ee462l, fall 2011 dc − dc buck converter

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EE462L, Fall 2011DC−DC Buck Converter

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Objective – to efficiently reduce DC voltage

DC−DC Buck

Converter

+

Vin

−

+

Vout

−

IoutIin

Lossless objective: Pin = Pout, which means that VinIin = VoutIout and

The DC equivalent of an AC transformer

out

in

in

outII

VV

3

Here is an example of an inefficient DC−DC converter

21

2RR

RVV inout

+

Vin

−

+

Vout

−

R1

R2

in

outVV

RRR

21

2

If Vin = 39V, and Vout = 13V, efficiency η is only 0.33

The load

Unacceptable except in very low power applications

4

Another method – lossless conversion of 39Vdc to average 13Vdc

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?

Rstereo+

39Vdc–

Switch state, Stereo voltage

Closed, 39Vdc

Open, 0Vdc

Switch openStereo voltage

39

0

Switch closed

DT

T

5

Convert 39Vdc to 13Vdc, cont.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+

39Vdc–

C

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.

By adding a “free wheeling” diode, the switch can open and the inductor current can continue to flow. With high-frequency switching, the load voltage ripple can be reduced to a small value.

Rstereo+

39Vdc–

C

L

Rstereo+

39Vdc–

C

L

A DC-DC Buck Converter

lossless

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C’s and L’s operating in periodic steady-stateExamine the current passing through a capacitor that is operating in periodic steady state. The governing equation is

dttdvCti )()( which leads to

tot

oto dtti

Ctvtv )(1)()(

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

),()( oo tvTtv

The conclusion is that

Tot

otoo dtti

CtvTtv )(10)()(or

0)( Tot

otdtti

the average current through a capacitor operating in periodic steady state is zero

which means that

Taken from “Waveforms and Definitions” PPT

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Now, an inductorExamine the voltage across an inductor that is operating in periodic steady state. The governing equation is

dttdiLtv )()( which leads to

tot

oto dttv

Ltiti )(1)()(

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

),()( oo tiTti

The conclusion is that

Tot

otoo dttv

LtiTti )(10)()(or

0)( Tot

otdttv

the average voltage across an inductor operating in periodic steady state is zero

which means that


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KVL and KCL in periodic steady-state

,0)(

loopAroundtv

,0)(

nodeofOutti

0)()()()( 321 tvtvtvtv N

Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL,

0)()()()( 321 titititi N

0)0(1)(1)(1)(1)(1321

dt

Tdttv

Tdttv

Tdttv

Tdttv

T

Tot

ot

Tot

otN

Tot

ot

Tot

ot

Tot

ot

0321 Navgavgavgavg VVVV

The same reasoning applies to KCL

0321 Navgavgavgavg IIII

KVL applies in the average sense

KCL applies in the average sense


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Vin

+ Vout

–

iL

LC iC

Iout iin

Buck converter

+ vL –

Vin

+ Vout

–

LC

Iout iin

+ 0 V –

What do we learn from inductor voltage and capacitor current in the average sense?

Iout

0 A

• Assume large C so that

Vout has very low ripple

• Since Vout has very low

ripple, then assume Iout has very low ripple

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The input/output equation for DC-DC converters usually comes by examining inductor voltages

Vin

+ Vout

–

LC

Iout iin + (Vin – Vout) –

iL

(iL – Iout)

Reverse biased, thus the diode is open

,dtdiLv L

L LVV

dtdi outinL

,dtdiLVV L

outin ,outinL VVv

for DT seconds

Note – if the switch stays closed, then Vout = Vin

Switch closed for DT seconds

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Vin

+ Vout

–

LC

Iout – Vout +

iL

(iL – Iout)

Switch open for (1 − D)T seconds

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.

,dtdiLv L

L LV

dtdi outL

,dtdiLV L

out ,outL Vv

for (1−D)T seconds

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Since the average voltage across L is zero

01 outoutinLavg VDVVDV

outoutoutin VDVVDDV

inout DVV

From power balance, outoutinin IVIV

DII in

out

, so

The input/output equation becomes

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

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Examine the inductor current

Switch closed,

Switch open,

LVV

dtdiVVv outinL

outinL

,

LV

dtdiVv outL

outL

,

sec/ ALVV outin

DT (1 − D)T

T

Imax

Imin

Iavg = Iout

From geometry, Iavg = Iout is halfway

between Imax and Imin

sec/ ALVout

ΔI

iL

Periodic – finishes a period where it started

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Effect of raising and lowering Iout while holding Vin, Vout, f, and L constant

iL

ΔI

ΔIRaise Iout

ΔI

Lower Iout

• ΔI is unchanged

• Lowering Iout (and, therefore, Pout ) moves the circuit toward discontinuous operation

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Effect of raising and lowering f while holding Vin, Vout, Iout, and L constant

iL

Raise f

Lower f

• Slopes of iL are unchanged

• Lowering f increases ΔI and moves the circuit toward discontinuous operation

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iL

Effect of raising and lowering L while holding Vin, Vout, Iout and f constant

Raise L

Lower L

• Lowering L increases ΔI and moves the circuit toward discontinuous operation

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RMS of common periodic waveforms, cont.

TTT

rms tTVdtt

TVdtt

TV

TV

03

3

2

0

23

2

0

22

31

T

V

0

3VVrms

Sawtooth


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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

0

V

0

V

0

0

-V

V

0

3VVrms

V

0

V

0


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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.

minmax II

0

)(tithe ripple

+

0

minI

the minimum value

)(timaxI

minI=

2

minmax IIIavg

avgI


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RMS of common periodic waveforms, cont. 2

min2 )( ItiAvgIrms

2minmin

22 )(2)( IItitiAvgIrms

2minmin

22 )( 2)( ItiAvgItiAvgIrms

2min

minmaxmin

2minmax2

22

3I

III

IIIrms

2minmin

22

3IIIII PP

PPrms

minmax IIIPP Define


21


2minPP

avgIII

222

223

PP

avgPPPP

avgPP

rmsIIIIIII

423

22

222 PP

PPavgavgPP

PPavgPP

rmsIIIIIIIII

222

243 avgPPPP

rms IIII

Recognize that

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222 PPavgrms

III

avgI

)(ti

minmax IIIPP

2

minmax IIIavg


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Inductor current rating

22222121

121 IIIII outppavgLrms

2222342

121

outoutoutLrms IIII

Max impact of ΔI on the rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout

outLrms II3

2

2Iout

0Iavg = Iout ΔI

iL

Use max

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Capacitor current and current rating

222223102

121

outoutavgCrms IIII

iL

LC

Iout

(iL – Iout)

Iout

−Iout

0ΔI

Max rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout

3out

CrmsII

Use max

iC = (iL – Iout) Note – raising f or L, which lowers ΔI, reduces the capacitor current

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MOSFET and diode currents and current ratings

iL

LC

Iout

(iL – Iout)

outrms II3

2

Use max

2Iout

0Iout

iin

2Iout

0Iout

Take worst case D for each

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Worst-case load ripple voltage

CfI

CIT

C

IT

CQV outoutout

4422

1

Iout

−Iout

0T/2

C chargingiC = (iL – Iout)

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.

Raising f or L reduces the load voltage ripple

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Vin

+ Vout

–

iL

LC iC

Iout

Vin

+ Vout

–

iL

LC iC

Iout iin

Voltage ratings

Diode sees Vin

MOSFET sees Vin

C sees Vout

• Diode and MOSFET, use 2Vin

• Capacitor, use 1.5Vout

Switch Closed

Switch Open

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There is a 3rd state – discontinuous

Vin

+ Vout

–

LC

Iout

• Occurs for light loads, or low operating frequencies, where the inductor current eventually hits zero during the switch-open 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

Iout

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Inductor voltage showing oscillation during discontinuous current operation

650kHz. With L = 100µH, this corresponds to net parasitic C = 0.6nF

vL = (Vin – Vout)

vL = –Vout

Switch open

Switch closed

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Onset of the discontinuous statesec/ A

LVout

fLDVTD

LVI

onset

out

onset

outout

112

2Iout

0

Iavg = Iout

iL

(1 − D)T

fIVLout

out2

guarantees continuous conductionuse max

use min

fIDVL

out

outonset 2

1

Then, considering the worst case (i.e., D → 0),

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Impedance matching

outout

load IVR

equivR

DC−DC Buck

Converter

+

Vin

−

+

Vout = DVin

−

Iout = Iin / DIin

+

Vin

−

Iin

22 D

R

DI

VDI

DV

IVR load

out

outout

out

inin

equiv

Equivalent from source perspective

Source

So, the buck converter makes the load resistance look larger to the source

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Example of drawing maximum power from solar panel

PV Station 13, Bright Sun, Dec. 6, 2002

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40 45

V(panel) - volts

I - a

mps

Isc

Voc

Pmax is approx. 130W (occurs at 29V, 4.5A)

44.65.4

29AVRload

For max power from panels at this solar intensity level, attach

I-V characteristic of 6.44Ω resistor

But as the sun conditions change, the “max power resistance” must also change

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Connect a 2Ω resistor directly, extract only 55W

PV Station 13, Bright Sun, Dec. 6, 2002

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40 45

V(panel) - volts

I - a

mps

130W

6.44Ω

resistor

2Ω re

sist

or

55W

56.044.62 ,2

equivloadload

equiv RRD

D

RR

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

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Vpanel

+ Vout

–

iL

LC iC

Iout ipanel

Buck converter for solar applications

+ vL –

Put a capacitor here to provide the ripple current required by the opening and closing of the MOSFET

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.

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

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Worst-Case Component Ratings Comparisons for DC-DC Converters

Converter Type

Input Inductor

Current (Arms)

Output Capacitor Voltage

Output Capacitor Current (Arms)

Diode and MOSFET Voltage

Diode and MOSFET Current (Arms)

Buck outI

32 1.5 outV

outI3

1 2 inV outI

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10A 10A10A 40V 40V

Likely worst-case buck situation

5.66A 200V, 250V 16A, 20AOur components

9A 250V

Our M (MOSFET). 250V, 20A

Our L. 100µH, 9AOur C. 1500µF, 250V, 5.66A p-p

Our D (Diode). 200V, 16A

BUCK DESIGN

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Comparisons of Output Capacitor Ripple Voltage

Converter Type Volts (peak-to-peak) Buck

CfIout4

10A

1500µF 50kHz

0.033V

BUCK DESIGN




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Minimum Inductance Values Needed to Guarantee Continuous Current

Converter Type For Continuous

Current in the Input Inductor

For Continuous Current in L2

Buck fI

VL

out

out2

–

40V

2A 50kHz

200µH

BUCK DESIGN




ee462l, fall 2011 dc − dc buck converter

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