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Fig. 1.4. Basic configuration of current-output converter.
1.4 A Review of Existing Methods to Analyze the Performance and Stability and Model Switched-Mode Converters
As it was stated earlier, the scientific research on the topic started in the 70s. The basic topologies (i.e. buck, boost and buck-boost) were analyzed and modeled for the first time in [19]. Averaged and linearized general power-stage models were also
state-space averaging (SSA) modeling technique, which produces both continuous-time steady state and dynamic linearized models. The basic idea behind the SSA is to average the switch on- and off-time state-space equations over one switching period.
Circuit averaging and hybrid modeling were considered as alternative modeling methods that give the same canonical circuit model as the SSA method. The SSA has become popular since its introduction basically due to its simple and clear methodology. It is commonly known that the SSA gives accurate open-loop models up to half the switching frequency, when the converter operates in continuous-conduction-mode (CCM) under direct-duty-ratio control or VMC [20]. The canonical circuit model introduced in [4] was argued for being a useful tool for analyzing small-signal dynamics of switched-mode converters regardless of the topology. However, it contains a resistive load as well as parasitic loss elements in the duty ratio dependent generators hiding the true internal dynamics.
In spite of the practicality and simplicity of the SSA, several other modeling approaches have also been developed. In circuit averaging, the voltage and current waveforms are averaged instead of averaging the state equations as in SSA [19], and [21]. Average models for the PWM-switch were introduced in [22] and [23].
Linearization of the averaged circuit and PWM-switch yields appropriate small-signal models of the converter. It is clear that the ripple information is lost in averaging.
However, the averaged models are also usable in time domain simulations and transient analyses. Switched circuits can be equally used, if the ripple information is needed [21]. Because the switching action is actually discrete, a sampled-data modeling has been proposed. The basis of the sampled data modeling is presented in [21]. The modeling is based on the continuous time state-space model and the standard matrix exponential expression for linear time-invariant (LTI) systems.
According to [21], [24] and [25], the prevailing method is to derive the discrete-time model from the continuous-time state and switching equations. In [26], the discrete-domain model is derived from the corresponding model in the Laplace-discrete-domain by using z-transformation. The sampled-data modeling is derived in [27]-[29] by using a discrete-time state-space model. The sampled-data modeling typically involves tedious calculations and therefore it is not widely adopted. However, the sampled-data models might become useful, when digital controllers replace the analog controllers.
It should be clear that the true internal or nominal dynamics can be derived from the power stage model and from the control circuit model. However, it seems that the definition of the nominal power stage or model is not clear among the scientists and engineers. There are numerous examples of modeling and analyzing switched-mode
DC-DC converters with a resistive load (these are only example papers, not the complete list: [4] , [18] and [30]-[35]). The actual load is very seldom a pure resistor but should be treated as an external system not included in the nominal model. The seminal paper [4] actually uses a resistive load when introducing the SSA method and the canonical equivalent circuit. It is obvious that including the load resistor in the so-called canonical model might lose the information of the nominal dynamics. Even the fundamental power electronics text books such as [36] and [18] use the resistive load in their analyses and provide incorrect information for the reader. So, what type of load should be used to get the nominal dynamics? A voltage-output DC-DC converter is known to have current source input and voltage source output ports [P5], so the natural nominal load connected to the voltage source output port is obviously a current-sink. Consequently, the nominal load for a current-output converter with a current source output port is a pure voltage-source. In spite of the prevailing technique to use the resistive load, a few attempts to define the nominal or general load have been presented. A general load impedance is treated as an alternative for the resistive load in [37]. First it is stated that the load can be seen as a current source, but later the load is replaced with the general load impedance. The idea of the general load (impedance) is actually correct, but the authors seem to lack the understanding of the true nominal dynamical behavior of switched-mode converters. It is explicitly stated in [38] that the nominal load refers to the use of either a resistive or dc current sink load. It is true that the internal output impedance can be measured either by using a resistive or current sink load, but when measuring or analyzing e.g. the loop gain this does not apply. It seems that the authors of [38] are confused with the terminology of the nominal dynamics or nominal load and provide vague information.
An approach known as an unterminated modeling was introduced in [39] treating a converter as a stand-alone module without considering the load impedance, but using a current sink load. The unterminated modeling method has been applied in [40]-[42]
for studying the load interactions. In [43], the method was successfully used for analyzing the input filter interactions. The unterminated model was derived in [44] by first constructing the models with the load resistor R and then letting Ro f. However, the most convenient way of getting the unterminated model is to use the constant-current-sink load as an initial load system as it was done e.g. in [P4] and [P5]. As a significant contribution of this thesis, it was found that the derivation of the nominal dynamics, and hence, the dynamical profile by using the correct initial load is a starting point for understanding the behavior of switched-mode converters under various conditions.
which may deteriorate the performance of the converter. The input filter interactions were first studied by Middlebrook in the seminal paper [3]. A converter with an input filter was modeled and design criteria for input filter were developed. The derivation of the input-filter-affected transfer functions were based on the method known later as an extra element theorem (EET) [18], and [45]-[47]. The EET provides a tool to analyze the change of transfer functions, when impedance is added to the network.
However, the EET involves tedious calculations, and therefore, may not be suitable for practical usage. The load and input filter interactions can be easily concluded from a two-port linear circuit representation of the converter with load and supply (e.g.
filter impedance) impedances [48] and [43]. The two-port modeling technique based on g-parameters [49] is reviewed and discussed more in detail in Chapter 2. The input-filter interactions have been under extensive research since the Middlebrook’s paper. It has been noticed that different topologies and control methods have different sensitivities for instability or performance degradation due to the input filter or supply impedance [33], and [50]-[56]. Obviously, the converter dynamics are also affected by the load. The load interactions have also been studied in various papers such as in [38]-[42], [44], [57], and [58]. The load interaction formalism is simpler to understand than the corresponding supply side formalism. In Chapter 2, it will be shown that the performance of a converter may be deteriorated if the load impedance and the open-loop output impedance of the converter overlap. Although, the supply and load-side-interaction formalisms are different, the stability of the converter with load and/or supply system can be concluded from the impedance ratio known as a minor-loop gain [3], [39] and [41]. If studying the supply interactions, the minor-loop gain is defined as the ratio of the supply impedance (e.g. input filter output impedance) and the closed-loop input impedance of the converter. The corresponding minor-loop gain at the load side is the ratio between the closed-loop output impedance of the converter and the load impedance. In order to guarantee the stability, the minor loop gain must satisfy the Nyquist stability criterion [3], and [59].
Various forbidden regions in the complex half plane, out of which the minor loop gain should stay, have been presented in the literature [41], and [60]-[63]. It is claimed that these forbidden regions provide certain phase (PM) and gain margins (GM) for the interconnected system. However, the PM and GM of the minor loop very seldom coincide with the corresponding margins in the load or supply-affected loop gain of the converter [P3] and [P7]. This means that the performance of the converter may be drastically deteriorated even if the corresponding margins of the minor loop gain are adequate. The minor-loop-gain analysis is only suitable for ensuring the stability but the performance and true margins should always be checked from the true “major” loop gain of the converter.
A typical method to analyze the converter performance in the time domain is a transient response analysis. There are numerous papers claiming that the higher the crossover frequency fc of the loop gain is the faster is the transient response [64]-[68]. According to the classical control theory this is true, because the reference is step-changed. However, in switched-mode converters the reference is usually kept constant but the load current is changed introducing a transient into the output voltage. In [P2], [69] and [70], it was demonstrated that even if the peak-current-mode controlled (PCMC) converter and PCMC converter with an output current feed-forward (OCF) have the same loop gain, the PCMC-OCF converter has a considerably faster transient response. The reason for this is the smaller open-loop output impedance of the PCMC-OCF converter. Therefore, the transient response actually relates to the open-loop output impedance of the converter and its behavior.
This was also noticed in [71], where a larger closed-loop output impedance of the PCMC converter at lower frequencies compared to the VMC converter yielded also a longer settling time. The transient response as a function of time using inverse Laplace transformation of the closed-loop output impedance, when a certain load step change occurs, was computed in [72]. The transient response of a parallel RLC-circuit was considered in [67] and [68] to mimic the transient response of a switched-mode DC-DC converter. The results of the above analyses seem to be, however, a bit unreliable because of the simplifications made in the analyses. The true relation between the frequency and time domain still seems to be fuzzy and needs further research in order to put the relation in a correct mathematical from. Actually, the challenge is the complex structure (i.e. the numerator and denominator are high-order polynomes in the Laplace variable s) of the closed-loop output impedance making the computation of the inverse Laplace transformation a difficult and tedious task.
Nevertheless, the relation can be implicitly studied as it was discussed above (i.e.
larger closed-loop output impedance at lower frequencies Æ longer settling time and small impedance Æ fast response). In addition, the amount of peaking in the closed-loop output impedance at the phase or gain crossover frequency dictates the PM and GM of the converter loop gain [72]: The peaking is related to sensitivity function (i.e.
1/ 1L s( ) ), which is a elementary part of the equation of the closed-loop output impedance by definition [11]. The peaking in the sensitivity function due to a low PM or GM would naturally be observable in the transient response.
The first basic courses on power electronics at universities are typically based on fundamental text books such as [18], [36] and [73]. They all present the fundamentals of switched-mode power conversion, but in [18] the study is done more in detail.
principles are discussed with the resistive load. The text book [18] is maybe the most often used introductory level book on switched-mode converters, but it loses the point of presenting the true canonical model (both steady-state and small-signal) and dynamical issues by incorporating the resistive load into the models. The contents of these fundamental text books may explain the reason, why the prevailing technique still strictly relies on the use of the resistive load in the analyses both among academia and industry.
Chapter 5 of this thesis is solely dedicated to the current-output converters, which are typically used in battery-powered applications. The modeling and analysis of the current output converter in [74]-[76] are based on the use of resistive load, although the real load typically consists of a back-up battery with low internal impedance [77].
The peculiar behavior of the increasing crossover frequency in the loop gain with battery-type load observed e.g. in [74] and [79] was shown to be due to the use of wrong initial load (i.e. resistive) in [P9].