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EE362L, Fall 2008
DCDC Buck Converter
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Objectiveto efficiently reduce DC voltage
DCDC Buck
Converter
+
Vin
+
Vout
IoutIin
Lossless objective: Pin
= Pout
, which means that Vin
Iin
= Vout
Iout
and
The DC equivalent of an AC transformer
out
in
in
out
I
I
V
V
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Here is an example of an inefficient DCDC
converter
21
2
RR
RVV inout
+
Vin
+
Vout
R1
R2
in
out
V
V
RR
R
21
2
If Vin= 39V, and Vout= 13V, efficiency is only 0.33
The load
Unacceptable except in very low power applications
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Another methodlosslessconversion 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 open
Stereo
voltage
39
0
Switch closed
DT
T
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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 currentspikes 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 toopen, the inductors 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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Cs and Ls 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 )(
)( which leads to totot
o dttiC
tvtv )(1
)()(
Since the capacitor is in periodic steady state, then the voltage at
time tois the same as the voltage one period T later, so
),()( oo tvTtv
The conclusion is that
Totot
oo dttiC
tvTtv )(1
0)()(or
0)( Totot
dtti
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 inductor
Examine the voltage across an inductor that is operating in
periodic steady state. The governing equation is
dt
tdiLtv )(
)( which leads to totot
o dttvL
titi )(1
)()(
Since the inductor is in periodic steady state, then the voltage at
time tois the same as the voltage one period T later, so
),()( oo tiTti
The conclusion is that
Totot
oo dttvL
tiTti )(1
0)()(or
0)( Totot
dttv
the average voltage across an inductor operating in periodic
steady state is zero
which means that
Taken from Waveforms and Definitions PPT
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KVL and KCL in periodic steady-state
,0)(
loopAround
tv
,0)(
nodeofOut
ti
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
)(1
321
dtT
dttvT
dttvT
dttvT
dttvT
Tot
ot
Tot
ot
N
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
Taken from Waveforms and Definitions PPT
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Vin
+Vout
iL
LC
iC
Ioutiin
Buck converter+ vL
Vin
+Vout
LC
Ioutiin
+ 0 V
What do we learn from inductor voltage and capacitor
current in the average sense?
Iout
0 A
Assume large C so that
Vouthas very low ripple
Since Vouthas 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
Ioutiin
+ (VinVout)iL
(iL Iout)
Reverse biased, thus the
diode is open
,dtdiLv LL
L
VV
dt
di outinL ,dt
diLVV Loutin ,outinL VVv
for DT seconds
Noteif 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
iLcontinues to flow, thus the diode is closed. This
is the assumption of continuous conduction in the
inductor which is the normal operating condition.
,dt
diLv LL
L
V
dt
di outL ,dt
diLV Lout,outL Vv
for (1D)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
D
II inout
, so
The input/output equation becomes
Noteeven though iinis not constant
(i.e., iinhas harmonics), the input power
is still simply Vin Iinbecause Vinhas no
harmonics
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Examine the inductor current
Switch closed,
Switch open,
LVV
dtdiVVv outinLoutinL
,
L
V
dt
diVv outLoutL
,
sec/AL
VV outin
DT (1 D)T
T
Imax
Imin
Iavg= Iout
From geometry, Iavg= Ioutis halfway
between Imaxand Iminsec/ALVout
I
iL
Periodicfinishes
a period where itstarted
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Effect of raising and lowering Ioutwhile
holding Vin, Vout, f, and L constant
iL
I
IRaise Iout
I
Lower Iout
I 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 iLare unchanged
Lowering f increasesI and moves the circuit toward
discontinuous operation
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iL
Effect of raising and lowering L while
holding Vin, Vout, Ioutand f constant
Raise L
Lower L
Lowering L increasesI and moves the circuit towarddiscontinuous operation
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RMS of common periodic waveforms, cont.
TTT
rms t
T
Vdtt
T
Vdtt
T
V
T
V
0
3
3
2
0
2
3
2
0
22
3
1
T
V
0
3
VVrms
Sawtooth
Taken from Waveforms and Definitions PPT
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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
0
V
0
V
0
0
-V
V
0
3
VVrms
V
0
V
0
Taken from Waveforms and Definitions PPT
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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.
minmax II
0
)(tithe ripple
+
0
minI
the minimum value
)(ti
maxI
minI=
2
minmax IIIavg
avgI
Taken from Waveforms and Definitions PPT
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RMS of common periodic waveforms, cont.
2min2 )( ItiAvgIrms
2minmin
22 )(2)( IItitiAvgIrms
2
minmin
22
)(2)( ItiAvgItiAvgIrms
2min
minmaxmin
2minmax2
22
3I
III
IIIrms
2minmin
22
3III
II PP
PPrms
minmax IIIPP Define
Taken from Waveforms and Definitions PPT
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RMS of common periodic waveforms, cont.
2min
PP
avg
III
222
223
PPavgPP
PPavg
PPrms
III
II
II
423
22222 PPPPavgavg
PPPPavg
PPrms
IIII
III
II
222
2
43 avgPPPP
rms III
I
Recognize that
12
222 PPavgrms
III
avgI
)(ti
minmax IIIPP
2
minmax IIIavg
Taken from Waveforms and Definitions PPT
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Inductor current rating
2222212
1
12
1 IIIII outppavgLrms
2222342
121
outoutoutLrms IIII
Max impact ofI on the rms current occurs at the boundary of
continuous/discontinuous conduction, whereI =2Iout
outLrms II3
2
2Iout
0
Iavg= IoutI
iL
Use max
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Capacitor current and current rating
222223
102
12
1outoutavgCrms IIII
iL
LC
Iout
(iL Iout)
Iout
Iout
0I
Max rms current occurs at the boundary of continuous/discontinuous
conduction, whereI =2Iout
3
outCrms
II
Use max
iC= (iL Iout) Noteraising f or L, which lowersI, 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
0
Iout
iin
2Iout
0
Iout
Take worst case D for each
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Worst-case load ripple voltage
Cf
I
C
IT
C
IT
C
QV outout
out
44
22
1
Iout
Iout
0T/2
C charging
iC= (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
Ioutiin
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 3rdstatediscontinuous
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 = 100H, this corresponds
to net parasitic C = 0.6nF
vL= (VinVout)
vL=Vout
Switch open
Switch
closed
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Onset of the discontinuous state
sec/AL
Vout
fLDVTD
LVI
onset
out
onset
outout 112
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),
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Impedance matching
out
outload
I
VR
equivR
DCDC Buck
Converter
+
Vin
+
Vout = DVin
Iout = Iin/ DIin
+
Vin
Iin
22D
R
DI
V
DI
D
V
I
VR load
out
out
out
out
in
inequiv
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-amps
Isc
Voc
Pmaxis approx. 130W
(occurs at 29V, 4.5A)
44.65.4
29
A
VRload
For max power frompanels at this solar
intensity level, attach
I-V characteristic of 6.44resistor
But as the sun conditionschange, the max power
resistance must also
change
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Connect a 2resistor 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-amps
130W55W
56.0
44.6
2,
2
equiv
loadloadequiv
R
RD
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
+V
out
iL
L
C iC
Ioutipanel
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 10F, 50V, 10A high-frequency bipolar (unpolarized) capacitor
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Worst-Case Component Ratings Comparisons
for DC-DC Converters
ConverterType
Input InductorCurrent(Arms)
OutputCapacitorVoltage
Output CapacitorCurrent (Arms)
Diode andMOSFETVoltage
Diode andMOSFETCurrent(Arms)
BuckoutI3
2
1.5 outV outI3
1
2 inV outI3
2
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. 100H, 9A
Our C. 1500F, 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
Cf
Iout
4
10A
1500F 50kHz
0.033V
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100H, 9A
Our C. 1500F, 250V, 5.66A p-p
Our D (Diode). 200V, 16A
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Minimum Inductance Values Needed toGuarantee Continuous Current
Converter Type For ContinuousCurrent in the Input
Inductor
For ContinuousCurrent in L2
BuckfI
VLout
out2
40V
2A 50kHz
200H
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100H, 9A
Our C. 1500F, 250V, 5.66A p-p
Our D (Diode). 200V, 16A
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