Chapter 2 Principles of Steady State Converter Analysis 2 1 Introduction 2 2 Inductor volt second balance capacitor charge balance and the small ripple approximation 2 3 Boost converter example 2 4 Cuk converter example 2 5 Estimating the ripple in converters containing twopole low pass filters 2 6 Summary of key points Fundamentals of Power Electronics 1 Chapter 2 Principles of steady state converter analysis 2 1 Introduction Buck converter 1 SPDT switch changes dc component Vg 2 vs t R Switch output voltage waveform Duty cycle D 0 D 1 complement D D 1 D Fundamentals of Power Electronics vs t v t Vg D Ts DTs 0 0 Switch position 2 DTs 1 Ts 2 t 1 Chapter 2 Principles of steady state converter analysis Dc component of switch output voltage vs t Vg vs DVg area DTsVg 0 0 DTs Ts t Fourier analysis Dc component average value vs 1 Ts Ts vs t dt 0 vs 1 DTsVg DVg Ts Fundamentals of Power Electronics 3 Chapter 2 Principles of steady state converter analysis Insertion of low pass filter to remove switching harmonics and pass only dc component L 1 Vg 2 C vs t R v t V v vs DVg Vg 0 Fundamentals of Power Electronics 0 4 1 D Chapter 2 Principles of steady state converter analysis Three basic dc dc converters a 1 L 1 Buck 2 Vg C R v M D iL t M D D 0 8 0 6 0 4 0 2 0 0 0 2 0 4 0 6 0 8 1 0 6 0 8 1 0 6 0 8 1 D b 5 2 iL t Vg 1 C R v M D 1 1 D 4 M D Boost L 3 2 1 0 0 0 2 0 4 D D c 0 0 2 0 4 0 Vg 2 iL t L C R v 1 M D 1 Buck boost 2 3 4 M D 1 DD 5 Fundamentals of Power Electronics 5 Chapter 2 Principles of steady state converter analysis Objectives of this chapter Develop techniques for easily determining output voltage of an arbitrary converter circuit Derive the principles of inductor volt second balance and capacitor charge amp second balance Introduce the key small ripple approximation Develop simple methods for selecting filter element values Illustrate via examples Fundamentals of Power Electronics 6 Chapter 2 Principles of steady state converter analysis 2 2 Inductor volt second balance capacitor charge balance and the small ripple approximation Actual output voltage waveform buck converter iL t 1 Buck converter containing practical low pass filter L vL t Vg 2 iC t C R v t Actual output voltage waveform Actual waveform v t V vripple t v t V v t V vripple t dc component V 0 t Fundamentals of Power Electronics 7 Chapter 2 Principles of steady state converter analysis The small ripple approximation Actual waveform v t V vripple t v t v t V vripple t V dc component V 0 t In a well designed converter the output voltage ripple is small Hence the waveforms can be easily determined by ignoring the ripple vripple V v t V Fundamentals of Power Electronics 8 Chapter 2 Principles of steady state converter analysis Buck converter analysis inductor current waveform iL t 1 L vL t original converter 2 Vg iC t C R v t switch in position 1 iL t L L vL t Vg switch in position 2 iC t C R vL t Vg v t iL t C R v t Fundamentals of Power Electronics iC t 9 Chapter 2 Principles of steady state converter analysis Inductor voltage and current Subinterval 1 switch in position 1 iL t Inductor voltage vL t vL Vg v t Small ripple approximation L Vg iC t C R v t vL Vg V Knowing the inductor voltage we can now find the inductor current via vL t L diL t dt Solve for the slope diL t vL t Vg V L L dt Fundamentals of Power Electronics The inductor current changes with an essentially constant slope 10 Chapter 2 Principles of steady state converter analysis Inductor voltage and current Subinterval 2 switch in position 2 L Inductor voltage vL t vL t v t Small ripple approximation Vg iL t iC t C R v t vL t V Knowing the inductor voltage we can again find the inductor current via vL t L diL t dt Solve for the slope diL t V L dt Fundamentals of Power Electronics The inductor current changes with an essentially constant slope 11 Chapter 2 Principles of steady state converter analysis Inductor voltage and current waveforms vL t Vg V DTs D Ts t V Switch position iL t 1 2 1 iL DTs I iL 0 0 Fundamentals of Power Electronics vL t L diL t dt iL Vg V L V L DTs 12 Ts t Chapter 2 Principles of steady state converter analysis Determination of inductor current ripple magnitude iL t iL DTs I iL 0 iL Vg V L 0 V L DTs Ts t change in iL slope length of subinterval Vg V 2 iL DTs L Vg V L DTs 2 iL Vg V iL DTs 2L Fundamentals of Power Electronics 13 Chapter 2 Principles of steady state converter analysis Inductor current waveform during turn on transient iL t Vg v t L v t L iL Ts iL 0 0 0 DTs Ts iL nTs 2Ts nTs iL n 1 Ts n 1 Ts t When the converter operates in equilibrium iL n 1 Ts iL nTs Fundamentals of Power Electronics 14 Chapter 2 Principles of steady state converter analysis The principle of inductor volt second balance Derivation Inductor defining relation di t vL t L L dt Integrate over one complete switching period iL Ts iL 0 1 L Ts vL t dt 0 In periodic steady state the net change in inductor current is zero Ts 0 vL t dt 0 Hence the total area or volt seconds under the inductor voltage waveform is zero whenever the converter operates in steady state An equivalent form 0 1 Ts Ts 0 vL t dt vL The average inductor voltage is zero in steady state Fundamentals of Power Electronics 15 Chapter 2 Principles of steady state converter analysis Inductor volt second balance Buck converter example vL t Vg V Inductor voltage waveform previously derived Total area t DTs V Integral of voltage waveform is area of rectangles Ts vL t dt Vg V DTs V D Ts 0 Average voltage is vL D Vg V D V Ts Equate to zero and solve for V 0 DVg D D V DVg V Fundamentals of Power Electronics 16 V DVg Chapter 2 Principles of steady state converter analysis The principle of capacitor charge balance Derivation Capacitor defining relation dv t iC t C C dt Integrate over one complete switching period vC Ts vC 0 1 C Ts iC t dt 0 In periodic steady state the net change in capacitor voltage is zero 0 1 Ts …