Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions1Chapter 8. Converter Transfer Functions8.1. Review of Bode plots8.1.1. Single pole response8.1.2. Single zero response8.1.3. Right half-plane zero8.1.4. Frequency inversion8.1.5. Combinations8.1.6. Double pole response: resonance8.1.7. The low-Q approximation8.1.8. Approximate roots of an arbitrary-degree polynomial8.2. Analysis of converter transfer functions8.2.1. Example: transfer functions of the buck-boost converter8.2.2. Transfer functions of some basic CCM converters8.2.3. Physical origins of the right half-plane zero in convertersFundamentals of Power Electronics Chapter 8: Converter Transfer Functions2Converter Transfer Functions8.3. Graphical construction of converter transferfunctions8.3.1. Series impedances: addition of asymptotes8.3.2. Parallel impedances: inverse addition of asymptotes8.3.3. Another example8.3.4. Voltage divider transfer functions: division of asymptotes8.4. Measurement of ac transfer functions andimpedances8.5. Summary of key pointsFundamentals of Power Electronics Chapter 8: Converter Transfer Functions3The Engineering Design Process1.Specifications and other design goals are defined.2.A circuit is proposed. This is a creative process that draws on thephysical insight and experience of the engineer.3.The circuit is modeled. The converter power stage is modeled asdescribed in Chapter 7. Components and other portions of the systemare modeled as appropriate, often with vendor-supplied data.4.Design-oriented analysis of the circuit is performed. This involvesdevelopment of equations that allow element values to be chosen suchthat specifications and design goals are met. In addition, it may benecessary for the engineer to gain additional understanding andphysical insight into the circuit behavior, so that the design can beimproved by adding elements to the circuit or by changing circuitconnections.5.Model verification. Predictions of the model are compared to alaboratory prototype, under nominal operating conditions. The model isrefined as necessary, so that the model predictions agree withlaboratory measurements.Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions4Design Process6.Worst-case analysis (or other reliability and production yieldanalysis) of the circuit is performed. This involves quantitativeevaluation of the model performance, to judge whetherspecifications are met under all conditions. Computersimulation is well-suited to this task.7. Iteration. The above steps are repeated to improve the designuntil the worst-case behavior meets specifications, or until thereliability and production yield are acceptably high.This Chapter: steps 4, 5, and 6Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions5Buck-boost converter modelFrom Chapter 7+–+–LRC1 : D D' : 1vg(s) Id(s) Id(s)i(s)+v(s)–(Vg – V)d(s)Zout(s)Zin(s)d(s) Control inputLineinputOutputGvg(s)=v(s)vg(s)d(s)=0Gvd(s)=v(s)d(s)vg(s)=0Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions6Bode plot of control-to-output transfer functionwith analytical expressions for important featuresf0˚–90˚–180˚–270˚|| Gvd ||Gd0 =|| Gvd ||∠ Gvd0 dBV–20 dBV–40 dBV20 dBV40 dBV60 dBV80 dBVQ =∠ Gvd10-1/2Qf0101/2Qf00˚–20 dB/decade–40 dB/decade–270˚fz /1010fz1 MHz10 Hz 100 Hz 1 kHz 10 kHz 100 kHzf0VDD'D'RCLD'2π LCD'2R2πDL(RHP)fzDVgω(D')3RCVgω2D'LCFundamentals of Power Electronics Chapter 8: Converter Transfer Functions7Design-oriented analysisHow to approach a real (and hence, complicated) systemProblems:Complicated derivationsLong equationsAlgebra mistakesDesign objectives:Obtain physical insight which leads engineer to synthesis of a good designObtain simple equations that can be inverted, so that element values canbe chosen to obtain desired behavior. Equations that cannot be invertedare useless for design!Design-oriented analysis is a structured approach to analysis, which attempts toavoid the above problemsFundamentals of Power Electronics Chapter 8: Converter Transfer Functions8Some elements of design-oriented analysis,discussed in this chapter•Writing transfer functions in normalized form, to directly expose salientfeatures•Obtaining simple analytical expressions for asymptotes, cornerfrequencies, and other salient features, allows element values to beselected such that a given desired behavior is obtained•Use of inverted poles and zeroes, to refer transfer function gains to themost important asymptote•Analytical approximation of roots of high-order polynomials•Graphical construction of Bode plots of transfer functions andpolynomials, toavoid algebra mistakesapproximate transfer functionsobtain insight into origins of salient featuresFundamentals of Power Electronics Chapter 8: Converter Transfer Functions98.1. Review of Bode plotsDecibelsGdB=20log10GTable 8.1. Expressing m agnitudes in decibelsActual magnitude Magnitude in dB1/2 – 6dB10 dB26 dB5 = 10/2 20 dB – 6 dB = 14 dB10 20dB1000 = 1033 ⋅ 20dB = 60 dBZdB=20log10ZRbaseDecibels of quantities havingunits (impedance example):normalize before taking log5Ω is equivalent to 14dB with respect to a base impedance of Rbase =1Ω, also known as 14dBΩ.60dBµA is a current 60dB greater than a base current of 1µA, or 1mA.Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions10Bode plot of fnG =ff0nBode plots are effectively log-log plots, which cause functions whichvary as fn to become linear plots. Given:Magnitude in dB isGdB=20log10ff0n=20n log10ff0ff0–2ff020dB–20dB–40dB–60dB20dB40dB60dBflog scalef00.1f010f0ff0ff0–1n = 1n = 2n = –2n = –120 dB/decade40dB/decade–20dB/decade–40dB/decade•Slope is 20n dB/decade•Magnitude is 1, or 0dB, atfrequency f = f0Fundamentals of Power Electronics Chapter 8: Converter Transfer Functions118.1.1. Single pole response+–RCv1(s)+v2(s)–Simple R-C exampleTransfer function isG(s)=v2(s)v1(s)=1sC1sC+ RG(s)=11+sRCExpress as rational fraction:This coincides with the normalizedformG(s)=11+sω0withω0=1RCFundamentals of Power Electronics Chapter 8: Converter Transfer Functions12G(jω) and || G(jω) ||Im(G(jω))Re(G(jω))G(jω)|| G(jω) ||∠G(jω)G(jω)=11+jωω0=1–jωω01+ωω02G(jω) =Re(G(jω))2+Im(G(jω))2=11+ωω02Let s = jω:Magnitude isMagnitude in dB:G(jω)dB=–20log101+ωω02dBFundamentals of Power Electronics Chapter 8: Converter Transfer Functions13Asymptotic behavior: low