Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...1Chapter 3. Steady-State Equivalent CircuitModeling, Losses, and Efficiency3.1. The dc transformer model3.2. Inclusion of inductor copper loss3.3. Construction of equivalent circuit model3.4. How to obtain the input port of the model3.5. Example: inclusion of semiconductor conductionlosses in the boost converter model3.6. Summary of key pointsFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...23.1. The dc transformer modelBasic equations of an idealdc-dc converter:Pin= PoutVgIg= VI(η = 100%)V = M(D) Vg(ideal conversion ratio)Ig= M(D)IThese equations are valid in steady-state. Duringtransients, energy storage within filter elements may causePin ≠ PoutSwitchingdc-dcconverterDControl inputPowerinputPoweroutputIg I+V–+Vg–Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...3Equivalent circuits corresponding toideal dc-dc converter equationsPin= PoutVgIg= VIV = M(D) VgIg= M(D)IDependent sources DC transformerPoweroutput+V–I+–M(D)VgPowerinput+Vg–IgM(D)IDControl inputPowerinputPoweroutput+V–+Vg–IgI1 : M(D)Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...4The DC transformer modelModels basic properties ofideal dc-dc converter:•conversion of dc voltagesand currents, ideally with100% efficiency•conversion ratio Mcontrollable via duty cycle•Solid line denotes ideal transformer model, capable of passing dc voltagesand currents•Time-invariant model (no switching) which can be solved to find dccomponents of converter waveformsDControl inputPowerinputPoweroutput+V–+Vg–IgI1 : M(D)Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...5Example: use of the DC transformer model1. Original system2. Insert dc transformer model3. Push source through transformer4. Solve circuitV = M(D) V1RR + M2(D) R1DRV1R1+–+Vg–+V–Switchingdc-dcconverter1 : M(D)RV1R1+–+Vg–+V–RM(D)V1M2(D)R1+–+V–Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...63.2. Inclusion of inductor copper lossDc transformer model can be extended, to include converter nonidealities.Example: inductor copper loss (resistance of winding):Insert this inductor model into boost converter circuit:LRLL+–CR+v–12iVgRLFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...7Analysis of nonideal boost converterswitch in position 1 switch in position 2L+–CR+v–12iVgRLLRL+–iCR+v–+ vL –iCVgLRL+–iCR+v–+ vL –iCVgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...8Circuit equations, switch in position 1Inductor current andcapacitor voltage:vL(t)=Vg– i(t) RLiC(t)=–v(t)/RSmall ripple approximation:vL(t)=Vg– IRLiC(t)=–V / RLRL+–iCR+v–+ vL –iCVgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...9Circuit equations, switch in position 2vL(t)=Vg– i(t) RL– v(t) ≈ Vg– IRL– ViC(t)=i(t)–v(t)/R ≈ I – V / RLRL+–iCR+v–+ vL –iCVgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...10Inductor voltage and capacitor current waveformsAverage inductor voltage:vL(t) =1TsvL(t)dt0Ts= D(Vg– IRL)+D'(Vg– IRL– V)Inductor volt-second balance:0=Vg– IRL– D'VAverage capacitor current:iC(t) = D (–V / R)+D'(I – V / R)Capacitor charge balance:0=D'I – V / RvL(t)tVg – IRLDTsD'TsVg – IRL – ViC(t)–V/RI – V/RtFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...11Solution for output voltageWe now have twoequations and twounknowns:0=Vg– IRL– D'V0=D'I – V / REliminate I andsolve for V:VVg=1D'1(1 + RL/ D'2R)DV/Vg0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.522.533.544.55RL/R = 0RL/R = 0.01RL/R = 0.02RL/R = 0.05RL/R = 0.1Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...123.3. Construction of equivalent circuit modelResults of previous section (derived via inductor volt-sec balance andcapacitor charge balance):vL=0=Vg– IRL– D'ViC=0=D'I – V / RView these as loop and node equations of the equivalent circuit.Reconstruct an equivalent circuit satisfying these equationsFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...13Inductor voltage equationvL=0=Vg– IRL– D'V• Derived via Kirchhoff’s voltagelaw, to find the inductor voltageduring each subinterval• Average inductor voltage thenset to zero•This is a loop equation: the dccomponents of voltage arounda loop containing the inductorsum to zero• IRL term: voltage across resistorof value RL having current I• D’V term: for now, leave asdependent source+–LRL+–+ 〈vL〉 – = 0+ IRL –ID'VVgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...14Capacitor current equation• Derived via Kirchoff’s currentlaw, to find the capacitorcurrent during each subinterval• Average capacitor current thenset to zero•This is a node equation: the dccomponents of current flowinginto a node connected to thecapacitor sum to zero• V/R term: current through loadresistor of value R having voltage V• D’I term: for now, leave asdependent sourceiC=0=D'I – V / RR+V–C〈iC〉 = 0NodeV/RD'IFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...15Complete equivalent circuitThe two circuits, drawn together:The dependent sources are equivalentto a D′ : 1 transformer:Dependent sources and transformers+–+V2–nV2nI1I1n : 1+V2–I1•sources have same coefficient•reciprocal voltage/currentdependence+–+–D'VRLID'IR+V–Vg+–RLIR+V–D' : 1VgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...16Solution of equivalent circuitConverter equivalent circuitRefer all elements to transformersecondary:Solution for output voltageusing voltage divider formula:V =VgD'RR +RLD'2=VgD'11+RLD'2R+–RLIR+V–D' : 1Vg+–D'IR+V–Vg/D'RL/D' 2Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...17Solution for input (inductor) currentI =VgD'2R + RL=VgD'211+RLD'2R+–RLIR+V–D' : 1VgFundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...18Solution for converter