Basic Population Model
POPULATION
SEX
FOOD
SUPPLY
4
May’s Population Model
POPULATION
FEEDBACK
SEX
FOOD
SUPPLY
5
Attractor for May model
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7
LINEAR CIRCUITS
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INDUCTOR
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Input Voltage: 5 V
10
Input Voltage: 10 V
11
Input Voltage: 15 V
12
Input Voltage: 20 V
13
Output Voltage (peak to peak)
Inductor (Output vs. Input)
Input Voltage (peak to Peak)
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DIODE
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Input Voltage: 5 V
16
Input Voltage: 10 V
17
Input Voltage: 15 V
18
Input Voltage: 20 V
19
Output Voltage (peak to peak)
Diode (Output vs. Input Voltage)
y = 0.4949 + 0.7762
Input Voltage (peak to peak)
20
NON-LINEAR CIRCUITS
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How can we tell if a circuit is
behaving non-linearly?
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Period 1
PM3394A
ch2
ch1 2
1
CH1
CH2
1 V~
1 V~
STOP
MTB5.00us
ch1+
23
Period 2
PM3394A
ch2
ch1
2
1
CH1
CH2
2 V~
2 V~
STOP
MTB5.00us
ch1+
24
Period 4
PM3394A
ch2
ch1
2
1
CH1
CH2
5 V~
2 V~
STOP
MTB5.00us
ch1+
25
Diode-Inductor Circuit
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Circuit Schematic
Inductor
5.89 mH
Variable
Voltage
Source
AC
Diode
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Period 1
28
Period 2
29
Period 4
30
Period 8
31
Period 16
32
Chaos
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Transistor Circuit
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Circuit Schematic
Inductor
Transistor
npn type
A.C.
Function
Generator
Variable
Resistor
Load inductor
Load
Variable
resistor
Feedback
Ground
D.C. Voltage Source
Ground
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Simplified Schematic
SIGNAL
FEEDBACK
AMPLIFIER
OUTPUT
POWER
SUPPLY
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Period 1
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Period 2
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Period 4
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Period 8
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Period 16
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Chaos
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BIFURCATION
DIAGRAMS
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Output Voltage (peak to peak)
Bifurcation Diagram (Inductor)
y = 1.3717x - 0.0724
Input Voltage (peak to Peak)
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Output Voltage (peak to peak)
Bifurcation Diagram (Diode-inductor)
Input Voltage (peak to peak)
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Output Voltage (peak to peak)
Bifurcation Diagram (Transistor)
Input Voltage (peak to peak)
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Mathematical model (May model)
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Output Voltage (peak to peak)
Chaotic Region (diode-inductor)
Input Voltage (peak to peak)
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Output Voltage (peak to peak)
Periodic Region Amplified (diode)
Input Voltage (peak to peak)
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55
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Feigenbaum Number
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4.7 V
Diode-Inductor Circuit
d1 = 4.7 / 1.1 = 4.27
Output Voltage (peak to peak)
1.1 V
Input Voltage (peak to peak)
58
Transistor Circuit
Transistor Circuit
Output Voltage (peak to peak)
1.2 V
d1 = 1.2 / 0.28 = 4.285
0.28 V
Input Voltage (peak to peak)
59
Feigenbaum Number
Theoretical value: d ∞ = 4.669201
Experimental value: d1 = 4.27 (diode-inductor)
Experimental value: d1 = 4.285 (transistor)
Other experimental values recorded:
Electrical circuit (varactor) d1 = 4.257
Fluid Mechanics (Convection) d1 = 4.4
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Why are these systems
chaotic ?
61
Water gate analogy of a diode
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Water rate corresponding to time
constant
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Simple Linear Amplifier
SIGNAL
AMPLIFIER
POWER
SUPPLY
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Chaotic Amplifier Circuit
SIGNAL
FEEDBACK
AMPLIFIER
POWER
SUPPLY
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ATTRACTORS
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Inductor
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Input Voltage: 3.23 V
PM3394A
ch1: pkpk= 3.23 V
1
CH1
2 V~
X= CH2
2 V~
70
Input Voltage: 4.44 V
PM3394A
ch1: pkpk= 4.44 V
1
CH1
2 V~
X= CH2
2 V~
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Input Voltage: 5.74 V
PM3394A
ch1: pkpk= 5.74 V
1
CH1
2 V~
X= CH2
2 V~
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Input Voltage: 7.11 V
PM3394A
ch1: pkpk= 7.11 V
1
CH1
2 V~
X= CH2
2 V~
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Input Voltage: 9.18 V
PM3394A
ch1: pkpk= 9.18 V
1
CH1
2 V~
X= CH2
2 V~
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Input Voltage: 11.0 V
PM3394A
ch1: pkpk= 11.0 V
1
CH1
2 V~
X= CH2
2 V~
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Input Voltage: 13.0 V
PM3394A
ch1: pkpk= 13.0 V
1
CH1
2 V~
X= CH2
2 V~
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Diode-Inductor
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Period 1
Input Voltage: 2.16 V
PM3394A
ch2: pkpk= 2.16 V
2
CH2
1 V~
X= CH1
1 V~
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Butterfly Diagram (Period 2)
Input Voltage: 2.75 V
PM3394A
ch2: pkpk= 3.10 V
2
CH2
1 V~
X= CH1
1 V~
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Period 4
Input Voltage: 7.45 V
PM3394A
ch2: pkpk= 6.82 V
2
CH2
1 V~
X= CH1
1 V~
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Period 8
Input Voltage: 8.55 V
PM3394A
ch2: pkpk= 7.23 V
2
CH2
1 V~
X= CH1
1 V~
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CHAOS
Input Voltage: 8.5 V
PM3394A
ch2: pkpk= 8.85 V
2
CH2
2 V~
STOP
X= CH1
2 V~
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Transistor
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Input Voltage: 0.259 V
PM3394A
ch2: pkpk= 259mV
2
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Input Voltage: 0.709 V
PM3394A
ch2: pkpk= 709mV
2
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Input Voltage: 1.63 V
PM3394A
ch2: pkpk= 1.63 V
2
86
Begin shifting phase
Input Voltage: 1.86 V
PM3394A
ch2: pkpk= 1.86 V
2
87
Phase shift complete
Input Voltage: 1.98 V
PM3394A
ch2: pkpk= 1.98 V
2
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Butterfly Diagram (Period 2)
Input Voltage: 3.06 V
PM3394A
ch2: pkpk= 3.06 V
2
89
Period 2 (anomaly)
Input Voltage: 3.34 V
PM3394A
ch2: pkpk= 3.34 V
2
90
Period 4
Input Voltage: 4.19 V
PM3394A
ch2: pkpk= 4.19 V
2
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Period 8
Input Voltage: 4.38 V
PM3394A
ch2: pkpk= 4.38 V
2
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Period 16 ?
Input Voltage: 4.54 V
PM3394A
ch2: pkpk= 4.54 V
2
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Approaching Chaos
Input Voltage: 4.96 V
PM3394A
ch2: pkpk= 4.96 V
2
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CHAOS
Input Voltage: 7.34 V
PM3394A
ch2: pkpk= 7.34 V
2
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More Chaos
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Acknowledgments
M.J. Murdock Charitable Trust
 Fluke Corporation
 Thomas J. Holthaus
 Pacific Lutheran University
 Dr. Keith Clay
 Lori Briggs
 Jana Steiner
 Christian Dilley

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