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CHAPTER 3: HF MODELLING STRATEGY Parameter Value Generation gap 0.9 Crossover rate 0.88 Mutation rate 0.36 Insertion rate 0.8 Number of individuals 150 Number of generations 60 Table 3.1: Used parameters for Genetic Algorithms problem. At the initialization, a specified number of individuals and generations are chosen, and the first population of individuals is generated randomly. At time k, the genetic operations of mutation and crossover are used to create the new generation of individuals starting from the selected individuals at the previous time k-1. In the optimization, the program will recursively calculate the model’s impedance versus the frequency, evaluating each of the individuals in the current population. Using the simulation results, a fitness value will be associated to all the individuals, proportionally to the difference between the simulated impedance and the reference one. Through the genetic operation of selection the individuals with the higher fitness value will have a higher probability to be selected in the population at the following step. Only the "most fit" individuals evolve to the next generation at time k+1. The algorithm will update at each step the best individual. The search procedure will ter- minate when the fixed target error or a maximum number of generations is reached. The final output of the procedure will be an optimum set of parameters with mini- mized fitness function. The fitness function is based on the error between the measured impedance of the system and the respective calculation with the current parameters based on the individual. Usually more attempts are needed to find an optimum solution since the system can converge to local minima. However, with genetic algorithms, this happens less fre- quently than with other optimization strategies. In fact the GA provides a stochastic optimization method where if it "gets stuck" at a local optimum, it tries to simultane- ously find other parts of the search space and "jump out" of the local optimum, aiming at a global one. The used parameters for the individuals generation are grouped in Table 3.1, and occa- sionally only the number of generation and individuals per generation were changed, if the found solution wasn’t close enough to the desired goal. 33
CHAPTER 3: HF MODELLING STRATEGY Re(Z) Fitness Function ref f 1 f 2 f 3 f 4 sim Figure 3.1: Example of error integration for impedance Log(f) The goodness of every individual is evaluated by the Fitness Function that associates a numerical value to each of them; this function is the way to map how far each individual is from the desired goal. It can weight in different ways many aspects of the final result, i.e. giving more importance to solutions generating a better match in a certain range of frequencies or penalizing more the individuals that do not produce a desired characteristic, such as a resonating frequency if the goal is to match an impedance with that behaviour. The Fitness Function used for the impedance matching in this work consists of the in- tegral of the error between the measured reference impedance versus the frequency and the relative simulated one with the parameters chosen by the GA, this is visually represented by the area between the reference and the simulated impedance. Because the impedance is a complex number, this area is calculated twice, once for the real and once for the imaginary part, these two numbers are successively added to get the final error. Fig. 3.1 shows the area so calculated, in this case just for the real part of the impedance. The reference and the simulated impedance points might not be sampled at the same frequency, so a process of interpolation takes place and subdivide the se- ries of data at all the vertex of either of the two curves, to obtain eventually two set of values sampled at the same frequencies. In this example the algorithm for the error integration calculates the total area given by the five subsections shown in alternate colors and patterns, every area can only be either a trapezium (third and fifth case) or the double opposite triangle (first, second and fourth case); this is because it is defined by 4 segments, two of witch have to be parallel, representing the frequency subdivision in the impedances, that have been adjusted to be the same. To distinguish between the two cases and pick the right formula, the algorithm simply checks if the reference and the simulated impedance values cross each other or not. 34
Page 1 and 2: EMI Filter Design for Matrix Conver
Page 3 and 4: that may involve long Finite Elemen
Page 5 and 6: CONTENTS 3.1 Introduction . . . . .
Page 7 and 8: CONTENTS B External DLL 118 B.1 C c
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CHAPTER 7: FILTER DESIGN AND REALIZ
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CHAPTER 8 Filter realization and ex
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CHAPTER 8: FILTER REALIZATION AND E
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CHAPTER 9 Conclusions and Further w
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APPENDIX A Scientific Publications
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B.1 C code APPENDIX B External DLL
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APPENDIX B: EXTERNAL DLL Tseq_Pulse
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APPENDIX B: EXTERNAL DLL B.2 MAST c
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References [1] RTCA Incorporated. R
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REFERENCES nious drives. In Europea
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REFERENCES [48] D.F. Knurek. Reduci
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