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SEA_uniovi_CC1_00

Universidad de Oviedo. Lección 4. Teoría básica de los convertidores CC / CC (I) (convertidores con un único transistor). Diseño de Sistemas Electrónicos de Potencia 4º Curso. Grado en Ingeniería en Tecnologías y Servicios de Telecomunicación . SEA_uniovi_CC1_00. Outline (I).

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SEA_uniovi_CC1_00

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  1. Universidad de Oviedo Lección 4 Teoría básica de los convertidores CC/CC (I) (convertidores con un único transistor) Diseño de Sistemas Electrónicos de Potencia 4º Curso. Grado en Ingeniería en Tecnologías y Servicios de Telecomunicación SEA_uniovi_CC1_00

  2. Outline (I) • Introducing switching regulators • Basis of their analysis in steady state • Detailed study of the basic DC/DC converters in continuous conduction mode • Buck, Boost and Buck-Boost converters • Common and different properties • Introduction to the synchronous rectification • Four-order converters SEA_uniovi_CC1_01

  3. Study of the basic DC/DC converters in discontinuous conduction mode DC/DC converters with galvanic isolation How and where to place a transformer in a DC/DC converter The Forward and Flyback converters Outline (II) SEA_uniovi_CC1_02

  4. Q iO iO ig ig RV RL vO RL vO vg vg vE vE - - Av Av Actual implementation Feedback loop Feedback loop Vref Vref Linear DC/DC conversion (analog circuitry) = (vOiO)/(vgig) iO ig   vO/vg First idea • Only a few components • Robust • No EMI generation • Only lower output voltage • Efficiency depends on input/output voltages • Low efficiency • Bulky SEA_uniovi_CC1_03

  5. S Q iO iO ig ig RL vO vg vE - Av RL vO PWM vg Vref Feedback loop vE - Av Feedback loop Vref vO vO_avg vg t Linear versus switching DC/DC conversion Linear Switching (provisional) Features: • 100% efficiency • Undesirable output voltage waveform SEA_uniovi_CC1_04

  6. S iO ig vO vO_avg vg RL vO vg t vE - Av PWM Vref Feedback loop S iO ig S iO ig C filter RL vO vg Filter RL vO vg C filter VO - Vg Av PWM vE t Vref Feedback loop Introducing the switching DC/DC conversion (I) The AC component must be removed!! Basic switching DC/DC converter (provisional) It doesn’t work!!! SEA_uniovi_CC1_05

  7. iL S iO ig L RL LC filter vO vg C LC filter Including a diode S iO ig iL S vD Filter iO VO ig RL vO vg Vg L + + vD vO RL iD vg t - C - - D Av PWM vE LC filter Vref Feedback loop Introducing the switching DC/DC conversion (II) Infinite voltage across L when S1 is opened It doesn’t work either!!! Basic switching DC/DC converter SEA_uniovi_CC1_06

  8. iL iS iO ig L C S + + vD vO RL vg - - iL iD D t iL S iO ig L + + iL vD vO RL iD vg DCM t CCM C - - D LC filter Introducing the switching DC/DC conversion (III) Buck converter Starting the analysis of the Buck converter in steady state: • L & C designed for negligible output voltage ripple (we are designing a DC/DC converter) • iL never reaches zero (Continuous Conduction Mode, CCM) • The study of the Discontinuous Conduction Mode (DCM) will done later SEA_uniovi_CC1_07

  9. vD vD_avg vg t iL iL iS iO iO ig + L L C S + C + + vD vO vO vD RL RL vg - - iD - D LC filter vD vg vO d: “duty cycle” t dT T First analysis of the Buck converter in CCM (In steady-state) Analysis based on the specific topology of the Buck converter = vO The AC component is removed by the filter • This procedure is only valid for converter with explicit LC filter vO = vD_avg = d·vg SEA_uniovi_CC1_08

  10. L1 ig C1 iD D - + Could we use the aforementioned analysis in the case of this converter (SEPIC)? iL2 iS R + VO Vg C2 L2 - S Introducing another analysis method (I) • Obviously, there is not an explicit LC filter • Therefore, we must use another method SEA_uniovi_CC1_09

  11. Introducing another analysis method (II) Powerful tools to analyze DC/DC converters in steady-state Step 1- To obtain the main waveforms (with no quantity values) using Faraday’s law and Kirchhoff’s current and voltage laws Step 2- To take into account the average value of the voltage across inductors and of the current through capacitors in steady-state Step 2 (bis)- To use the volt·second balance Step 3- To apply Kirchhoff’s current and voltage laws in average values Step 4- Input-output power balance SEA_uniovi_CC1_10

  12. + L vL_avg = 0 Circuit in steady-state - Vg iC_avg = 0 C Introducing another analysis method (III) Any electrical circuit that operates in steady-state satisfies: • The average voltage across an inductor is zero. Else, the net current through the inductor always increases and, therefore, steady-state is not achieved • The average current through a capacitor is zero. Else, the net voltage acrossthe capacitoralways increasesand, therefore, steady-state is not achieved SEA_uniovi_CC1_11

  13. Same areas vL v1 + + L t vL Circuit in steady-state - - -v2 dT Vg iC T C vL_avg = 0 iC_avg = 0 iC + t - Same areas Introducing another analysis method (IV) Particular case of many DC/DC converters in steady-state: • Voltage across the inductors are rectangular waveforms • Current through the capacitors are triangular waveforms Volt·second balance: V1dT – V2(1-d)T = 0 SEA_uniovi_CC1_12

  14. iL1 iC1 L1 Node1 C1 Example - + - + vL1 vC1 iS + vL2 Vg L2 S - Loop1 Introducing another analysis method (V) Any electrical circuit of small dimensions (compared with the wavelength associated to the frequency variations) satisfies: • Kirchhoff’s current law (KCL) is not only satisfied for instantaneous current values, but also for average currentvalues • Kirchhoff’s voltage law (KVL) is not only satisfied for instantaneous voltage values, but also for average voltagevalues • KVL applied to Loop1 yields: • vg - vL1 - vC1 - vL2 = 0 • vg - vL1_avg - vC1_avg - vL2_avg = 0 • Therefore: vC1_avg = vg • KCL applied to Node1 yields: • iL1 - iC1 - iS = 0 • iL1_avg - iC1_avg - iS_avg = 0 • Therefore: iS_avg =iL1_avg SEA_uniovi_CC1_13

  15. ig iO Switching-mode DC/DC converter + vO RL vg - ig_avg iO + vO RL vg - 1:N DC Transformer Introducing another analysis method (VI) A switching converter is (ideally) a lossless system • Input power: • Pg = vgig_avg • Output power: • PO = vOiO = vO2/RL • Power balance: • Pg =PO Therefore: vgig_avg = vO2/RL • A switching-mode DC/DC converter as an ideal DC transformer being N = vO/vg ig_avg = iOvO/vg = N·iO Important concept!! SEA_uniovi_CC1_14

  16. ig iS iO iL vS - + Driving signal L iD C + + S vO vD RL t vg - - D iL iO iL S on, D off S off, D on t L C + iS vO RL vg - t During dT iO iD iL t L C + vO dT RL - T During (1-d)T Steady-state analysis of the Buck converter in CCM (I) Step 1: Main waveforms. Remember that the output voltage remains constant during a switching cycle if the converter has been properly designed SEA_uniovi_CC1_15

  17. Driving signal t DiL iL iL_avg iO iL t S on, D off, dT vL L C + vg-vO vO RL vg t - iO iL - vO dT L C + S off, D on, (1-d)T vO T RL - vL vL + + - - Steady-state analysis of the Buck converter in CCM (II) Step 1: Main waveforms (cont’) vL + - iO iL vS - + L C + + S vO vD RL vg - - D • From Faraday’s law: • DiL = vO(1-d)T/L SEA_uniovi_CC1_16

  18. ig vL iS iL + - iO L iC iD + Node1 vO S RL vg - D C Driving signal t iL iL_avg t vL vg-vO t + - - vO dT T Steady-state analysis of the Buck converter in CCM (III) Step 2 and 2 (bis): Average inductor voltage and capacitor current • Average value of iC: • iC_avg = 0 • Volt·secondbalance over L: • (vg- vO)dT - vO(1-d)T = 0 • Therefore: vO =d·vg (always vO < vg) Step 3: Average KCL and KVL: • KCL applied to Node1 yields: • iL - iC - iO = 0 • iL_avg - iC_avg - iO = 0 • Therefore: iL_avg = iO = vO/RL Step 4: Power balance: ig_avg = iS_avg = iOvO/vg = d·iO SEA_uniovi_CC1_17

  19. ig iS iO iL vS - Driving signal + L iD C t + + S vO vD RL vg vD - - D vg t iL iO t iS DiL t iD t dT T Steady-state analysis of the Buck converter in CCM (IV) Summary vO =d·vg (always vO < vg) vSmax =vDmax = vg iL_avg = iO = vo/RL ig_avg = iS_avg = d·iO iD_avg = iL_avg - iS_avg = (1-d)·iO DiL = vO(1-d)T/L iL_peak = iL_avg + DiL/2 = iO + vO(1-d)T/(2L) iS_peak = iD_peak = iL_peak SEA_uniovi_CC1_18

  20. Step 1: Main waveforms ig vL iD + - iO iL Driving signal L DiL D iS + + t vO RL vg C - S - iL t iL iS S on, D off, during dT L t vg iO iL iD L C + vO S off, D on, during (1-d)T t RL - dT vL vL + + - - T Steady-state analysis of the Boost converter in CCM (I) Can we obtain vO > vg? ÞBoost converter • From Faraday’s law: • DiL = vgdT/L SEA_uniovi_CC1_19

  21. ig vL iD + - iO iL Node1 iC L D iS + + vO RL vg C - S - Driving signal t iD DiL iD_avg t vL vg t dT -(vO-vg) T Steady-state analysis of the Boost converter in CCM (II) Step 2 and 2 (bis): Average values • Average value of iC: • iC_avg = 0 • Volt·secondbalance over L: • vgdT - (vO- vg)(1-d)T = 0 • Therefore: vO =vg/(1-d) (always vO > vg) Step 3: Average KCL and KVL: • KCL applied to Node1 yields: • iD - iC - iO = 0 • iD_avg - iC_avg - iO = 0 • Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL Step 4: Power balance: ig_avg = iL_avg = iOvO/vg = iO/(1-d) SEA_uniovi_CC1_20

  22. ig vL vD + - iO iL - + Driving signal L iC iD D iS + t vO RL vg C S - vD vO + t vS - iL t iS DiL t iD iO t dT T Steady-state analysis of the Boost converter in CCM (III) Summary vO =vg/(1-d) (always vO > vg) vSmax =vDmax = vO iL_avg = ig_avg = iO/(1-d) = vo/[RL(1-d)] iS_avg = d·iL_avg = d·vo/[RL(1-d)] DiL = vgdT/L iD_avg = iO iL_peak = iL_avg + DiL/2 = iL_avg + vgdT/(2L) iS_peak = iD_peak = iL_peak SEA_uniovi_CC1_21

  23. ig iD iS iO iL D - - + Driving signal S vO vL RL DiL + C + vg L - t iL ig t + iL iS vL S on, D off, during dT vg L - t Charging stage iO iD + S off, D on, during (1-d)T t iL - C - vO vL RL dT + + L - T Discharging stage Steady-state analysis of the Buck-Boost converter in CCM (I) Can we obtain either vO < vg or vO > vg? ÞBuck-Boost converter • From Faraday’s law: • DiL = vgdT/L SEA_uniovi_CC1_22

  24. Node1 ig iS iD iO iL D iC - - + vO S vL RL + + C vg L - Driving signal t iD DiL iD_avg t vL vg t -vO dT T Steady-state analysis of the Buck-Boost converter in CCM (II) Step 2 and 2 (bis): Average values • Average value of iC: • iC_avg = 0 • Volt·secondbalance over L: • vgdT - vO(1-d)T = 0 • Therefore: vO =vgd/(1-d) Step 3: Average KCL and KVL: • KCL applied to Node1 yields: • iD - iC - iO = 0 • iD_avg - iC_avg - iO = 0 • Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL Step 4: Power balance: ig_avg = iS_avg = iOvO/vg = iOd/(1-d) SEA_uniovi_CC1_23

  25. vS vD + + - - ig iD iS iO Driving signal iL t D - + - S vO vL vD RL + vg L vO + vg + C - t iL t iS DiL t iD iO t dT T Steady-state analysis of the Buck-Boost converter in CCM (III) Summary vO =vgd/(1-d) (both vO < vg and vO > vg) vSmax =vDmax = vO + vg iD_avg = iO DiL = vgdT/L iL_avg = iD_avg/(1-d) = iO/(1-d) = vo/[RL(1-d)] iS_avg = ig_avg = d·iL_avg = d·vo/[RL(1-d)] iL_peak = iL_avg + DiL/2 = iL_avg + vgdT/(2L) iS_peak = iD_peak = iL_peak SEA_uniovi_CC1_24

  26. L + + Complementary switches + inductor S vO RL vg C - - D + vO Buck - L D + + + + vO vO RL RL D - vg C S vg - - C S - d 1-d Boost L D Voltage source - S RL + C vg L Buck-Boost Common issues in basic DC/DC converters (I) The inductor is a energy buffer to connect two voltage sources SEA_uniovi_CC1_25

  27. vg L + + vO RL S vg C - - D Buck + vO - L D + C + vO RL vg - vO S - Boost D C - vO + vg S RL Buck-Boost + vg L Common issues in basic DC/DC converters (II) Diode turn-off • The diode turns off when the transistor turns on • The diode reverse recovery time is of primary concern evaluating switching losses • Schottky diodes are desired from this point of view • In the range of line voltages, SiC diodes are very appreciated SEA_uniovi_CC1_26

  28. ig iO L + + + S vO vO RL C vg - - - D Buck ig iO L D + + vO RL vg C - S - Boost ig_avg iO ig iO + D vO RL - vg - S RL 1:N + C vg L DC Transformer Buck-Boost Comparing basic DC/DC converters (I) Generalized study as DC transformer (I) • Buck:N=d (only vO < vg) • Boost: N=1/(1-d) (only vO > vg) • Buck-Boost: N= -d/(1-d) (both vO < vg and vO > vg) SEA_uniovi_CC1_27

  29. ig_avg = iON = iOd/(1-d) ig_avg iO + vO RL vg - 1:N DC Transformer Comparing basic DC/DC converters (II) Generalized study as DC transformer (II) • Buck: ig_avg = iON = iOd • Boost: ig_avg = iON = iO/(1-d) • Buck-Boost: ig_avg = iON = - iOd/(1-d) SEA_uniovi_CC1_28

  30. iS ig iO vS - + iD + + S vO vD RL vg - - D DC/DC converter Comparing basic DC/DC converters (III) Electrical stress on components (I) • Buck: vSmax =vDmax = vg iS_avg = ig_avg iL_avg = iO iD_avg = iL_avg - iS_avg • Buck-Boost: vSmax =vDmax = vO + vg iS_avg = ig_avg iD_avg = iO iL_avg = iS_avg + iD_avg • Boost: vSmax =vDmax = vO iL_avg = ig_avg iD_avg = iO iS_avg = iL_avg - iD_avg SEA_uniovi_CC1_29

  31. 1 A (avg) 2 A L + + S 50 V RL C - - 100 V D 100 W Buck, 100% efficiency 2 A 1 A (avg) D - - S RL + + C L 50 V 100 V 100 W Buck-Boost, 100% efficiency Comparing basic DC/DC converters (IV) Example of electrical stress on components (I) vS_max = vD_max = 100 V iS_avg = iD_avg = 1 A iL_avg = 2 A FOMVA_S = FOMVA_D = 100 VA vS_max = vD_max = 150 V iS_avg = 1 A iD_avg = 2 A iL_avg = 3 A FOMVA_S = 150 VA FOMVA_D = 300 VA • Higher electrical stress in the case of Buck-Boost converter • Therefore, lower actual efficiency SEA_uniovi_CC1_30

  32. 4 A (avg) 2 A 50 V 25 V L D 100 W Boost, 100% efficiency + + RL C - S - 2 A 4 A (avg) D - - S RL + + C L 50 V 25 V 100 W Buck-Boost, 100% efficiency Comparing basic DC/DC converters (V) Example of electrical stress on components (II) vS_max = vD_max = 50 V iS_avg = iD_avg = 2 A iL_avg = 4 A FOMVA_S = FOMVA_D = 100 VA vS_max = vD_max = 75 V iS_avg = 4 A iD_avg = 2 A iL_avg = 6 A FOMVA_S = 300 VA FOMVA_D = 150 VA • Higher electrical stress in the case of Buck-Boost converter • Therefore, lower actual efficiency SEA_uniovi_CC1_31

  33. Comparing basic DC/DC converters (VI) • Price to pay for simultaneous step-down and step-up capability: Higher electrical stress on components and, therefore, lower actual efficiency • Converters with limited either step-down or step-up capability: Lower electrical stress on components and, therefore, higher actual efficiency SEA_uniovi_CC1_32

  34. 6.12 A (avg) 5 A L D 1.12 A (avg) + + 60 V RL C - 50 V S - 300 W Boost, 98% efficiency Comparing basic DC/DC converters (VII) Example of power conversion between similar voltage levels based on a Boost converter vS_max = vD_max = 60 V iS_avg = 1.12 A iD_avg = 5 A iL_avg = 6.12 A FOMVA_S = 67.2 VA FOMVA_D = 300 VA Very high efficiency can be achieved!!! SEA_uniovi_CC1_33

  35. 20 - 2 A (avg) 5 A 60 V 20 - 200 V D - - S 300 W Buck-Boost, 75% efficiency RL + + C L Comparing basic DC/DC converters (VIII) The opposite case: Example of power conversion between very different and variable voltage levels based on a Buck-Boost converter vS_max = vD_max = 260 V iS_avg_max = 20 A iD_avg_max = 5 A iL_avg = 25 A FOMVA_S_max = 5200 VA FOMVA_D = 1300 VA Remember previous example: FOMVA_S = 67.2 VA FOMVA_D = 300 VA High efficiency cannot be achieved!!! SEA_uniovi_CC1_34

  36. L D + + vg vO RL S C - - Boost Comparing basic DC/DC converters (IX) One disadvantage exhibited by the Boost converter: The input current has a “direct path” from the input voltage source to the load. No switch is placed in this path. As a consequence, two problems arise: • Large peak input current in start-up • No over current or short-circuit protection can be easily implemented (additional switch needed) Buck and Buck-Boost do not exhibit these problems SEA_uniovi_CC1_35

  37. idevice L L S1 S L MOSFET S2 S1 D Diode S2 vdevice Synchronous rectification (I) • To use controlled transistors (MOSFETs) instead of diodes to achieve high efficiency in low output-voltage applications • This is due to the fact that the voltage drop across the device can be lower if a transistor is used instead a diode • The conduction takes place from source terminal to drain terminal • In practice, the diode (Schottky) is not removed SEA_uniovi_CC1_36

  38. L S2 + + vO L S1 vO vg RL C S1 - - D Q’ - S2 Av PWM Synchronous Buck Q Vref Feedback loop Synchronous rectification (II) • In converters without a transformer, the control circuitry must provide proper driving signals • In converters with a transformer, the driving signals can be obtained from the transformer (self-driving synchronous rectification) • Nowadays, very common technique with low output-voltage Buck converters SEA_uniovi_CC1_37

  39. iRC ig iS vS - + + + vO RL iD - vg C + - S vD - D DC/DC converter Desired current Desired current ig iRC t t Input current and current injected into the output RC cell (I) • If a DC/DC converter were an ideal DC transformer, the input and output currents should also be DC currents • As a consequence, no pulsating current is desired in the input and output ports and even in the current injected into the RC output cell SEA_uniovi_CC1_38

  40. iRC ig iRC ig L + + S vO + RL C vg - vO - t D t - Low noise Buck Noisy iRC ig iRC L D ig + + vO RL t C vg - S - t Noisy Boost Low noise ig iRC iRC ig D - t S t RL + C vg L Noisy Noisy Buck-Boost Input current and current injected into the output RC cell (II) SEA_uniovi_CC1_39

  41. iRC ig L LF + + + S vO RL C CF - - vg - D Buck Filter iRC ig L LF D + + + vO RL - CF - - vg C S Boost Filter iRC ig D LF LF + - - - S vO CF CF RL - vg C + + + L Buck-Boost Filter Filter Input current and current injected into the output RC cell (III) Adding EMI filters SEA_uniovi_CC1_40

  42. Same vO/vgas Buck-Boost • Same stress as Buck-Boost • vC1 = vg • Filtered input L1 ig iD C1 D - + vC1 iL2 iS RL + vO vg L2 C2 - S ig L1 L2 SEPIC iL2 C1 • Same vO/vg as Buck-Boost • Same stress as Buck-Boost • vC1 = vg + vO • Filtered input and output - + vC1 iD RL iS - vO vg + C2 D S iL2 iS Cuk C1 - + L2 vC1 S • Same vO/vg as Buck-Boost • Same stress as Buck-Boost • vC1 = vO • Filtered output RL + D vO L1 C2 vg - iL1 iD Zeta Four-order converters (converters with integrated filters) SEA_uniovi_CC1_41

  43. ig iL iL vg L iL_avg iO t S + Driving signal D vO t RL - dT DC/DC converter T DC/DC converters operating in DCM (I) • Only one inductor in basic DC/DC converters • The current passing through the inductor decreases when the load current decreases (load resistance increases) • Buck: iL_avg = iO • Boost: iL_avg = iO/(1-d) • Buck-Boost: iL_avg = iS_avg + iD_avg = diO/(1-d) + iO = iO/(1-d) SEA_uniovi_CC1_42

  44. RL_1 iL iL_avg t RL_2 > RL_1 Decreasing load iL iL_avg t RL_crit > RL_2 Boundary between CCM and DCM iL iL_avg t DC/DC converters operating in DCM (II) • When the load decreases, the converter goes toward Discontinuous Conduction Mode (DCM) Operation in CCM It corresponds to RL = R L_crit SEA_uniovi_CC1_43

  45. RL_crit iL iL_avg t RL_3 > RL_crit iL iL_avg Decreasing load t CCM w. SR RL_3 > RL_crit iL iL_avg t DCM w. diode DC/DC converters operating in DCM (III) What happens when the load decreases below the critical value? • DCM starts if a diode is used as rectifier • If a synchronous rectifier (SR) is used, the operation depends on the driving signal • CCM operation is possible with synchronous rectifier with a proper driving signal (synchronous rectifier with signal almost complementary to the main transistor) SEA_uniovi_CC1_44

  46. RL > RL_crit CCM w. SR iL iL_avg t RL > RL_crit DCM w. diode iL iL_avg t DC/DC converters operating in DCM (IV) Remember: iL_avg = iO (Buck) or iL_avg = iO/(1-d) (Boost and Buck-Boost) • For a given duty cycle, lower average value (due to the negative area) Þ lower output current for a given load Þ lower output voltage • For a given duty cycle, higher average value (no negative area) Þ higher output current for a given load Þ higher output voltage The voltage conversion ratio vO/vg is always higher in DCM than in CCM (for a given load and duty cycle) SEA_uniovi_CC1_45

  47. iL After decreasing the inductor inductance t iL After decreasing the switching frequency t iL After decreasing the load (increasing the load resistance) t DC/DC converters operating in DCM (V) How can we get DCM (of course, with a diode as rectifier) ? SEA_uniovi_CC1_46

  48. ig Driving signal iO + + iL iL - C - t vL vO vL iL vg RL + L + L - - iL_avg t iD iD_avg + iL ig iS iD iO vL t vL vg L - iL D + - - + vO S vL - RL + + t C -vO vg L - d·T d’·T T During (1-d-d’)T During d’·T During d·T DC/DC converters operating in DCM (VI) Three sub-circuits instead of two: • The transistor is on. During d·T • The diode is on. During d’·T • Both the transistor and the diode are off. During (1-d-d’)T Example: Buck-Boost converter SEA_uniovi_CC1_47

  49. Driving signal ig + iL t iL vL vg iL_avg L - iL_max During d·T t iD iD_avg iL_max iO + iL - C - t vL vO vL RL vg + + L - + During d’·T - t -vO d·T d’·T T DC/DC converters operating in DCM (VII) Voltage conversion ratio vO/vg for the Buck-Boost converter in DCM From Faraday’s law: vg = LiL_max/(dT) And also: vO = LiL_max/(d’T) Also: iD_avg = iL_maxd’/2, iD_avg = vO/R And finally calling M = vO/vg we obtain: M =d/(k)1/2 where k =2L/(RT) SEA_uniovi_CC1_48

  50. RL = RL_crit iL iL_avg t DC/DC converters operating in DCM (VIII) The Buck-Boost converter just on the boundary between DCM and CCM • Due to being in DCM: M = vO/vg = d/(k)1/2, where: k = 2L/(RT) • Due to being in CCM: N = vO/vg = d/(1-d) • Just on the boundary: M = N, R = Rcrit, k = kcrit • Therefore: kcrit = (1-d)2 • The converter operates in CCM if: k > kcrit • The converter operates in DCM if: k < kcrit SEA_uniovi_CC1_49

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