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8/18/2019 Slides PECh3 July03
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© Ned Mohan, 2003
3- 1
3-1 DC-DC Converters
3-2 Switching Power-Pole in DC Steady State
3-3 Simplifying Assumptions
3-4 Common Operating Principles
3-5 Buck Converter Switching Analysis in DC Steady State
3-6 Boost Converter Switching Analysis in DC Steady State
3-7 Buck-Boost Converter Switching Analysis in DC Steady State
3-8 Topology Selection
3-9 Worst-Case Design
3-10 Synchronous-Rectified Buck Converter for Very Low Output Voltages
3-11 Interleaving of Converters
3-12 Regulation of DC-DC Converters by PWM
3-13 Dynamic Average Representation of Converters in CCM
3-14 Bi-Directional Switching Power-Pole
3-15 Discontinuous-Conduction Mode (DCM)
ReferencesProblems
Appendix 3A Discontinuous-Conduction Mode (DCM) in DC-DC Converters
Chapter 3
SWITCH-MODE DC-DC CONVERTERS:
SWITCHING ANALYSIS, TOPOLOGY
SELECTION AND DESIGN
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3- 2
Regulated switch-mode dc power supplies
Figure 3-1 Regulated switch-mode dc power supplies.
inV oV
,o ref V controller
dc-dcconverter topology
,in oV V
,in o I I
(a) (b)
inV oV
,o ref V controller
dc-dcconverter topology
,in oV V
,in o I I
(a) (b)
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3- 3
Switching power-pole as the building block of dc-dc converters
( ) ( ) L L si t i t T = −
0
area area
10
s s
s
DT T
L L L
s DT
A B
V v d v d T
τ τ
= ⋅ + ⋅ =
∫ ∫
( ) ( )C C sv t v t T = −
Figure 3-2 Switching power-pole as the building block of dc-dc converters.
inV
Lv
Li
q
A Lv
Li
t
t
B
0
0
s DT
sT
( )b( )a
inV
Lv
Li
q
inV
Lv
Li
q
A Lv
Li
t
t
B
0
0
s DT
sT
Lv
Li
t
t
B
0
0
s DT
sT
( )b( )a
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3- 4
• Simplifying Assumptions
• Two-Step Process
• Common Operating Principles
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3- 5
BUCK CONVERTER SWITCHING ANALYSIS IN DC STEADY STATE
o A inV V DV = =
(1 )in o o L s sV V V
i DT D T L L
−∆ = = −
o L o
V I I = =
,( ) ( )C L ripplei t i t
in L o DI DI = =
in in o oV I V I =
Figure 3-3 Buck dc-dc converter.
inV
Li
Av
L in ov V V = −
Lv
oV
1q =
inV
Li Av
L ov V = −
oV
0q =
0 Av =
inV
ini
Li
Av Lv
oV
q
C i o I
(a)
(b)
q
Av
Lv
, L ripplei
Li
ini
inV A oV V =
( )in o
V V −
( )oV −
Li∆
L o I I =
in I
A
B
t
t
t
t
t
t 0
0
0
0
0
0
1
(c)(d)
inV
Li
Av
L in ov V V = −
Lv
oV
1q =
Li
Av
L in ov V V = −
Lv
oV
1q =
inV
Li
Av
L ov V = −
oV
0q =
0 Av =
inV
Li
Av
L ov V = −
oV
0q =
0 Av =
inV
ini
Li
Av Lv
oV
q
C i o I
(a)
(b)
q
Av
Lv
, L ripplei
Li
ini
inV A oV V =
( )in o
V V −
( )oV −
Li∆
L o I I =
in I
A
B
t
t
t
t
t
t 0
0
0
0
0
0
1q
Av
Lv
, L ripplei
Li
ini
inV A oV V =
( )in o
V V −
( )oV −
Li∆
L o I I =
in I
A
B
t
t
t
t
t
t 0
0
0
0
0
0
1
(c)(d)
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3- 6
PSpice Modeling: C:\FirstCourse_PE_Book03\Buckconv.sch
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3- 7
Simulation Results
Time
450us 455us 460us 465us 470us 475us 480us 485us 490us 495us 500us
I(C1) I(L1) V(L1:1,L1:2)
-8
-4
0
4
8
12
16
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3- 8
BOOST CONVERTER SWITCHING ANALYSIS IN DC STEADY STATE
Figure 3-4 Boost dc-dc converter.
oV
inV
q p
C Lv
Li
inV
oV
p
C Lv
q
Li
(a) (b)
oV
inV
q p
C Lv
Li
oV
inV
q p
C Lv
Li
inV
oV
p
C Lv
q
Li
inV
oV
p
C Lv
q
Li
(a) (b)
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© Ned Mohan, 2003
3- 9
Boost converter: operation and waveforms
1
1
o
in
V
V D=
− ( )o inV V >
(1 )in o in L s sV V V
i DT D T L L
−∆ = = −
in in o oV I V I = 11 1
o o o L in o
in
V I V I I I
V D D R= = = =
− −
,( ) ( )C diode ripple diode oi t i t i I = −
Figure 3-5 Boost converter: operation and waveforms.
inV oV
L inv V =
Li 0 Av =
1q =
inV
oV
L in ov V V = −
Li
0q =
A ov V =
Lv
Av
q
, L ripplei
Li
diodei
C i
t
t
t
t
t
t
t 0
0
0
0
0
0
0
oV
A inv V =
A
B( )o inV V − −
Li∆
L I
( )diode o
I I =
inV
(a)
(b) (c)
inV oV
L inv V =
Li 0 Av =
1q =
inV oV
L inv V =
Li 0 Av =
1q =
inV
oV
L in ov V V = −
Li
0q =
A ov V =
Lv
Av
q
, L ripplei
Li
diodei
C i
t
t
t
t
t
t
t 0
0
0
0
0
0
0
oV
A inv V =
A
B( )o inV V − −
Li∆
L I
( )diode o
I I =
inV L
v
Av
q
, L ripplei
Li
diodei
C i
t
t
t
t
t
t
t 0
0
0
0
0
0
0
oV
A inv V =
A
B( )o inV V − −
Li∆
L I
( )diode o
I I =
inV
(a)
(b) (c)
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PSpice Modeling: C:\FirstCourse_PE_Book03\Boost.sch
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3- 11
Time
1.950ms 1.955ms 1.960ms 1.965ms 1.970ms 1.975ms 1.980ms 1.985ms 1.990ms 1.995ms 2.000ms
I(L1) V(L1:1,L1:2)
-15
-10
-5
0
5
10
15
Simulation Results
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3- 12
Boost converter: voltage transfer ratio
Figure 3-6 Boost converter: voltage transfer ratio.
0
1
1 D−
, L crit I DCM CCM
L I
o
in
V
V
1
0
1
1 D−
, L crit I DCM CCM
L I
o
in
V
V
1
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3- 13
BUCK-BOOST CONVERTER ANALYSIS IN DC
STEADY STATE
Figure 3-7 Buck-Boost dc-dc converter.
q
A
Av
Lv
Li inV
oV
diodei
o I Lv
Av
oV inV o
I
(a) (b)
Li
q
A
Av
Lv
Li inV
oV
diodei
o I
q
A
Av
Lv
Li inV
oV
diodei
o I Lv
Av
oV inV o
I
(a) (b)
Li
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3- 14
Figure 3-8 Buck-Boost converter: operation and waveforms.
ini
L inv V =
A in ov V V = +
oV inV Li
inV Li L ov V = −
0 Av =
oV
(a)
(b)
Lv
Av
q
, L ripplei
Li
diodei
C i
t
t
t
t
t
t
t 0
0
0
0
0
0
0
oV −
A oV V =
A
B
( )in oV V +
Li∆
L I
( )diode o I I =
inV
(c)
inio I
o I
ini
L inv V =
A in ov V V = +
oV inV Li
L inv V =
A in ov V V = +
oV inV Li
inV Li L ov V = −
0 Av =
oV
(a)
(b)
Lv
Av
q
, L ripplei
Li
diodei
C i
t
t
t
t
t
t
t 0
0
0
0
0
0
0
oV −
A oV V =
A
B
( )in oV V +
Li∆
L I
( )diode o I I =
inV
(c)
inio I
o I
Buck-Boost converter: operation and waveforms
1
o
in
V D
V D
=
−
(1 )in o L s sV V
i DT D T L L
∆ = = −
L in o I I = +
in in o oV I V I =
1
oin o o
in
V D
I I I V D= = −1 1
1 1
o L in o o
V I I I I
D D R= + = =
− −
,( ) ( )C diode ripplei t i t
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PSpice Modeling: C:\FirstCourse_PE_Book03\Buck-Boost_Switching.sch
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© Ned Mohan, 2003
3- 16
Simulation Results
Time
2.950ms 2.955ms 2.960ms 2.965ms 2.970ms 2.975ms 2.980ms 2.985ms 2.990ms 2.995ms 3.000ms
I(L1) V(L1:1,L1:2)
-30
-20
-10
0
10
20
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3- 17
Buck-Boost converter: voltage transfer ratio
Figure 3-9 Buck-Boost converter: voltage transfer ratio.
0
1
D
D−
, L crit I DCM CCM
L I
o
in
V
V
1
0
1
D
D−
, L crit I DCM CCM
L I
o
in
V
V
1
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Other Buck-Boost Topologies
• SEPIC Converters (Single-Ended Primary Inductor Converters)
• Cuk Converters
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SEPIC Converters (Single-Ended Primary Inductor Converters)
(1 )in oV D V = −1
o
in
V D
V D
=
−
Figure 3-10 SEPIC converter.
inV
2 Li
q
C v
oV
Li diodei
2 Lv(a)
inV
C v
oV
2 L C v v=1q =
2 LvoV
inV
0q =
C v2 Lv
2 L ov V = −
(b) (c)
inV
2 Li
q
C v
oV
Li diodei
2 Lv(a) inV
2 Li
q
C v
oV
Li diodei
2 LvinV
2 Li
q
C v
oV
Li diodei
2 Lv(a)
inV
C v
oV
2 L C v v=1q =
2 LvoV
inV
0q =
C v2 Lv
2 L ov V = −
(b) (c)inV
C v
oV
2 L C v v=1q =
2 LvinV
C v
oV
2 L C v v=1q =
2 LvoV
inV
0q =
C v2 Lv
2 L ov V = −
oV inV
0q =
C v2 Lv
2 L ov V = −
(b) (c)
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Cuk Converter
(1 )o in I D I = − 1in
o
I D
I D=
− 1o
in
V D
V D=
−
Figure 3-11 Cuk converter.
inV
q
C v
oV
Li oi
o I
C 1 L 2 L
(a)
inV
1q =
C v
oV ini
oi
inV
0q =
C v
oV ini oi
(b) (c)
inV
q
C v
oV
Li oi
o I
C 1 L 2 L
(a) inV
q
C v
oV
Li oi
o I
C 1 L 2 L
inV
q
C v
oV
Li oi
o I
C 1 L 2 L
(a)
inV
1q =
C v
oV ini
oi
inV
0q =
C v
oV ini oi
(b) (c)
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TOPOLOGY SELECTION
Criterion Buck Boost Buck-Boost
Transistor
ˆ
V inV oV ( )in oV V +
Transistor ˆ I o I in I in o I I +
rms I Transistor o DI in DI ( )in o D I I +
Transistoro DI in DI ( )in o D I I + avg I
Diode (1 ) o D I − (1 ) in D I − ( )(1 ) in o D I I − +
L I o I in I in o I I +
Effect of L on C significant little little
Pulsating Current input output both
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WORST-CASE DESIGN
The worst-case design should consider the ranges in which the input voltage and the
output load vary. As mentioned earlier, often converters above a few tens of watts are
designed to operate in CCM. To ensure CCM even under very light load conditions
would require prohibitively large inductance. Hence, the inductance value chosen is
often no larger than three times the critical inductance ( )3 c L L< , where, as discussed in
section 3-15, the critical inductance c L is the value of the inductor that will make the
converter operate at the border of CCM and DCM at full-load.
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SYNCHRONOUS-RECTIFIED BUCK CONVERTER FOR
VERY LOW OUTPUT VOLTAGES
Figure 3-12 Buck converter: synchronous rectified.
inV
oV Av
T +
T −
q+
q−
Li
( )a
q+ q−
Av
Li
t
t
t
0t =
s DT sT
inV
oV 0
0
0
0 L I
( )b
inV
oV Av
T +
T −
q+
q−
Li
( )a
inV
oV Av
T +
T −
q+
q−
Li
( )a
q+ q−
Av
Li
t
t
t
0t =
s DT sT
inV
oV 0
0
0
0 L I
( )b
q+ q−
Av
Li
t
t
t
0t =
s DT sT
inV
oV 0
0
0
0 L I
( )b
3 24
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INTERLEAVING OF CONVERTERS
Figure 3-13 Interleaving of converters.
1q2q
1
q
2q
0
0
t
t
(a) (b)
inV
+
−
−
+
oV
1 Li
2 Li
1q2q
1
q
2q
0
0
t
t
(a) (b)
inV
+
−
−
+
oV
1 Li
2 Li
3 25
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REGULATION OF DC-DC CONVERTERS BY PWM
( )( )
ˆc
r
v t d t
V =
Figure 3-14 Regulation of output by PWM.
inV oV
,o ref V controller
dc-dcconverter
topology
(a) (b)
0 sd T
sT t
ˆr V
( )cv t
t
r v
( )q t
0
1
inV oV
,o ref V controller
dc-dcconverter
topology
(a) (b)
0 sd T
sT t
ˆr V
( )cv t
t
r v
( )q t
0
1
3 26
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Figure 3-15 Average dynamic model of a switching power-pole.
( )q t
( )r v t
( )cv t
vpv
cpv
cpi
vpi
vpV
cpV
cp I
vp I
1: D
cpv
1: ( )d t
( )cv t ^
1
r V
(c)(a) (b)
vpi
vpv
cpi
( )q t
( )r v t
( )cv t
vpv
cpv
cpi
vpi
vpV
cpV
cp I
vp I
1: D
cpv
1: ( )d t
( )cv t ^
1
r V
(c)(a) (b)
vpi
vpv
cpi
DYNAMIC AVERAGE REPRESENTATION OF
CONVERTERS IN CCM
( ) ( ) ( )cp vpv t d t v t =
( ) ( ) ( )vp cpi t d t i t =
cp vpV DV =
vp o I D I =
3 27
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Average dynamic models of three converters
Figure 3-16 Average dynamic models of three converters.
q
inV
ov Lv
Li
oV
inV
q p
A A
qoV
inV
inV
1: ( )d t
inV
1: (1 ( ))d t −
p 1: ( )d t
inV
(a)
(b)
Li Li
L
i
Li Li
ov ov
ov
⇓ ⇓ ⇓q
inV
ov Lv
Li
oV
inV
q p
A A
qoV
inV
inV
1: ( )d t
inV
1: (1 ( ))d t −
p 1: ( )d t
inV
(a)
(b)
Li Li
L
i
Li Li
ov ov
ov
⇓ ⇓ ⇓
3 28
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PSpice Modeling: C:\FirstCourse_PE_Book03\Buck-Boost_Avg_CCM.sch
3 29
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PSpice Modeling: C:\FirstCourse_PE_Book03\Buck-Boost_Switching_LoadTransient.sch
3 30
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3- 30
Simulation Results
Time
0s 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms
I(L1) V(L1:1,L1:2)
-40
-20
0
20
40
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3- 32
Average dynamic model of the switching power-pole with
bi-directional power flow
Figure 3-18 Average dynamic model of the switching power-pole with bi-directional power
flow.
q
inV
Li
Buck Boost
(1 )q q−
= − 1: d
Li
inV
(a) (b)q
inV
Li
Buck Boost
(1 )q q−
= − 1: d
Li
inV
(a) (b)
3- 33
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DISCONTINUOUS-CONDUCTION MODE (DCM)
Figure 3-19 Inductor current at various loads; duty-ratio is kept constant.
1 Li
2 Li, L crii
1 L I
2 L I
, L crit I
t
Li
0
1 Li
2 Li, L crii
1 L I
2 L I
, L crit I
t
Li
0
3- 34
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Figure 3-20 Waveforms for the three converters at the border of CCM/DCM.
Li
ˆ L I
t
t
Av
in oV V −
0
0
0
S DT (1 ) S D T −
S T
(a)
Liˆ L I
t
t
AvoV
0
0
0
S DT (1 ) S D T −
S T
(b) (c)
Li
ˆ L I
t
t
Av
in oV V +
0
0
0
S DT (1 ) S D T −
S T
Li
ˆ L I
t
t
Av
in oV V −
0
0
0
S DT (1 ) S D T −
S T
(a)
Li
ˆ L I
t
t
Av
in oV V −
0
0
0
S DT (1 ) S D T −
S T
Li
ˆ L I
t
t
Av
in oV V −
0
0
0
S DT (1 ) S D T −
S T
(a)
Liˆ L I
t
t
AvoV
0
0
0
S DT (1 ) S D T −
S T
(b)
Liˆ L I
t
t
AvoV
0
0
0
S DT (1 ) S D T −
S T
(b) (c)
Li
ˆ L I
t
t
Av
in oV V +
0
0
0
S DT (1 ) S D T −
S T
(c)
Li
ˆ L I
t
t
Av
in oV V +
0
0
0
S DT (1 ) S D T −
S T
Li
ˆ L I
t
t
Av
in oV V +
0
0
0
S DT (1 ) S D T −
S T
, , , , -2
in L crit Boost L crit Buck Boost
V I I D f
= =, , (1 )2
in L crit Buck
V I D D Lf
= −
Critical Inductor Currents
3- 35
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Buck converter in DCM
Figure 3A-1 Buck converter in DCM.
Liˆ L I s
t
T
Av
oV inV
0
0
D ,1off D ,2off D
1
o
in
V
V
, L crit I DCM CCM
L I
(a)
0
D
1
s
t T
(b)
Liˆ L I s
t
T
Av
oV inV
0
0
D ,1off D ,2off D
1
o
in
V
V
, L crit I DCM CCM
L I
(a)
0
D
1
s
t T
(b)
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© Ned Mohan, 2003
3 36
Boost Converters in DCM
Figure 3A-2 Boost converter in DCM.
Liˆ L I
s
t
T
AvoV
inV
0
0
D ,1off D ,2off D
1(a)
0
1
1 D−
, L crit
I DCM CCM L I
o
in
V
V
1
(b)
s
t
T
Liˆ L I
s
t
T
AvoV
inV
0
0
D ,1off D ,2off D
1(a)
0
1
1 D−
, L crit
I DCM CCM L I
o
in
V
V
1
(b)
s
t
T
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© Ned Mohan, 2003
3 37
Buck-Boost converter in DCM
Figure 3A-3 Buck-Boost converter in DCM.
Li
ˆ L I s
t
T
Av
oV in oV V +
0
0
D ,1off D ,2off D
1
(a)
0
1
D
D−
, L crit I DCM CCM L
I
o
in
V
V
(b)
1
s
t
T
Li
ˆ L I s
t
T
Av
oV in oV V +
0
0
D ,1off D ,2off D
1
(a)
0
1
D
D−
, L crit I DCM CCM L
I
o
in
V
V
(b)
1
s
t
T
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© Ned Mohan, 2003
AVERAGE REPRESENTATION OF DC-DC
CONVERTERS IN DCM
, 0
2[1 ]
( )
s Lk Buck
in o
Lf iv v
V v d = −
−
2
, 0( )2
k Buck in Ld i V v di Lf
= − −
• Dependent Sources in Buck and Buck-Boost Converters in DCM
,
21 s Lk Buck Boost o
in
Lf iv v
V d −
= −
2
,2
k Buck Boost in Ld i V di Lf
− = −
Figure 3A-4 Average dynamic model for Buck and Buck-Boost converters.
vpv cpv
vpicpi
k v
k i
1: ( )d t
vpv cpv
vpicpi
k v
k i
1: ( )d t
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© N d M h 2003
• Dependent Sources in Boost Converters in DCM
( ), 02
1 s Lk Boost inin
Lf iv V vV d
= − −
2
,
2k Boost in L
d i V di
Lf
= −
Figure 3A-5 Average dynamic model for Boost converters.
(1 ) :1d −
cpv
cpi k v
k i
vpi
vpv
(1 ) :1d −
cpv
cpi k v
k i
vpi
vpv
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