Classical Averaged Model
This chapter deals with methodologies of obtaining the so-called averaged model, which focuses on capturing the low-frequency behavior of power electronic converters while neglecting high-frequency variations due to circuit switching. This appears to be a
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Classical Averaged Model
This chapter deals with methodologies of obtaining the so-called averaged model, which focuses on capturing the low-frequency behavior of power electronic converters while neglecting high-frequency variations due to circuit switching. This appears to be a natural action, as every converter employs filters in order to limit the ripple of various variables. The result is a continuous-time model, one which is easier to handle by classical analysis and control formalisms. This chapter is organized as follows. It starts by presenting the basics of averaging methodology and states some theoretical fundamentals. Then the methodology of obtaining small-signal and large-signal averaged models and their equivalent averaged diagrams are given. The error introduced by averaging, computed with respect to the exact sampled-data model, is also analyzed. A case study will serve at illustrating the various approaches. The chapter ends with some problems and their solutions and some proposed problems.
4.1
Introduction
In the previous chapter it was shown that electrical circuits containing static converters can be mathematically described by a cyclic set of state equations, corresponding to different electrical configurations listed along the entire converter operation cycle. The switched model is the product of such an analysis and it is particularly suitable for designing nonlinear control laws, such as variable-structure or hysteresis control. However, in most control applications it is the low-frequency behavior that is interesting. In this context, the various high-frequency switching phenomena are parasitic and must be neglected. When certain control laws (e.g., linear control) need to be implemented, the designer must transform the original discontinuous model in a continuous invariant model that provides the best representation of the system macroscopic behavior. The obtained model should be easy to employ; to this end, the averaging method is strongly recommended. S. Bacha et al., Power Electronic Converters Modeling and Control: with Case Studies, 55 Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-1-4471-5478-5_4, © Springer-Verlag London 2014
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4 Classical Averaged Model
Because of its undoubted utility, this kind of model – named the averaged model – has been studied since early 1970s, either by circuit averaging or statespace averaging (Wester and Middlebrook 1973; Middlebrook and C´uk 1976) or by averaged equivalent electrical diagram analysis (Pe´rard et al. 1979). The utility of the averaged model for simulation purposes, using dedicated software products such as SPICE®, SABER®, MATLAB®, has also been largely proved (Sanders and Verghese 1991; Ben-Yakoov 1993; Vuthchhay and Bunlaksananusorn 2008). This kind of model is useful for analytically expressing the essential dynamical behavior of power electronic circuits, both in continuoustime (Middlebrook 1988; Rim et al. 1988; Lehman and Bass 1996) and discretetime domain (Maksimovic´ and Zane 2007). Averaging techni
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