1 DC−DC Buck Converter. 2 Objective – to efficiently reduce DC voltage DC−DC Buck Converter +...

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1
DC−DC Buck Converter
2
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
out
I
I
V
V
!
3
Here is an example of an inefficient DC−DC 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 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 open
Stereo voltage
39
0
Switch closed
DT
T
Taken from “Course Overview” PPT !
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 highfrequency switching, the load voltage ripple can be reduced to a small value.
Rstereo
+39Vdc
–C
L
Rstereo
+39Vdc
–C
L
A DCDC Buck Converter
lossless
Taken from “Course Overview” PPT
6
C’s and L’s operating in periodic steadystate
Examine the current passing through a capacitor that is operating in periodic steady state. The governing equation is
dt
tdvCti
)()( which leads to
tot
oto dtti
Ctvtv )(
1)()(
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
),()( 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
Taken from “Waveforms and Definitions” PPT !
7
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
tot
oto dttv
Ltiti )(
1)()(
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
),()( 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
Taken from “Waveforms and Definitions” PPT !
8
KVL and KCL in periodic steadystate
,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
)(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
Taken from “Waveforms and Definitions” PPT !
9
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 ashortcircuit)
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
)()(
!
10
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
!
11
The input/output equation for DCDC converters usually comes by examining inductor voltages
Vin
+ Vout
–
LC
Iout iin + (Vin – Vout) –
iL
(iL – Iout)
Reverse biased, thus the diode is open
,dt
diLv L
L L
VV
dt
di outinL ,
dt
diLVV L
outin ,outinL VVv
for DT seconds
Note – if the switch stays closed, then Vout = Vin
Switch closed for DT seconds
12
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.
,dt
diLv L
L L
V
dt
di outL ,
dt
diLV L
out ,outL Vv
for (1−D)T seconds
13
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
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
!
14
Examine the inductor current
Switch closed,
Switch open,
L
VV
dt
diVVv outinLoutinL
,
L
V
dt
diVv outLoutL
,
sec/ AL
VV outin
DT (1 − D)T
T
Imax
Imin
Iavg = Iout
From geometry, Iavg = Iout is halfway
between Imax and Imin
sec/ AL
Vout
ΔI
iL
Periodic – finishes a period where it started
15
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
16
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
17
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
18
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
T
V
0
3
VVrms
Sawtooth
Taken from “Waveforms and Definitions” PPT !
19
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 !
20
RMS of common periodic waveforms, cont.
Now, consider a useful example, based upon a waveform that is often seen in DCDC 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 !
21
RMS of common periodic waveforms, cont.
2min
2 )( ItiAvgIrms
2minmin
22 )(2)( IItitiAvgIrms
2minmin
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
22
RMS of common periodic waveforms, cont.
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
Taken from “Waveforms and Definitions” PPT
23
Inductor current rating
22222
12
1
12
1IIIII outppavgLrms
2222
3
42
12
1outoutoutLrms 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
24
Capacitor current and current rating
22222
3
102
12
1outoutavgCrms IIII
iL
LC
Iout
(iL – Iout)
Iout
−Iout
0ΔI
Max rms current occurs at the boundary of continuous/discontinuous
conduction, where ΔI =2Iout
3out
CrmsI
I
Use max
iC = (iL – Iout) Note – raising f or L, which lowers ΔI, reduces the capacitor current
25
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
26
Worstcase load ripple voltage
Cf
I
C
IT
C
IT
C
QV outout
out
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 peaktopeak ripple voltage.
Raising f or L reduces the load voltage ripple
!
27
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
28
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 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
Iout
MOSFET
DIODE
!
29
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
30
Onset of the discontinuous state
sec/ AL
Vout
fL
DVTD
L
VI
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),
!
31
Impedance matching
out
outload I
VR
equivR
DC−DC Buck Converter
+
Vin
−
+
Vout = DVin
−
Iout = Iin / DIin
+
Vin
−
Iin
22 D
R
DI
V
DIDV
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
!
32
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
mp
s
Isc
Voc
Pmax is approx. 130W
(occurs at 29V, 4.5A)
44.65.4
29
A
VRload
For max power from panels at this solar intensity level, attach
IV characteristic of 6.44Ω resistor
But as the sun conditions change, the “max power resistance” must also change
33
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
mp
s
130W
6.44Ω
resistor
2Ω r
esis
tor
55W
56.044.6
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
!
34
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 ripplefree 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 highfrequency bipolar (unpolarized) capacitor
35
WorstCase Component Ratings Comparisons for DCDC 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
10A 10A10A 40V 40V
Likely worstcase 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 pp
Our D (Diode). 200V, 16A
BUCK DESIGN
36
Comparisons of Output Capacitor Ripple Voltage
Converter Type Volts (peaktopeak) Buck
Cf
Iout4
10A
1500µF 50kHz
0.033V
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9A
Our C. 1500µF, 250V, 5.66A pp
Our D (Diode). 200V, 16A
37
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
–
40V
2A 50kHz
200µH
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9A
Our C. 1500µF, 250V, 5.66A pp
Our D (Diode). 200V, 16A