1 Exp. 4, Part 2 Converter Transfer Functions The engineering design process is comprised of several major steps: 1. Specifications and other design goals are defined.2. A circuit is proposed . This is a creative process that draws on the physical insight and experience of theengineer.3. The circuit is modeled . The converter power stage is modeled as described in Chapter 7. Components andother portions of the system are modeled as appropriate, often with vendor-supplied data.4. Design-oriented analysis of the circuit is performed. This involves development of equations that allowelement values to be chosen such that specifications and design goals are met. In addition, it may be neces-sary for the engineer to gain additional understanding and physical insight into the circuit behavior, so thatthe design can be improved by adding elements to the circuit or by changing circuit connections.5. Model verification . Predictions of the model are compared to a laboratory prototype, under nominal oper-ating conditions. The model is refined as necessary, so that the model predictions agree with laboratorymeasurements.6. Worst-case analysis (or other reliability and production yield analysis) of the circuit is performed. Thisinvolves quantitative evaluation of the model performance, to judge whether specifications are met underall conditions. Computer simulation is well-suited to this task.7. Iteration . The above steps are repeated to improve the design until the worst-case behavior meets specifi-cations, or until the reliability and production yield are acceptably high. Part 3 of this experiment is concerned with the modeling, simulation, and verification steps that arerequired to design the feedback system of a switched-mode converter.2 Converter Transfer Functions 2.1 INTRODUCTION Converter systems invariably require feedback. For example, in a typical dc–dc converter application, theoutput voltage v ( t ) must be kept constant, regardless of changes in the input voltage v g ( t ) or in the effec-tive load resistance R . This is accomplished by building a circuit that varies the converter control input[i.e., the duty cycle d ( t )] in such a way that the output voltage v ( t ) is regulated to be equal to a desired ref-erence value v ref . In inverter systems, a feedback loop causes the output voltage to follow a sinusoidalreference voltage. In modern low-harmonic rectifier systems, a control system causes the converter inputcurrent to be proportional to the input voltage, such that the input port presents a resistive load to the acsource. So feedback is commonly employed.A typical dc–dc system incorporating a buck converter and feedback loop block diagram isillustrated in Fig. 2.1. It is desired to design this feedback system in such a way that the output voltage isaccurately regulated, and is insensitive to disturbances in v g ( t ) or in the load current. In addition, thefeedback system should be stable, and properties such as transient overshoot, settling time, and steady-state regulation should meet specifications.To design the system of Fig. 2.1, we need a dynamic model of the switching converter. How dovariations in the power input voltage, the load current, or the duty cycle affect the output voltage? Whatare the small-signal transfer functions? To answer these questions, we will derive an equivalent circuitmodel of the converter, which predicts the dynamics introduced by the inductors and capacitors of theconverter. Modeling is the representation of physical phenomena by mathematical means. In engineering,it is desired to model the important dominant behavior of a system, while neglecting other insignificantphenomena. Simplified terminal equations of the component elements are used, and many aspects of thesystem response are neglected altogether, that is, they are “unmodeled.” The resulting simplified modelyields physical insight into the system behavior, which aids the engineer in designing the system to oper-ate in a given specified manner. Thus, the modeling process involves use of approximations to neglectsmall but complicating phenomena, in an attempt to understand what is most important. Once this basicinsight is gained, it may be desirable to carefully refine the model, by accounting for some of the previ-ously ignored phenomena. It is a fact of life that real, physical systems are complex, and their detailedanalysis can easily lead to an intractable and useless mathematical mess. Approximate models are animportant tool for gaining understanding and physical insight.The switching ripple is small in a well-designed converter operating in continuous conductionmode (CCM). Hence, we should ignore the switching ripple, and model only the underlying ac variationsin the converter waveforms. For example, suppose that some ac variation is introduced into the converterduty cycle d ( t ), such that (2.1) where D and D m are constants, | D m | < D , and the modulation frequency ω m is much smaller than theconverter switching frequency ω s = 2 π f s . The resulting transistor gate drive signal is illustrated inFig. 2.2(a), and a typical converter output voltage waveform v ( t ) is illustrated in Fig. 2.2(b). The spec-trum of v ( t ) is illustrated in Fig. 2.3. This spectrum contains components at the switching frequency aswell as its harmonics and sidebands; these components are small in magnitude if the switching ripple issmall. In addition, the spectrum contains a low-frequency component at the modulation frequency ω m .The magnitude and phase of this component depend not only on the duty cycle variation, but also on thefrequency response of the converter. If we neglect the switching ripple, then this low-frequency compo-d(t)=D + Dmcosωmt2.1 Introduction 3 nent remains [also illustrated in Fig. 2.2(b)]. The objective of our ac modeling efforts is to predict thislow-frequency component.A simple method for deriving the small-signal model of CCM converters is explained here. Theswitching ripples in the inductor current and capacitor voltage waveforms are removed by averaging over+–+v(t)–vg(t)Switching converterPowerinputLoad–+RCompensatorGc(s)vrefVoltagereferencevFeedbackconnectionPulse-widthmodulatorvcTransistorgate driverδ(t)δ(t)TsdTsttvc(t)ControllerFig. 2.1 A simple dc–dc regulator system, including a buck converter power stage and a feedback network.ttGatedriveActual waveform v(t)including rippleAveraged waveform
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