Excitation Optimization of a Standard LRC Circuit with Impulsive Forces Selected via Simulated

88 Pages·2017·1.16 MB·English

by  Ryan M. Spies

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Departmental Honors Projects College of Liberal Arts
Spring 2013
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Ryan M. Spies
Hamline University
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Excitation Optimization of a Standard LRC
Circuit with Impulsive Forces Selected via
Simulated Annealing
Ryan M. Spies
An Honors Thesis
Submitted for partial fulfillment of the requirements for
graduation with honors in Physics from Hamline University
April 26, 2013
ii
Abstract
For an unknown oscillator, it is sometimes useful to know what the
potential energy function associated with it is. An argument for
using a method of determining the optimal sequence of impulsive
forces in order to find the potential energy function is made using
principles of energy. Global optimization via simulated annealing
is discussed, and various parameters that can be adjusted across
experiments are established. A method for determining the optimal
sequence of impulsive forces for the excitation of a standard LRC
circuit is established using the methodology of simulated annealing.
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iv
Contents
1 Motivation 1
1.1 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 2
1.2 General Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 LRC Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Methodology 9
2.1 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Acceptance Probability . . . . . . . . . . . . . . . . . . . 10
2.1.2 Annealing Schedule . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Neighborhood Selection . . . . . . . . . . . . . . . . . . . 11
2.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Arduino Program . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Main Python Program . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Results 17
3.1 How to Interpret Simulation Results . . . . . . . . . . . . . . . . 17
3.2 Results of Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 1st Simulation Run . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 2nd Simulation Run . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 3rd Simulation Run . . . . . . . . . . . . . . . . . . . . . 21
3.2.4 4th Simulation Run . . . . . . . . . . . . . . . . . . . . . 22
3.2.5 5th Simulation Run . . . . . . . . . . . . . . . . . . . . . 23
3.2.6 6th Simulation Run . . . . . . . . . . . . . . . . . . . . . 24
3.2.7 7th Simulation Run . . . . . . . . . . . . . . . . . . . . . 25
3.2.8 8th Simulation Run . . . . . . . . . . . . . . . . . . . . . 26
3.2.9 9th Simulation Run . . . . . . . . . . . . . . . . . . . . . 27
3.2.10 10th Simulation Run . . . . . . . . . . . . . . . . . . . . . 28
3.2.11 11th Simulation Run . . . . . . . . . . . . . . . . . . . . . 29
3.2.12 12th Simulation Run . . . . . . . . . . . . . . . . . . . . . 30
3.2.13 13th Simulation Run . . . . . . . . . . . . . . . . . . . . . 31
3.2.14 14th Simulation Run . . . . . . . . . . . . . . . . . . . . . 32
3.2.15 15th Simulation Run . . . . . . . . . . . . . . . . . . . . . 33
3.2.16 16th Simulation Run . . . . . . . . . . . . . . . . . . . . . 34
3.2.17 17th Simulation Run . . . . . . . . . . . . . . . . . . . . . 35
3.2.18 18th Simulation Run . . . . . . . . . . . . . . . . . . . . . 36
3.2.19 19th Simulation Run . . . . . . . . . . . . . . . . . . . . . 37
3.2.20 20th Simulation Run . . . . . . . . . . . . . . . . . . . . . 38
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vi CONTENTS
3.2.21 21st Simulation Run . . . . . . . . . . . . . . . . . . . . . 39
3.2.22 22nd Simulation Run. . . . . . . . . . . . . . . . . . . . . 40
3.2.23 23rd Simulation Run . . . . . . . . . . . . . . . . . . . . . 41
3.2.24 24th Simulation Run . . . . . . . . . . . . . . . . . . . . . 42
3.2.25 25th Simulation Run . . . . . . . . . . . . . . . . . . . . . 43
3.2.26 26th Simulation Run . . . . . . . . . . . . . . . . . . . . . 44
3.2.27 27th Simulation Run . . . . . . . . . . . . . . . . . . . . . 45
3.2.28 28th Simulation Run . . . . . . . . . . . . . . . . . . . . . 46
3.2.29 29th Simulation Run . . . . . . . . . . . . . . . . . . . . . 47
3.2.30 30th Simulation Run . . . . . . . . . . . . . . . . . . . . . 48
3.2.31 31st Simulation Run . . . . . . . . . . . . . . . . . . . . . 49
3.2.32 32nd Simulation Run. . . . . . . . . . . . . . . . . . . . . 50
3.3 General Remarks on Simulation Results . . . . . . . . . . . . . . 51
3.4 Results of Physical Experiment . . . . . . . . . . . . . . . . . . . 51
3.5 Analysis of Physical Experiment . . . . . . . . . . . . . . . . . . 54
4 Discussion 57
4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Next Steps With this Methodology . . . . . . . . . . . . . . . . . 58
4.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A Simulated Annealing Test Algorithm 59
B Main Experiment Python Code 67
C Arduino Module Code 73
D Arduino Sketch Code 75
E Mathematica Analysis for Experiment 77
List of Figures
2.1 Pseudo-Python implementation of the standard simulated an-
nealing algorithm for finding a global minimum.. . . . . . . . . . 10
2.2 Circuit diagram of the follower circuit attached to the RLC circuit. 13
3.1 The graph of the interpolated function that was explored in the
algorithm in Appendix A. The x-axis is the solution space, and
the y-axis is the corresponding fitness for any given part of the
solution space. Note the global maximum at approximately 8. . . 18
3.2 Results of 1st Experiment . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Results of 2nd Experiment. . . . . . . . . . . . . . . . . . . . . . 20
3.4 Results of 3rd Experiment . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Results of 4th Experiment . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Results of 5th Experiment . . . . . . . . . . . . . . . . . . . . . . 23
3.7 Results of 6th Experiment . . . . . . . . . . . . . . . . . . . . . . 24
3.8 Results of 7th Experiment . . . . . . . . . . . . . . . . . . . . . . 25
3.9 Results of 8th Experiment . . . . . . . . . . . . . . . . . . . . . . 26
3.10 Results of 9th Experiment . . . . . . . . . . . . . . . . . . . . . . 27
3.11 Results of 10th Experiment . . . . . . . . . . . . . . . . . . . . . 28
3.12 Results of 11th Experiment . . . . . . . . . . . . . . . . . . . . . 29
3.13 Results of 12th Experiment . . . . . . . . . . . . . . . . . . . . . 30
3.14 Results of 13th Experiment . . . . . . . . . . . . . . . . . . . . . 31
3.15 Results of 14th Experiment . . . . . . . . . . . . . . . . . . . . . 32
3.16 Results of 15th Experiment . . . . . . . . . . . . . . . . . . . . . 33
3.17 Results of 16th Experiment . . . . . . . . . . . . . . . . . . . . . 34
3.18 Results of 17th Experiment . . . . . . . . . . . . . . . . . . . . . 35
3.19 Results of 18th Experiment . . . . . . . . . . . . . . . . . . . . . 36
3.20 Results of 19th Experiment . . . . . . . . . . . . . . . . . . . . . 37
3.21 Results of 20th Experiment . . . . . . . . . . . . . . . . . . . . . 38
3.22 Results of 21st Experiment . . . . . . . . . . . . . . . . . . . . . 39
3.23 Results of 22nd Experiment . . . . . . . . . . . . . . . . . . . . . 40
3.24 Results of 23rd Experiment . . . . . . . . . . . . . . . . . . . . . 41
3.25 Results of 24th Experiment . . . . . . . . . . . . . . . . . . . . . 42
3.26 Results of 25th Experiment . . . . . . . . . . . . . . . . . . . . . 43
3.27 Results of 26th Experiment . . . . . . . . . . . . . . . . . . . . . 44
3.28 Results of 27th Experiment . . . . . . . . . . . . . . . . . . . . . 45
3.29 Results of 28th Experiment . . . . . . . . . . . . . . . . . . . . . 46
3.30 Results of 29th Experiment . . . . . . . . . . . . . . . . . . . . . 47
3.31 Results of 30th Experiment . . . . . . . . . . . . . . . . . . . . . 48
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viii LIST OF FIGURES
3.32 Results of 31st Experiment . . . . . . . . . . . . . . . . . . . . . 49
3.33 Results of 32nd Experiment . . . . . . . . . . . . . . . . . . . . . 50
3.34 1st pulse timing selection . . . . . . . . . . . . . . . . . . . . . . 52
3.35 2nd pulse timing selection . . . . . . . . . . . . . . . . . . . . . . 52
3.36 3rd pulse timing selection . . . . . . . . . . . . . . . . . . . . . . 53
3.37 4th pulse timing selection . . . . . . . . . . . . . . . . . . . . . . 53
3.38 5th pulse timing selection . . . . . . . . . . . . . . . . . . . . . . 54
3.39 Electric potential fit . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Plot of voltage for this LRC . . . . . . . . . . . . . . . . . . . . . 57
Chapter 1
Motivation
We want to develop a methodology that will allow us to determine the optimal
sequence of pulse timings for an oscillator, which will in turn determine the
optimal motion for the oscillator as well. This methodology must also allow us
tofindthepotentialenergyfunctionofagivenoscillator. Inordertounderstand
whatitmeanstodeterminetheoptimalsequenceofpulsetimings,wemustfirst
understand the dynamics of oscillators. In particular, we want to understand
the dynamics of oscillators after they are hit by an impulsive force. Also, we
want to know what it means to find the potential energy function for a given
oscillator, and to figure out how that is related to the problem of finding the
optimal sequence of pulse timings. Finally, we want to understand what it
means to take the principles of these classical oscillator problems and apply
them to electronic oscillators.
For a thought experiment, let us consider a pendulum. If we gave this
pendulum a slight push, and if there is no friction present, then the pendulum
will move and the motion will not decrease. If we want to get this pendulum
to swing out further we want to push it more, but then comes the matter of
determining when it is best to push the pendulum again. Assuming that the
pendulum can only be pushed from one direction, the best moment for pushing
it again would be when it is back at its equilibrium position and when it is
going in the same direction as the push. However, since the pendulum has
a periodic behavior that is nonlinear for sufficiently large angles (so that the
approximation for the angular position θ of the pendulum, θ ≈ sin(θ), or the
angles for which simple harmonic motion occurs, no longer holds) the timing
of these pushes becomes a problem. The amount of time that one would have
to wait before pushing the pendulum again at the best time changes, so this
motivates a method that can find when to push the pendulum without prior
knowledge of the pendulum’s behavior.
In order to understand how our thought experiment relates to the larger
problem let us consider the dynamics of oscillators in general, and start with
the example of the simple harmonic oscillator.
1