# Chapter 7 DC-DC Switch-Mode Converters

```Chapter 7
DC-DC Switch-Mode Converters
• dc-dc converters for switch-mode dc power
supplies and dc-motor drives
Chapter 7 DC-DC Switch-Mode Converters
7-1
Block Diagram of DC-DC Converters
• Functional block diagram
Chapter 7 DC-DC Switch-Mode Converters
7-2
Stepping Down a DC Voltage
• A simple approach that shows the evolution
Chapter 7 DC-DC Switch-Mode Converters
7-3
Pulse-Width Modulation in
DC-DC Converters
• Role of PWM
Chapter 7 DC-DC Switch-Mode Converters
7-4
Duty cycle
• Switching frequency given by the sawtooth
• Control voltage ucontrol is the diffence
between reference and measured voltages
amplified by the controller
– ucontrol gives the on-time of the switch ton
• Relative on time, duty cycle
ton uohj
D

Ts Uˆ st
Chapter 7 DC-DC Switch-Mode Converters
7-5
Step-Down DC-DC Converter
• Pulsating input to
the low-pass filter
Chapter 7 DC-DC Switch-Mode Converters
7-6
Output voltage
• In continuous conduction mode (CCM)
Ts
Vo = 1
Ts
0
t
v o t dt = on Vd = DVd
Ts
• Output depends linearly from the
– Duty cycle
– And therefore also from control voltage ucontrol
• Discontinuous conduction mode (DCM)
– Will be discussed in next slides
– Also output current (power) has an effect on voltage
average
– Nonlinear dependence on the duty cycle
Chapter 7 DC-DC Switch-Mode Converters
7-7
Step-Down DC-DC Converter: Waveforms
• Steady state; inductor current flows continuously
Chapter 7 DC-DC Switch-Mode
Converters
7-8
Continuous Conduction Mode, CCM
• Voltage integral over the inductor must be
Vd – Vo ton = Vo Ts – ton
=>
Vo ton
=
=D
Vd Ts
• If there are no losses in the converter
VdId = VoIo
=>
Io Vd 1
=
=
Id Vo D
Chapter 7 DC-DC Switch-Mode Converters
7-9
”Transformer”
• In CCM operates like transformer without
galvanic isolation
– Duty cycle D acts like a portabless turns ratio
in a transformer, changes from 0 to 1
• Input current id changes also as the switch
operates
– In many cases filtering needed in the input
side too, depends on the supplying voltage
source
Chapter 7 DC-DC Switch-Mode Converters
7-10
FilteringYtytt
• Filter inductance ensures that
output is current source as
input normally is voltage
source
• Filter corner frequency ƒC
– Selected to be much smaller
than ƒs
– Attenuation of harmonics is high
– Output voltage can be assumed
to be constant in many cases
Chapter 7 DC-DC Switch-Mode Converters
7-11
Step-Down DC-DC Converter: Waveforms at
the boundary of Cont./Discont. Conduction
• Critical current below which inductor current
becomes discontinuous I  iLpeak  U d  U o t  DTs
LB
2
2L
on
Chapter 7 DC-DC Switch-ModeConverters
2L
U d  Uo   IoB
7-12
Step-Down DC-DC Converter:
Discontinuous Conduction Mode
• Steady state; inductor current discontinuous
Chapter 7 DC-DC Switch-Mode Converters
7-13
DCM with constant Ud
• Typical in motor drives
• Replacing Uo = DUd we
end up to
I LB 
ILB = IoB
Vd = vakio
DU d Ts
1  D   IoB
2L
ILB ,max = Ts Vd
8L
• Highest output current
for CCM needed at D =
0,5
D
0.5
Chapter 7 DC-DC Switch-Mode Converters
1.0
7-14
Uo/Ud in DCM with constant Ud
• Voltage integral over the inductor
U d  U o  DTs  U o 1Ts  0 
• Average of output current
Uo
D

U d D  1
D  1 U o
D  1 U d Ts

1Ts

D1
2
L
2
2L
Io
1
 1 =
4 I LB max D
I o  iLpeak
• Replacing we obtain
Uo
D2

U d D2  1 Io
4 I LB max
Chapter 7 DC-DC Switch-Mode Converters
7-15
Step-Down DC-DC Converter: Limits of
Cont./Discont. Conduction
• The duty-ratio of 0.5 has the highest value of the
critical current
Chapter 7 DC-DC Switch-Mode Converters
7-16
DCM with constant Uo
• Typical in power supplies
• Replacing Ud = Uo /D
I LB 
iLpeak
2
U d  Uo
TsU o

ton 
1  D 
2L
2L
• Maximum required current for CCM when
D=0
TsU o
I LB max 
2L
Chapter 7 DC-DC Switch-Mode Converters
7-17
Duty cycle in DCM with constant Uo
• Using previous equations
Uo
U
D
D
D



 o
U
Io
U d D  1 D  2 LI o
Ud
D o
U d DTs
U d D I LB max
• Duty cycle is
D
Uo
Ud
• Ratio of voltages is
nonlinear
 2 Uo
Io 
2
D


D
U d I LB max 

I o I LB max
1  Uo Ud
2
2
U o  D  D D  4 I o I LB max

Ud
2 I o I LB max
Chapter 7 DC-DC Switch-Mode Converters
7-18
Step-Down DC-DC Converter: Limits of
Cont./Discont. Conduction
• Output voltage is kept constant
Chapter 7 DC-DC Switch-Mode Converters
7-19
Step-Down Conv.: Output Voltage Ripple
• ESR is assumed to be zero
Chapter 7 DC-DC Switch-Mode Converters
7-20
Ripple (sykkeisyys in Finnish)
• Peak-to-peak
Q
1 I L Ts
U o 

C
2C 2 2
Uo
Uo
Uo
I L 
toff 
Ts  ton   1  D  Ts
L
L
L
π2

U o 1 Ts2

1  D   1  D  
Uo
8 LC
2

fc 

fs 
2
jossa fc 
• Ripple reduces when ƒC « ƒs
1
2π LC
– Compare to Fig 7-4
• In CCM doesn’t depend on output power
• Allowed ripple often < 1 %
– uo(t) = Uo can be assumed to be constant
Chapter 7 DC-DC Switch-Mode Converters
7-21
Step-Up DC-DC Converter (Boost)
• Output voltage must be greater than the input
Chapter 7 DC-DC Switch-Mode Converters
7-22
Step-Up DC-DC Converter Waveforms
U d ton  U d  U o  toff  0 
U d I d  Uo Io 
Uo
T
1
 s 
U d toff 1  D
Io U d

 1 D
Id Uo
• Continuous current conduction mode
Chapter 7 DC-DC Switch-Mode Converters
7-23
Step-Up DC-DC Converter: Limits of
Cont./Discont. Conduction, constant Uo
I LB 
iLpeak
2
U
TU
 d ton  s o D 1  D 
2L
2L
•Note IL ≠ Io
TU
2
I oB  I LB 1  D   s o D 1  D 
2L
I oB max 
2 TsU o
27 L
Chapter 7 DC-DC Switch-Mode Converters
7-24
Step-Up DC-DC Converter: DCM, constant Uo
U d DTs  U d  U o  1Ts  0

Io
1

I d D  1
U o D  1

Ud
1
U
TU
D  1
I d  d DTs
jolloin I o  s d D1
2L
1
2L
D
Uo
4 Uo
1  1 
Ud
27 U d
Chapter 7 DC-DC Switch-Mode Converters
 Uo
 Io

1


 Ud
 I oB max
7-25
Step-Up DC-DC Converter: Limits of
Cont./Discont. Conduction
• The output voltage is held constant
Chapter 7 DC-DC Switch-Mode Converters
7-26
Step-Up DC-DC Converter: Effect of
Parasitics
• The duty-ratio is generally limited before the
parasitic effects become significant
Chapter 7 DC-DC Switch-Mode Converters
7-27
Effect of Parasitics
• Especially nonidealities, i.e. resistive
losses in L and C
• At large D, peak value of current is large
when compared to the average value
– Major part of input voltage is needed to
overcome the voltage drop of parasitics
– Losses are increasing too
– Output voltage ripple and capacitor current
increase too
Chapter 7 DC-DC Switch-Mode Converters
7-28
Effect of Parasitics, Uo/Ud
• Losses of inductor and capacitor, rL , rc
– Resistors are percentages of load resistor R
• It can be shown that
Uo
1

Ud 1 D
1  D 2 R , jossa
R´
R 2 1  D 
RrC
´
R  rL 
1  D  
R  rC
R  rC
2
• Analysis of transfer functions in Chapter
10 gives this
Chapter 7 DC-DC Switch-Mode Converters
7-29
Uo/Ud when parasitics are 1 %
10
1
1 D
9
8
7
6
Uo
Ud
rL  rC  0
5
4
3
2
rL  rC  0, 01R
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D
Chapter 7 DC-DC Switch-Mode Converters
7-30
Effect of Parasitics, Uo/Ud
• Up to D ≈ 0,6 ideal and practical curve match
well
– Maximum increase is about 4,7 with 1 % resistors
– In practice always less than 10, i.e. infinite gain
impossible
• E.g. operating at D ≈ 0,88
– Supply voltage reduces
– Feedback increases D in order to increas Uo Because
of parasitics Uo actually drops and finally D = 1 and
short circuit
– D needs to be limited to some practical value
Chapter 7 DC-DC Switch-Mode Converters
7-31
Efficiency with parasitics
• For simplicity parasitics of L considered only
R´  rL  R 1  D 
Uo
1 D
1 D


,
2
2
U d R r  1  D 
  1  D 
L
2
R

rL
Uo
Ud
D
1 
1
Chapter 7 DC-DC Switch-Mode Converters
7-32
Efficiency with parasitics
• Only inductor
losses

Po
U o2 R
1


,
2
2
2
Po  PrL U o R  I L rL 1   1  D 
• The higher D the
smaller efficiency is

R
rL

1
D
1
Chapter 7 DC-DC Switch-Mode Converters
7-33
Step-Up DC-DC Converter Output Ripple
• ESR is assumed
to be zero
•Inductance has no
effect as it is in the
input side
U o 
U o DTs DTs
Q I o DTs U o DTs





C
C
R C
Uo
RC

Chapter 7 DC-DC Switch-Mode Converters
7-34
Step-Down/Up DC-DC Converter,
Buck-Boost
• How to derive Buck-Boost
• Series connection of Buck and Boost
I
K1
D2
+
L
+
Vd
D1
K2
Vo
C
–
–
• Buck, K2 is always open
• Boost, K1 is always closed
Chapter 7 DC-DC Switch-Mode Converters
7-35
Common current source
• Same current source, Boost turned around
• D1 needs to be removed, it only shortcircuits the current source
las keva katkoja ylös alais in käännetty
nos tava katkoja
K1
K1
D2
–
+
Vd
–
D1
I
I
K2
+
–
+
Vo
C
D2
Vd
I
–
Chapter 7 DC-DC Switch-Mode Converters
K2
Vo
C
+
7-36
+
Buck-Boost
• K2 can be removed too, it is only shortcircuiting current source
K1
id
D2
+
Vo
+
–
Vd
–
I
K2
Vo
C
+
Vd
–
+
vL
iL
L
-
Chapter 7 DC-DC Switch-Mode Converters
–
Vo
C
io
R
+
7-37
Step-Down/Up DC-DC Converter, BuckBoost
• The output voltage can be higher or lower than
the input voltage
Chapter 7 DC-DC Switch-Mode Converters
7-38
Step-Up DC-DC Converter: Waveforms
U d ton  U otoff  0
 U d DTs  U o 1  D  Ts  0

Uo
I
D
1 D

ja o 
Ud 1 D
Id
D
• CCM, Continuous conduction mode
Chapter 7 DC-DC Switch-Mode
Converters
7-39
Step-Up DC-DC Converter: Limits of
Cont./Discont. Conduction, Vo constant
I LB 
iLpeak
2
U
TU
 d DTs  s o 1  D 
2L
2L
TU
2
I o  I L  I d ja I oB  I LB  I dB  1  D  I LB  s o 1  D 
2L
TU
I oB max  s o
2L
Chapter 7 DC-DC Switch-Mode Converters
7-40
Step-Up DC-DC Converter: Discontinuous
Conduction Mode, Vo constant
U d DTs  U o 1Ts  0 
Uo D
I


ja o  1
U d 1
Id D
U
D  1
I L  d DTs
2L
1
U
T
U
U
2L
I d  I o  o 1Ts 1 s  1 
I o eli D  o 1  o
2L
Ts
U oTs
Ud
Ud
Chapter 7 DC-DC Switch-Mode Converters
Io
I oB max
7-41
Step-Up DC-DC Converter: Limits of
Cont./Discont. Conduction
• The output voltage is held constant
Chapter 7 DC-DC Switch-Mode Converters
7-42
Buck-Boost: Effect of Parasitics
• The duty-ratio is limited to avoid these parasitic
effects from becoming significant
Chapter 7 DC-DC Switch-Mode Converters
7-43
Step-Up DC-DC Converter: Output
Voltage Ripple
• ESR is assumed to be
zero
•Inductance not part of
ripple equation
•Note IL ≠ Io ≠ Io
U o DTs DTs
Q I o DTs U o DTs
U o 





C
C
R C
Uo
RC

Chapter 7 DC-DC Switch-Mode Converters
7-44
Cuk DC-DC Converter
• Name is based on the family name of the invertor
•The output voltage can be higher or lower than the
input voltage, dualism with buck-boost
• Capacitor C1 acts as intermediate energy storage
Chapter 7 DC-DC Switch-Mode Converters
7-45
Cuk DC-DC Converter: Waveforms
• The capacitor
voltage is assumed
constant
U d DTs  U d  UC1 1  D  Ts  0  UC1 
Ud
1 D
UC1  Uo  DTs   Uo 1  D  Ts  0  UC1 
Uo
D

Ud 1 D
Uo
D
Io U d 1  D


Id Uo
D
Chapter 7 DC-DC Switch-Mode Converters
7-46
Sepic Converter
• Single-Ended Primary Inductance
Converter
– L1:n correspons to Boost
– L2:n correspons to Buck-Boost
L1
+
Ud
+
u L1
iL1
+
–
uC 1
C1
T
–
D
i L2
–
L2
u L2
+
–
Chapter 7 DC-DC Switch-Mode Converters
+
C
R
Uo
–
7-47
Sepic comparison
– Continuous input current as in boost
– Step up is possible
– L1 and L2 can be connected magnetically, possible to
compensate ripple in one of the inductors
– Reduces start and short-circuit currents
– Galvanic isolation by adding a second winding to L2
– Switch and diode current and voltage higher than in
Boost
Chapter 7 DC-DC Switch-Mode Converters
7-48
Sepic, voltages
• Similar to Cúk
L1
U d D  1  D U o  U C1  U d 
L2
1 D
U C1D  1  D U o  0  U C1 
Uo
D
1 D
D

 U
U d D  1  D   U o 
Uo  U d   o 
D

 Ud 1 D
Chapter 7 DC-DC Switch-Mode Converters
7-49
Sepic, currents
• In steady state average current of C1
needs to be zero
iC1  iL 2 kytkimen
johtaessa
Switch conducts
Switch
is noteiconducting
iC1  iL1 kun
kytkimen
johda
1 D
 I L 2 D  1  D  I L1  1  D  I d  I L 2 
Id  Io
D
• IL1 = Id and IL2 = Io
Chapter 7 DC-DC Switch-Mode Converters
7-50
Full-bridge dc-dc converter
• Converter for UPS and DC-Motor Drives
• Either dc or ac output depending on modulation
• Four quadrant operation is possible
Chapter 7 DC-DC Switch-Mode Converters
7-51
Bi-polar voltage
switching
• Switches
• TA+ and TB–
• TA– and TB+
•Are controlled simultaneously
•Output changes between
+Ud:n and –Ud:n
Chapter 7 DC-DC Switch-Mode Converters
7-52
Control of output
• Triangle
utri  Uˆ tri
t
Ts 4
• On time of TA+ and TB–
0  t  Ts 4
T
ton  2t1  s
2
ucontrol Ts
t1 
Uˆ tri 4
ton 1  ucontrol 
D1 
 1 

ˆ
Ts 2 
U tri 
• On time of TA– and TB+
D2 =1  D1
U o  U AN  U BN   D1  D2 U d   2 D1  1U d
U
 d ucontrol  vakio  ucontrol
Uˆ tri
Chapter 7 DC-DC Switch-Mode Converters
7-53
Uni-polar voltage
switching
•Two control voltages with
opposite signs
ton 1  ucontrol 
D1 
 1 

Ts 2 
Uˆ tri 
U o   2 D1  1U d 
Ud
ucontrol  vakio  ucontrol
Uˆ tri
Chapter 7 DC-DC Switch-Mode Converters
7-54
RMS of output voltage
• Bipolar
1 Ts 2
U o, rms 
uo dt  U d

0
Ts
U r , rms  U o2, rms  U o2  U d 1   2D1  1  2U d D1  D12
2
• Unipolar
U o, rms 
ucontrol
1 Ts 2
4t1 2
u
dt

U

U
 U d 2 D1  1
o
d
d

ˆ
0
Ts
Ts
Utri
U r , rms  U o2, rms  U o2  U d
 2D1  1   2D1  12  U d
Chapter 7 DC-DC Switch-Mode Converters
6D1  4D12  2
7-55
Output Ripple in Converters for DCMotor Drives
• bi-polar and uni-polar voltage switching
Chapter 7 DC-DC Switch-Mode Converters
7-56
Switch Utilization in DC-DC Converters
• Ratio
• UT = maximum
voltage over switch
• IT = maximum current
Po U o I o

PT UT IT
•It varies significantly in
various converters
Chapter 7 DC-DC Switch-Mode Converters
7-57
Equivalent Circuits in DC-DC Converters
• replacing inductors
and capacitors by
current and voltage
sources, respectively
Chapter 7 DC-DC Switch-Mode Converters
7-58
Reversing the Power Flow in DC-DC
Conv.
• For power flow from right to left, the input
current direction should also reverse
Chapter 7 DC-DC Switch-Mode Converters
7-59
Efficiency and losses of power
semiconductor devices
• Losses of power semiconductor devices
– Conduction losses
– Switching losses, turn-on and turn-off
• For simplicity following investigation is on
Buck converter
– Can be generalized to other converters
Chapter 7 DC-DC Switch-Mode Converters
7-60
Conduction losses
• On-state voltage drop
– Assumed to be the same in the switch and diode
• Losses
toff
ton
PDC  PQ  PD  U on I o
 U on I o
 U on I o ,
Ts
Ts
• Efficiency

ton  toff  Ts
Po
Uo Io
Uo


Po  PDC U o I o  U on I o U o  U on
• Example
– Uo = 3 V, Uon = 0,5 V =>  = 85,7%
Chapter 7 DC-DC Switch-Mode Converters
7-61
Switching losses
• Infinite rise and fall time when switches
are turning on and off
• An ideal switch would
without voltage drop
• Current and voltage are assumed to
change linearly during turn-on and -off
Chapter 7 DC-DC Switch-Mode Converters
7-62
Switching inductive current
Chapter 7 DC-DC Switch-Mode Converters
7-63
Turning off and on
• T- conducts Io and it is turned off
– Voltage over it increases and when it is Ud diode D+
starts to conduct
– Because of parasitic inductances voltage exceeds Ud
• D+ conducts Io and T- is turned on
– Current inceares and exceeds Io because of diode
reverse recovery current
– After recovery of the diode voltage over T- drops to
nearly zero
Chapter 7 DC-DC Switch-Mode Converters
7-64
Switching losses, best case
• It is assumed that voltage and current are changing
simultaneously
ton ton
ton U d I o
Pon 
uidt

Ts 0
Ts 6
Poff 
Vd
Switch voltage
Turning on
toff
toff
Ts 0
toff U d I o
uidt 
Ts
6
Io
Turning off
Switch current
ton
Ts
toff
Chapter 7 DC-DC Switch-Mode Converters
7-65
Conduction and switching losses, best case
• It is assumed that ton = toff
ton U d I o
PAC  Pon  Poff 
Ts 3
• Efficiency including conduction and switching
losses

Po
Po  PDC  PAC

U o Io
Uo

U o I o  U on I o  U d I oton 3Ts U o  U on  U d ton 3Ts
• Example (cont.)
– Lets assume Ud = 48 V, Uo = 3 V, Uon = 0,5 V, fs = 50 kHz and ton
= toff = 0,3 µs and Ts = 20 µs
3


100%  80, 2%
– With these numbers
3  0,5  48  0,3 3  20
Chapter 7 DC-DC Switch-Mode Converters
7-66
Conduction and switching losses, worst case
• Current reaches final value before voltage drops
t Ri U d I o t FU U d I o
Pon 

Ts 2
Ts
2
Vd
t
U I
t U I
Poff  RU d o  Fi d o
Ts
2
Ts 2
Turning off
Switch voltage
Turning on
Switch current
Ts
tRi
tFu
tRu
Chapter 7 DC-DC Switch-Mode Converters
7-67
Conduction and switching losses, worst case
•
tRi
P

P

P

2
U
I
tRi = tFU = tRU = tFi
AC
on
off
d o
Ts
Po
Uo


Po  PDC  PAC U o  U on  2U d t Ri Ts
• Other values same as before, additionally, tRi =
3
0,3 µs  
100%  60, 7%
3  0,5  2  48  0,3 20
• Corresponding linear power supply
Uo
3

100%  100%  6, 25%
Ud
48
Chapter 7 DC-DC Switch-Mode Converters
7-68
```
Symbols

28 Cards

Transducers

13 Cards

Monorails

23 Cards