Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
Systems
(IJPEDS)
V
ol.
7,
No.
2,
June
2016,
pp.
279
–
293
ISSN:
2088-8694
279
Single-Switch
Soft-Switched
Boost
P
o
wer
F
actor
Corr
ector
f
or
Modular
A
pplications
T
oms
A.
Gonzlez
*
,
Daniel
O.
Mer
curi
*
,
Her
nn
E.
T
acca
*
,
and
Mximo
E.
Pupar
eli
**
*
Department
of
Electronics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Buenos
Aires.
**
HT
.
S.A.,
Ar
gentina.
Article
Inf
o
Article
history:
Recei
v
ed
No
v
14,
2015
Re
vised
Feb
13,
2016
Accepted
Feb
27,
2016
K
eyw
ord:
Po
wer
f
actor
corrector
Boost
con
v
erter
Soft-switching
P
assi
v
e
non-dissipati
v
e
snubber
ABSTRA
CT
Modern
dc
po
wer
supplies
pro
vide
po
wer
f
actor
correction
b
ut
t
he
classical
tw
o-stage
ap-
proach,
using
hard-switched
prere
gulators,
has
detrimental
ef
fects
on
ef
ficienc
y
and
reli-
ability
,
particularly
for
high
po
wer
applications.
W
ith
some
circuit
modifications
and
the
addition
of
a
fe
w
magnetic
components,
diodes
and
capacitors,
we
ha
v
e
turned
a
classical
boost
po
wer
f
actor
corrector
into
a
high
ef
ficienc
y
soft-switc
hed
v
ersion.
The
proposed
con
v
erter
turns
on
its
single
switch
with
zero
current
and
turns
it
of
f
with
zero
v
oltage.
In
this
paper
we
e
xplain
the
proposed
changes,
we
study
the
w
a
v
eforms
and
equations
and
we
v
erify
them
with
an
e
xperimental
prototype.
W
e
also
sho
w
ho
w
the
con
v
erter
can
be
used
for
modular
single-
and
three-phase
high
po
wer
applications.
Copyright
c
2016
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
T
oms
A.
Gonzlez.
Department
of
Electronics,
F
aculty
of
Engineering,
UB
A
A
v
.
P
aseo
Coln
850,
C1063A
CV
,
Buenos
Aires,
Ar
gentina.
T
el:
+54
(11)
4343-0893
/
4343-0092
tgonzalez@fi.uba.ar
1.
INTR
ODUCTION
Con
v
entional
po
wer
supplies
e
xhibit
lo
w
po
wer
f
actor
and
produce
harmonic
pollution
of
the
mains.
In
high
po
wer
applications,
passi
v
e
filtering
becomes
non
viable
due
to
the
increasing
size
and
weight
of
the
required
components,
such
as
lo
w
frequenc
y
line
filters
and
capacitor
banks.
No
w
adays,
po
wer
electronic
con
v
ersion
techniques
f
acilitate
the
use
of
acti
v
e
po
wer
f
actor
correction
methods
and
circuits,
kno
wn
as
PWM
rectifiers
[1].
The
boost
con
v
erter
is
one
of
the
most
used
topologies
in
PWM
rectifiers[2,
3,
4].
Single
phase
dc
po
wer
supplies
are
usually
composed
of
the
boost
front-end
or
prere
gulator
produces
near
unity
po
wer
f
actor
and
an
inter
-
mediate
high
v
oltage
b
us
and
an
isolated
dc-dc
con
v
erter
to
supply
the
desired
output
v
oltage
re
gulation.
At
high
po
wer
,
the
ef
ficienc
y
of
the
front
end
is
a
k
e
y
characteris
tic.
Man
y
high
ef
ficienc
y
boost
con
v
erters
ha
v
e
been
de
v
el-
oped
in
the
past
using
soft-switching
techniques
[5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16];
some
of
them
use
auxiliary
switches[5,
6,
7,
8,
9,
10,
11,
12],
making
them
more
comple
x
and
less
reliable.
P
assi
v
e
soft-switching
techniques
combine
higher
reliability
and
simplicity
of
design
[13,
14,
15].
The
po
wer
supply
manuf
acturer
sa
v
es
not
only
the
cost
of
auxiliary
switches,
b
ut
also
the
de
v
elopment
of
a
specific
inte
grated
circuit
to
dri
v
e
them
[17].
Modularization
is
an
attr
acti
v
e
technique
for
po
wer
con
v
erter
manuf
acturers
and
users
since
it
pro
vides
re-
dundanc
y
,
fle
xibility
,
reduction
of
manuf
acturing
cost
and
time
[18];it
also
applies
to
po
wer
supplies
with
po
wer
f
actor
correction.
P
aralleling
single
phase
con
v
erters
in
order
to
get
a
higher
po
wer
supply
w
as
sho
wn
in
[19]
with
boost
con
v
erters.
Se
v
eral
fields
e
xist
where
this
is
a
useful
practice:
single
phase
rail
w
ay
systems
[20]
or
L1/L2
battery
char
gers
for
plug-in
electric
and
h
ybrid
v
ehicles
[21].
Another
e
xample
of
modularization
are
three-phase
po
wer
supplies
manuf
actured
from
single
phase
modules
[22,
9,
13].
This
paper
introduces
a
boost-deri
v
ed
prere
gulator
with
no
v
el
passi
v
e
soft-switching
netw
orks
and
sho
ws
its
application
to
modular
high
po
wer
con
v
erters.
Zero
V
oltage
Switching
(ZVS)
at
turn-of
f
and
Zero
Current
Switching
(ZCS)
at
turn-on
pro
vide
a
reduction
in
losses
compared
to
a
classic
boost
con
v
erter
[1].
In
Section
2.
we
present
the
proposed
circuit
and
we
mak
e
qualitati
v
e
and
quantitati
v
e
analyses
to
deri
v
e
its
principle
of
operation,
w
a
v
eforms
and
characterizing
equations.
Ne
xt,
we
describe
the
e
xperimental
results
of
a
single
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJPEDS
Evaluation Warning : The document was created with Spire.PDF for Python.
280
ISSN:
2088-8694
D
A
D
B
C
O
L
B
L
C
D
D
D
C
L
S
Q
Load
C
S
v
ac
D
E
C
S
N
D
D
2
V
X
N
1
D
1
N
2
(a)
Schematic
circuit.
D
A
D
B
L
B
L
C
D
D
D
C
L
S
Q
C
S
D
E
V
X
v
I
V
O
v
C
S
v
C
E
i
C
i
L
S
i
L
C
v
L
B
L
M
i
L
M
(b)
Simplified
model
with
v
oltage
and
current
reference
direc-
tions.
Figure
1.
Boost
po
wer
f
actor
corrector
basic
module.
module
in
Section
3..
Finally
,
in
Section
4.,
we
e
xplain
the
considerations
that
shoul
d
be
accounted
for
in
order
to
use
the
con
v
erter
in
a
modular
approach.
2.
BOOST
PFC
SOFT
-SWITCHED
MODULE
AND
ITS
PRINCIPLE
OF
OPERA
TION.
Figure
1a
sho
ws
t
he
considered
module.
The
circui
t
composed
of
D
1
,
D
2
,
and
C
S
N
D
acts
as
a
lossless
snubber
for
diode
D
A
;
it
allo
ws
the
dischar
ge
of
the
s
tored
ener
gy
remaining
in
the
secondary
leakage
inductance
of
inductor
L
B
.
D
B
,
D
C
,
D
D
,
C
S
,
L
S
and
L
C
comprises
the
basic
soft-switching
netw
orks.
A
v
oltage
source,
V
X
,
connected
through
diode
D
E
enables
the
transistor
to
swit
ch
softly
throughout
the
entire
input
v
oltage
c
ycle.
W
e
will
discuss
the
need
of
this
source,
its
implementation
and
v
alue
in
the
follo
wing
sections.
The
circuit
is
based
on
the
tapped
boost
con
v
erter
,
with
a
transfer
function
gi
v
en
by:
V
O
v
I
=
1
+
d
(
t
)
N
2
N
1
1
d
(
t
)
;
(1)
which
approximates
the
classical
boost
transfer
function
when
N
1
>
N
2
and
still
produces
a
step
up
characteristic.
Thus,
it
is
possible
to
use
boost
PFC
commercial
controllers
[23]
with
a
v
erage
current
mode
control
[1]
in
order
to
re
gulate
the
output
v
oltage
and
mak
e
the
input
current
follo
w
the
input
v
oltage
w
a
v
eform.
The
simplified
circuital
model
of
Figure
1b
allo
ws
us
to
perform
the
follo
wing
qualitati
v
e
and
quantitati
v
e
analyses
of
the
switching
process.
2.1.
Qualitati
v
e
analysis
If
v
I
<
V
X
,
the
circuit
is
in
mode
I.
When
the
IGBT
turn-of
f
occurs,
at
t
=
t
0
,
the
circuit
transitions
from
the
conducti
on
state
sho
wn
in
Figure
2a
to
the
first
state
of
the
turn-of
f
process
sho
wn
in
Figure
2b.
The
inductance
of
both
L
B
and
L
C
force
diode
D
C
to
carry
the
magnetizing
current.
Starting
from
v
I
v
C
S
(
t
0
)
,
v
C
E
rises
linearly
due
to
the
constant
current
dischar
ging
of
C
S
.
In
order
to
ensure
zero
v
oltage
switching
v
C
S
(
t
0
)
must
be
equal
to
v
I
.
At
t
=
t
1
,
v
L
B
reaches
V
O
v
I
and
turns
diode
D
A
on.
C
S
k
eeps
dischar
ging
through
L
C
and
v
C
E
further
rises
(Figure2c)
until
v
C
E
=
V
O
and
v
C
S
=
v
I
V
O
,
at
t
=
t
2
;
D
B
turns
on
and
L
C
dischar
ges
the
stored
ener
gy
to
the
load
(Figure2f).
When
i
L
C
=
0
A
at
t
=
t
3
,
the
turn-of
f
ends
and
the
state
of
Figure
2g
starts.
After
the
switch
turns
on
at
t
=
t
4
,
its
current
starts
rising
gradually
due
to
L
C
;
zero
current
switching
occurs.
Simultaneously
,
the
LC
resonanc
e
of
C
S
and
L
S
in
v
erts
the
polarity
of
v
C
S
(Figure2h).
If
V
O
v
I
>
v
I
,
the
diode
D
C
turns
on
before
the
complete
di
schar
ge
of
L
S
,
v
C
S
clamps
at
v
I
,
and
the
state
pictured
in
Figure
2j
starts.
The
pre
vious
condition
is
equi
v
alent
to
v
I
<
V
O
=
2
and
it
is
of
paramount
importance
because
it
guarantees
the
starting
v
alue
of
v
C
S
in
the
ne
xt
switching
c
ycle
to
produce
the
ZVS
turn-of
f
of
the
con
v
erter
.
The
purpose
of
the
v
oltage
source
V
X
is
to
pro
vide
ZVS
turn-of
f
for
v
I
>
V
O
=
2
.
The
idea
is
to
clamp
the
v
oltage
of
C
S
to
a
v
alue
lo
wer
than
v
I
,
since
diode
D
C
will
not
be
able
t
o
turn
on.
In
order
to
clarify
this,
we
will
e
xplore
mode
II
that
occurs
for
v
I
>
V
X
.
At
t
=
t
0
,
the
transistor
turns
of
f
and
the
con
v
erter
enter
s
the
state
sho
wn
in
Figure
2d.
In
this
mode,
the
magnetizing
current
flo
ws
through
D
E
and
transfers
ener
gy
to
V
X
.
As
in
mode
I,
v
C
E
and
v
C
S
start
rising
linearly
b
ut
v
C
S
(
t
0
)
should
be
equal
to
V
X
so
as
to
ensure
ZVS
turn-of
f.
At
t
=
t
1
,
the
state
pictured
in
Figure
2e
starts:
v
L
B
turns
IJPEDS
V
ol.
7,
No.
2,
June
2016:
279
–
293
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
281
2*
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(a)
ON
state
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(b)
State
A
mode
I
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(c)
State
B
mode
I
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(d)
State
A
mode
II
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(e)
State
B
mode
II
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(f)
State
C
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(g)
OFF
state
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(h)
State
D
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(i)
State
E
D
A
D
B
L
B
L
C
v
I
D
D
D
C
L
S
Q
C
S
D
E
V
X
V
O
(j)
State
F
mode
I
Figure
2.
Operating
states
in
mode
I
(
v
I
<
V
X
)
and
mode
II
(
v
I
>
V
X
).
D
A
on,
taking
the
magnetizing
current,
and
C
S
continues
its
char
ge
through
L
C
.
The
final
state
of
turn-of
f,
sho
wn
in
Figure
2f,
starts
at
t
2
when
v
C
E
=
V
O
.
D
B
turns
on
and
L
C
dischar
ges
to
the
output
as
in
mode
I.
The
ZCS
turn-on
process
is
similar
as
before
and
presented
in
Figures
2h
and
2i.
The
dif
ference
is
that
the
last
state
is
missing,
because
the
resonant
oscillation
of
v
C
S
lasts
until
D
C
turns
of
f.
v
C
S
re
v
erts
its
polarity
to
V
X
,
the
required
v
alue
for
ZVS
turn-of
f
to
occur
.
In
order
to
guarantee
ZVS
for
all
v
alues
of
input
v
oltage,
V
X
should
tak
e
a
v
alue
that
i
ntroduces
the
circuit
to
mode
II
if
v
I
>
V
O
=
2
:
V
X
V
O
2
:
(2)
If
V
X
=
V
O
=
2
,
mode
I
occurs
for
v
I
<
V
O
=
2
and
mode
II
for
v
I
>
V
O
=
2
.
Selec
ting
a
smaller
v
alue
for
V
X
produces
an
additional
transfer
of
ener
gy
to
this
source
when
v
I
>
V
O
=
2
.
As
this
ener
gy
should
be
dissipated
or
sent
back
to
the
source,
it
is
important
to
minimize
it
by
adopting
V
X
=
V
O
=
2
.
This
condition
will
be
better
e
xpl
ained
in
the
quantitati
v
e
analysis
of
the
follo
wing
section.
The
tapped
boost
configuration
allo
ws
in
both
modes
the
dischar
ge
of
the
ener
gy
stored
in
L
C
at
the
end
of
the
switching
c
ycle,
thus
pro
viding
ZCS
turn-on.
Single-Switc
h
Soft-Switc
hed
Boost
P
ower
F
actor
Corr
ector
for
Modular
Applications
(T
.A.
Gonzlez)
Evaluation Warning : The document was created with Spire.PDF for Python.
282
ISSN:
2088-8694
2.2.
Quantitati
v
e
analysis:
equations
and
wa
v
ef
orms
By
considering
that
the
switching
frequenc
y
is
much
higher
than
the
li
ne
frequenc
y
,
we
emplo
y
an
approxi-
mately
constant
v
alue
of
v
I
for
each
switching
period.
Also,
we
ne
glect
output
v
oltage
ripple:
v
O
'
V
O
'
constant.
The
controller
modulates
the
transistor
duty-c
ycle
d
(
t
)
to
produce
i
L
M
(
t
)
'
I
I
max
j
sin(
!
t
)
j
.
As
a
result,
we
re
g
ard
i
L
M
as
constant
during
the
switching
c
ycle,
also
ne
glecting
the
current
ripple.
W
e
also
ne
glect
transistor
and
diode
v
oltage
drops
and
parasitic
capacitances.
2.2.1.
Mode
I
t
0
<
t
t
1
(Figure2b):
The
solution
for
v
C
S
is
v
C
S
(
t
)
=
v
C
S
(
t
0
)
+
1
C
S
Z
t
t
0
i
C
S
d
=
v
C
S
(
t
0
)
i
L
M
C
S
(
t
t
0
)
:
(3)
The
transistor
v
oltage
rises
follo
wing
v
C
E
(
t
)
=
v
I
v
C
S
(
t
)
=
v
I
v
C
S
(
t
0
)
+
i
L
M
C
S
(
t
t
0
)
;
(4)
where
the
ZVS
condition
arises
v
C
E
(
t
0
)
=
0
(
)
v
C
S
(
t
0
)
=
v
I
;
(5)
and
it
follo
ws
that
v
C
E
(
t
)
=
i
L
M
C
S
(
t
t
0
)
:
(6)
During
this
t
ime
interv
al
i
C
(
t
)
=
0
and
i
L
C
=
i
L
M
,
and
because
we
consider
i
L
M
almost
constant
consequently
v
LC
=
0
.
v
L
B
rises
follo
wing
the
changes
in
v
C
S
until
diode
D
A
turns
on,
reaching
v
L
B
(
t
1
)
=
V
O
v
I
=
N
1
+
N
2
N
1
v
C
S
(
t
1
)
:
(7)
Using
this
result
and
Equation
(4)
gi
v
es
v
C
S
(
t
1
)
=
(
V
O
v
I
)
N
1
N
1
+
N
2
(8)
and
v
C
E
(
t
1
)
=
v
I
N
1
N
1
+
N
2
(
V
O
v
I
)
;
(9)
which
we
can
solv
e
for
t
1
i
L
M
C
S
(
t
1
t
0
)
=
v
I
N
1
N
1
+
N
2
(
V
O
v
I
)
:
(10)
t
1
<
t
t
2
(Figure2c):
When
diode
D
A
turns
on,
the
boost
inductor
imposes
a
v
oltage
to
the
series
connection
of
L
C
and
C
S
.
C
S
char
ges
resonantly
according
to
d
2
v
C
S
dt
2
+
1
L
C
C
S
v
C
S
=
(
v
I
V
O
)
L
C
C
S
N
1
N
1
+
N
2
(11)
with
initial
conditions
v
C
S
(
t
1
)
=
(
v
I
V
O
)
N
1
N
1
+
N
2
;
(12)
dv
C
S
dt
(
t
1
)
=
i
L
M
C
S
:
(13)
W
e
solv
e
Equation
(11)
in
order
to
find
v
C
S
(
t
)
=
(
v
I
V
O
)
N
1
N
1
+
N
2
i
L
M
r
L
C
C
S
sin
t
t
1
p
L
C
C
S
(14)
IJPEDS
V
ol.
7,
No.
2,
June
2016:
279
–
293
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
283
and
v
C
E
(
t
)
=
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
+
i
L
M
r
L
C
C
S
sin
t
t
1
p
L
C
C
S
:
(15)
The
state
e
v
olv
es
as
calculated
until
D
B
turns
on
at
time
t
2
,
when
v
C
E
(
t
2
)
=
V
O
;
(16)
and
v
C
S
(
t
2
)
=
v
I
V
O
:
(17)
W
e
can
solv
e
Equation
(16)
for
t
2
i
L
M
r
L
C
C
S
sin
t
2
t
1
p
L
C
C
S
=
(
V
O
v
I
)
N
2
N
1
+
N
2
:
(18)
During
this
state
i
C
=
0
and
i
L
C
oscillates
according
to
i
L
C
(
t
)
=
C
S
dv
C
S
dt
=
i
L
M
cos
t
t
1
p
L
C
C
S
:
(19)
t
2
<
t
t
3
(Figure2f):
Through
the
clamping
action
of
D
B
,
the
ener
gy
stored
in
L
C
transfers
to
the
output
at
constant
v
oltage
i
L
C
(
t
)
=
i
L
C
(
t
2
)
+
1
L
C
Z
t
t
2
(
v
I
V
O
)
N
2
N
1
+
N
2
d
=
i
L
M
cos
t
2
t
1
p
L
C
C
S
(
V
O
v
I
)
N
2
N
1
+
N
2
(
t
t
2
)
L
C
(20)
Meanwhile,
i
C
(
t
)
=
0
,
v
C
E
(
t
)
=
V
O
,
and
v
C
S
(
t
)
=
v
I
V
O
.
This
state
ends
at
t
3
,
when
D
B
turns
of
f;
i
L
C
(
t
3
)
=
0
.
W
e
find
t
3
using
i
L
M
cos
t
2
t
1
p
L
C
C
S
(
V
O
v
I
)
N
2
N
1
+
N
2
(
t
3
t
2
)
L
C
=
0
:
(21)
t
3
<
t
t
4
(Figure2g).
This
interv
al
corresponds
to
the
con
v
entional
of
f
state
of
a
tapped
boost.
t
4
<
t
t
5
(Figure2h):
T
ransistor
Q
turns
on
and
its
current
has
tw
o
components
i
C
(
t
)
=
i
L
S
(
t
)
+
i
L
C
(
t
)
:
(22)
W
e
find
the
first
component
solving
the
dif
ferential
equation
d
2
v
C
S
dt
2
+
1
L
S
C
S
v
C
S
=
0
;
(23)
v
C
S
(
t
4
)
=
v
I
V
O
;
(24)
dv
C
S
dt
(
t
4
)
=
0
;
(25)
which
results
in
v
C
S
(
t
)
=
(
v
I
V
O
)
cos
t
t
4
p
L
S
C
S
;
(26)
and
lets
us
compute
i
L
S
(
t
)
=
(
v
I
V
O
)
r
C
S
L
S
sin
t
t
4
p
L
S
C
S
:
(27)
Simultaneously
,
L
C
starts
taking
the
boost
inductor
current
at
constant
v
oltage:
i
L
C
(
t
)
=
i
L
C
(
t
4
)
+
1
L
C
Z
t
t
4
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
d
=
1
L
C
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
(
t
t
4
)
:
(28)
Single-Switc
h
Soft-Switc
hed
Boost
P
ower
F
actor
Corr
ector
for
Modular
Applications
(T
.A.
Gonzlez)
Evaluation Warning : The document was created with Spire.PDF for Python.
284
ISSN:
2088-8694
This
state
ends
when
L
C
tak
es
the
output
current,
the
whole
magnetizing
current
of
the
tapped
boost
inductor
,
and
D
A
turns
of
f
i
L
C
(
t
5
)
=
i
L
M
=
1
L
C
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
(
t
5
t
4
)
:
(29)
In
practice,
we
ha
v
e
b
uilt
L
C
as
a
swinging
inductor
in
order
to
pro
vide
a
greater
inductance
for
lo
w
currents,
at
the
be
ginning
of
the
turn-on
process.
At
higher
currents,
the
ung
apped
part
of
the
core
saturates,
and
the
inductance
reduces.
This
structure
lets
us
use
a
smaller
core
than
a
non-saturable
inductor
w
ould
require.
It
can
also
pre
v
ent
the
destruction
of
the
inductor
due
to
e
xcessi
v
e
heat
dissipation
if
a
fully
saturable
core
were
to
be
used:
the
non-s
aturable
mass
of
the
core
acts
as
a
heat
sink
if
only
the
partial
g
ap
material
is
saturable.
As
a
result,
the
current
does
not
rise
linearly
as
stated
in
Equation
(28).
t
5
<
t
t
6
(Figure2i):
The
state
e
v
olv
es
with
current
i
L
S
oscillating
according
to
Equation
(27)
until
diode
D
C
turns
on,
when
v
C
S
=
v
I
.
T
o
find
t
6
we
solv
e
v
I
=
(
v
I
V
O
)
cos
t
6
t
4
p
L
S
C
S
;
(30)
which
has
a
solution
only
if
V
O
v
I
>
v
I
,
and
leads
to
the
condition
v
I
<
V
O
=
2
:
(31)
If
V
X
and
D
E
were
not
present,
ZVS
turn-of
f
w
ould
not
be
possible
for
v
I
>
V
O
=
2
:
the
v
oltage
of
C
S
w
ould
not
be
clamped
to
v
I
at
the
end
of
this
state,
b
ut
to
V
O
v
I
>
v
I
and
the
additional
v
oltage
w
ould
produce
a
hard
turn-of
f
in
the
follo
wing
switching
c
ycle.
If
we
someho
w
fix
V
X
=
V
O
=
2
,
D
E
w
ould
turn
on
for
v
alues
of
v
I
lar
ger
than
V
O
=
2
,
instead
of
D
C
,
and
v
C
S
w
ould
be
clamped
to
V
O
=
2
.
This
situation
corresponds
to
mode
II.
t
6
<
t
t
7
(Figure2j):
The
remaining
ener
gy
stored
in
L
S
dischar
ges
at
constant
v
oltage
v
I
i
L
S
(
t
)
=
i
L
S
(
t
6
)
1
L
S
Z
t
t
6
v
I
d
=
(
V
O
v
I
)
r
C
S
L
s
sin
t
6
t
4
p
L
S
C
S
v
I
L
S
(
t
t
6
)
:
(32)
It
ends
when
diodes
D
C
and
D
D
turn-of
f,
at
t
7
,
found
from
i
L
S
(
t
7
)
=
0
(
V
O
v
I
)
r
C
S
L
s
sin
t
6
t
4
p
L
S
C
S
v
I
L
S
(
t
7
t
6
)
=
0
:
(33)
t
7
<
t
T
S
+
t
0
(Figure2a)
ON-State.
The
pre
viously
deri
v
ed
w
a
v
eforms
of
v
C
E
,
i
C
,
v
C
S
,
i
L
C
,
and
i
L
S
are
depicted
in
Figure
3.
2.2.2.
Mode
II
As
pre
viously
deri
v
ed,
we
choose
V
X
=
V
O
=
2
.
This
will
be
considered
for
the
follo
wing
theoretical
analysis
and
w
a
v
eform
deri
v
ations.
t
0
<
t
t
1
(Figure2d):
The
solution
for
v
C
S
is
v
C
S
(
t
)
=
v
C
S
(
t
0
)
+
1
C
S
Z
t
t
0
i
C
S
d
=
v
C
S
(
t
0
)
i
L
M
C
S
(
t
t
0
)
:
(34)
The
transistor
v
oltage
rises
follo
wing
v
C
E
(
t
)
=
V
O
2
v
C
S
(
t
)
=
V
O
2
v
C
S
(
t
0
)
+
i
L
M
C
S
(
t
t
0
)
;
(35)
where
the
ZVS
condition
arises
v
C
E
(
t
0
)
=
0
(
)
v
C
S
(
t
0
)
=
V
O
2
;
(36)
IJPEDS
V
ol.
7,
No.
2,
June
2016:
279
–
293
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
285
t
v
C
E
t
0
t
1
t
2
t
3
V
O
v
I
N
2
+
V
O
N
1
N
1
+
N
2
i
C
t
v
C
S
t
i
L
C
t
v
I
N
1
(
v
I
V
O
)
N
1
+
N
2
v
I
N
1
(
v
I
V
O
)
N
1
+
N
2
v
I
V
O
i
L
S
t
t
4
t
5
t
6
t
7
(
V
O
v
I
)
q
C
S
L
S
i
L
M
+
(
V
O
v
I
)
q
C
S
L
S
i
L
M
i
L
M
ON
States
A
B
C
OFF
D
E
F
Figure
3.
Main
v
oltage
and
current
w
a
v
eforms
for
a
mode
I
switching
c
ycle.
A
dashed
line
sho
ws
a
better
approxima-
tion
to
the
real
w
a
v
eforms,
including
the
current
ripple
of
i
C
.
and
it
follo
ws
that
v
C
E
(
t
)
=
i
L
M
C
S
(
t
t
0
)
:
(37)
During
this
time
interv
al
i
C
(
t
)
=
0
and
i
L
C
=
i
L
M
,
and
because
we
consider
i
L
M
almost
constant
consequently
v
LC
=
0
.
v
L
B
rises
follo
wing
the
changes
in
v
C
S
until
diode
D
A
turns
on,
reaching
v
L
B
(
t
1
)
=
V
O
v
I
=
N
1
+
N
2
N
1
V
O
2
v
I
v
C
S
(
t
1
)
:
(38)
Using
this
result
and
Equation
(35)
gi
v
es
v
C
S
(
t
1
)
=
V
O
2
v
I
(
V
O
v
I
)
N
1
N
1
+
N
2
;
(39)
and
v
C
E
(
t
1
)
=
v
I
+
N
1
N
1
+
N
2
(
V
O
v
I
)
;
(40)
which
we
can
solv
e
for
t
1
i
L
M
C
S
(
t
1
t
0
)
=
v
I
+
N
1
N
1
+
N
2
(
V
O
v
I
)
:
(41)
t
1
<
t
t
2
(Figure2e):
When
diode
D
A
turns
on,
the
boost
inductor
imposes
a
v
oltage
to
the
series
connection
of
L
C
and
C
S
.
C
S
char
ges
resonantly
according
to
d
2
v
C
S
dt
2
+
1
L
C
C
S
v
C
S
=
1
L
C
C
S
V
O
2
v
I
(
V
O
v
I
)
N
1
N
1
+
N
2
(42)
with
initial
conditions
v
C
S
(
t
1
)
=
V
O
2
v
I
(
V
O
v
I
)
N
1
N
1
+
N
2
;
(43)
dv
C
S
dt
(
t
1
)
=
i
L
M
C
S
:
(44)
Single-Switc
h
Soft-Switc
hed
Boost
P
ower
F
actor
Corr
ector
for
Modular
Applications
(T
.A.
Gonzlez)
Evaluation Warning : The document was created with Spire.PDF for Python.
286
ISSN:
2088-8694
W
e
can
solv
e
Equation
(42)
in
order
to
find
v
C
S
(
t
)
=
V
O
2
v
I
(
V
O
v
I
)
N
1
N
1
+
N
2
i
L
M
r
L
C
C
S
sin
t
t
1
p
L
C
C
S
;
(45)
and
v
C
E
(
t
)
=
v
I
+
(
V
O
v
I
)
N
1
N
1
+
N
2
+
i
L
M
r
L
C
C
S
sin
t
t
1
p
L
C
C
S
:
(46)
The
state
e
v
olv
es
as
calculated
until
D
B
turns
on
at
time
t
2
,
when
v
C
E
(
t
2
)
=
V
O
;
(47)
and
v
C
S
(
t
2
)
=
V
O
2
V
O
=
V
O
2
:
(48)
W
e
can
solv
e
Equation
(47)
for
t
2
V
O
=
v
I
+
(
V
O
v
I
)
N
1
N
1
+
N
2
+
i
L
M
r
L
C
C
S
sin
t
2
t
1
p
L
C
C
S
:
(49)
During
this
state
i
C
=
0
and
i
L
C
oscillates
according
to
i
L
C
(
t
)
=
C
S
dv
C
S
dt
=
i
L
M
cos
t
t
1
p
L
C
C
S
:
(50)
t
2
<
t
t
3
(Figure2f):
Through
the
clamping
action
of
D
B
,
the
ener
gy
stored
in
L
C
transfers
to
the
output
at
constant
v
oltage
i
L
C
(
t
)
=
i
L
C
(
t
2
)
+
1
L
C
Z
t
t
2
(
v
I
V
O
)
N
2
N
1
+
N
2
d
=
i
L
M
cos
t
2
t
1
p
L
C
C
S
(
V
O
v
I
)
N
2
N
1
+
N
2
(
t
t
2
)
L
C
:
(51)
Meanwhile,
i
C
(
t
)
=
0
,
v
C
E
(
t
)
=
V
O
,
and
v
C
S
(
t
)
=
V
O
=
2
.
This
state
ends
at
t
3
,
when
D
B
turns
of
f;
i
L
C
(
t
3
)
=
0
.
W
e
find
t
3
using
i
L
M
cos
t
2
t
1
p
L
C
C
S
(
V
O
v
I
)
N
2
N
1
+
N
2
(
t
3
t
2
)
L
C
=
0
:
(52)
t
3
<
t
t
4
(Figure2g)
This
interv
al
corresponds
to
the
con
v
entional
of
f
state
of
a
tapped
boost.
t
4
<
t
t
5
(Figure2h):
T
ransistor
Q
turns
on
and
its
current
has
tw
o
components
i
C
(
t
)
=
i
L
S
(
t
)
+
i
L
C
(
t
)
:
(53)
W
e
find
the
first
component
solving
the
dif
ferential
equation
d
2
v
C
S
dt
2
+
1
L
S
C
S
v
C
S
=
0
;
(54)
v
C
S
(
t
4
)
=
V
O
2
;
(55)
dv
C
S
dt
(
t
4
)
=
0
;
(56)
which
results
in
v
C
S
(
t
)
=
V
O
2
cos
t
t
4
p
L
S
C
S
;
(57)
and
lets
us
compute
i
L
S
(
t
)
=
V
O
2
r
C
S
L
S
sin
t
t
4
p
L
S
C
S
:
(58)
IJPEDS
V
ol.
7,
No.
2,
June
2016:
279
–
293
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
287
T
able
1.
Specifications
and
components
of
the
e
xperimental
module.
Specification
V
alue
De
vice
V
alue
P
O
3
kW
Q
IRGP50B60PD1
(600
V/33
A)
v
ac
220
V/
50
Hz
D
A
HF
A16TB120
(1200
V/16
A)
V
O
500
V
D
B
;C
;D
;
1
;
2
;
3
;
5
MUR860
(600
V/8
A)
f
S
70
kHz
D
4
1.5KE16
Simultaneously
,
L
C
starts
taking
current
at
constant
v
oltage
from
the
boost
inductor
i
L
C
(
t
)
=
i
L
C
(
t
4
)
+
1
L
C
Z
t
t
4
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
d
=
1
L
C
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
(
t
t
4
)
:
(59)
This
state
ends
when
L
C
tak
es
the
output
current,
the
whole
magnetizing
current
of
the
tapped
boost
inductor
,
and
D
A
turns
of
f
i
L
C
(
t
5
)
=
i
L
M
=
1
L
C
v
I
(
v
I
V
O
)
N
1
N
1
+
N
2
(
t
5
t
4
)
:
(60)
t
5
<
t
t
6
(Figure2i):
The
state
e
v
olv
es
with
current
i
L
S
oscillating
according
to
Equation
(58)
until
diode
D
E
turns
of
f.
W
e
solv
e
0
=
V
O
2
r
C
S
L
S
sin
t
6
t
4
p
L
S
C
S
;
(61)
in
order
to
find
t
6
t
4
=
p
L
S
C
S
:
(62)
Thus,
capacitor
C
S
ends
char
ged
with
v
C
S
(
t
6
)
=
V
O
2
;
(63)
which
is
the
v
oltage
needed
in
the
ne
xt
mode
II
turn-on
switching
to
produce
ZVS.
t
6
<
t
T
S
+
t
0
(Figure2a)
ON-State.
The
w
a
v
eforms
for
this
mode
are
similar
to
those
pict
ured
in
Figure
3.
The
dif
ferences
lie
in
the
characteristic
v
alues
and
in
the
i
L
S
(
t
)
plot,
which
presents
a
complete
resonant
half
c
ycle
up
to
t
6
instead
of
the
linear
dischar
ge.
3.
EXPERIMENT
AL
RESUL
TS
Figure
4
sho
ws
the
complete
schematic
of
the
boost
PFC
module.
In
order
to
control
the
po
wer
f
actor
corrector
,
we
used
the
UC3854
inte
grated
circuit
[24,
23],
which
implements
a
v
erage
input
current
control
[1].
The
controller
pro
vides
both
output
v
oltage
re
gulation
and
sinusoidal
input
current.
The
input
current
is
indirectly
measured
with
current
transformers
C
T
1
and
C
T
2
as
proposed
in
[24].
Ex-
tra
diodes
D
3
and
transient
v
oltage
suppressor
D
4
were
included
to
clamp
o
v
erv
oltages
produced
by
C
T
2
leakage
inductance.
The
v
oltage
source
V
X
is
implemented
as
proposed
i
n
Figure
10b
with
a
47
F
capacitor
,
a
1
k
resistor
and
diode
D
5
,
in
order
to
return
some
of
the
ener
gy
to
the
input
and
impro
v
e
the
ef
fi
cienc
y
.
The
v
alue
of
the
clamp
resistor
w
as
e
xperimentally
adjusted
to
set
the
desired
v
alue
of
V
X
=
V
O
=
2
,
as
we
deri
v
ed
in
Section
2..
T
able
1
lists
the
complete
specifications
and
significant
components
of
the
prototype.
Figure
5
is
a
photograph
of
the
prototype
and
sho
ws
the
relati
v
e
sizing
of
the
main
inducti
v
e
components.
3.1.
Switching
W
a
v
ef
orms
W
e
ha
v
e
captured
switch
v
oltage
and
current
w
a
v
eforms
for
output
po
wers
of
1
:
5
kW
(Figure6)
and
3
kW
(Figure7).
Both
were
measured
near
the
peak
input
v
oltage.
Some
features
that
we
described
in
the
theoretical
w
a
v
eforms
of
Figure
3
are
present:
the
slo
wly
rising
current
during
turn-on,
the
resonant
oscillation
of
capacitor
C
S
with
inductor
L
S
,
and
the
linearly
increasing
v
oltage
during
turn-of
f.
Single-Switc
h
Soft-Switc
hed
Boost
P
ower
F
actor
Corr
ector
for
Modular
Applications
(T
.A.
Gonzlez)
Evaluation Warning : The document was created with Spire.PDF for Python.
288
ISSN:
2088-8694
D
A
D
B
L
B
L
C
D
D
D
C
L
S
Q
Load
C
S
v
ac
D
E
C
S
N
D
D
2
N
1
D
1
N
2
:
1
F
:
147
F
47
F
1
k
UC3854
Vss
1
14
Ct
12
Rset
Vg
16
V
ref
9
PKLMT
2
CAout
3
Isens
4
MulOut
5
Vsens
V
Aout
7
11
Vcc
15
ENA
10
SS
13
8
V
rms
6
IA
C
D
5
D
4
2
:
7
k
330
k
330
k
470
F
470
F
1
F
270
k
270
k
10
k
4
:
7
k
10
10
2
:
7
k
22
2
:
2
nF
8
:
2
k
1
F
10
k
270
pF
1
:
5
k
10
F
22
k
10
F
:
1
F
150
k
680
k
820
k
180
k
22
k
:
1
F
:
47
F
10
F
1
:
5
nF
20
pF
20
nF
22
k
6
:
8
k
2
:
2
1
:
2
k
1
nF
5
:
6
k
47
nF
180
k
2
:
2
k
1
F
4
:
7
k
v
O
v
O
V
R
E
F
v
P
V
C
C
=
+18
V
V
C
C
v
P
V
R
E
F
v
I
v
I
v
I
v
G
v
G
TIP31
TIP32
1N5819
1N5819
1N4740
1N4007
KBPC5010
D
3
1N5819
1N5819
C
T
2
C
T
1
v
I
(36)
22
nF
80
H
250
H
20
H
(7)
Figure
4.
Complete
circuit
schematic
of
the
boost
PFC
module.
Figure
5.
Photograph
of
the
3
kW
prototype.
The
main
inducti
v
e
components
are
labeled.
Additional
non-ideal
features
appear:
IGBT
tail
current
and
transistor
turn-of
f
o
v
erv
oltage
in
Figure
6c
—
limited
by
the
non-dissipati
v
e
snubber
of
Figure
1a—,
inductor
current
ripple
and
o
v
erdamped
oscillation
of
switch
v
oltage
after
L
C
dischar
ges
(Figure6a).
Unmodeled
parasitic
components
produce
some
of
the
features
present
in
Figure
7a,
which
are
not
justified
in
the
theoretical
w
a
v
eforms
of
Figure
3
or
observ
able
in
Figure
6a.
F
or
e
xample,
we
added
diodes
D
3
and
D
4
in
order
to
reach
the
final
nominal
po
wer
of
3
kW.
These
diodes
were
remo
v
ed
to
measure
the
w
a
v
eforms
of
Figure
6a.
IJPEDS
V
ol.
7,
No.
2,
June
2016:
279
–
293
Evaluation Warning : The document was created with Spire.PDF for Python.