DC−DC Buck Converter 1. DC-DC switch mode converters 2

DC−DC Buck Converter

1

DC-DC switch mode converters

2

Basic DC-DC converters

3

• Step-down converter • Step-up converter

• Derived circuits• Step-down/step-up converter

(flyback)• (Ćuk-converter)• Full-bridge converter

Applications•DC-motor drives•SMPS

Objective – to efficiently reduce DC voltage

out

in

in

out

I

I

V

V

4


+

Vin

−

+

Vout

−

IoutIin

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

The DC equivalent of an AC transformer

Inefficient DC−DC converter

21

2

RR

RVV inout

in

out

V

V

RR

R

21

2

5

+

Vin

−

+

Vout

−

R1

R2

If Vin = 15V, and Vout = 5V, efficiency η is only 0.33

The load

Unacceptable except in very low power applications

A lossless conversion of 15Vdc to average 5Vdc

6

If the duty cycle D of the switch is 0.33, then the average voltage to the expensive car stereo is 15 ● 0.33 = 5Vdc. This is lossless conversion, but is it acceptable?

R+

15Vdc–

Switch state, voltage

Closed, 15Vdc

Open, 0Vdc

Switch open

voltage

15

0

Switch closed

DT

T

Convert 15Vdc to 5Vdc, cont.

7

Try adding a large C in parallel with the load to control ripple. But if the C has 5Vdc, then when the switch closes, the source current spikes to a huge value and burns out the switch.

Rstereo

+15Vdc

–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

+15Vdc

–C

L

Rstereo

+15Vdc

–C

L

A DC-DC Buck Converter

lossless

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

dt

tdvCti

)()(

tot

oto dtti

Ctvtv )(

1)()(

8

which leads 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

),()( oo tvTtv

The conclusion is that

Tot

otoo dtti

CtvTtv )(

10)()(or

0)( Tot

ot

dtti

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

which means that

Now, an inductor

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

dt

tdiLtv

)()(

tot

oto dttv

Ltiti )(

1)()(

9

which leads 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

),()( oo tiTti

The conclusion is that

Tot

otoo dttv

LtiTti )(

10)()(or

0)( Tot

ot

dttv

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

which means that

KVL and KCL in periodic steady-state

,0)(

loopAroundtv

,0)(

nodeofOutti

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

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

10

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

0)0(1

)(1

)(1

)(1

)(1

321

dtT

dttvT

dttvT

dttvT

dttvT

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

11

Capacitors and Inductors

In capacitors:dt

tdvCti

)()(

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 ashort-circuit)

The voltage cannot change instantaneously

In inductors:

Inductors tend to keep the current constant (current “inertia”). An idealinductor 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)

The current cannot change instantaneouslydt

tdiLtv

)()(

12

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

,dt

diLv L

L

L

VV

dt

di outinL

,dt

diLVV L

outin ,outinL VVv

13

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

for DT seconds

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

Switch closed for DT seconds

,dt

diLv L

L

L

V

dt

di outL

,dt

diLV L

out

14

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.

,outL Vv

for (1−D)T seconds

Since the average voltage across L is zero

01 outoutinLavg VDVVDV

outoutoutin VDVVDDV

inout DVV

outoutinin IVIV

15

From power balance,

D

II inout

, 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

L

VV

dt

diVVv outinLoutinL

,

L

V

dt

diVv outLoutL

,

sec/ AL

VV outin

sec/ AL

Vout

16

Examine the inductor current

Switch closed,

Switch open,

DT (1 − D)T

T

Imax

Imin

Iavg = Iout

From geometry, Iavg = Iout is halfway

between Imax and Imin

ΔI

iL

Periodic – finishes a period where it started

17

Effect of raising and lowering Iout while

holding Vin, Vout, f, and L constant

iL

ΔI

ΔI

Raise Iout

ΔI

Lower Iout

• ΔI is unchanged

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

toward discontinuous operation

18

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

19

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

RMS of common periodic waveforms, cont.

TTT

rms tT

Vdtt

T

Vdtt

T

V

TV

0

33

2

0

23

2

0

22

3

1

3

VVrms

20

T

V

0

Sawtooth


3

VVrms

21

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

V

0

V

0


22

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

)(ti

maxI

minI=

2

minmax IIIavg

avgI


2min

2 )( ItiAvgIrms

2minmin

22 )(2)( IItitiAvgIrms

2minmin

22 )( 2)( ItiAvgItiAvgIrms

23

2min

minmaxmin

2minmax2

22

3I

III

IIIrms

2minmin

22

3III

II PP

PPrms

minmax IIIPP Define


24

2minPP

avgI

II

222

223

PP

avgPPPP

avgPP

rmsI

III

II

I

423

22

222 PP

PPavgavgPP

PPavgPP

rmsI

IIII

III

I

222

2

43 avgPPPP

rms III

I

Recognize that

12

222 PPavgrms

III

avgI

)(ti

minmax IIIPP

2

minmax IIIavg

Inductor current rating

22222

12

1

12

1IIIII outppavgLrms

2222

3

42

12

1outoutoutLrms IIII

outLrms II3

2

25

Max impact of ΔI on the rms current occurs at the boundary of

continuous/discontinuous conduction, where ΔI =2Iout

2Iout

0Iavg = Iout ΔI

iL

Use max

Capacitor current and current rating

22222

3

102

12

1outoutavgCrms IIII

3out

CrmsI

I

26

iL

LC

Iout

(iL – Iout)

Iout

−Iout

0ΔI

Max rms current occurs at the boundary of continuous/discontinuous

conduction, where ΔI =2Iout Use max

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

MOSFET and diode currents and current ratings

outrms II3

2

27

iL

LC

Iout

(iL – Iout)

Use max

2Iout

0Iout

iin

2Iout

0Iout

Take worst case D for each

Worst-case load ripple voltage

Cf

I

C

IT

C

IT

C

QV outout

out

4422

1

28

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

29

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

There is a 3rd state – discontinuous

30

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

MOSFET

DIODE

Onset of the discontinuous state

sec/ AL

Vout

fL

DVTD

L

VI

onset

out

onset

outout

112

31

2Iout

0

Iavg = Iout

iL

(1 − D)T

fI

VL

out

out

2 guarantees continuous conduction

use max

use min

fI

DVL

out

outonset 2

1

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

Impedance matching

out

outload I

VR equivR

22 D

R

DI

V

DIDV

I

VR load

out

out

out

out

in

inequiv

32


+

Vin

−

+

Vout = DVin

−

Iout = Iin / DIin

+

Vin

−

Iin

Equivalent from source perspective

Source

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

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

3

2

1.5 outV outI

3

1

2 inV outI

3

2

33

10A 10A10A 40V 40V

Likely worst-case buck situation

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

Our components

9A 250V

Our M (MOSFET). 250V, 20A

Our L. 100µH, 9A

Our C. 1500µF, 250V, 5.66A p-p

Our D (Diode). 200V, 16A

BUCK DESIGN

Comparisons of Output Capacitor Ripple Voltage

Converter Type Volts (peak-to-peak) Buck

Cf

Iout4

34

10A

1500µF 50kHz

0.033V

BUCK DESIGN


Our L. 100µH, 9A

Our C. 1500µF, 250V, 5.66A p-p


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

out

2

–

35

40V

2A 50kHz

200µH

BUCK DESIGN


Our L. 100µH, 9A

Our C. 1500µF, 250V, 5.66A p-p


Documents

DC−DC Buck Converter 1. DC-DC switch mode converters 2