Chapter 3 Steady State Equivalent Circuit Modeling Losses and Efficiency 3 1 The dc transformer model 3 2 Inclusion of inductor copper loss 3 3 Construction of equivalent circuit model 3 4 How to obtain the input port of the model 3 5 Example inclusion of semiconductor conduction losses in the boost converter model 3 6 Summary of key points 1 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling 3 1 The dc transformer model Ig Basic equations of an ideal dc dc converter Pin Pout input 100 I Switching Vg dc dc V converter Power Vg I g V I Power output D V M D Vg ideal conversion ratio I g M D I Control input These equations are valid in steady state During transients energy storage within filter elements may cause Pin Pout 2 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Equivalent circuits corresponding to ideal dc dc converter equations Pin Pout Vg I g V I V M D Vg Dependent sources I g M D I DC transformer Ig Ig I 1 M D I Vg V Power Power Power input Vg M D I M D Vg V input Power output output D Control input Fundamentals of Power Electronics 3 Chapter 3 Steady state equivalent circuit modeling The DC transformer model Ig I 1 M D Vg V Power Power input output Models basic properties of ideal dc dc converter conversion of dc voltages and currents ideally with 100 efficiency D conversion ratio M controllable via duty cycle Control input Solid line denotes ideal transformer model capable of passing dc voltages and currents Time invariant model no switching which can be solved to find dc components of converter waveforms 4 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Example use of the DC transformer model 1 Original system 3 Push source through transformer R1 M 2 D R1 Switching V1 dc dc Vg V R M D V1 converter V R D 4 Solve circuit 2 Insert dc transformer model R1 V1 1 M D Vg V V M D V1 R R M 2 D R 1 R 5 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling 3 2 Inclusion of inductor copper loss Dc transformer model can be extended to include converter nonidealities Example inductor copper loss resistance of winding L RL Insert this inductor model into boost converter circuit L RL 2 i Vg 1 C R v Fundamentals of Power Electronics 6 Chapter 3 Steady state equivalent circuit modeling Analysis of nonideal boost converter RL L 2 i Vg 1 C v R switch in position 1 i L L i vL Vg switch in position 2 RL iC C R RL vL Vg v iC C R 7 Fundamentals of Power Electronics v Chapter 3 Steady state equivalent circuit modeling Circuit equations switch in position 1 L i Inductor current and capacitor voltage RL vL vL t Vg i t RL Vg iC t v t R iC C R v Small ripple approximation vL t Vg I RL iC t V R 8 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Circuit equations switch in position 2 i L RL vL Vg iC C R v vL t Vg i t RL v t Vg I RL V iC t i t v t R I V R Fundamentals of Power Electronics 9 Chapter 3 Steady state equivalent circuit modeling Inductor voltage and capacitor current waveforms vL t Average inductor voltage Vg IRL T s vL t 1 v t dt Ts 0 L D Vg I RL D Vg I RL V D Ts DTs t Vg IRL V Inductor volt second balance iC t I V R 0 Vg I RL D V Average capacitor current t V R iC t D V R D I V R Capacitor charge balance 0 D I V R 10 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Solution for output voltage 5 We now have two equations and two unknowns 3 5 0 D I V R V 1 1 Vg D 1 RL D 2R RL R 0 01 4 0 Vg I RL D V RL R 0 02 3 V Vg Eliminate I and solve for V RL R 0 4 5 2 5 2 RL R 0 05 1 5 RL R 0 1 1 0 5 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 D Fundamentals of Power Electronics 11 Chapter 3 Steady state equivalent circuit modeling 3 3 Construction of equivalent circuit model Results of previous section derived via inductor volt sec balance and capacitor charge balance vL 0 Vg I RL D V iC 0 D I V R View these as loop and node equations of the equivalent circuit Reconstruct an equivalent circuit satisfying these equations Fundamentals of Power Electronics 12 Chapter 3 Steady state equivalent circuit modeling Inductor voltage equation vL 0 Vg I RL D V Derived via Kirchhoff s voltage law to find the inductor voltage during each subinterval Vg Average inductor voltage then set to zero This is a loop equation the dc components of voltage around a loop containing the inductor sum to zero L RL vL 0 IRL I D V IR L term voltage across resistor of value RL having current I D V term for now leave as dependent source 13 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Capacitor current equation Node iC 0 D I V R V R Derived via Kirchoff s current law to find the capacitor current during each subinterval D I Average capacitor current then set to zero iC 0 C V R This is a node equation the dc components of current flowing into a node connected to the capacitor sum to zero V R term current through load resistor of value R having voltage V D I term for now leave as dependent source 14 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Complete equivalent circuit Dependent sources and transformers The two circuits drawn together I1 RL Vg D V I nV2 V D I R nI1 V2 n 1 The dependent sources are equivalent to a D 1 transformer RL I Vg I1 V R sources have same coefficient reciprocal voltage current dependence Fundamentals of Power Electronics V2 D 1 15 Chapter 3 Steady state equivalent circuit modeling Solution of equivalent circuit Converter equivalent circuit RL D 1 I Vg V R Refer all elements to transformer secondary Solution for output voltage using voltage divider formula RL D 2 D I Vg D V V Vg D R R R RL D 2 Vg D 1 1 RL D 2 R 16 Fundamentals of Power Electronics Chapter 3 Steady state equivalent circuit modeling Solution for input inductor current RL D 1 I Vg V R I Vg Vg 1 2 D 2 R RL D 1 RL D 2 R 17 Fundamentals of Power Electronics …