Power Magnetic Devices. Scott D. Sudhoff

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a maximum value hmx. If this is the first constraint and if we use h(x) to represent the height in terms of our design parameters, our first constraint could be calculated as

      (1.9-3)equation

      Next, let us consider our design metrics. The design metrics will also be a function of the parameter vector x. In this text, common metrics will be mass and loss. Let the number of metrics be denoted M, and the value of the ith metric be denoted mi. It will henceforth be assumed that all metric values are greater than zero.

      As we will see, metrics and objectives are closely related but not synonymous. Metrics will be based on attributes of interest. Objectives will be based on the metrics, but also be influenced by constraints.

      Let us now discuss the definition of the objective or fitness function. In keeping with the usual practice of GAs, we will assume that our fitness function (which is synonymous with the objective function) is of a form to be maximized. One approach to creating a fitness function begins with first forming a combined constraint. This can be done by averaging the constraints as

      Next, the elements of the fitness function are defined as either

      or

      where ε is a small positive number. The value has no impact on results but is convenient for plotting the fitness versus generation. It is commonly chosen to be on the order of 10−10.

calculate constraint ci CI = CI + 1 CS = CS + ci if (CS < CI) images return end

      The code block shown in Table 1.6 is repeated sequentially for every constraint. After all constraints are tested, the metrics may be calculated and the fitness assigned as in (1.9-5) and (1.9-6) for the case when all constraints are met. There are also variations of this procedure. For example, it may be convenient to calculate and test several constraints at a time.

      In this section, we will conclude this chapter with a design example. In particular, we will endeavor to design a UI‐core inductor using an optimization‐based design process. At this point, we have not studied any magnetics or magnetic devices. This will be the subject of the remainder of this book. Since we are not yet in a position to derive or understand the relationships needed, an elementary analysis will be provided and the reader is asked to simply accept these relationships at face value for the time being. It should be observed that the analysis used to derive the needed relationships is very simplistic, but this does not matter because our purpose here is only to look at the design process. We will conduct a much more detailed analysis and design in subsequent chapters.

Schematic illustration of uI-core inductor.

      The

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