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CU-Boulder ECEN 5817 - Sinusoidal Approximations

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CHAPTER 2Sinusoidal Approximationsn this chapter, the properties of the series, parallel, and other resonant converters areinvestigated using the sinusoidal approximation. Harmonics of the switching frequency areneglected, and the tank waveforms are assumed to be purely sinusoidal. This allows simpleequivalent circuits to be derived for the bridge inverter, tank, rectifier, and output filter portions ofthe converter, whose operation can be understood and solved using standard linear ac analysis.This intuitive approach is quite accurate for operation in the continuous conduction mode with ahigh-Q response, but becomes less accurate when the tank is operated with a low Q-factor or foroperation in or near the discontinuous conduction mode.The important result of this approach is that the dc voltage conversion ratio of a continuousconduction mode resonant converter is given approximately by the ac transfer function of the tankcircuit, evaluated at the switching frequency. The tank is loaded by the effective output resistance,nearly equal to the output voltage divided by the output current. It is thus quite easy to determinehow the tank components and circuit connections affect the converter behavior. The influence oftank component losses, transformer nonidealities, etc., on the output voltage and converterefficiency can also be found.It is found that the series resonant converter operates with a step-down voltage conversionratio. With a 1:1 transformer turns ratio, the dc output voltage is ideally equal to the dc inputvoltage when the transistor switching frequency is equal to the tank resonant frequency. Theoutput voltage is reduced as the switching frequency is increased or decreased away fromresonance. On the other hand, the parallel resonant converter is capable of both step-up and step-down of voltage levels, depending on the switching frequency and the effective tank Q-factor.Switching loss mechanisms are also considered in this chapter. “Zero voltage switching” isa property that can be obtained in resonant converters whenever the tank presents a lagging(inductive) load to the switch network. This occurs for operation above resonance in the seriesIPrinciples of Resonant Power Conversion2resonant converter, and it can lead to elimination of the switching loss which arises from the switchoutput capacitances. Likewise, “zero current switching” can be obtained when the tank presents aleading (capacitive) load to the switch network, as in the series resonant converter operation belowresonance. This property allows natural commutation of thyristors, and elimination of switchingloss mechanisms associated with package and other parasitic inductances.2.1. First Order Network ModelsConsider the class of resonant converters which contain a controlled switch network NSand drive a linear resonant tank network NT. The latter in turn is connected to an uncontrolledrectifier NR, filter NF and load R, which is illustrated in Fig. 2.1. Many well-known converterscan be represented in this form, including the series, parallel, LCC, et al.controlledswitch networklow passfilterresonanttank network+–→powerinputNS→I→RloaduncontrolledrectifierNTNRNFiR+ vR –+ V –+ –→i (t)ViSvSggFig. 2.1.A class of resonant converters which consist of cascaded switch, tank,rectifier, and filter networks. A series resonant converter example isshown.In the most common modes of operation, the controlled switch network produces a square wavevoltage output vS(t) whose frequency fS is close to the tank network resonant frequency f0. Inresponse, the tank network rings with approximately sinusoidal waveforms of frequency fS. Thetank output waveform vR or iR is then rectified by network NR and filtered by network NF, toproduce the dc load voltage V and current I. By changing the switching frequency fS (closer to orfarther from resonance f0), the magnitude of the tank ringing response can be modified, and hencethe dc output voltage can be controlled.Chapter 2. Sinusoidal Approximations3In the case where the resonant tankresponds primarily to the fundamentalcomponent fS of the switch waveformvS(t), and has negligible response at theharmonic frequencies nfS, n = 3, 5, 7, ...,then the tank waveforms are wellapproximated by their fundamentalcomponents. As shown in Fig. 2.2, this isindeed the case when the tank networkcontains a high-Q resonance at or near theswitching frequency, and a low-passcharacteristic at higher frequencies. Hence,let us neglect harmonics, and compute therelationships between the fundamentalcomponents of the tank terminal waveformsvS(t), iS(t), iR(t), and vR(t), and theconverter dc terminal quantities Vg, V, and I.Controlled Switch NetworkIf the switch of Fig. 2.3 is controlled to produce a square wave of frequency fS as in Fig.2.4, then its output voltage waveform vS(t) can be expressed in the Fourier seriesvS(t) = 4 Vgπ ∑n=1,3,5,... 1n sin (2nπfSt) (2-1)switchoutputvoltagespectrumresonanttankresponsef 3f 5f S SSf 3f 5f S SSffftank current ispectrumsf 3f 5f S SSFig. 2.2.The tank responds primarily to thefundamental component of the appliedwaveforms.1212+–→→+–vSiSigvgFig. 2.3. An ideal switch network.tswitchposition:12v (t)S1v (t)STSVg-VgFig. 2.4. Controlled switch network outputvoltage vS(t) and its fundamentalcomponent vS1(t).Principles of Resonant Power Conversion4The fundamental component is vS1(t) = 4 Vgπ sin (2π fSt) (2-2)which has a peak amplitude of (4/π) times the dc input voltage Vg, and is in phase with the originalsquare wave vS(t). Hence, the switch network output terminal is modeled as a sinusoidal voltagegenerator, vS1(t).It is interesting to model the converter dc input. This requires computation of the dccomponent Ig of the switch input current ig(t). The switch input current ig(t) is equal to the outputcurrent iS(t) when the switches are in position 1, and its inverse -iS(t) when the switches are inposition 2. Under the conditions described above, the tank rings sinusoidally and iS(t) is wellapproximated by a sinusoid of some peak amplitude IS1 and phase ϕS:iS(t) ≅ IS1 sin (2πfSt – ϕS)(2-3)The input current waveform is shown in Fig. 2.5.The dc component, or average value, of theinput current can be found by averaging ig(t) overone half switching period:ig = 2TS ig(t)dt0TS/2 ≅ 2TS IS1 sin (2πfSt - ϕS ) dt 0TS/2 =


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CU-Boulder ECEN 5817 - Sinusoidal Approximations

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