Switched Model
This chapter focuses on methodologies for obtaining the so-called switched model. This model describes basic low-frequency dynamics, as given by the energy accumulation variations, and it captures the switching dynamics of power electronic converters as w
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Switched Model
This chapter focuses on methodologies for obtaining the so-called switched model. This model describes basic low-frequency dynamics, as given by the energy accumulation variations, and it captures the switching dynamics of power electronic converters as well. The switched model is a useful analysis tool, which emphasizes the presence of the external control action. In its bilinear form this model can be directly used for simulation and control design purposes. The switched model offers a starting point for obtaining other types of models, such as averaged or reduced-order models. The chapter first states the mathematical framework, then provides a general modeling methodology. Some illustrative examples and a case study complete the presentation of switched-model-related topics. Problems with solutions, as well as proposed problems, can be found at the end of chapter.
3.1 3.1.1
Mathematical Modeling General Mathematical Framework
Because of its multiple combinations of switch states, a power electronic converter exhibits a periodically repeated sequence of possible configurations during its operation time interval, also called switching period. Each such configuration represents in fact a unique circuit containing sources and passive elements, which can be mathematically described by a set of differential equations. Under the assumptions already stated in Chap. 2, a generic power electronic converter, represented in Fig. 3.1 which switches between N distinct configurations,
S. Bacha et al., Power Electronic Converters Modeling and Control: with Case Studies, 27 Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-1-4471-5478-5_3, © Springer-Verlag London 2014
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3
Fig. 3.1 Conventional symbols for power electronic converter representation (Kassakian et al. 1991)
Switched Model
vC iL
ie
vS
ve
iS
i0
is described as a dynamical system (Tymerski et al. 1989; Sun and Grotstollen 1992; Maksimovic´ et al. 2001): d xðtÞ ¼ Ai xðtÞ þ Bi eðtÞ, ti t tiþ1 , dt
(3.1)
with N X
ðti ti1 Þ ¼ T,
i¼1
where T is the switching time, ti are different time points defining the switching between N configurations, Ai and Bi are the n n state matrix and n p input matrix respectively, corresponding to configuration i, x(t) is the n-length state vector and e(t) is the p-length vector of the independent sources of the system. Note that in Eq. (3.1) the control input does not appear explicitly. A more compact form of (3.1) is N X d xðtÞ ¼ ðAi xðtÞ þ Bi eðtÞÞ hi , dt i¼1
(3.2)
where hi are respective validation functions associated to configurations. These functions take values 1 or 0 depending on whether their respective configurations are activated or not. Comments on power electronic converter classification In general, power electronic converters are classified according to the following criteria: • conversion mode: DC-DC, DC-AC, etc.;
3.1 Mathematical Modeling
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• innermost control type (at switching level): pulse width modulation (PWM), hysteresis control, sliding-m
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