Chapter 3. Steady-State Equivalent Circuit Modeling .

Chapter 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 Electronics1Chapter 3: Steady-state equivalent circuit modeling, .

3.1. The dc transformer modelIgBasic equations of an idealdc-dc converter:Pin PoutPower(η 100%)inputVg I g V IV M(D) VgI Switching Vgdc-dcV–converter–PoweroutputD(ideal conversion ratio)I g M(D) IControl inputThese equations are valid in steady-state. Duringtransients, energy storage within filter elements may causePin PoutFundamentals of Power Electronics2Chapter 3: Steady-state equivalent circuit modeling, .

Equivalent circuits corresponding toideal dc-dc converter equationsPin PoutVg I g V IV M(D) VgDependent sourcesI g M(D) IDC transformerIgIg1 : M(D) II PowerPowerPowerinput Vg M(D)IM(D)Vg– –VinputPoweroutput–VgV––outputDControl inputFundamentals of Power Electronics3Chapter 3: Steady-state equivalent circuit modeling, .

The DC transformer modelIg1 : M(D) I PowerPowerinputVgV––DoutputModels basic properties ofideal dc-dc converter: conversion of dc voltagesand currents, ideally with100% efficiency conversion ratio Mcontrollable via duty cycleControl input 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 waveformsFundamentals of Power Electronics4Chapter 3: Steady-state equivalent circuit modeling, .

Example: use of the DC transformer model1. Original system3. Push source through transformerR1M 2 (D) R1 SwitchingV1 –dc-dcVgVRM (D) V1 –converter––VR–D4. Solve circuit2. Insert dc transformer modelR1V1 –1 : M(D) VgV––Fundamentals of Power ElectronicsV M(D) V1RR M 2(D) R 1R5Chapter 3: Steady-state equivalent circuit modeling, .

3.2. Inclusion of inductor copper lossDc transformer model can be extended, to include converter nonidealities.Example: inductor copper loss (resistance of winding):LRLInsert this inductor model into boost converter circuit:LRL2 iVg1 –CRv–Fundamentals of Power Electronics6Chapter 3: Steady-state equivalent circuit modeling, .

Analysis of nonideal boost converterLRL2 iVg1 –CvR–switch in position 1iLRL vL –Vgswitch in position 2 –i iCCR vL –Vgv –RL iCCRv––Fundamentals of Power ElectronicsL7Chapter 3: Steady-state equivalent circuit modeling, .

Circuit equations, switch in position 1Inductor current andcapacitor voltage:vL(t) Vg – i(t) RLLiRL vL –VgiC(t) –v(t) / R – iCCRv–Small ripple approximation:vL(t) Vg – I RLiC(t) –V / RFundamentals of Power Electronics8Chapter 3: Steady-state equivalent circuit modeling, .

Circuit equations, switch in position 2iLRL vL –Vg iC –CRv–vL(t) Vg – i(t) RL – v(t) Vg – I RL – ViC(t) i(t) – v(t) / R I – V / RFundamentals of Power Electronics9Chapter 3: Steady-state equivalent circuit modeling, .

Inductor voltage and capacitor current waveformsAverage inductor voltage:vL(t)Vg – IRLTsvL(t) 1v (t)dtTs 0 L D(Vg – I RL) D'(Vg – I RL – V)Inductor volt-second balance:DTsD'TstVg – IRL – ViC(t)I – V/R0 Vg – I RL – D'VAverage capacitor current:–V/RtiC(t) D ( – V / R) D' (I – V / R)Capacitor charge balance:0 D'I – V / RFundamentals of Power Electronics10Chapter 3: Steady-state equivalent circuit modeling, .

Solution for output voltage5We now have twoequations and twounknowns:3.50 D'I – V / RV 11Vg D' (1 RL / D' 2R)RL /R 0.0140 Vg – I RL – D'VRL /R 0.023V/ VgEliminate I andsolve for V:RL /R 04.52.52RL /R 0.051.5RL /R 0.110.5000.10.20.30.40.50.60.70.80.91DFundamentals of Power Electronics11Chapter 3: Steady-state equivalent circuit modeling, .

3.3. Construction of equivalent circuit modelResults of previous section (derived via inductor volt-sec balance andcapacitor charge balance):vL 0 Vg – I RL – 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 Electronics12Chapter 3: Steady-state equivalent circuit modeling, .

Inductor voltage equationvL 0 Vg – I RL – D'V Derived via Kirchhoff’s voltagelaw, to find the inductor voltageduring each subintervalVg Average inductor voltage thenset to zero This is a loop equation: the dccomponents of voltage arounda loop containing the inductorsum to zero –LRL 〈vL 〉 – 0 IRL –I –D'V IRL term: voltage across resistorof value RL having current I D’V term: for now, leave asdependent sourceFundamentals of Power Electronics13Chapter 3: Steady-state equivalent circuit modeling, .

Capacitor current equationNodeiC 0 D'I – V / RV/R Derived via Kirchoff’s currentlaw, to find the capacitorcurrent during each subintervalD'I Average capacitor current thenset to zero〈iC 〉 0 CVR– 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 sourceFundamentals of Power Electronics14Chapter 3: Steady-state equivalent circuit modeling, .

Complete equivalent circuitDependent sources and transformersThe two circuits, drawn together:I1RL Vg –D'VI –nV2VD'IR –nI1 V2––n:1The dependent sources are equivalentto a D′ : 1 transformer:RL – –VR sources have same coefficient reciprocal voltage/currentdependence–Fundamentals of Power Electronics V2D' : 1IVgI115Chapter 3: Steady-state equivalent circuit modeling, .

Solution of equivalent circuitConverter equivalent circuitRLD' : 1 IVg –VR–Refer all elements to transformersecondary:Solution for output voltageusing voltage divider formula:RL /D' 2D'IVg /D' – V VRVgD'RR RLD' 2 VgD'11 RLD' 2 R–Fundamentals of Power Electronics16Chapter 3: Steady-state equivalent circuit modeling, .

Solution for input (inductor) currentRLD' : 1 IVg –VR–VgVg1I 22D' R RLD' 1 RLD' 2 RFundamentals of Power Electronics17Chapter 3: Steady-state equivalent circuit modeling, .

Solution for converter efficiencyRLPin (Vg) (I)Pout (V) (D'I) IVgD' : 1 –VR–η Pout (V) (D'I) V D'PinVg(Vg) (I)1η 1 RLD' 2 RFundamentals of Power Electronics18Chapter 3: Steady-state equivalent circuit modeling, .

Efficiency, for various values of RL100%η 190%R1 2LD' R80%0.0020.010.0270%0.0560%η50%RL /R damentals of Power Electronics19Chapter 3: Steady-state equivalent circuit modeling, .

3.4. How to obtain the input port of the modelBuck converter example —use procedure of previous section toderive equivalent circuitig1LiLRL vL –Vg –2CvCR–Average inductor voltage and capacitor current:vL 0 DVg – I LRL – VCFundamentals of Power ElectronicsiC 0 I L – VC/R20Chapter 3: Steady-state equivalent circuit modeling, .

Construct equivalent circuit as usualvL 0 DVg – I LRL – VCiC 0 I L – VC/RRL 〈vL〉 – 0DVg –〈iC〉 0ILVCVC /RR–What happened to the transformer? Need another equationFundamentals of Power Electronics21Chapter 3: Steady-state equivalent circuit modeling, .

Modeling the converter input portInput current waveform ig(t):ig(t)iL (t) ILarea DTs IL00TsDTstDc component (average value) of ig(t) isIg 1TsFundamentals of Power ElectronicsTsig(t) dt DI L022Chapter 3: Steady-state equivalent circuit modeling, .

Input port equivalent circuitIg 1TsTsig(t) dt DI L0VgFundamentals of Power Electronics –IgDIL23Chapter 3: Steady-state equivalent circuit modeling, .

Complete equivalent circuit, buck converterInput and output port equivalent circuits, drawn together:IgILRL Vg – –DILVCDVgR–Replace dependent sources with equivalent dc transformer:IgIL1:DRL Vg –VCR–Fundamentals of Power Electronics24Chapter 3: Steady-state equivalent circuit modeling, .

3.5. Example: inclusion of semiconductorconduction losses in the boost converter modelBoost converter exampleLi iCVg –DTsTsC –Rv–Models of on-state semiconductor devices:MOSFET: on-resistance RonDiode: constant forward voltage VD plus on-resistance RDInsert these models into subinterval circuitsFundamentals of Power Electronics25Chapter 3: Steady-state equivalent circuit modeling, .

Boost converter example: circuits duringsubintervals 1 and 2Li iCVg –DTsTsC –RL vL –Vg –switch in position 2i iCRonCRv–Fundamentals of Power Electronics26LRLRD –Lv–switch in position 1iR vL –Vg –VD iCCRv–Chapter 3: Steady-state equivalent circuit modeling, .

Average inductor voltage and capacitor currentvL(t)Vg – IRL – IRonDTsD'TstVg – IRL – VD – IRD – ViC(t)I – V/Rt–V/RvL D(Vg – IRL – IRon) D'(Vg – IRL – VD – IRD – V) 0iC D(–V/R) D'(I – V/R) 0Fundamentals of Power Electronics27Chapter 3: Steady-state equivalent circuit modeling, .

Construction of equivalent circuitsVg – IRL – IDRon – D'VD – ID'RD – D'V 0D'VDDRonD'RD –RL IRL – IDRon –Vg – ID'RD – –ID'I – V/R 0D'VV/R D'IVR–Fundamentals of Power Electronics28Chapter 3: Steady-state equivalent circuit modeling, .

Complete equivalent circuitD'VDDRonD'RD –RLVg – D'VI –D'IVR–D'VDDRonD'RD –RLVg –D' : 1 VIR–Fundamentals of Power Electronics29Chapter 3: Steady-state equivalent circuit modeling, .

Solution for output voltageD'VDDRonD'RDD' : 1 –RLVg – VIR–V 1D'Vg – D'VDV 1VgD'1–Fundamentals of Power ElectronicsD'VDVgD' 2RD' 2R RL DRon D'RD1R DRon D'RD1 LD' 2R30Chapter 3: Steady-state equivalent circuit modeling, .

Solution for converter efficiencyD'VDDRon –RLPin (Vg) (I)Vg –IPout (V) (D'I)1 D' : 1 VR–1–η D' V VgD'RDD'VDVgRL DRon D'RDD' 2RConditions for high efficiency:Vg/D' V DD' 2R RL DRon D'RDFundamentals of Power Electronics31Chapter 3: Steady-state equivalent circuit modeling, .

Accuracy of the averaged equivalent circuitin prediction of losses Model uses averagecurrents and voltages To correctly predict powerloss in a resistor, use rmsvalues Result is the same,provided ripple is smallMOSFET current waveforms, for variousripple magnitudes:i(t)2I(c)(b)(a)I0DTs0Inductor current ripple(a) i 0(b) i 0.1 I(c) i IFundamentals of Power Electronics1.1 IMOS FET rms currentID(1.00167) I(1.155) I32TstAverage power loss in R onD I2 R onDD(1.0033) D I2 R on(1.3333) D I2 R onChapter 3: Steady-state equivalent circuit modeling, .

Summary of chapter 31. The dc transformer model represents the primary functions of any dc-dcconverter: transformation of dc voltage and current levels, ideally with100% efficiency, and control of the conversion ratio M via the duty cycle D.This model can be easily manipulated and solved using familiar techniquesof conventional circuit analysis.2. The model can be refined to account for loss elements such as inductorwinding resistance and semiconductor on-resistances and forward voltagedrops. The refined model predicts the voltages, currents, and efficiency ofpractical nonideal converters.3. In general, the dc equivalent circuit for a converter can be derived from theinductor volt-second balance and capacitor charge balance equations.Equivalent circuits are constructed whose loop and node equationscoincide with the volt-second and charge balance equations. In convertershaving a pulsating input current, an additional equation is needed to modelthe converter input port; this equation may be obtained by averaging theconverter input current.Fundamentals of Power Electronics33Chapter 3: Steady-state equivalent circuit modeling, .

Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, .14 Capacitor current equation Derived via Kirchoff’s current law, to find the capacitor current during each subinterval Average capacitor current then set to zero This is a node equation: the