ECE1750, Spring 2018, p g

Week 5 – Buck ConverterWeek 5 – Buck Converter

1

Objective – to efficiently reduce DC voltageObjective to efficiently reduce DC voltageThe DC equivalent of an AC transformer

+ +

IoutIin

DC−DC Buck ConverterVin

−

Vout−

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

inoutII

VV

2

outin IV

Here is an example of an inefficient DC−DC tconverter

R1 The load

+

Vin−

+

Vout−

R2

21

2RR

RVV inout

− −

21 RR

outVV

RRR

21

2inVRR 21

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

3Unacceptable except in very low power applications

Another method – lossless conversion of 39Vd t 13Vd39Vdc to average 13Vdc

Switch openStereo voltage

Switch closed

Rstereo+

39Vdc–

39

0

Switch state, Stereo voltage

Closed 39Vdc

0

DT

TClosed, 39Vdc

Open, 0Vdc

T

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

, p

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

Rstereo+

39Vdc–

Csp es to a uge a ue a d bu s out t eswitch.

Try adding an L to prevent the huge t ik B t if th L h

Lcurrent 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.

Rstereo+

39Vdc–

C

By adding a “free wheeling” diode, the switch can open and the inductor current+

Llossless

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

A DC DC Buck Converter

5

A DC-DC Buck Converter

Capacitors and Inductors

In capacitors:dt

tdvCti )()(

C it t d t k th lt t t ( lt “i ti ”) A id l

The voltage cannot change instantaneously

Capacitors tend to keep the voltage constant (voltage “inertia”). An idealcapacitor with infinite capacitance acts as a constant voltage source.Thus, a capacitor cannot be connected in parallel with a voltage sourceor a switch (otherwise KVL would be violated i e there will be aor a switch (otherwise KVL would be violated, i.e. there will be ashort-circuit)

In inductors: The current cannot change instantaneouslytdiLtv )()( In inductors:

Inductors tend to keep the current constant (current “inertia”). An idealinductor with infinite inductance acts as a constant current source

The current cannot change instantaneouslydt

Ltv )(

inductor with infinite inductance acts as a constant current source.Thus, an inductor cannot be connected in series with a current sourceor a switch (otherwise KCL would be violated)

6

Buck converterA l C th t

iL Ioutiin

+ vL –• Assume large C so that

Vout has very low ripple

Si V h l

Vin

+Vout

–

LC iC

• Since Vout has very low ripple, then assume Iouthas very low ripple

What do we learn from inductor voltage and capacitor

I+ 0 V –

current in the average sense?

I t

Vin

+Vout

LC

Ioutiin Iout

0 A

7

–0 A

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

Ii+ (Vin – Vout) –

i

Vin

+Vout

LC

Ioutiin iL

(i I )Switch closed for

DT d in–

C (iL – Iout)

Reverse biased, thus the di d i

DT seconds

diode is open

diL VVdi iL diLVV,dtdiLv L

L LVV

dtdi outinL

,dtdiLVV L

outin ,outinL VVv

for DT seconds

8Note – if the switch stays closed, then Vout = Vin

Switch open for (1 − D)T seconds

– Vout +

Switch open for (1 D)T seconds

Vi

+Vout

LC

IoutiL

Vin Vout–

C (iL – Iout)

iL continues to flow, thus the diode is closed. This Lis the assumption of “continuous conduction” in the inductor which is the normal operating condition.

,dtdiLv L

L L

Vdtdi outL

,dtdiLV L

out ,outL Vv

for (1−D)T seconds

9

( )

Since the average voltage across L is zeroS ce t e a e age o tage ac oss s e o

01 outoutinLavg VDVVDV g

outoutoutin VDVVDDV outoutoutin

inout DVV The input/output equation becomes inout

From power balance, outoutinin IVIV , so

p p q

DII in

out Note – even though iin is not constant (i.e., iin has harmonics), the input power is still simply Vin • Iin because Vin has no

10

D is still simply Vin Iin because Vin has no harmonics

Examine the inductor current

Switch closed,LVV

dtdiVVv outinL

outinL

,

Switch open,

Ldt

LV

dtdiVv outL

outL

,

From geometry, Iavg = Iout is halfway

between Imax and Iminsec/ AL

VoutiL

sec/ ALVV outin

Imax

Imin

Iavg = Iout

max minL

∆I Periodic – finishes a period where it L

DT (1 − D)T

min pstarted

11T

The state variables (inductor currents and capacitor voltages) transition from a transient state to another transient state but in average they reach steady state

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

iL

∆IRaise Iout

∆I

Lower Iout

∆I

• ∆I is unchanged

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

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toward discontinuous operation (when the current in the inductor is zero for part of the period)

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

iL

Lower f

Raise f

• Slopes of iL are unchanged

• Lowering f increases ∆I and moves the circuit toward discontinuous operation

13

discontinuous operation

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

iLLower L

Raise L

• Lowering L increases ∆I and moves the circuit toward• Lowering L increases ∆I and moves the circuit toward discontinuous operation

14

Inductor current ratingducto cu e t at g

22222121

121 IIIII outppavgLrms

1212 outppavgLrms

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

2IoutIavg = Iout

iL

2222 41

0avg out ∆I

2222342

121

outoutoutLrms IIII

II 2 Use max

15

outLrms II3

Capacitor current and current ratingi IiL

LC

Iout

(i I )C (iL – Iout)

IiC = (iL – Iout) Note – raising f or L, which lowers

∆I d th it tIout

−Iout

0∆I

∆I, reduces the capacitor current

Max rms current occurs at the boundary of continuous/discontinuous conduction, where ∆I =2Iout Use max

222223102

121

outoutavgCrms IIII 3

outCrms

II

16

MOSFET and diode currents and current ratingsiL

L

Ioutiin

C (iL – Iout)

2I t2Iout

0Iout

2Iout

0Iout

2

Use max

0

17

outrms II3

2Take worst case D for each

Worst-case load ripple voltageo st case oad pp e o tageNow, consider that the assumption that the output voltage is perfectly constant is no longer valid so you have some output voltage ripple

Iout

0T/2

C chargingiC = (iL – Iout)

−IoutT/2

During the charging period, the C voltage moves from the min to the max.

IITITQ t

1

The area of the triangle shown above gives the peak-to-peak ripple voltage.

CfI

CIT

C

I

CQV outoutout

4422

18

Raising f or L reduces the load voltage ripple

Buck converter waveformsConsider a Buck converter with f = 20 kHz, Vin = 50 V, D = 0.4, R=2 ohms, L=100 μH, C=20 μF (horizontal axis is time in seconds).

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Buck converter waveformsConsider a Buck converter with f = 20 kHz, Vin = 50 V, D = 0.4, R=2Consider a Buck converter with f 20 kHz, Vin 50 V, D 0.4, R 2 ohms, L=100 μH, C=20 μF (horizontal axis is time in seconds).

20

Buck converter waveformsConsider a Buck converter with f = 20 kHz, Vin = 50 V, D = 0.4, R=2

h L 100 H C 20 F (h i t l i i ti i d )ohms, L=100 μH, C=20 μF (horizontal axis is time in seconds).

T T iLI

,maxLI

outI

,min( )L outI I

,max( )L outI I

TT ,minLI

Assume output current is constantC L outi i I ,max ,min( ) ( )

2L

L out out Li I I I I

C Li i

T

From the circuit:

,max ,min( ) ( )L out L outI I T I I T 2TT T

(1 )D TV (1 )1 1 outL D TVi TT

From the circuit, when the switch is open:

21

(1 ) outL

D TViL

2

(1 )1 1(1 )2 2 2 2 2

8

outLout

D TVi TT D VQ LVC C C LCf

From the blue triangle

i I

Voltage ratings

Vin

+Vout

iL

LC

IoutiinC sees Vout

Switch ClosedVin out

–C iC

Diode sees Vin

iL Iout

MOSFET sees Vin

Vin

+Vout

–

LC iC

Switch Open

–

• Diode and MOSFET use 2Vi

22

• Diode and MOSFET, use 2Vin• Capacitor, use 1.5Vout

There is a another mode of operation: discontinuous

+L

IoutMOSFET

Vin Vout–

C

O f li ht l d l ti f i h

IoutDIODE

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

Th di d t t b k d t fl• 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 oscillationinteract 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

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phenomenon

Onset of the discontinuous statesec/ A

LVout

2IoutI = I

iL

0

Iavg = Iout

(1 − D)T

fLDVTD

LVI

onset

out

onset

outout

112

fIDVL

out

outonset 2

1

VL out guarantees continuous conductionuse max

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

24

fIout2g

use min