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DC DC Converter

Basics about DC DC Converter

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DC DC Converter

  1. 1. DC-DC Converters
  2. 2. DC-DC Converters DC-DC converters are power electronic circuits that convert a DC voltage to a different DC voltage level, often providing a regulated output. A BASIC SWITCHING CONVERTER An efficient alternative to the linear regulator Uses Power electronics switches like BJT,MOSFET IGBT… Also known as DC Chopper
  3. 3. DC-DC Converters Figure: (a) A basic DC-DC switching converter; (b) Switching equivalent; ( c) Output voltage. οƒΌ Assuming the switch is ideal β€’ The output is the same as the input when the switch ON β€’ And the output is zero when the switch OFF οƒΌ Periodic opening and closing of the switch gives the pulsed output waveform. οƒΌ The average or DC component of the output voltage is π‘‰π‘œ = 1 𝑇 0 𝑇 𝑣 π‘œ 𝑑 𝑑𝑑 = 1 𝑇 0 𝐷𝑇 𝑉𝑠 𝑑𝑑 = 𝑉𝑠 𝐷
  4. 4. DC-DC Converters 𝐷 ≑ 𝑑 π‘œπ‘› 𝑑 π‘œπ‘› + 𝑑 π‘œπ‘“π‘“ The DC component of the output voltage will be less than or equal to the input voltage for this circuit. ideal switch Zero loss Zero voltage across when ON Zero current through it when OFF But real switch has some power loss Considerable because it creates heat on the switch.
  5. 5. THE BUCK (STEP-DOWN) CONVERTERApplication Example: Controlling the speed of DC Motor Low-pass filter will be placed at the output β€’ The diode provides a path for the inductor current when the switch is opened and is reverse-biased when the switch is closed.
  6. 6. THE BUCK (STEP-DOWN) CONVERTERVOLTAGE AND CURRENT RELATIONSHIPS: οƒ˜π‘£ π‘₯ in Fig. above, is 𝑉𝑠 when the switch is closed and is zero when the switch is open, provided that the inductor current remains positive, keeping the diode on. οƒ˜If the switch is closed periodically at a duty ratio D, the average voltage at the filter input is 𝑉𝑠 𝐷 οƒ˜An inductor current that remains positive throughout the switching period is known as continuous current. οƒ˜Conversely, discontinuous current is characterized by the inductor current’s returning to zero during each period.
  7. 7. THE BUCK (STEP-DOWN) CONVERTERBuck converters and DC-DC converters in general, have the following properties when operating in the steady state 1. The inductor current is periodic. π’Š 𝑳 𝒕 + 𝑻 = π’Š 𝑳 𝒕 2. The average inductor voltage is zero 𝑽 𝑳 = 𝟏 𝑻 𝒕 𝒕+𝑻 𝒗 𝑳 𝝀 𝒅𝝀 = 𝟎 3. The average capacitor current is zero 𝑰 π‘ͺ = 𝟏 𝑻 𝒕 𝒕+𝑻 π’Š π‘ͺ 𝝀 𝒅𝝀 = 𝟎 4. The power supplied by the source is the same as the power delivered to the load. For non-ideal components, the source also supplies the losses. 𝑷 𝒔 = 𝑷 𝒐 π’Šπ’…π’†π’‚π’ 𝑷 𝒔= 𝑷 𝒐 + 𝒍𝒐𝒔𝒔𝒆𝒔 π’π’π’π’Šπ’…π’†π’‚π’
  8. 8. THE BUCK (STEP-DOWN) CONVERTERοƒ˜Assumptions for Analysis of Buck converter: β€’ The circuit is operating in the steady state. β€’ The inductor current is continuous (always positive) β€’ The capacitor is very large, and the output voltage is held constant at voltage Vo. This restriction will be relaxed later to show the effects of finite capacitance. β€’ The switching period is 𝑇; the switch is closed for time 𝐷𝑇 and open for time (1 βˆ’ 𝐷)𝑇. β€’ The components are ideal. The key to the analysis for determining the output π‘‰π‘œ is to examine the inductor current and inductor voltage first for the switch closed and then for the switch open.
  9. 9. THE BUCK (STEP-DOWN) CONVERTERοƒ˜ANALYSIS FOR THE SWITCH CLOSED: β€’ The voltage across the inductor is β€’ 𝑣 𝐿 = 𝑉𝑆 βˆ’ π‘‰π‘œ = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 β€’ Rearranging, β€’ 𝑑𝑖 𝐿 𝑑𝑑 = 𝑉 π‘†βˆ’π‘‰π‘œ 𝐿 π‘ π‘€π‘–π‘‘π‘β„Ž π‘π‘™π‘œπ‘ π‘’π‘‘ β€’ 𝑑𝑖 𝐿 𝑑𝑑 = βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 𝐷𝑇 = 𝑉 π‘†βˆ’π‘‰π‘œ 𝐿 β€’ βˆ†π‘– 𝐿 π‘π‘™π‘œπ‘ π‘’π‘‘ = 𝑉 π‘†βˆ’π‘‰π‘œ 𝐿 𝐷𝑇
  10. 10. THE BUCK (STEP-DOWN) CONVERTERοƒ˜ANALYSIS FOR THE SWITCH OPEN: β€’ When the switch is open, the diode becomes forward-biased to carry the inductor current β€’ The voltage across the inductor when the switch is open is 𝑣 𝐿 = βˆ’π‘‰π‘œ = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 β€’ Rearranging, 𝑑𝑖 𝐿 𝑑𝑑 = βˆ’π‘‰π‘œ 𝐿 π‘ π‘€π‘–π‘‘π‘β„Ž π‘œπ‘π‘’π‘› β€’ The derivative of current in the inductor is a negative constant, and the current decreases linearly β€’ The change in inductor current when the switch is open is βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 1βˆ’π· 𝑇 = βˆ’π‘‰π‘œ 𝐿 βˆ†π‘– 𝐿 π‘œπ‘π‘’π‘› = βˆ’ π‘‰π‘œ 𝐿 (1 βˆ’ 𝐷)𝑇
  11. 11. THE BUCK (STEP-DOWN) CONVERTERοƒ˜At steady state βˆ†π‘– 𝐿 π‘π‘™π‘œπ‘ π‘’π‘‘ + βˆ†π‘– 𝐿 π‘œπ‘π‘’π‘› = 0 𝑉 π‘†βˆ’π‘‰π‘œ 𝐿 𝐷𝑇 βˆ’ π‘‰π‘œ 𝐿 1 βˆ’ 𝐷 = 0 β€’ Solving for π‘‰π‘œ, π‘‰π‘œ = 𝑉𝑠 𝐷 β€’ Therefore, the buck converter produces an output voltage that is less than or equal to the input. β€’ An alternative derivation of the output voltage is based on the inductor voltage. β€’ Since the average inductor voltage is zero for periodic operation 𝑉𝐿 = 𝑉𝑆 βˆ’ 𝑉𝑂 𝐷𝑇 + βˆ’π‘‰π‘‚ 1 βˆ’ 𝐷 𝑇 = 0 β€’ Solving for 𝑉𝑂 𝑉𝑂= 𝑉𝑆 𝐷.
  12. 12. THE BUCK (STEP-DOWN) CONVERTERοƒ˜The average inductor current must be the same as the average current in the load resistor, since the average capacitor current must be zero for steady-state operation. 𝐼𝐿 = 𝐼 𝑅 = 𝑉𝑂 𝑅 οƒ˜The maximum and minimum values of the inductor current are computed as βˆ†π‘– 𝐿 π‘π‘™π‘œπ‘ π‘’π‘‘ = 𝑉𝑆 βˆ’ π‘‰π‘œ 𝐿 𝐷𝑇 βˆ†π‘– 𝐿 π‘œπ‘π‘’π‘› = βˆ’ π‘‰π‘œ 𝐿 (1 βˆ’ 𝐷)𝑇  𝐼 π‘šπ‘Žπ‘₯= 𝐼𝐿 + βˆ†π‘– 𝐿 2 = 𝑉𝑂 + 1 𝑉𝑂 1 βˆ’ 𝐷 𝑇 = 𝑉𝑂 1 + 1 βˆ’ 𝐷
  13. 13. THE BUCK (STEP-DOWN) CONVERTER 𝐼 π‘šπ‘–π‘› = 𝐼𝐿 βˆ’ βˆ†π‘– 𝐿 2 = 𝑉𝑂 𝑅 βˆ’ 1 2 𝑉𝑂 𝐿 1 βˆ’ 𝐷 𝑇 = 𝑉𝑂 1 𝑅 βˆ’ 1 βˆ’ 𝐷 2𝐿𝑓 where 𝑓 = 1/𝑇 is the switching frequency. β€’ The above Eq. can be used to determine the combination of L and f that will result in continuous current. Since 𝐼 π‘šπ‘–π‘› = 0 is the boundary between continuous and discontinuous current, 𝐼 π‘šπ‘–π‘› = 0 = 𝑉𝑂 1 𝑅 βˆ’ 1 βˆ’ 𝐷 2𝐿𝑓 (𝐿𝑓) π‘šπ‘–π‘›= (1 βˆ’ 𝐷)𝑅 2
  14. 14. THE BUCK (STEP-DOWN) CONVERTERβ€’ If the desired switching frequency is established, 𝐿 π‘šπ‘–π‘› = (1 βˆ’ 𝐷)𝑅 2𝑓 π‘“π‘œπ‘Ÿ π‘π‘œπ‘›π‘‘π‘–π‘›π‘’π‘œπ‘’π‘  π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘ where 𝐿 π‘šπ‘–π‘› is the minimum inductance required for continuous current. β€’ In practice inductance is chosen greater than 𝐿 π‘šπ‘–π‘› β€’ The peak-to-peak variation in the inductor current is often used as a design criterion in buck converter. β€’ to determine the value of inductance for a specified peak-to-peak inductor current for continuous-current operation: βˆ†π‘– 𝐿 = 𝑉𝑠 βˆ’ π‘‰π‘œ 𝐿 𝐷𝑇 = 𝑉𝑠 βˆ’ π‘‰π‘œ 𝐿𝑓 𝐷 = 𝑉𝑂 1 βˆ’ 𝐷 𝐿𝑓 π‘œπ‘Ÿ 𝐿 = 𝑉𝑠 βˆ’ π‘‰π‘œ βˆ†π‘– 𝐿 𝑓 𝐷 = 𝑉𝑂 1 βˆ’ 𝐷 βˆ†π‘– 𝐿 𝑓
  15. 15. THE BUCK (STEP-DOWN) CONVERTERβ€’ Since the converter components are assumed to be ideal, the power supplied by the source must be the same as the power absorbed by the load resistor. 𝑃𝑆 = 𝑃𝑂 𝑉𝑠 𝐼𝑠 = π‘‰π‘œ 𝐼 π‘œ π‘œπ‘Ÿ π‘‰π‘œ 𝑉𝑠 = 𝐼𝑠 𝐼 π‘œ β€’ Basically from this expression we can say that DC-DC converter is DC transformer.
  16. 16. THE BUCK (STEP-DOWN) CONVERTERβ€’ OUTPUT VOLTAGE RIPPLE β€’ In the preceding analysis, the capacitor was assumed to be very large to keep the output voltage constant. β€’ The variation in output voltage is computed from the voltage- current relationship of the capacitor. β€’ The current in the capacitor is 𝑖 𝐢 = 𝑖 𝐿 βˆ’ 𝑖 𝑅 β€’ While the capacitor current is positive, the capacitor is charging. From the definition of capacitance, 𝑄 = πΆπ‘‰π‘œ βˆ†π‘„ = πΆβˆ†π‘‰π‘œ βˆ†π‘‰π‘œ = βˆ†π‘„ 𝐢
  17. 17. THE BUCK (STEP-DOWN) CONVERTERβ€’ The change in charge βˆ†π‘„ is the area of the triangle above the time axis βˆ†π‘„ = 1 2 𝑇 2 βˆ†π‘– 𝐿 2 = π‘‡βˆ†π‘– 𝐿 8 β€’ resulting in βˆ†π‘‰π‘œ = π‘‡βˆ†π‘– 𝐿 8𝐢
  18. 18. THE BUCK (STEP-DOWN) CONVERTER β€’ Using βˆ†π‘– 𝐿 π‘œπ‘π‘’π‘› = βˆ’ π‘‰π‘œ 𝐿 (1 βˆ’ 𝐷)𝑇 = βˆ†π‘–πΏ βˆ†π‘‰π‘œ = π‘‡π‘‰π‘œ 8𝐢𝐿 1 βˆ’ 𝐷 𝑇 = π‘‰π‘œ(1 βˆ’ 𝐷) 8𝐿𝐢𝑓2 β€’ It is also useful to express the ripple as a fraction of the output voltage βˆ†π‘‰π‘œ π‘‰π‘œ = 1 βˆ’ 𝐷 8𝐿𝐢𝑓2 β€’ 𝐢 = 1βˆ’π· 8𝐿(βˆ†π‘‰π‘œ/π‘‰π‘œ)𝑓2
  19. 19. THE BUCK (STEP-DOWN) CONVERTER β€’ Using βˆ†π‘– 𝐿 π‘œπ‘π‘’π‘› = βˆ’ π‘‰π‘œ 𝐿 (1 βˆ’ 𝐷)𝑇 for βˆ†π‘–πΏ βˆ†π‘‰π‘œ = π‘‡π‘‰π‘œ 8𝐢𝐿 1 βˆ’ 𝐷 𝑇 = π‘‰π‘œ(1 βˆ’ 𝐷) 8𝐿𝐢𝑓2 β€’ It is also useful to express the ripple as a fraction of the output voltage βˆ†π‘‰π‘œ π‘‰π‘œ = 1 βˆ’ 𝐷 8𝐿𝐢𝑓2 β€’ 𝐢 = 1βˆ’π· 8𝐿(βˆ†π‘‰π‘œ/π‘‰π‘œ)𝑓2
  20. 20. CAPACITOR RESISTANCEβ€”THE EFFECT ON RIPPLE VOLTAGE The ESR may have a significant effect on the output voltage rippleA real capacitor can be modelled as a capacitance with an equivalent series resistance (ESR) and an equivalent series inductance (ESL). ESR, often producing a ripple voltage greater than that of the ideal capacitance. The inductance in the capacitor is usually not a significant factor at typical switching frequencies. The ripple due to ESR can be approximated by first determining the ripple current assuming the capacitor ideal βˆ†π‘‰π‘œ,𝐸𝑆𝑅 = βˆ†π‘– 𝐢 π‘ŸπΆ = βˆ†π‘– 𝐿 π‘ŸπΆ βˆ†π‘‰π‘œ < βˆ†π‘‰π‘œ,𝐢 + βˆ†π‘‰π‘œ,𝐸𝑆𝑅
  21. 21. CAPACITOR RESISTANCEβ€”THE EFFECT ON RIPPLE VOLTAGE β€’ The ripple voltage due to the ESR can be much larger than the ripple due to the pure capacitance. β€’ In that case, the output capacitor is chosen on the basis of the equivalent series resistance rather than capacitance only. βˆ†π‘‰π‘œ β‰ˆ βˆ†π‘‰π‘œ,𝐸𝑆𝑅 = βˆ†π‘– 𝐢 π‘ŸπΆ
  22. 22. SYNCHRONOUS RECTIFICATION FOR THE BUCK CONVERTER Figure: A synchronous buck converter. The MOSFET S2 carries the inductor current when S1 is off to provide a lower voltage drop than a diode.
  23. 23. THE BOOST CONVERTER Assumptions 1. Steady-state conditions exist. 2. The switching period is 𝑇, and the switch is closed for time 𝐷𝑇 and open for (1 βˆ’ 𝐷)𝑇. 3. The inductor current is continuous (always positive). 4. The capacitor is very large, and the output voltage is held constant at voltage π‘‰π‘œ. 5. The components are ideal.
  24. 24. THE BOOST CONVERTER β€’ ANALYSIS FOR THE SWITCH CLOSED: 𝑣 𝐿 = 𝑉𝑆 = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 π‘œπ‘Ÿ 𝑑𝑖 𝐿 𝑑𝑑 = 𝑉𝑆 𝐿 βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 𝐷𝑇 = 𝑉𝑆 𝐿 β€’ Solving for βˆ†π‘– 𝐿 for the switch closed, βˆ†π‘– 𝐿 π‘π‘™π‘œπ‘ π‘’π‘‘ = 𝑉𝑆 𝐷𝑇 𝐿
  25. 25. THE BOOST CONVERTER β€’ ANALYSIS FOR THE SWITCH OPEN: 𝑣 𝐿 = 𝑉𝑠 βˆ’ π‘‰π‘œ = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 𝑑𝑖 𝐿 𝑑𝑑 = 𝑉𝑠 βˆ’ π‘‰π‘œ 𝐿 βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 1 βˆ’ 𝐷 𝑇 = 𝑉𝑠 βˆ’ π‘‰π‘œ 𝐿 β€’ Solving for βˆ†π‘– 𝐿 (βˆ†π‘– 𝐿) π‘œπ‘π‘’π‘›= (𝑉𝑠 βˆ’ π‘‰π‘œ) 1 βˆ’ 𝐷 𝑇 𝐿
  26. 26. THE BOOST CONVERTER For steady-state operation, the net change in inductor current must be zero. (βˆ†π‘– 𝐿) π‘π‘™π‘œπ‘ π‘’π‘‘ + (βˆ†π‘– 𝐿) π‘œπ‘π‘’π‘› = 0 𝑉𝑠 𝐷𝑇 𝐿 + (π‘‰π‘ βˆ’π‘‰π‘œ)(1 βˆ’ 𝐷)𝑇 𝐿 = 0 Solving for π‘‰π‘œ, 𝑉𝑠 𝐷 + 1 βˆ’ 𝐷 βˆ’ π‘‰π‘œ 1 βˆ’ 𝐷 = 0 π‘‰π‘œ= 𝑉𝑠 1 βˆ’ 𝐷 Expressing the average inductor voltage over one switching period, 𝑉𝐿 = 𝑉𝑠 𝐷 + 𝑉𝑠 βˆ’ π‘‰π‘œ 1 βˆ’ 𝐷 = 0
  27. 27. THE BOOST CONVERTER Average inductor current can be obtained by assuming 𝑃𝑠 = π‘ƒπ‘œ Output power is π‘ƒπ‘œ = π‘‰π‘œ 2 𝑅 = π‘‰π‘œ 𝐼 π‘œ 𝑉𝑆 𝐼𝑆= 𝑉𝑠 𝐼𝐿 Equating input and output powers 𝑉𝑠 𝐼𝐿 = π‘‰π‘œ 2 𝑅 = [𝑉𝑠/(1 βˆ’ 𝐷)]2 𝑅 = 𝑉𝑠 2 (1 βˆ’ 𝐷)2 𝑅 𝐼𝐿 = 𝑉𝑠 (1βˆ’π·)2 𝑅 = π‘‰π‘œ 2 𝑉𝑠 𝑅 = π‘‰π‘œ 𝐼 π‘œ 𝑉𝑠 β€’ 𝐼 π‘šπ‘Žπ‘₯ = 𝐼𝐿 + βˆ†π‘– 𝐿 2 = 𝑉𝑠 (1βˆ’π·)2 𝑅 + 𝑉𝑠 𝐷𝑇 2𝐿 β€’ 𝐼 π‘šπ‘–π‘› = 𝐼𝐿 βˆ’ βˆ†π‘– 𝐿 2 = 𝑉𝑠 1βˆ’π· 2 𝑅 βˆ’ 𝑉𝑠 𝐷𝑇 2𝐿
  28. 28. THE BOOST CONVERTER A condition necessary for continuous inductor current is for 𝐼 π‘šπ‘–π‘› to be positive. Therefore, the boundary between continuous and discontinuous inductor current is determined from 𝐼 π‘šπ‘–π‘› = 0 = 𝑉𝑠 1 βˆ’ 𝐷 2 𝑅 βˆ’ 𝑉𝑠 𝐷𝑇 2𝐿 π‘œπ‘Ÿ 𝑉𝑠 1 βˆ’ 𝐷 2 𝑅 = 𝑉𝑠 𝐷𝑇 2𝐿 = 𝑉𝑠 𝐷 2𝐿𝑓  𝐿𝑓 π‘šπ‘–π‘› = 𝐷 1βˆ’π· 2 𝑅 2 π‘œπ‘Ÿ 𝐿 π‘šπ‘–π‘› = 𝐷 1βˆ’π· 2 𝑅 2𝑓 From a design perspective, it is useful to express 𝐿 in terms of a desired βˆ†π‘– 𝐿, 𝐿 = 𝑉𝑠 𝐷𝑇 βˆ†π‘– 𝐿 = 𝑉𝑠 𝐷 βˆ†π‘– 𝐿 𝑓
  29. 29. THE BOOST CONVERTER The peak-to-peak output voltage ripple can be calculated from the capacitor current waveform. The change in capacitor charge can be calculated from βˆ†π‘„ = π‘‰π‘œ 𝑅 𝐷𝑇 = πΆβˆ†π‘‰π‘œ An expression for ripple voltage is then βˆ†π‘‰π‘œ= π‘‰π‘œ 𝐷𝑇 𝑅𝐢 = π‘‰π‘œ 𝐷 𝑅𝐢𝑓 π‘œπ‘Ÿ βˆ†π‘‰π‘œ π‘‰π‘œ = 𝐷 𝑅𝐢𝑓 where 𝑓 is the switching frequency. 𝐢 = 𝐷 𝑅(βˆ†π‘‰π‘œ/π‘‰π‘œ)𝑓
  30. 30. THE BOOST CONVERTER As Buck Converter the voltage ripple due to the ESR is βˆ†π‘‰π‘œ,𝐸𝑆𝑅= βˆ†π‘– 𝐢 π‘ŸπΆ = 𝐼𝐿,π‘šπ‘Žπ‘₯ π‘ŸπΆ Effects of Inductor Resistance β€’ Inductors should be designed to have small resistance to minimize power loss and maximize efficiency. β€’ Inductor resistance affects performance of the boost converter, especially at high duty ratios. β€’ For the boost converter, recall that the output voltage for the ideal case is π‘‰π‘œ = 𝑉𝑠 1 βˆ’ 𝐷 β€’ The power supplied by the source must be the same as the power absorbed by the load and the inductor resistance, neglecting other losses. 𝑃𝑠= π‘ƒπ‘œ + π‘ƒπ‘ŸπΏ 𝑉𝑠 𝐼𝐿 = π‘‰π‘œ 𝐼 𝐷 + 𝐼𝐿 2 π‘ŸπΏ where π‘ŸπΏ is the series resistance of the inductor.
  31. 31. THE BOOST CONVERTER β€’ The average diode current is 𝐼 𝐷 = 𝐼𝐿 1 βˆ’ 𝐷 β€’ Then 𝑉𝑠 𝐼𝐿 = π‘‰π‘œ 𝐼𝐿 1 βˆ’ 𝐷 + 𝐼𝐿 2 π‘ŸπΏ β€’ which becomes 𝑉𝑠 = π‘‰π‘œ 1 βˆ’ 𝐷 + 𝐼𝐿 π‘ŸπΏ 𝐼𝐿 = 𝐼 𝐷 1 βˆ’ 𝐷 = π‘‰π‘œ/𝑅 1 βˆ’ 𝐷 β€’ Substituting for 𝐼𝐿 𝑉𝑠 = π‘‰π‘œ π‘ŸπΏ 1 βˆ’ 𝐷 𝑅 + π‘‰π‘œ 1 βˆ’ 𝐷 β€’ Solving for π‘‰π‘œ, π‘‰π‘œ = 𝑉𝑠 1 βˆ’ 𝐷 1 1 + π‘ŸπΏ 𝑅 1 βˆ’ 𝐷 2
  32. 32. THE BOOST CONVERTER Figure: Boost converter for a nonideal inductor. (a) Output voltage; (b) Boost converter efficiency. πœ‚ = π‘ƒπ‘œ π‘ƒπ‘œ + π‘ƒπ‘™π‘œπ‘ π‘  = π‘‰π‘œ 2 /𝑅 π‘‰π‘œ 2 /𝑅 + 𝐼𝐿 2 π‘ŸπΏ = π‘‰π‘œ 2 /𝑅 π‘‰π‘œ 2 /𝑅 + [(π‘‰π‘œ 2 /𝑅)2/ 1 βˆ’ 𝐷 2] π‘ŸπΏ = 1 1 + π‘ŸπΏ[𝑅 1 βˆ’ 𝐷 2]
  33. 33. BUCK-BOOST CONVERTER β€’ The output voltage of the buck-boost converter can be either higher or lower than the input voltage. β€’ Assumptions : 1. The circuit is operating in the steady state. 2. The inductor current is continuous. 3. The capacitor is large enough to assume a constant output voltage. 4. The switch is closed for time 𝐷𝑇 and open for (1 βˆ’ 𝐷)𝑇. 5. The components are ideal.
  34. 34. BUCK-BOOST CONVERTER Figure: Buck-boost converter. (a) Circuit; (b) Equivalent circuit for the switch closed; (c) Equivalent circuit for the switch open.
  35. 35. BUCK-BOOST CONVERTER
  36. 36. BUCK-BOOST CONVERTER β€’ ANALYSIS FOR THE SWITCH CLOSED: 𝑣 𝐿 = 𝑉𝑠 = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 𝑑𝑖 𝐿 𝑑𝑑 = 𝑉𝑠 𝐿 βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 𝐷𝑇 = 𝑉𝑠 𝐿 (βˆ†π‘– 𝐿) π‘π‘™π‘œπ‘ π‘’π‘‘ = 𝑉𝑠 𝐷𝑇 𝐿 ANALYSIS FOR THE SWITCH OPEN: 𝑣 𝐿 = π‘‰π‘œ = 𝐿 𝑑𝑖 𝐿 𝑑𝑑 𝑑𝑖 𝐿 𝑑𝑑 = π‘‰π‘œ 𝐿 βˆ†π‘– 𝐿 βˆ†π‘‘ = βˆ†π‘– 𝐿 1 βˆ’ 𝐷 𝑇 = π‘‰π‘œ 𝐿 (βˆ†π‘– 𝐿) π‘œπ‘π‘’π‘›= π‘‰π‘œ 1βˆ’π· 𝑇 𝐿
  37. 37. BUCK-BOOST CONVERTER β€’ For steady-state operation, the net change in inductor current must be zero over one period. (βˆ†π‘– 𝐿) π‘π‘™π‘œπ‘ π‘’π‘‘ + (βˆ†π‘– 𝐿) π‘œπ‘π‘’π‘›= 0 𝑉𝑠 𝐷𝑇 𝐿 + π‘‰π‘œ 1 βˆ’ 𝐷 𝑇 𝐿 = 0 β€’ Solving for π‘‰π‘œ, π‘‰π‘œ = βˆ’π‘‰π‘  𝐷 1 βˆ’ 𝐷 β€’ The required duty ratio for specified input and output voltages can be expressed as 𝐷 = π‘‰π‘œ 𝑉𝑠 + π‘‰π‘œ
  38. 38. BUCK-BOOST CONVERTER β€’ The average inductor voltage is zero for periodic operation, resulting in 𝑉𝐿 = 𝑉𝑠 𝐷 + π‘‰π‘œ 1 βˆ’ 𝐷 = 0 β€’ Solving for π‘‰π‘œ yields π‘‰π‘œ = βˆ’π‘‰π‘  𝐷 1 βˆ’ 𝐷 β€’ The output voltage has opposite polarity from the source voltage. β€’ If 𝐷 > 0.5, the output voltage is larger than the input; and if 𝐷 < 0.5, the output is smaller than the input.
  39. 39. BUCK-BOOST CONVERTER β€’ Power absorbed by the load must be the same as that supplied by the source, where 𝑃0 = π‘‰π‘œ 2 𝑅 𝑃𝑠 = 𝑉𝑠 𝐼𝑠 π‘‰π‘œ 2 𝑅 = 𝑉𝑠 𝐼𝑠 β€’ Average source current is related to average inductor current by 𝐼𝑠 = 𝐼𝐿 𝐷 β€’ resulting in π‘‰π‘œ 2 𝑅 = 𝑉𝑠 𝐼𝐿 𝐷 β€’ Substituting for π‘‰π‘œ and solving for 𝐼𝐿, we find 𝐼𝐿 = π‘‰π‘œ 2 𝑉𝑠 𝑅𝐷 = 𝑃0 𝑉𝑠 𝐷 = 𝑉𝑠 𝐷 𝑅(1 βˆ’ 𝐷)2
  40. 40. BUCK-BOOST CONVERTER 𝐼 π‘šπ‘Žπ‘₯ = 𝐼𝐿 + βˆ†π‘– 𝐿 2 = 𝑉𝑠 𝐷 𝑅(1 βˆ’ 𝐷)2 + 𝑉𝑠 𝐷𝑇 2𝐿 𝐼 π‘šπ‘–π‘› = 𝐼𝐿 βˆ’ βˆ†π‘– 𝐿 2 = 𝑉𝑠 𝐷 𝑅 1 βˆ’ 𝐷 2 βˆ’ 𝑉𝑠 𝐷𝑇 2𝐿 (𝐿𝑓) π‘šπ‘–π‘›= (1 βˆ’ 𝐷)2 𝑅 2 𝐿 π‘šπ‘–π‘› = (1 βˆ’ 𝐷)2 𝑅 2𝑓 where 𝑓 is the switching frequency.
  41. 41. BUCK-BOOST CONVERTER β€’ OUTPUT VOLTAGE RIPPLE β€’ The output voltage ripple for the buck-boost converter is computed from the capacitor current . βˆ†π‘„ = π‘‰π‘œ 𝑅 𝐷𝑇 = πΆβˆ†π‘‰π‘œ β€’ Solving for βˆ†π‘‰π‘œ, βˆ†π‘‰π‘œ = π‘‰π‘œ 𝐷𝑇 𝑅𝐢 = π‘‰π‘œ 𝐷 𝑅𝐢𝑓 π‘œπ‘Ÿ βˆ†π‘‰π‘œ π‘‰π‘œ = 𝐷 𝑅𝐢𝑓
  42. 42. BUCK-BOOST CONVERTER β€’ As is the case with other converters, the equivalent series resistance of the capacitor can contribute significantly to the output ripple voltage. The peak-to-peak variation in capacitor current is the same as the maximum inductor current. βˆ†π‘‰π‘œ,𝐸𝑆𝑅= βˆ†π‘– 𝐢 π‘ŸπΆ = 𝐼𝐿,π‘šπ‘Žπ‘₯ π‘ŸπΆ
  43. 43. BUCK-BOOST CONVERTER β€’As is the case with other converters, the equivalent series resistance of the capacitor can contribute significantly to the output ripple voltage. The peak-to-peak variation in capacitor current is the same as the maximum inductor current. βˆ†π‘‰π‘œ,𝐸𝑆𝑅= βˆ†π‘– 𝐢 π‘ŸπΆ = 𝐼𝐿,π‘šπ‘Žπ‘₯ π‘ŸπΆ
  44. 44. Other ConverterTopologies CÚk Converter Sepic Converter
  45. 45. Other ConverterTopologies Full-bridge DC-DC Converter
  46. 46. Other ConverterTopologies IsolatedFull-bridge DC-DC Converter
  47. 47. Other ConverterTopologies IsolatedHalf-bridge DC-DC Converter

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