EE462L, Spring 2014 DC−DC Buck Converter 1

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

! C’s and L’s operating in periodic steady-state Taken from “Waveforms and Definitions” PPT 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 t o is the same as the voltage one period T later, so v ( to T ) v ( to ), or The conclusion is that t 1 o T v ( to T ) v ( to ) 0 i ( t )dt C to T 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 ( t o ) v ( t )dt L to Since the inductor is in periodic steady state, then the voltage at time t o is the same as the voltage one period T later, so i ( to T ) i ( to ), or The conclusion is that t 1 o T i ( to T ) i ( to ) 0 v ( t )dt L to T 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 V Navg 0 to to KVL applies in the average sense The same reasoning applies to KCL i (t ) Out of node 0, i1 ( t ) i2 ( t ) i3 ( t ) i N ( t ) 0 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 Assume large C so that Iout L Vin C Vout has very low ripple iC Vout – Since V out 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 Vin Iout Iout L C Vout 0A – 10

The input/output equation for DC-DC converters usually comes by examining inductor voltages iin Switch closed for DT seconds (Vin – Vout) – iL Iout L Vin C (iL – Iout) Vout – Reverse biased, thus the diode is open v L 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 L Vin C Iout 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. v L 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, v L Vin Vout , diL Vin Vout dt L diL Vout vL Vout , dt L Switch open, Vout A / sec L iL Imax From geometry, Iavg Iout is halfway between Imax and Imin Iavg Iout Vin Vout A / sec L Imin DT 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 I avg Imin 0 the minimum value Imin I Imin I avg max 2 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 i 2 (t ) 2i (t ) I min I min 2 2 I rms Avg i 2 (t ) 2 I min Avg i (t ) I min 2 I rms I max 3 I min 2 2 I min I max 2 I min 2 I min Define I PP I max I min 2 I PP 2 2 I rms 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 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 PP 3 2 I PP 4 2 I avg 2 I PP 2 2 I rms I avg i (t ) I avg I I I avg max 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 0 2 2 I Lrms I out ΔI 1 2 2 I 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 Note – raising f or L, which lowers ΔI, reduces the capacitor current iC (iL – Iout) 0 (iL – Iout) ΔI Iout Max rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI 2Iout 2 2 I Crms I avg 1 2 2 I out 2 02 1 I out 12 3 Use max 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 Switch Closed Iout C sees Vout L Vin C iC Vout – Diode sees Vin MOSFET sees Vin iL Switch Open Vin Iout L C iC Vout – Diode and MOSFET, use 2Vin Capacitor, use 1.5Vout 27

! There is a 3rd state – discontinuous Iout MOSFET Vin L 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 V R Requiv in D out 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 or 5 2Ω res ist I - amps 4 3 2 1 4Ω or 4 . 6 s is t re 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 Rload 2 Requiv load , D 0.56 2 Requiv 6.44 D 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 Vpanel vL – iL Iout L 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 for DC-DC Converters Our components 9A Converter Type Buck Input Inductor Current (Arms) 2 I out 3 10A 250V Output Capacitor Voltage 5.66A 200V, 250V Output Capacitor Current (Arms) Diode and MOSFET Voltage 2 Vin 1.5 Vout 1 3 I out 40V 10A 40V Likely worst-case buck situation 16A, 20A Diode and MOSFET Current (Arms) 2 I out 3 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