Fundamentals of Power Electronics1Appendix C: MiddlebrookÕs Extra Element TheoremAppendix CMiddlebrookÕs Extra Element TheoremC.1 Basic ResultC.2 DerivationC.3 DiscussionC.4 ExamplesC.4.1 A simple transfer functionC.4.2 An unmodeled elementC.4.3 Addition of an input filter to a converterC.4.4 Dependence of transistor current on load in aresonant inverterFundamentals of Power Electronics2Appendix C: MiddlebrookÕs Extra Element Theoremvout(s)vin(s)= G(s)Z(s) → ∞1+ZN(s)Z(s)1+ZD(s)Z(s)C.1 Basic ResultPortOpen-circuitLinear circuitInput Output{+–vin(s)+vout(s)–G(s)Z(s) →∞Transfer functionObject: find how addition of an element changes a transfer function G(s)PortLinear circuitInput Output+–vin(s)+vout(s)–Transfer functionG(s)Z(s)Original conditions: Addition of element Z(s):vout(s)vin(s)= G(s)Z(s) →∞Fundamentals of Power Electronics3Appendix C: MiddlebrookÕs Extra Element Theoremvout(s)vin(s)= G(s)Z(s) → 01+Z(s)ZN(s)1+Z(s)ZD(s)Dual Form of Basic ResultWhen the added impedance replaces a short circuit:PortLinear circuitInput Output+–vin(s)+vout(s)–Transfer functionG(s)Z(s)Original conditions: Addition of element Z(s):G(s)Z(s) → 0PortShort-circuitLinear circuitInput Output{+–vin(s)+vout(s)–Transfer functionFundamentals of Power Electronics4Appendix C: MiddlebrookÕs Extra Element TheoremComparison of formsvout(s)vin(s)= G(s)Z(s) → ∞1+ZN(s)Z(s)1+ZD(s)Z(s)vout(s)vin(s)= G(s)Z(s) → 01+Z(s)ZN(s)1+Z(s)ZD(s)The two forms of the extra element theorem:These equations describe the same transfer function, referenced tothe limiting cases of Z = 0 and Z = ∞. Upon equating them, oneobtains the reciprocity relationship:G(s)Z(s) → ∞G(s)Z(s) → 0=ZD(s)ZN(s)The quantities ZN and ZD are the same in both forms.Fundamentals of Power Electronics5Appendix C: MiddlebrookÕs Extra Element TheoremFinding ZDZD(s)=v(s)i(s)vin(s)=0PortShort-circuitLinear circuitInput Output{vin(s) = 0+vout(s)–i(s)+ v(s) –ZD is the driving-point impedance (i.e., the Thevenin-equivalentimpedance) at the port where the new element is connected. Formally,it is found by setting independent sources to zero, and injecting acurrent i(s) at the port. ZD(s) is the ratio of v(s) to i(s).Fundamentals of Power Electronics6Appendix C: MiddlebrookÕs Extra Element TheoremFinding ZNZN is the impedance seen at the port when the output is nulled. In thepresence of the input vin(s), a current i(s) is injected at the port. Thiscurrent is adjusted such that the output vout(s) is nulled to zero. Underthese conditions, ZN(s) is the ratio of v(s) to i(s). Note: nulling is not thesame as shorting.ZN(s)=v(s)i(s)vout(s)o0PortLinear circuitInput Output+vout(s) o 0–i(s)+ v(s) –+–vin(s)Fundamentals of Power Electronics7Appendix C: MiddlebrookÕs Extra Element TheoremC.2 Derivationu(s) y(s)i(s)+ v(s) –PortOpen-circuitLinear networkInput Output{u(s) y(s)i(s)+ v(s) –PortLinear networkInput OutputZ(s)Original system:With extra element:Gold(s)=y(s)u(s)i(s)=0G(s)=y(s)u(s)v(s)=–i(s)Z(s)[The input and output need not bevoltages, and are denoted here bythe general names u(s) and y(s)]Fundamentals of Power Electronics8Appendix C: MiddlebrookÕs Extra Element TheoremCurrent injection at the portu(s) y(s)i(s)+ v(s) –PortLinear networkInput OutputThere are now two independent inputs:u(s) andi(s)The dependent quantities y(s) and v(s)can be expressed as functions of theindependent inputs using superposition:y(s)=Gold(s)u(s)+Gi(s)i(s)v(s)=Gv(s)u(s)+ZD(s)i(s)with:Gold(s)=y(s)u(s)i(s)=0Gi(s)=y(s)i(s)u(s)=0ZD(s)=v(s)i(s)u(s)=0Gv(s)=v(s)u(s)i(s)=0Fundamentals of Power Electronics9Appendix C: MiddlebrookÕs Extra Element TheoremSolution for G(s)Now eliminate v(s) and i(s) from equations of previous slide, andsolve from transfer function G(s):G(s)=y(s)u(s)= Gold(s)–Gv(s)Gi(s)Z(s)+ZD(s)Gold(s) and ZD(s) are found using definitions on previous slide.We could stop at this point, and use the above equation to evaluateG(s). The quantities Gi(s) and Gv(s) would be evaluated using thedefinitions on the previous slide. However, it is preferable to eliminateGi(s) and Gv(s), and instead express G(s) in terms of impedancesmeasured at the given port. This can be accomplished with analternate thought experiment involving null double injection.Fundamentals of Power Electronics10Appendix C: MiddlebrookÕs Extra Element TheoremNull double injectionu(s) y(s)i(s)+ v(s) –PortLinear networkInput OutputIn the presence of the input u(s),inject current i(s) at the port.Adjust i(s) in the special way thatcauses the output y(s) to benulled to zero. Under theseconditions, the impedance ZN(s) isdefined as:ZN(s)=v(s)i(s)y(s)o 0y(s)=Gold(s)u(s)+Gi(s)i(s)Nulling: Note thatTherefore, the value of i(s) thatachieves the null condition y(s) o 0 isgiven byGold(s)u(s)+Gi(s)i(s) o0So the output is nulled wheni(s) is chosen to satisfyu(s)y(s) o 0=–Gi(s)Gold(s)i(s)y(s) o 0Fundamentals of Power Electronics11Appendix C: MiddlebrookÕs Extra Element TheoremExpression for ZN(s)v(s)y(s) o 0= Gv(s)u(s)y(s) o 0+ ZD(s)i(s)y(s) o 0=–Gv(s)Gi(s)Gold(s)+ ZD(s) i(s)y(s) o 0u(s)y(s) o 0=–Gi(s)Gold(s)i(s)y(s) o 0v(s)=Gv(s)u(s)+ZD(s)i(s)v(s)y(s) o 0= ZN(s)i(s)y(s) o 0=–Gv(s)Gi(s)Gold(s)+ ZD(s) i(s)y(s) o 0ZN(s)=ZD(s)–Gv(s)Gi(s)Gold(s)Now substitute result from previous slide,into previous expression for output voltageThe result is:Use definition of ZN(s):Hence:Fundamentals of Power Electronics12Appendix C: MiddlebrookÕs Extra Element TheoremExpression for G(s)Now, eliminate Gi(s) and Gv(s) from expression for G(s), using ZNresult:G(s)=Gold(s)–ZD(s)–ZN(s)Z(s)+ZD(s)Gold(s)Simplify:G(s)=Gold(s)1+ZN(s)Z(s)1+ZD(s)Z(s)Or,G(s)= G(s)Z(s) → ∞1+ZN(s)Z(s)1+ZD(s)Z(s)(Desired result)Fundamentals of Power Electronics13Appendix C: MiddlebrookÕs Extra Element Theorem• Find• Express result infactored pole-zeroform.Example:A simple transfer function+–R1C+v2(s)–v1(s)R2R3R4G(s)=v2(s)v1(s)Fundamentals of Power Electronics14Appendix C: MiddlebrookÕs Extra Element TheoremC.4.3 Addition of an Input Filter to a Converter+–InputfilterConverterT(s)ControllervgZo(s) Zi(s)H(s)dvAddition of aninput filter changesthe small-signaltransfer functionsof a converterControl-to-output transfer function Gvd(s):Gvd(s)=v(s)d(s)vg(s)=0ConverterdvZo(s)Gvd(s)Set vg = 0. Input filter effectively becomesan impedance Zo(s), added to the converterpower input port.Fundamentals of Power Electronics15Appendix C: MiddlebrookÕs Extra Element TheoremApplication of Extra