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EE462L, Spring 2014DC−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
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
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 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
Taken from “Course Overview” PPT
6
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 !
7
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
Taken from “Waveforms and Definitions” PPT !
8
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
Taken from “Waveforms and Definitions” PPT !
9
Capacitors and Inductors
In capacitors:dttdvCti )()(
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 instantaneouslydttdiLtv )()(
!
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 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
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.
,dtdiLv L
L LV
dtdi outL
,dtdiLV 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
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
!
14
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
15
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
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 tTVdtt
TVdtt
TV
TV
03
3
2
0
23
2
0
22
31
T
V
0
3VVrms
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
3VVrms
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 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
Taken from “Waveforms and Definitions” PPT !
21
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
Taken from “Waveforms and Definitions” PPT
22
RMS of common periodic waveforms, cont.
2minPP
avgIII
222
223
PP
avgPPPP
avgPP
rmsIIIIIII
423
22
222 PP
PPavgavgPP
PPavgPP
rmsIIIIIIIII
222
243 avgPPPP
rms IIII
Recognize that
12
222 PPavgrms
III
avgI
)(ti
minmax IIIPP
2
minmax IIIavg
Taken from “Waveforms and Definitions” PPT
23
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
24
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
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
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
!
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 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
!
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 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),
!
31
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
!
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
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
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
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
!
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 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
35
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
32
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
36
Comparisons of Output Capacitor Ripple Voltage
Converter Type Volts (peak-to-peak) Buck
CfIout4
10A
1500µF 50kHz
0.033V
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9AOur C. 1500µF, 250V, 5.66A p-p
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
out2
–
40V
2A 50kHz
200µH
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9AOur C. 1500µF, 250V, 5.66A p-p
Our D (Diode). 200V, 16A