EE462L, Spring 2014 DC−DC Buck Converter 1 ! Objective – to efficiently reduce DC voltage The DC equivalent of an AC transformer Iin + Vin Iout DC−DC Buck Converter − + Vout − Lossless objective: Pin = Pout, which means that VinIin = VoutIout and Vout I in Vin I out 2 Here is an example of an inefficient DC−DC converter The load R1 + Vin + R2 − Vout − R2 Vout Vin R1 R2 Vout R2 R1 R2 Vin If Vin = 39V, and Vout = 13V, efficiency η is only 0.33 Unacceptable except in very low power applications 3 Taken from “Course Overview” PPT Another method – lossless conversion of 39Vdc to average 13Vdc Stereo voltage ! Switch closed Switch open 39 + 39Vdc – Rstereo 0 Switch state, Stereo voltage Closed, 39Vdc DT T Open, 0Vdc If the duty cycle D of the switch is 0.33, then the average voltage to the expensive car stereo is 39 ● 0.33 = 13Vdc. This is lossless conversion, but is it acceptable? 4 Taken from “Course Overview” PPT Convert 39Vdc to 13Vdc, cont. + 39Vdc – Try adding a large C in parallel with the load to control ripple. But if the C has 13Vdc, then when the switch closes, the source current spikes to a huge value and burns out the switch. Rstereo C L + 39Vdc – C Rstereo Try adding an L to prevent the huge current spike. But now, if the L has current when the switch attempts to open, the inductor’s current momentum and resulting Ldi/dt burns out the switch. lossless L + 39Vdc – C Rstereo By adding a “free wheeling” diode, the switch can open and the inductor current can continue to flow. With highfrequency switching, the load voltage ripple can be reduced to a small value. A DC-DC Buck Converter 5 Taken from “Waveforms and Definitions” PPT ! C’s and L’s operating in periodic steady-state Examine the current passing through a capacitor that is operating in periodic steady state. The governing equation is i(t ) C dv ( t ) dt which leads to t 1 o t v ( t ) v ( to ) i ( t )dt C to Since the capacitor is in periodic steady state, then the voltage at time to is the same as the voltage one period T later, so v ( to T ) v ( to ), or t 1 o T v ( to T ) v ( to ) 0 i ( t )dt C t o T The conclusion is that i ( t )dt 0 to which means that to the average current through a capacitor operating in periodic steady state is zero 6 Taken from “Waveforms and Definitions” PPT Now, an inductor ! Examine the voltage across an inductor that is operating in periodic steady state. The governing equation is v(t ) L di ( t ) dt which leads to t 1 o t i ( t ) i ( to ) v ( t )dt L to Since the inductor is in periodic steady state, then the voltage at time to is the same as the voltage one period T later, so i ( to T ) i ( to ), or t 1 o T i ( to T ) i ( to ) 0 v ( t )dt L t o T The conclusion is that v ( t )dt 0 to which means that to the average voltage across an inductor operating in periodic steady state is zero 7 Taken from “Waveforms and Definitions” PPT KVL and KCL in periodic steady-state ! Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL, v(t ) 0, v1 ( t ) v2 ( t ) v3 ( t ) v N ( t ) 0 Around loop t t t t t 1 o T 1 o T 1 o T 1 o T 1 o T v1 ( t )dt v2 ( t )dt v3 ( t )dt v N ( t )dt (0)dt 0 T T T T T to to to V1avg V2avg V3avg VNavg 0 to to KVL applies in the average sense The same reasoning applies to KCL i(t ) 0, i1 ( t ) i2 ( t ) i3 ( t ) i N ( t ) 0 Out of node I1avg I 2avg I 3avg I Navg 0 KCL applies in the average sense 8 Capacitors and Inductors In capacitors: i(t ) C dv ( t ) dt ! The voltage cannot change instantaneously Capacitors tend to keep the voltage constant (voltage “inertia”). An ideal capacitor with infinite capacitance acts as a constant voltage source. Thus, a capacitor cannot be connected in parallel with a voltage source or a switch (otherwise KVL would be violated, i.e. there will be a short-circuit) In inductors: v( t ) L di ( t ) dt The current cannot change instantaneously Inductors tend to keep the current constant (current “inertia”). An ideal inductor with infinite inductance acts as a constant current source. Thus, an inductor cannot be connected in series with a current source or a switch (otherwise KCL would be violated) 9 ! Buck converter + vL – iL iin Iout L Vin C • Assume large C so that Vout has very low ripple iC + Vout – • Since Vout has very low ripple, then assume Iout has very low ripple What do we learn from inductor voltage and capacitor current in the average sense? +0V– iin Iout Iout L Vin C + Vout 0A – 10 The input/output equation for DC-DC converters usually comes by examining inductor voltages + (Vin – Vout) – iin Switch closed for DT seconds iL Iout + V (iL – Iout) out – L Vin C Reverse biased, thus the diode is open vL L diL , dt vL Vin Vout , Vin Vout L diL , dt diL Vin Vout dt L for DT seconds Note – if the switch stays closed, then Vout = Vin 11 Switch open for (1 − D)T seconds – Vout + iL Iout L Vin C + Vout (iL – Iout) – iL continues to flow, thus the diode is closed. This is the assumption of “continuous conduction” in the inductor which is the normal operating condition. vL L diL , dt vL Vout , Vout L diL , dt diL Vout dt L for (1−D)T seconds 12 ! Since the average voltage across L is zero VLavg D Vin Vout 1 D Vout 0 DVin D Vout Vout D Vout The input/output equation becomes Vout DVin From power balance, Vin I in Vout I out , so I in I out D Note – even though iin is not constant (i.e., iin has harmonics), the input power is still simply Vin • Iin because Vin has no harmonics 13 Examine the inductor current Switch closed, vL Vin Vout , diL Vout vL Vout , dt L Switch open, Vout A / sec L iL Imax Iavg = Iout Vin Vout A / sec L Imin DT diL Vin Vout dt L From geometry, Iavg = Iout is halfway between Imax and Imin ΔI Periodic – finishes a period where it started (1 − D)T T 14 Effect of raising and lowering Iout while holding Vin, Vout, f, and L constant iL ΔI Raise Iout ΔI Lower Iout ΔI • ΔI is unchanged • Lowering Iout (and, therefore, Pout ) moves the circuit toward discontinuous operation 15 Effect of raising and lowering f while holding Vin, Vout, Iout, and L constant iL Lower f Raise f • Slopes of iL are unchanged • Lowering f increases ΔI and moves the circuit toward discontinuous operation 16 Effect of raising and lowering L while holding Vin, Vout, Iout and f constant iL Lower L Raise L • Lowering L increases ΔI and moves the circuit toward discontinuous operation 17 Taken from “Waveforms and Definitions” PPT RMS of common periodic waveforms, cont. ! Sawtooth V 0 T T 2 2T 2 1 V V V 2 2 3T Vrms t dt t dt t 3 3 0 T T T 3 T 0 0 V Vrms 3 18 Taken from “Waveforms and Definitions” PPT RMS of common periodic waveforms, cont. ! Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example V V 0 0 0 -V V V V 0 0 0 V 0 V Vrms 3 19 Taken from “Waveforms and Definitions” PPT RMS of common periodic waveforms, cont. ! Now, consider a useful example, based upon a waveform that is often seen in DC-DC converter currents. Decompose the waveform into its ripple, plus its minimum value. i (t ) Imax Imin the ripple i (t ) Imax 0 I avg = Imin + the minimum value I avg Imax Imin 2 Imin 0 20 Taken from “Waveforms and Definitions” PPT RMS of common periodic waveforms, cont. 2 I rms Avg i (t ) I min 2 2 2 I rms Avg i2 (t ) 2i (t ) I min I min 2 2 I rms Avg i2 (t ) 2 I min Avg i (t ) I min 2 I rms I max I min 2 2I 3 I max I min I 2 min min 2 Define I PP I max I min 2 I rms 2 I PP 2 I min I PP I min 3 21 Taken from “Waveforms and Definitions” PPT RMS of common periodic waveforms, cont. I Recognize that I min I avg PP 2 2 2 I PP I PP I PP 2 I rms I avg I PP I avg 3 2 I rms 2 2 2 2 2 I PP I PP I PP 2 I avg I PP I avg I avg I PP 3 2 4 2 I rms 2 I PP 3 2 I PP 2 2 I rms I avg 4 2 I avg 2 I PP i (t ) I avg I avg I max I min 2 I PP I max I min 12 22 Inductor current rating 2 2 I Lrms I avg 1 2 1 2 I pp I out I 2 12 12 Max impact of ΔI on the rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout 2Iout iL Iavg = Iout ΔI 0 2 2 I Lrms I out 1 2 2I out 2 4 I out 12 3 2 I Lrms I out 3 Use max 23 Capacitor current and current rating iL Iout L C Iout iC = (iL – Iout) 0 −Iout (iL – Iout) Note – raising f or L, which lowers ΔI, reduces the capacitor current ΔI Max rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout Use max 2 2 I Crms I avg 1 2 2 I out 2 02 1 I out 12 3 I I Crms out 3 24 MOSFET and diode currents and current ratings iL iin Iout L C (iL – Iout) 2Iout Iout 0 2Iout Iout 0 Use max Take worst case D for each I rms 2 I out 3 25 ! Worst-case load ripple voltage Iout 0 −Iout iC = (iL – Iout) C charging T/2 During the charging period, the C voltage moves from the min to the max. The area of the triangle shown above gives the peak-to-peak ripple voltage. 1 T Q 2 2 I out T I out I out V C C 4C 4Cf Raising f or L reduces the load voltage ripple 26 Voltage ratings iL iin Iout C sees Vout Switch Closed L Vin C iC + Vout – Diode sees Vin MOSFET sees Vin iL Switch Open Iout L Vin C iC + Vout – • Diode and MOSFET, use 2Vin • Capacitor, use 1.5Vout 27 ! There is a 3rd state – discontinuous Iout MOSFET L Vin DIODE C Iout + Vout – • Occurs for light loads, or low operating frequencies, where the inductor current eventually hits zero during the switchopen state • The diode opens to prevent backward current flow • The small capacitances of the MOSFET and diode, acting in parallel with each other as a net parasitic capacitance, interact with L to produce an oscillation • The output C is in series with the net parasitic capacitance, but C is so large that it can be ignored in the oscillation phenomenon 28 Inductor voltage showing oscillation during discontinuous current operation vL = (Vin – Vout) Switch closed vL = –Vout Switch open 650kHz. With L = 100µH, this corresponds to net parasitic C = 0.6nF 29 Onset of the discontinuous state 2Iout Iavg = Iout iL ! Vout A / sec L 0 (1 − D)T Vout Vout 1 D 2 I out 1 D T Lonset Lonset f Vout 1 D Lonset 2 I out f Then, considering the worst case (i.e., D → 0), Vout L 2 I out f use max guarantees continuous conduction use min 30 ! Impedance matching Iout = Iin / D Iin + + Source DC−DC Buck Converter Vin Vout = DVin − − V Rload out I out Iin + Vin Equivalent from source perspective Requiv − Vout V Vout R D Requiv in load I in I out D I out D 2 D2 So, the buck converter makes the load resistance look larger to the source 31 Example of drawing maximum power from solar panel PV Station 13, Bright Sun, Dec. 6, 2002 6 Isc Pmax is approx. 130W (occurs at 29V, 4.5A) 5 I - amps 4 For max power from panels at this solar intensity level, attach 3 2 Rload 1 29V 6.44 4.5 A 0 0 5 10 15 20 25 V(panel) - volts 30 35 40 Voc I-V characteristic of 6.44Ω resistor 45 But as the sun conditions change, the “max power resistance” must also change 32 Connect a 2Ω resistor directly, extract only 55W ! PV Station 13, Bright Sun, Dec. 6, 2002 55W 6 130W 5 I - amps 4 3 2 1 0 0 5 10 15 20 25 30 35 40 45 V(panel) - volts To draw maximum power (130W), connect a buck converter between the panel and the load resistor, and use D to modify the equivalent load resistance seen by the source so that maximum power is transferred R Requiv load , D D2 Rload 2 0.56 Requiv 6.44 33 Buck converter for solar applications The panel needs a ripple-free current to stay on the max power point. Wiring inductance reacts to the current switching with large voltage spikes. ipanel + vL – iL Iout L Vpanel C iC + Vout – Put a capacitor here to provide the ripple current required by the opening and closing of the MOSFET In that way, the panel current can be ripple free and the voltage spikes can be controlled We use a 10µF, 50V, 10A high-frequency bipolar (unpolarized) capacitor 34 BUCK DESIGN Worst-Case Component Ratings Comparisons Our components for DC-DC Converters 9A Converter Type Buck Input Inductor Current (Arms) 2 I out 3 10A 250V Output Capacitor Voltage 5.66A Output Capacitor Current (Arms) 1.5 Vout 1 3 I out 200V, 250V 16A, 20A Diode and MOSFET Voltage 2 Vin Diode and MOSFET Current (Arms) 2 I out 3 40V 10A 40V Likely worst-case buck situation 10A Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 35 BUCK DESIGN Comparisons of Output Capacitor Ripple Voltage Converter Type Buck Volts (peak-to-peak) I out 10A 4Cf 0.033V 1500µF 50kHz Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 36 BUCK DESIGN Minimum Inductance Values Needed to Guarantee Continuous Current Converter Type Buck For Continuous For Continuous Current in the Input Current in L2 Inductor V 40V L out – 200µH 2 I out f 2A 50kHz Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 37