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

Hamline University DigitalCommons@Hamline Departmental Honors Projects College of Liberal Arts Spring 2013 Excitation Optimization of a Standard LRC Circuit with Impulsive Forces Selected via Simulated Annealing Ryan M Spies Hamline University Follow this and additional works at: https://digitalcommons.hamline.edu/dhp Part of the Physics Commons Recommended Citation Spies, Ryan M., "Excitation Optimization of a Standard LRC Circuit with Impulsive Forces Selected via Simulated Annealing" (2013) Departmental Honors Projects https://digitalcommons.hamline.edu/dhp/3 This Honors Project is brought to you for free and open access by the College of Liberal Arts at DigitalCommons@Hamline It has been accepted for inclusion in Departmental Honors Projects by an authorized administrator of DigitalCommons@Hamline For more information, please contact digitalcommons@hamline.edu, lterveer01@hamline.edu 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 iii iv Contents Motivation 1.1 Simple Harmonic Oscillator 1.2 General Oscillator 1.3 LRC Extensions Methodology 2.1 Simulated Annealing 2.1.1 Acceptance Probability 2.1.2 Annealing Schedule 2.1.3 Neighborhood Selection 2.2 Electronics 2.2.1 Circuitry 2.2.2 Arduino Program 2.3 Main Python Program Results 3.1 How to Interpret Simulation 3.2 Results of Simulation 3.2.1 1st Simulation Run 3.2.2 2nd Simulation Run 3.2.3 3rd Simulation Run 3.2.4 4th Simulation Run 3.2.5 5th Simulation Run 3.2.6 6th Simulation Run 3.2.7 7th Simulation Run 3.2.8 8th Simulation Run 3.2.9 9th Simulation Run 3.2.10 10th Simulation Run 3.2.11 11th Simulation Run 3.2.12 12th Simulation Run 3.2.13 13th Simulation Run 3.2.14 14th Simulation Run 3.2.15 15th Simulation Run 3.2.16 16th Simulation Run 3.2.17 17th Simulation Run 3.2.18 18th Simulation Run 3.2.19 19th Simulation Run 3.2.20 20th Simulation Run 9 10 11 11 12 12 14 15 Results 17 17 19 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 v vi CONTENTS 3.3 3.4 3.5 3.2.21 21st Simulation Run 3.2.22 22nd Simulation Run 3.2.23 23rd Simulation Run 3.2.24 24th Simulation Run 3.2.25 25th Simulation Run 3.2.26 26th Simulation Run 3.2.27 27th Simulation Run 3.2.28 28th Simulation Run 3.2.29 29th Simulation Run 3.2.30 30th Simulation Run 3.2.31 31st Simulation Run 3.2.32 32nd Simulation Run General Remarks on Simulation Results Results of Physical Experiment Analysis of Physical Experiment 39 40 41 42 43 44 45 46 47 48 49 50 51 51 54 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 2.2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 Pseudo-Python implementation of the standard simulated annealing algorithm for finding a global minimum 10 Circuit diagram of the follower circuit attached to the RLC circuit 13 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 Results of 1st Experiment Results of 2nd Experiment Results of 3rd Experiment Results of 4th Experiment Results of 5th Experiment Results of 6th Experiment Results of 7th Experiment Results of 8th Experiment Results of 9th Experiment Results of 10th Experiment Results of 11th Experiment Results of 12th Experiment Results of 13th Experiment Results of 14th Experiment Results of 15th Experiment Results of 16th Experiment Results of 17th Experiment Results of 18th Experiment Results of 19th Experiment Results of 20th Experiment Results of 21st Experiment Results of 22nd Experiment Results of 23rd Experiment Results of 24th Experiment Results of 25th Experiment Results of 26th Experiment Results of 27th Experiment Results of 28th Experiment Results of 29th Experiment Results of 30th Experiment vii 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 viii LIST OF FIGURES 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 Results of 31st Experiment Results of 32nd Experiment 1st pulse timing selection 2nd pulse timing selection 3rd pulse timing selection 4th pulse timing selection 5th pulse timing selection Electric potential fit 49 50 52 52 53 53 54 55 4.1 Plot of voltage for this LRC 57 Chapter 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 to find the potential energy function of a given oscillator In order to understand what it means to determine the optimal sequence of pulse timings, we must first 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 .. .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... sequence of pulses used to excite the oscillator is simulated annealing Simulated annealing is an optimization method that mimics the thermodynamic process of annealing with the goal of finding a particular... implementation of the standard simulated annealing algorithm for finding a global minimum 10 Circuit diagram of the follower circuit attached to the RLC circuit 13 The graph of the interpolated