Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
7,
No.
5,
October
2017,
pp.
2374
–
2381
ISSN:
2088-8708
2374
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Accurate
Symbolic
Steady
State
Modeling
of
Buck
Con
v
erter
Ek
o
Setiawan
1
,
T
akuya
Hirata
2
,
and
Ichijo
Hodaka
3
1
Interdisciplinary
Graduate
School
of
Agriculture
and
Engineering,
Uni
v
ersity
of
Miyazaki,
Japan
2
Department
of
Electronic
Mechanical
Engineering,
National
Institute
of
T
echnology
,
Oshima
Colle
ge,
Japan
3
Department
of
En
vironmental
of
Robotics,
Uni
v
ersity
of
Miyazaki,
Japan
Article
Inf
o
Article
history:
Recei
v
ed:
Apr
6,
2017
Re
vised:
Jun
9,
2017
Accepted:
Jun
23,
2017
K
eyw
ord:
Steady
state
analysis
F
ourier
series
Switching
circuit
Buck
con
v
erter
ABSTRA
CT
Steady
state
analysis
is
fundamental
to
an
y
electric
and
electronic
circuit
design.
Buck
con
v
erter
is
one
of
most
popular
po
wer
electronics
circuit
and
has
been
analyzed
in
v
arious
situations.
Although
the
beha
vior
of
b
uck
con
v
erters
can
be
understood
approximately
by
the
well-kno
wn
state
space
a
v
eraging
method,
little
is
kno
wn
in
the
sense
of
detailed
beha
vior
or
e
xa
ct
solution
to
equations.
In
this
paper
a
steady
state
analys
is
of
b
uck
con
v
erter
is
proposed
which
allo
ws
the
e
xact
cal
culation
of
steady
state
response.
Our
e
xact
solution
is
e
xpressed
as
a
F
ourier
series.
Our
result
is
compared
with
numerical
calculation
to
be
v
erified.
Our
method
copes
with
more
complicated
problems
such
as
describing
a
v
erage
po
wer
and
root-mean-square
po
wer
that
are
most
critical
issues
in
po
wer
electronics
circuit.
Copyright
c
2017
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ichijo
Hodaka
Department
of
En
vironmental
Robotics,
Uni
v
ersity
of
Miyazaki
1-1,
Gakuen
Kibanadai-nishi,
Miyazaki,
889-2192,
Japan
Phone:
+81
985
587
352
E-mail:
hijhodaka[at]cc.miyazaki-u.ac.jp
1.
INTR
ODUCTION
An
analysis
of
steady-state
re
sponse
of
a
system
is
important
k
e
y
in
circuit
design
and
control,
included
dc-dc
con
v
erter
.
In
con
v
ential
m
ethod,
the
steady
state
of
dc-dc
con
v
erter
is
assumed
as
the
constant
v
alue.
Man
y
researches
based
on
state
space
a
v
eraging
method
and
high
switching
frequenc
y
assumpt
ion
observ
e
steady
state
response
[1,
2,
3,
4,
5].
The
method
gi
v
es
simple
w
ay
to
analyze
b
ut
some
ripples
are
undescribed
clearly
.
The
po
wer
electronic
handbook
approximate
linear
ripple
to
analyze
dc-dc
con
v
erter
more
accurately
[6].
The
approximation
may
be
correct
if
the
switching
frequenc
y
is
high.
Since
there
are
some
limitation
in
component,
some
high
frequenc
y
is
not
al
w
ays
reached.
The
importance
of
accurate
steady-state
analysis
has
already
noticed
in
man
y
researc
h
e
s
[7,
8,
9,
10,
11,
12].
A
significant
part
of
the
design
of
circuits
requires
the
simulation
of
the
steady-state
response.
P
arameters
such
as
the
g
ain,
harmonic
distortion
and
the
input
and
output
impedances
are
studied
in
the
steady-state
mode
of
operation
[7].
Using
con
v
entional
time-stepping
simulations
and
w
aiting-time
for
possible
steady
state
is
often
not
practical
because
in
most
cases
the
time
constants
of
the
modes
are
much
lar
ger
than
the
switching
period
[8].
In
con
v
entional
method
of
dc-dc
con
v
erter
analysis,
steady
state
ripple
v
alues
are
ne
gligible,
compared
to
the
steady
state
v
alues
themselv
es.
Switching
po
wer
con
v
erters
are
inherently
nonlinear
and
consequently
it
is
v
ery
dif
ficult
to
calculate
the
root-mean-square
(RMS)
v
alues
of
the
state
v
ariable
ripple.
These
RMS
v
alues
are
important
in
order
to
calculate
the
current
stresses
of
the
dif
ferent
po
wer
con
v
erter
de
vices
as
well
as
to
filter
design
in
order
to
meet
the
gi
v
en
specifications
[9].
Though
po
wer
electronic
handbook
[6]
sho
ws
the
RMS
calculation
using
approximation
linear
ripple,
the
result
is
not
absolutely
correct
due
to
linear
approximation.
In
order
to
achie
v
e
a
high
performance,
proper
design
and
control,
it
is
necessary
to
ha
v
e
an
e
xact
model
of
con
v
erter
[10,
11].
High
accurac
y
is
one
of
major
features
of
a
good
modeling
[11].
The
dc
-dc
con
v
erter
analysis
can
be
classified
into
tw
o
cate
gories,
numeric
and
symbolic
analysis.
The
numeric
analysis
is
found
in
[7,
8,
10,
12,
13,
14,
15].
The
numeric
analysis
observ
es
the
system
response
by
inputting
para
meter
v
alues
i
nto
model.
The
numeric
analysis
needs
some
computation-time
to
sho
w
the
output
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v7i5.pp2374-2381
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2375
response.
Relation
between
each
parameter
is
not
describe
in
the
equation.
The
relation
between
parameter
and
output
is
al
w
ays
observ
ed
by
comparison
between
parameter
change
to
output
change.
The
[7]
construct
impedance
or
admittance
matrix
of
dc-dc
con
v
erter
.
The
output
response
of
the
con
v
erter
is
calculated
by
Ne
wton-Raphson.
The
steady-state
output
is
solv
ed
per
fix
ed
time-step
(fix
ed
sampling
interv
al).
The
accurac
y
of
analysis
is
dependent
on
time-step.
Fe
wer
sampling
point
causes
less
accurac
y
.
On
the
other
side,
more
sampling
points
need
more
calculation
time.
The
[13]
substitute
original
circuit
with
periodically
switched
linear
(PSL)
circuit.
The
PSL
can
be
observ
ed
by
F
ourier
series.
The
paper
use
110
as
sequence
number
in
F
ourier
series
summation.
The
[14]
also
use
F
ourier
series
to
simulate
steady-state
response.
Comparing
with
[13],
the
[14]
only
uses
21
sequences.
The
[13]
applies
F
ourier
series
of
current
switching
part.
The
F
ourier
series
of
switching
part
substitutes
original
part
to
be
analyzed
by
Kirchhof
f
V
oltage
and
Current
La
w
.
The
paper
uses
180
sequence
number
to
dra
w
a
steady
state
response
of
system.
The
[12]
analyzes
b
uck
con
v
erter
in
frequenc
y
domain
due
to
high
accurac
y
comparing
with
con
v
entional
time-domain.
The
paper
simulate
in
three
dif
ferent
sequences
number
that
0,
10
and
100.
The
paper
sho
ws
that
10th
sequences
order
is
enough
to
describe
steady
state
response.
In
F
ourier
series
based
method,
greater
number
send
the
accurate
steady-state
model
b
ut
it
need
more
calculation.
High
order
sequence
number
doesnt
bring
significant
accurac
y
.
Determination
of
the
proper
sequence
number
is
another
problem
beside
the
main
steady-state
analysis
problem.
Contrary
with
numerical
analysis,
symbolic
analysis
describes
relation
between
parameter
and
output
in
the
equation.
The
relation
between
parameter
and
output
can
be
observ
ed
roughly
by
the
equation.
The
symbolic
analysis
is
found
in
[9]
and
[11].
Symbolic
analysis
is
complicated
to
be
done
although
it
sho
w
relation
between
parameter
and
output
in
equation.
The
[9]
perform
steady
state
symbolic
analysis
to
calculate
rms
v
alue.
The
paper
sho
w
solution
in
a
matrix
form
that
is
more
complicated
than
o
r
dinary
equation.
The
a
v
erage
and
rms
calculation
include
term
which
is
obtained
from
zero
deri
v
ati
v
e
state
assumption.
Since
the
ripple
is
e
xist
and
cannot
be
ne
glected,
this
assumption
is
contrast
with
early
definition.
The
[11]
solv
e
s
teady-state
equation
by
Laplace
transform
and
re
v
ert
back
i
nto
time-domain
by
in
v
erse
of
Laplace
transform.
The
solution
need
the
kno
wn
initial
v
alue.
The
Z-transform
is
applied
due
to
similarity
between
initial
v
alue
and
last
v
alue
in
one
period.
Though
the
paper
sho
w
symbolic
analysis,
the
proses
is
long
enough
due
to
calculation
of
three
transformations
(Laplace
transform,
Z-transform
and
in
v
erse
of
Laplace
transform).
In
this
paper
,
an
alternati
v
e
method
is
proposed
that
accurately
predict
and
analyze
the
steady
state
of
switching
po
wer
con
v
erter
.
The
proposed
method
based
on
F
ourier
series
since
man
y
references
sho
w
the
accurac
y
[12,
13,
14].
The
reco
v
ery
function
is
also
proposed
to
generate
analysis
without
dependent
in
number
of
sequences
orders.
The
paper
is
subdi
vided
in
se
v
eral
sections
to
present
clearly
e
xplanation.
Section
2
sho
ws
the
basic
idea
of
proposed
method.
Section
3
discusses
about
implementation
of
proposed
m
ethod
in
b
uck
con
v
erter
and
the
reco
v
ery
function.
Complete
steady
state
output
function
of
dc-dc
con
v
erter
is
sho
wn
in
this
section.
Section
4
v
erify
proposed
steady
state
function
by
circuit
simulation.
Finally
,
the
conclusion
of
this
paper
is
declared
in
Section
5
2.
PR
OPOSED
FOURIER
SERIES
METHOD
Man
y
phenomena
studied
in
electric
circuit
are
periodic
in
nature
when
time
goes
to
infinit
y
.
These
periodic
functions
in
time
t
can
be
e
xpressed
as
F
ourier
series
as
in
the
follo
wing
equation
[16].
f
(
t
)
=
0
+
X
n
6
=0
n
e
j
n!
t
=
:::
+
2
e
j
2
!
t
+
1
e
j
!
t
+
0
+
1
e
j
!
t
+
2
e
j
2
!
t
+
:::
(1)
with
F
ourier
coef
ficients
0
(
f
)
=
1
T
Z
T
0
f
(
t
)
dt;
n
(
f
)
=
1
T
Z
T
0
f
(
t
)
e
j
n!
t
dt
(2)
where
T
is
the
period
and
!
is
the
angular
frequenc
y
defined
by
!
=
2
T
.
In
order
to
help
understanding
ho
w
F
ourier
series
w
orks,
let
us
gi
v
e
an
input
square-w
a
v
e
function
as
follo
ws.
u
(
t
)
=
(
g
(0
t
<
T
d
)
0
(
T
d
t
<
T
)
u
(
t
)
=
u
(
t
+
T
)
(3)
F
ourier
coef
ficients
of
(3)
are
calculated
as
follo
ws.
0
(
u
)
=
g
d;
n
(
u
)
=
g
j
n!
T
1
e
j
n
2
d
(4)
Accur
ate
Symbolic
Steady
State
Modeling
of
Buc
k
Con
verter
(Ek
o
Setiawan)
Evaluation Warning : The document was created with Spire.PDF for Python.
2376
ISSN:
2088-8708
Then
the
F
ourier
series
of
the
equation
(3)
is
written
by
equation
(5).
u
(
t
)
=
g
d
+
X
n
6
=0
g
j
n
2
1
e
j
n
2
d
e
j
n!
t
(5)
An
electric
system
can
be
described
in
a
sim
ple
block
diagram
as
sho
wn
in
Figure
1.
It
has
been
e
xplained
that
a
periodic
input
has
possibility
to
be
analyzed
with
F
ourier
series.
A
periodic
input
for
a
transfer
function
G
(
s
)
will
generate
a
periodic
output.
By
kno
wing
its
transfer
function,
the
output
al
so
can
be
analyzed
into
F
ourier
series.
This
paper
analyzes
steady
state
output
in
the
equation
(6).
Buck
con
v
erter
can
be
modelled
as
a
transfer
function.
Then
as
we
e
xplained
the
abo
v
e,
b
uck
con
v
erter
can
be
e
xpressed
as
the
equation
(6).
x
ss
(
t
)
=
G
(0)
0
(
u
)
+
X
n
6
=0
G
(
j
n!
)
n
(
u
)
e
j
n!
t
(6)
Figure
1.
Block
diagram
Figure
2.
Buck
con
v
erter
circuit
3.
B
UCK
CONVER
TER
AN
AL
YSIS
Buck
con
v
erter
circuit
is
sho
wn
in
Figure
2.
Principally
,
b
uck
con
v
erter
is
dri
v
en
by
tw
o
contrary
switching
ON
and
OFF
.
In
continuous
conduction
mode
(CCM),
b
uck
con
v
e
rter
has
only
tw
o
modes.
Each
mode
can
be
arranged
as
non-switching
circuit
by
assuming
switching
part
connected
(ON)
or
disconnected
(OFF).
Figure
3
sho
ws
the
tw
o
modes.
Let
us
assume
v
oltage
at
switching
parts
is
equal
to
input
function
u
(
t
)
as
sho
wn
in
Figure
4.
Input
function
u
(
t
)
can
be
described
as
in
the
equation
(3).
(a)
Mode-1
(b)
Mode-2
Figure
3.
Modes
of
Buck
con
v
erter
Based
on
Figure
4,
let
G
(
s
)
be
a
transfer
function
from
the
input
v
oltage
u
(
t
)
to
the
output
v
oltage
x
(
t
)
as
in
the
follo
wing
equation.
G
(
s
)
=
1
LC
s
2
+
1
R
C
s
+
1
LC
(7)
The
transfer
function
(7)
has
tw
o
poles.
Here
we
assume
the
poles
are
comple
x-conjug
ate
pairs
(
s
=
!
j
!
),
that
is,
1
R
2
C
2
4
LC
<
0
:
(8)
W
ith
s
=
j
!
,
the
frequenc
y
transfer
function
can
be
written
as
in
the
follo
wing
equation.
IJECE
V
ol.
7,
No.
5,
October
2017:
2374
–
2381
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2377
Figure
4.
Equi
v
alent
circuit
G
(
j
n!
)
=
1
LC
(
j
n!
!
)
2
+
(
!
)
2
(9)
Moreo
v
er
,
the
transfer
function
at
s
=
0
is
called
a
DC-g
ain
as
follo
ws.
G
(0)
=
1
LC
0
2
+
1
R
C
0
+
1
LC
=
1
(10)
In
the
pre
vious
section,
F
ourier
series
of
square-w
a
v
e
function
w
as
e
xplained
by
the
equation
(5).
Then,
F
ourier
coef
ficients
of
the
input
function
(3)
were
calculated
in
(4).
By
assuming
that
q
=
j
n
and
substituting
(2),
(9),
and
(10)
into
(6),
the
proposed
equation
of
b
uck
con
v
erter
can
be
e
xpressed
as
follo
ws.
x
ss
(
t
)
=
g
d
+
X
n
6
=0
1
!
2
LC
(
q
)
2
+
2
!
g
2
q
(1
e
2
q
d
)
e
q
!
t
=
g
d
+
X
n
6
=0
g
2
!
2
LC
1
(
q
)
2
+
2
1
q
(1
e
2
q
d
)
e
q
!
t
(11)
3.1.
Reco
v
ery
function
The
infinite
series
of
equation
(11)
can
be
represented
as
follo
ws.
f
saw
(
t
)
=
!
t;
(0
t
<
T
)
;
f
saw
(
t
)
=
f
saw
(
t
+
T
)
f
c
(
t
)
=
2
+
2
+
e
(
!
t
2
)
cos(
!
t
)
e
!
t
cos(
(
!
t
2
))
cosh(2
)
cos(2
)
;
(0
t
<
T
)
;
f
c
(
t
)
=
f
c
(
t
+
T
)
f
s
(
t
)
=
e
(
!
t
2
)
sin(
!
t
)
e
!
t
sin(
(
!
t
2
))
cosh(2
)
cos(2
)
2
+
2
;
(0
t
<
T
)
;
f
s
(
t
)
=
f
s
(
t
+
T
)
(12)
F
ourier
coef
ficients
of
reco
v
ery
functions
are
calculated
by
(2).
By
assuming
q
=
j
n
,
F
ourier
series
of
proposed
reco
v
ery
function
is
described
as
follo
ws.
f
saw
(
t
)
=
X
n
6
=0
n
(
f
saw
)
e
q
!
t
f
c
(
t
)
=
X
n
6
=0
n
(
f
c
)
e
q
!
t
f
s
(
t
)
=
X
n
6
=0
n
(
f
s
)
e
q
!
t
(13)
where
n
(
f
saw
)
=
1
q
,
0
(
f
saw
)
=
0
n
(
f
c
)
=
q
(
q
)
2
+
2
,
0
(
f
c
)
=
0
n
(
f
s
)
=
(
q
)
2
+
2
,
0
(
f
s
)
=
0
(14)
Accur
ate
Symbolic
Steady
State
Modeling
of
Buc
k
Con
verter
(Ek
o
Setiawan)
Evaluation Warning : The document was created with Spire.PDF for Python.
2378
ISSN:
2088-8708
3.2.
T
ime-delay
function
Let
time-delay
function
be
described
as
in
the
follo
wing.
~
f
(
t
)
=
f
(
t
T
h
)
(15)
where
T
is
periodic
t
ime
and
h
is
delay
constant.
F
ourier
coef
ficients
of
time-delay
function
are
simply
described
as
n
(
~
f
)
=
e
q
2
h
n
(
f
)
:
(16)
3.3.
Pr
oposed
steady-state
function
W
e
can
reco
v
er
function
of
time
e
xpressed
by
the
e
xponential
and
trigonometric
functions
by
using
our
reco
v
ery
functions
(13).
The
proposed
steady
state
b
uck
con
v
erter
equation
can
be
also
solv
ed
by
the
reco
v
ery
functions.
The
summation
part
of
equation
(11)
can
be
partially
decomposed
as
follo
ws.
1
(
q
)
2
+
2
1
q
=
1
(
2
+
2
)
1
q
+
q
+
2
(
q
)
2
+
2
=
1
2
+
2
1
q
q
(
q
)
2
+
2
+
(
q
)
2
+
2
(17)
In
the
ne
xt
step,
we
substitute
(17)
into
(11).
x
ss
(
t
)
=
g
d
+
g
2
!
2
LC
(
2
2
)
X
n
6
=0
"
1
q
q
(
q
)
2
+
2
+
(
q
)
2
+
2
(1
e
2
q
d
)
e
q
!
t
#
(18)
Let
us
assume
that
m
=
g
2
!
2
LC
(
2
2
)
;
=
:
(19)
By
assuming
q
=
j
n
,
the
equation
(18)
may
be
written
as
in
the
follo
wing.
x
ss
(
t
)
=
g
d
+
m
X
n
6
=0
1
q
(1
e
2
q
d
)
e
q
!
t
m
X
n
6
=0
q
(
q
)
2
+
2
(1
e
2
q
d
)
e
q
!
t
+
m
X
n
6
=0
(
q
)
2
+
2
(1
e
2
q
d
)
e
q
!
t
=
g
d
+
m
0
B
B
B
B
B
@
X
n
6
=0
n
(
f
saw
)
e
q
!
t
|
{z
}
f
saw
(
t
)
X
n
6
=0
n
(
f
saw
)
e
2
q
d
e
q
!
t
|
{z
}
f
saw
(
t
T
d
)
1
C
C
C
C
C
A
m
0
B
B
B
B
B
@
X
n
6
=0
n
(
f
c
)
e
q
!
t
|
{z
}
f
c
(
t
)
X
n
6
=0
n
f
c
e
2
q
d
e
q
!
t
|
{z
}
f
c
(
t
T
d
)
1
C
C
C
C
C
A
+
m
0
B
B
B
B
B
@
X
n
6
=0
n
(
f
s
)
e
q
!
t
|
{z
}
f
s
(
t
)
X
n
6
=0
n
(
f
s
)
e
2
q
d
e
q
!
t
|
{z
}
f
s
(
t
T
d
)
1
C
C
C
C
C
A
(20)
The
equation
(20)
is
easy
to
be
understood
if
we
use
the
reco
v
ery
functions
as
(21).
x
ss
(
t
)
=
g
d
+
m
f
saw
(
t
)
f
saw
(
t
T
d
)
m
f
c
(
t
)
f
c
(
t
T
d
)
+
m
f
s
(
t
)
f
s
(
t
T
d
)
(21)
IJECE
V
ol.
7,
No.
5,
October
2017:
2374
–
2381
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2379
The
proposed
equation
(21)
co
v
ers
infinite
series
with
the
reco
v
ery
functions.
Furthermore,
the
calculation
of
the
a
v
erage
and
RMS
po
wer
are
traceable
by
the
reco
v
ery
functions
as
follo
ws
accurately
.
P
av
g
=
1
T
Z
T
0
(
x
ss
(
t
))
2
R
dt
P
r
ms
=
s
1
T
Z
T
0
(
x
ss
(
t
))
4
R
2
dt
(22)
(a)
Complete
response
(b)
Steady
state
response
Figure
5.
Comparison
between
SPICE
and
proposed
analysis
of
parameter
-1
4.
SIMULA
TION
RESUL
T
This
section
v
alidates
our
proposed
function
(21)
result
by
comparing
with
SPICE
(Simulation
Program
with
Inte
grated
Circuit
Emphasis).
Numerical
parameter
of
dc-dc
con
v
erter
is
determined
as
sho
wn
in
T
able
1.
Numerical
parameter
of
resistance
(
R
),
inductance
(
L
)
and
capacitance
(
C
)
gi
v
e
comple
x-conjug
ate
poles.
Numerical
calculation
utilized
mathematical
softw
are
to
plot
the
proposed
steady
state
response.
SPICE
simulates
the
actual
circuit
responses.
Complete
response
of
capacitor
v
oltage
by
parameter
-1
is
sho
wn
in
Figure
5a.
Numer
-
ical
calculation
of
proposed
method
is
plotted
in
solid-line
while
SPICE
result
in
dashed-line.
The
magnification
of
steady
state
response
is
sho
wn
in
Figure
5b.
Comparison
between
numerical
calculation
of
proposed
function
and
steady
state
of
SPICE
result
has
similarity
in
shape
and
v
alue.
The
other
complete
response
of
circuit
simulation
using
parameter
-2
is
sho
wn
in
Figure
6a
and
steady
state
of
SPICE
and
proposed
result
is
descr
ibed
Figure
6b.
Figure
5b
and
6b
describe
that
numerical
calculation
of
proposed
methods
gi
v
e
consistent
result
with
SPICE
result.
The
proposed
method
has
adv
antage
in
obtaining
steady
state
response
without
w
aiting
transient
time.
Accur
ate
Symbolic
Steady
State
Modeling
of
Buc
k
Con
verter
(Ek
o
Setiawan)
Evaluation Warning : The document was created with Spire.PDF for Python.
2380
ISSN:
2088-8708
T
able
1.
Numerical
parameter
V
ariable
P
arameter
-1
P
arameter
-2
Resistance
(R)
6.35
1.81
Inductance(L)
100
H
285
H
Capacitance
(C)
62.7
F
21.9
F
T
ime-period
(T)
50
s
20
s
V
oltage
source
(g)
10
V
15
V
Duty-ratio
(d)
0.5
0.5
Pole
(s)
-0.0101
j0.1
-0.0402
j0.0034
5.
CONCLUSION
W
e
ha
v
e
sho
wn
that
the
proposed
method
describes
steady-s
tate
response
directly
without
calculating
transient
response.
It
gi
v
es
e
xact
calculation
and
symbolic
complete
solution
of
steady
state
output.
T
ransition
between
each
mode
is
desc
ribed
clearly
by
proposed
method.
Reco
v
ery
function
gi
v
es
an
accurate
solution
of
F
ourier
series
without
depending
on
numerical
calculation
of
summation.
Proposed
steady
state
analysis
of
b
uck
con
v
erter
has
been
clarifie
d
by
comparing
with
SPICE.
Moreo
v
er
,
proposed
method
mak
es
calculation
of
po
wer
traceable.
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ariable
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5,
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2017:
2374
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Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2381
(a)
Complete
response
(b)
Steady
state
response
Figure
6.
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and
proposed
analysis
of
parameter
-2
Frequenc
y-Domain
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Modeling
of
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k
Con
verter
(Ek
o
Setiawan)
Evaluation Warning : The document was created with Spire.PDF for Python.