Chapter 7
DC-DC Switch-Mode Converters
• dc-dc converters for switch-mode dc power
supplies and dc-motor drives
• these dc-dc converters are studied:
• step-down (buck)
• step-up (boost)
• step-up/ step-down (buck-boost)
• Cúk
• Full-bridge
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-1
Block Diagram of DC-DC Converters
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-2
Stepping Down a DC Voltage
• switching at constant frequency:
• pulse-width modulation (PWM) switching
vo
Ts  ton  toff
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-3
Pulse-Width Modulation (1)
• signal-level control voltage vcontrol generated by amplifying
the difference between actual output voltage and desired
output voltage
• switch control signal generated by comparing vcontrol with
repetitive waveform
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-4
Pulse-Width Modulation (2)
• switch
duty-cycle D is
Copyright © 2008
by Jose Bastos
ton vcontrol
D

Ts
Vˆst
Chapter 7 DC-DC Switch-Mode
Converters
7-5
Step-Down (Buck) DC-DC Converter (1)
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-6
Step-Down (Buck) DC-DC Converter (2)
• average output voltage Vo
ton
Ts

 t
1
Vo    Vd dt   0 dt  on Vd  D Vd
Ts  0
 Ts
ton
or
Vd
Vo 
vcontrol  k vcontrol
Vˆst
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-7
R L C low-pass filter
1
40 log
 40 log
LC
fc 
Copyright © 2008
by Jose Bastos
1
2
LC
Chapter 7 DC-DC Switch-Mode
Converters
7-8
Continuous conduction mode (1)
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-9
Continuous conduction mode (2)
Ts
t on
Ts
0
0
t on
v
dt

v
dt

v
dt

0
L
L
L



(Vd  Vo ) ton  Vo (Ts  ton )
or
Vo ton

D
Vd Ts
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-10
Continuous conduction mode (3)
• input power equals output power:
Pd  Po
Vd I d  Vo I o
I o Vd 1


I d Vo D
• step-down converter is equivalent to a dc
transformer where the turns ratio is in the range 0-1
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-11
Edge of Cont./Discont. Conduction
• Critical current below which inductor current becomes discontinuous:
V
1
1t
1 D Ts
1 D Ts
I LB  iL , peak  on (Vd  Vo ) 
(Vd  Vo ) 
Vd (1  o )
2
2 L
2 L
2 L
Vd

1 D Ts
Vd (1  D)
2 L
Copyright © 2008
by Jose Bastos

I LB ,max 
TsVd
8L
Chapter 7 DC-DC Switch-Mode
Converters
7-12
Discontinuous Conduction Mode
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-13
Discontinuous Conduction Mode (2)
• Vo/Vd in the discontinuous mode
• integrating the inductor voltage over one time period,
(Vd  Vo ) DTs  (Vo )1Ts  0
Vo
D
 
Vd D  1
(1)
• From the figure,
iL , peak 
Vo
1Ts
L
and
DT
1Ts

iL , peak
1  s iL , peak
Io   
t dt  
t dt
Ts  0 DTs

T
1 s
0

Copyright © 2008
by Jose Bastos
2
2
1  iL , peak DTs  iL , peak 1Ts  
 


Ts  DTs
2
1Ts
2 
D  1
 iL , peak
2
Chapter 7 DC-DC Switch-Mode
substituti
ng i
Converters L , peak
7-14
D  1

i
L , peak
Discontinuous
2
substituting iL , peak
Conduction Mode (3)
VoTs
( D   1 ) 1
2L
and using (1)
I0 
Vd Ts
I0 
D 1
2L
2 LI o
Io
 1 

Vd Ts D 4 I LB ,max D
I LB ,max
TsVd

8L
Substit uting in (1)
Vo
D2

Vd D 2  I o
4 I LB ,max
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-15
Limits of Cont./Discont. Conduction with
constant D
TV
I LB ,max 
I LB
s d
8L
/ I LB ,max  4 D (1  D)
• The duty-ratio of 0.5 has the highest value of the critical
current
• The boundary between the cont/discont mode is shown by
the dashed curve
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-16
Discont. Conduction mode with constant Vo
• in regulated dc power supplies Vo is kept constant by
adjusting the duty ratio D
• since Vd=Vo/D the average inductor current at the edge
of cont/discont mode is
I LB
DTs

Vd (1  D )
2L
TsVo

(1  D )
2L
• when D=0 the maximum ILB,max is
I LB ,max
Copyright © 2008
by Jose Bastos
TsVo

2L
Chapter 7 DC-DC Switch-Mode
Converters
7-17
Discont. Conduction mode with constant Vo
I LB ,max
Copyright © 2008
by Jose Bastos
TsVo

2L
Vo
D
Vd
 I o / I LB ,max 


 1  Vo / Vd 
Chapter 7 DC-DC Switch-Mode
Converters
1/ 2
7-18
Step-Down Conv.: Output Voltage Ripple (1)
Peak-peak voltage ripple:
Q 1 1 I L Ts
Vo 

C
C2 2 2
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-19
Step-Down Conv.: Output Voltage Ripple (2)
Computing
I L :
I L
Vo  L
t
During toff:
1
I L  Vo toff
L
1
I L  Vo (Ts  ton )
L
ton
D
Ts
1
I L  Vo Ts (1  D )
L
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-20
Step-Down Conv.: Output Voltage Ripple (3)
1
Vo Ts (1  D )
L
subst it ut ing I L int o
I L 
Q 1 1 I L Ts
Vo 

C
C2 2 2
gives
 fc 
Vo 1 Ts (1  D ) 


(1  D ) 
Vo
8
LC
2
 fs 
2
2
2
fc 
1
2 LC
• ripple can be minimized by making fc of the low pass
filter fc << fs
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-21
Step-Up (Boost) DC-DC Converter
• Output voltage is greater than the input
• main application: regulated dc power supplies
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-22
Step-Up DC-DC Converter Waveforms (1)
Vd ton  (Vo  Vd )toff
Vd
ton
T t
 (Vo  Vd ) s on
Ts
Ts
Vd D  (Vo  Vd )(1  D)
Vd D  Vo (1  D)  Vd  Vd D
Vd  Vo (1  D)
Vo
1

Vd 1  D
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-23
Effect of Parasitics
• The duty-ratio is generally limited before the
parasitic effects become significant
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-24
Step-Up DC-DC Converter Waveforms (2)
• Assuming a lossless circuit Pd=Po
Vd I d  Vo I o
Io
 (1  D)
Id
Thus:
• power remains the same
• voltage increases
• current decreases
• equivalent to a DC transformer
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-25
Edge of Cont./Discont. Conduction (1)
1
I LB  iL , peak
2
1 Vd

ton
2 L
using
Vo
1

Vd 1  D
I LB
Copyright © 2008
by Jose Bastos
and D 
ton
Ts
TsVo

D (1  D )
2L
Chapter 7 DC-DC Switch-Mode
Converters
7-26
Edge of Cont./Discont. Conduction (2)
I LB 
TsVo
D (1  D )
2L
• recognizing that the inductor current IL and the input
current Id are the same Id= IL
• and I o  I d (1  D)
I oB
• highest ILB at D=0.5
TsVo
2

D1  D 
2L
I LB ,max 
• highest IOB at D=1/3
Copyright © 2008
by Jose Bastos
TsVo
8L
2 TsVo
I oB ,max 
27 L
Chapter 7 DC-DC Switch-Mode
Converters
(1)
7-27
Edge of Cont./Discont. Conduction (3)
I oB
TsVo
2

D1  D 
2L
Copyright © 2008
by Jose Bastos
I LB
Chapter 7 DC-DC Switch-Mode
Converters
TsVo

D (1  D )
2L
7-28
Discont. Conduction (1)
• Occurs at light loads
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-29
Discont. Conduction (2)
Vd DTs  (Vd  Vo )1Ts  0
Vo 1  D

Vd
1
( 2)
and since power remainsconst ant
Io
1

I d 1  D
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
(3)
7-30
Discont. Conduction (3)
iL , peak 
Vd
DTs
L
and
1Ts
 DT s iL , peak

iL , peak
t dt  
t dt

DT

T
 0

s
1 s
0
2
iL , peak 1Ts 2 
1  iL , peak DTs 
 


Ts  DTs
2
1Ts
2 
D  1
 iL , peak
2
subst it ut ing iL , peak
1
Id 
Ts
Vd Ts
( D  1 ) D
2L
and using (3)
VT
I 0  d s D 1
2L
Id 
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-31
Discont. Conduction (4)
Vo 1  D

Vd
1
Solving Io in order to 1
2 LI o
1 
Vd Ts D
Substituting above
2 LI o
 D2
Vo Vd Ts

2 LI o
Vd
Vd Ts
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-32
Edge of Cont./Discont. Conduction with
Vd constant
2 LI o
 D2
Vo Vd Ts

2 LI o
Vd
Vd Ts
TsVo
2
I oB 
D 1  D  
27
2L

2


I
/
I

D
1

D
 oB oB ,max
2 TsVo
4

I oB ,max 

27 L
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-33
Discont. Conduction with constant Vo (5)
substituting IoB,max
I oB ,max
2 TsVo

27 L
We get
4 Vo I o
 D2
Vo 27 Vd I oB ,max

4 Vo I o
Vd
27 Vd I oB ,max
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-34
Discont. Conduction with Vo constant (6)
Solving for D
 4 Vo
D
 27 Vd
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
 Vo  I o 
  1

 Vd
 I oB ,max 
1/ 2
7-35
Boost Converter Output Ripple (1)
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-36
Boost Converter Output Ripple (2)
• the ripple current flows through the capacitor and
the average through R
idiode
Io
Q I o DTs
Vo 

C
C
Vo DTs
Vo 
R C
Vo DTs
Ts

D
Vo
RC

(where =RC is the time const)
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-37
Step-Down/Up (Buck-Boost) Converter
• The output voltage can be higher or lower than the input
voltage
•The output voltage is negative
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-38
Buck-Boost DC-DC Converter: Waveforms
• equating the integral of inductor
voltage over one period:
Vd DTs  (Vo )(1  D)Ts  0
D
Vo 
Vd
1 D
• D>0.5 means Vo>Vd
• D<0.5 means Vo<Vd
•power is conserved Pd=Po:
Io 
Copyright © 2008
by Jose Bastos
1 D
Id
D
Chapter 7 DC-DC Switch-Mode
Converters
7-39
Limits of Cont./Discont. Conduction (1)
• The average inductor current is
1
I LB  iL , peak
2
1 TsVd

D
2 L
1 TsVo

(1  D)
2 L
• Noting that average capacitor current
is zero:
D
I oB  I LB  I d  I LB 
re-arranging:
Substituting:
Copyright © 2008
by Jose Bastos
1 D
I oB
I oB  I LB (1  D )
I oB
TsVo

(1  D ) 2
2L
Chapter 7 DC-DC Switch-Mode
Converters
7-40
Limits of Cont./Discont. Conduction (2)
• Maximum values obtained
when D=0
TsVo
2L
TsVo
I oB ,max 
(1)
2L
TsVo
I LB 
(1  D)
2L
TV
I oB  s o (1  D) 2
2L
I LB ,max 
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-41
Discontinuous Conduction Mode (1)
• integrating the inductor
voltage over one period
Vd DTs  (Vo )1Ts  0
Vo D

Vd 1
• and since Pd=Po
I o 1

( 2)
Id D
• This occurs at light loads
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-42
Discontinuous Conduction Mode (2)
iL , peak 
Vd
DTs
L
and
1Ts
DT

iL , peak
1  s iL , peak
dt
t

dt
t
IL 

0 1Ts 
Ts  0 DTs

2
iL , peak  1Ts  2 
1  iL , peak  DTs 




2 
1Ts
2
Ts  DTs
D  1
 iL , peak
2
subst it ut ing iL , peak
IL 
Copyright © 2008
by Jose Bastos
Vd Ts
( D  1 ) D
2L
Chapter 7 DC-DC Switch-Mode
Converters
7-43
Limits of Cont./Discont. Conduction
Vd
Io  Id  I L 
DTs ( D  1 )
2L
TsVo
Io
I oB ,max 
(1) 1  D
( 2)
2L
Id
Io  Id 
I oB ,maxVd
Vo
D( D  D
Io
)
Id
I oB ,maxVd D 2
(Io  Id ) 
(Io  Id )
Vo
Id
I oB ,maxVd D 2
1
Vo
Id
Copyright © 2008
by Jose Bastos
; I d  Vo I o / Vd
Vo  D Vd
Chapter 7 DC-DC Switch-Mode
Converters
I oB ,max
Io
7-44
Limits of Cont./Discont. Conduction with Vo
constant
Vo
D
Vd
Copyright © 2008
by Jose Bastos
Io
I oB ,max
,
I oB ,max
Chapter 7 DC-DC Switch-Mode
Converters
TsVo

2L
7-45
Buck-Boost Converter: Effect of Parasitics
• The duty-ratio is limited to avoid these parasitic
effects from becoming significant
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-46
Buck-Boost Converter:
Output Voltage Ripple
Q I o DTs
Vo 

C
C
Vo DTs
Vo 
R C
Vo DTs
Ts

D
Vo
RC

(where =RC is the time const)
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-47
Cuk DC-DC Converter
• The output voltage can be higher or lower than the input
voltage
• The output voltage is negative
• Capacitor C1 stores and transfers energy
• in steady state average inductor voltages VL1, VL2 are zero
• VC1 is larger than Vd and Vo:
Copyright © 2008
by Jose Bastos
VC1  Vd  Vo
Chapter 7 DC-DC Switch-Mode
Converters
7-48
Cuk DC-DC Converter: Waveforms (1)
• when switch T is off
• VL1=Vd-VC1
• VL2=-Vo
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-49
Cuk DC-DC Converter: Waveforms (2)
• when switch T is on
• VL1=Vd
• VL2=VC1-Vo
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-50
Cuk DC-DC Converter: Waveforms (3)
• equating the integral voltages of L1, L2
over one period:
L1 :
(Vd  VC1 )(1  D)Ts  Vd DTs  0
1
Vd
(1  D)
L 2 : (Vo )(1  D)Ts  (VC1  Vo ) DTs  0
VC1 
VC1 
• thus
1
Vo
D
Vo
D

Vd 1  D
• and since Po=Pd
Io 1 D

Id
D
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-51
Cuk DC-DC Converter: pros and cons
Advantages
• input current and output current have small ripple
•Disadvantages
• requirement of capacitor C1 with large ripple current
capability
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-52
Full-Bridge DC-DC Converter
• Four quadrant operation is possible
• (TA+,TB-) (TA-,TB+) are switch pairs
• bipolar voltage switching: switches in each pair are
activated simultaneously
• unipolar voltage switching: switches are activated
independently
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-53
Bipolar voltage switching
vtri (t )  Vˆtri
t
Ts / 4
at t  t1 vtri  vcontrol
thus
vcontrol Ts
t1 
Vˆtri 4
ton of switch pair (TA+,TB-) is
v
T 1
1
ton  2t1  Ts  control s  Ts
2
Vˆtri 2 2
D1 

ton 1  vcontrol
 
 1
Ts 2  Vˆtri

Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-54
Bipolar voltage switching (2)
• Duty-rate of pair (TA+,TB-)

ton 1  vcontrol

D1 
 
 1
Ts 2  Vˆtri

• Duty-rate of pair (TA-,TB+)
D2  1  D1
• noting that
Vo  VAN  VBN  D1Vd  D2Vd  (2D1 1)Vd
Vo  (2D1 1)Vd
Copyright © 2008
by Jose Bastos
• D1 can vary between 0 and 1
• Vo can vary between -Vd and +Vd
Chapter 7 DC-DC Switch-Mode
Converters
7-55
unipolar voltage switching
TA , TB  on if  vcontrol  vtri  vcontrol
TA , TB  on if vtri  vcontrol
TA , TB  on if vtri  vcontrol
• Duty-rate of switch TA+

ton 1  vcontrol
D1 
 
 1
ˆ
Ts 2  Vtri

• Duty-rate of switch TB+
D2  1  D1
• noting that
Vo  VAN  VBN  (2D1 1)Vd
Copyright © 2008
by Jose Bastos
• D1 can vary between 0.5 and 1
• Vo can vary between 0 and Vd
Chapter 7 DC-DC Switch-Mode
Converters
7-56
Output Ripple in full-bridge DC converters
• unipolar voltage switching vs
bipolar voltage switching:
• switching frequency is
doubled
ripple
bipolar
unipolar
• ripple is reduced
• better frequency response
• better output voltage
• more complex switch control
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-57
Converter Comparison
• Buck, Boost, Buck-Boost, Cuk transfer energy in only one
direction
• full-bridge is capable of bi-direccional power flow
• in Buck, Boost, switch utilization is good
• in Buck-Boost, Cuk, full-bridge switch utilization is poor
CONCLUSION
• Prefer Buck, Boost converters
• If higher and lower voltages needed, use Buck-Boost, Cuk
• If four-quadrant operation needed, use full-bridge
Copyright © 2008
by Jose Bastos
Chapter 7 DC-DC Switch-Mode
Converters
7-58