This document discusses DC-DC buck converters. It begins by introducing different types of DC-DC converters and their applications. It then explains the objective of a buck converter is to efficiently reduce DC voltage. It discusses how a simple inefficient converter can achieve only 33% efficiency. Through the addition of an inductor and diode, lossless conversion becomes possible. The document explains the operating principles of the buck converter through examination of the inductor voltage and capacitor current in steady state. It derives the input-output voltage relationship and discusses how varying different circuit parameters affects the inductor current waveform. Finally, it covers RMS calculations for common periodic waveforms seen in converter circuits.

1. DC−DC Buck Converter
1

2. DC-DC switch mode converters
2

3. Basic DC-DC converters
3
• Step-down converter
• Step-up converter
• Derived circuits
• Step-down/step-up converter (flyback)
• (Ćuk-converter)
• Full-bridge converter
Applications
• DC-motor drives
• SMPS

4. Objective – to efficiently reduce DC voltage
out
in
in
out
I
I
V
V

4
DC−DC Buck
Converter
+
Vin
−
+
Vout
−
Iout
Iin
Lossless objective: Pin = Pout, which means that VinIin = VoutIout and
The DC equivalent of an AC transformer

5. Inefficient DC−DC converter
2
1
2
R
R
R
V
V in
out



in
out
V
V
R
R
R



2
1
2

5
+
Vin
−
+
Vout
−
R1
R2
If Vin = 15V, and Vout = 5V, efficiency η is only 0.33
The load
Unacceptable except in very low power applications

6. A lossless conversion of 15Vdc to average 5Vdc
6
If the duty cycle D of the switch is 0.33, then the average
voltage to the expensive car stereo is 15 ● 0.33 = 5Vdc. This is
lossless conversion, but is it acceptable?
R
+
15Vdc
–
Switch state, voltage
Closed, 15Vdc
Open, 0Vdc
Switch open
voltage
15
0
Switch closed
DT
T

7. Convert 15Vdc to 5Vdc, cont.
7
Try adding a large C in parallel with the load to
control ripple. But if the C has 5Vdc, then
when the switch closes, the source current
spikes to a huge value and burns out the
switch.
Rstereo
+
15Vdc
–
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
+
15Vdc
–
C
L
Rstereo
+
15Vdc
–
C
L
A DC-DC Buck Converter
lossless

8. C’s and L’s operating in periodic steady-state
Examine the current passing through a capacitor that is operating in
periodic steady state. The governing equation is
dt
t
dv
C
t
i
)
(
)
(  



t
o
t
o
t
o dt
t
i
C
t
v
t
v )
(
1
)
(
)
(
8
which leads to
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
),
(
)
( o
o t
v
T
t
v 

The conclusion is that






T
o
t
o
t
o
o dt
t
i
C
t
v
T
t
v )
(
1
0
)
(
)
(
or
0
)
( 

T
o
t
o
t
dt
t
i
the average current through a capacitor operating in periodic
steady state is zero
which means that

9. Now, an inductor
Examine the voltage across an inductor that is operating in periodic steady
state. The governing equation is
dt
t
di
L
t
v
)
(
)
(  



t
o
t
o
t
o dt
t
v
L
t
i
t
i )
(
1
)
(
)
(
9
which leads to
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
),
(
)
( o
o t
i
T
t
i 

The conclusion is that






T
o
t
o
t
o
o dt
t
v
L
t
i
T
t
i )
(
1
0
)
(
)
(
or
0
)
( 

T
o
t
o
t
dt
t
v
the average voltage across an inductor operating in periodic
steady state is zero
which means that

10. KVL and KCL in periodic steady-state
,
0
)
( 

loop
Around
t
v
,
0
)
( 

node
of
Out
t
i
0
)
(
)
(
)
(
)
( 3
2
1 



 t
v
t
v
t
v
t
v N

0
)
(
)
(
)
(
)
( 3
2
1 



 t
i
t
i
t
i
t
i N

10
Since KVL and KCL apply at any instance, then they must also be valid
in averages. Consider KVL,
0
)
0
(
1
)
(
1
)
(
1
)
(
1
)
(
1
3
2
1 




 









dt
T
dt
t
v
T
dt
t
v
T
dt
t
v
T
dt
t
v
T
T
o
t
o
t
T
o
t
o
t
N
T
o
t
o
t
T
o
t
o
t
T
o
t
o
t

0
3
2
1 



 Navg
avg
avg
avg V
V
V
V 
The same reasoning applies to KCL
0
3
2
1 



 Navg
avg
avg
avg I
I
I
I 
KVL applies in the average sense
KCL applies in the average sense

11. 11
Capacitors and Inductors
In capacitors:
dt
t
dv
C
t
i
)
(
)
( 
Capacitors tend to keep the voltage constant (voltage “inertia”). An ideal
capacitor with infinite capacitance acts as a constant voltage source.
Thus, a capacitor cannot be connected in parallel with a voltage source
or a switch (otherwise KVL would be violated, i.e. there will be a
short-circuit)
The voltage cannot change instantaneously
In inductors:
Inductors tend to keep the current constant (current “inertia”). An ideal
inductor with infinite inductance acts as a constant current source.
Thus, an inductor cannot be connected in series with a current source
or a switch (otherwise KCL would be violated)
The current cannot change instantaneously
dt
t
di
L
t
v
)
(
)
( 

12. 12
Vin
+
Vout
–
iL
L
C iC
Iout
iin
Buck converter
+ vL –
Vin
+
Vout
–
L
C
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

13. ,
dt
di
L
v L
L 
L
V
V
dt
di out
in
L 

,
dt
di
L
V
V L
out
in 

,
out
in
L V
V
v 

13
The input/output equation for DC-DC converters
usually comes by examining inductor voltages
Vin
+
Vout
–
L
C
Iout
iin
+ (Vin – Vout) –
iL
(iL – Iout)
Reverse biased, thus the
diode is open
for DT seconds
Note – if the switch stays closed, then Vout = Vin
Switch closed for
DT seconds

14. ,
dt
di
L
v L
L 
L
V
dt
di out
L 

,
dt
di
L
V L
out 

14
Vin
+
Vout
–
L
C
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.
,
out
L V
v 

for (1−D)T seconds

15. Since the average voltage across L is zero
      0
1 






 out
out
in
Lavg V
D
V
V
D
V
out
out
out
in V
D
V
V
D
DV 




in
out DV
V 
out
out
in
in I
V
I
V 
15
From power balance,
D
I
I 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

16. L
V
V
dt
di
V
V
v out
in
L
out
in
L



 ,
L
V
dt
di
V
v out
L
out
L



 ,
sec
/
A
L
V
V out
in 
sec
/
A
L
Vout

16
Examine the inductor current
Switch closed,
Switch open,
DT (1 − D)T
T
Imax
Imin
Iavg = Iout
From geometry, Iavg = Iout is halfway
between Imax and Imin
ΔI
iL
Periodic – finishes
a period where it
started

17. 17
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

18. 18
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

19. 19
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

20. RMS of common periodic waveforms, cont.
T
T
T
rms t
T
V
dt
t
T
V
dt
t
T
V
T
V
0
3
3
2
0
2
3
2
0
2
2
3
1








 

3
V
Vrms 
20
T
V
0
Sawtooth

21. RMS of common periodic waveforms, cont.
3
V
Vrms 
21
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
V
0
V
0

22. RMS of common periodic waveforms, cont.
22
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.
 
min
max I
I 
0
)
(t
i
the ripple
+
0
min
I
the minimum value
)
(t
i
max
I
min
I
=
 
2
min
max I
I
Iavg


avg
I

23. RMS of common periodic waveforms, cont.
 
 
2
min
2
)
( I
t
i
Avg
Irms 
 
 
2
min
min
2
2
)
(
2
)
( I
I
t
i
t
i
Avg
Irms 


 

    2
min
min
2
2
)
(
2
)
( I
t
i
Avg
I
t
i
Avg
Irms 


 

23
    2
min
min
max
min
2
min
max
2
2
2
3
I
I
I
I
I
I
Irms 





2
min
min
2
2
3
I
I
I
I
I PP
PP
rms 


min
max I
I
IPP 

Define

24. RMS of common periodic waveforms, cont.
24
2
min
PP
avg
I
I
I 

2
2
2
2
2
3
















 PP
avg
PP
PP
avg
PP
rms
I
I
I
I
I
I
I
4
2
3
2
2
2
2
2 PP
PP
avg
avg
PP
PP
avg
PP
rms
I
I
I
I
I
I
I
I
I 





2
2
2
2
4
3
avg
PP
PP
rms I
I
I
I 


Recognize that
12
2
2
2 PP
avg
rms
I
I
I 

avg
I
)
(t
i
min
max I
I
IPP 

 
2
min
max I
I
Iavg



25. Inductor current rating
 
2
2
2
2
2
12
1
12
1
I
I
I
I
I out
pp
avg
Lrms 




  2
2
2
2
3
4
2
12
1
out
out
out
Lrms I
I
I
I 

 
out
Lrms I
I
3
2

25
Max impact of ΔI on the rms current occurs at the boundary of
continuous/discontinuous conduction, where ΔI =2Iout
2Iout
0
Iavg = Iout
ΔI
iL
Use max

26. Capacitor current and current rating
  2
2
2
2
2
3
1
0
2
12
1
out
out
avg
Crms I
I
I
I 



3
out
Crms
I
I 
26
iL
L
C
Iout
(iL – Iout)
Iout
−Iout
0
ΔI
Max rms current occurs at the boundary of continuous/discontinuous
conduction, where ΔI =2Iout Use max
iC = (iL – Iout) Note – raising f or L, which lowers
ΔI, reduces the capacitor current

27. MOSFET and diode currents and current ratings
out
rms I
I
3
2

27
iL
L
C
Iout
(iL – Iout)
Use max
2Iout
0
Iout
iin
2Iout
0
Iout
Take worst case D for each

28. Worst-case load ripple voltage
Cf
I
C
I
T
C
I
T
C
Q
V out
out
out
4
4
2
2
1









28
Iout
−Iout
0
T/2
C charging
iC = (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

29. 29
Vin
+
Vout
–
iL
L
C iC
Iout
Vin
+
Vout
–
iL
L
C 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

30. There is a 3rd state – discontinuous
30
Vin
+
Vout
–
L
C
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

31. Onset of the discontinuous state
sec
/
A
L
Vout

   
f
L
D
V
T
D
L
V
I
onset
out
onset
out
out





1
1
2
31
2Iout
0
Iavg = Iout
iL
(1 − D)T
f
I
V
L
out
out
2
 guarantees continuous conduction
use max
use min
 
f
I
D
V
L
out
out
onset
2
1

Then, considering the worst case (i.e., D → 0),

32. Impedance matching
out
out
load
I
V
R 
equiv
R
2
2
D
R
D
I
V
D
I
D
V
I
V
R load
out
out
out
out
in
in
equiv 





32
DC−DC Buck
Converter
+
Vin
−
+
Vout = DVin
−
Iout = Iin / D
Iin
+
Vin
−
Iin
Equivalent from
source perspective
Source
So, the buck converter
makes the load
resistance look larger
to the source

33. 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
out
I
3
2 1.5 out
V
out
I
3
1 2 in
V
out
I
3
2
33
10A 10A
10A 40V 40V
Likely worst-case 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 p-p
Our D (Diode). 200V, 16A
BUCK DESIGN

34. Comparisons of Output Capacitor Ripple Voltage
Converter Type Volts (peak-to-peak)
Buck
Cf
Iout
4
34
10A
1500µF 50kHz
0.033V
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9A
Our C. 1500µF, 250V, 5.66A p-p
Our D (Diode). 200V, 16A

35. Minimum Inductance Values Needed to
Guarantee Continuous Current
Converter Type For Continuous
Current in the Input
Inductor
For Continuous
Current in L2
Buck
f
I
V
L
out
out
2

–
35
40V
2A 50kHz
200µH
BUCK DESIGN
Our M (MOSFET). 250V, 20A
Our L. 100µH, 9A
Our C. 1500µF, 250V, 5.66A p-p
Our D (Diode). 200V, 16A