Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.
6, N
o
. 1
,
Mar
c
h
20
15
,
pp
. 32
~44
I
S
SN
: 208
8-8
6
9
4
32
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
A Comparison Between Two Av
erage Modelling Techniques of
AC-AC Power Converters
Pawe
ł
Sz
cz
e
ś
ni
ak
Institute of
El
ect
rical Eng
i
neerin
g, Universi
t
y
of
Zielon
a Gór
a
, Podgorn, Poland
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 22, 2014
R
e
vi
sed Dec 4,
2
0
1
4
Accepted Dec 21, 2014
In this
paper, a
com
p
arativ
e eva
l
ua
tion of two modelling too
l
s for switching
AC-AC power converters is pres
ented
.
Both of them are based
on averag
e
modelling techn
i
ques. The f
i
rst approach
is
bas
e
d on the cir
c
ui
t averag
ing
techn
i
que and
consists in the topol
ogical manipulations,
ap
plied to a
converter states. The second ap
proach
makes use of a state-space aver
aged
model of the converter and is based
on analy
t
ical manipulations using the
differen
t
stat
e representations of a
convert
er.
The two m
odelling techn
i
ques
are app
lied
to a same AC-AC
called
matrix-
r
eactance frequen
c
y
conver
t
er
based on bu
ck-b
oost topolog
y
.
These techniqu
es
are
compared o
n
the b
a
sis
of their rap
i
dit
y
, qu
anti
t
y
of
calcu
lations and transformations and its
lim
itat
i
ons.
Keyword:
A
v
er
ag
e
d
mo
d
e
l
Bu
ck-
boo
st topo
log
y
Matrix
conv
ert
e
r
Mo
d
e
lling
Po
wer co
n
v
ert
e
rs
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Pawe
ł
Szcze
ś
niak
,
In
stitu
te
o
f
Electrical Eng
i
n
e
ering
,
Uni
v
ersity
of Zielona Gora,
6
5
-2
46
Zielon
a Gó
r
a
, Po
dgo
rn1
5
0
, Po
land
.
Em
a
il: P.Szczesniak@iee.uz.z
gora.pl
1.
INTRODUCTION
Mo
d
e
lling
a
p
o
wer electron
i
c conv
erter
is a co
m
p
le
x
issu
e d
u
e
to
th
e fact
of d
i
gital
co
n
t
ro
l
com
p
l
e
xi
t
y
and hi
gh
n
u
m
b
er of
p
o
we
r c
o
n
v
ert
e
r
com
p
o
n
e
nt
s. F
u
rt
herm
ore
,
gi
ven t
h
e
i
n
creasi
n
g
n
u
m
ber of
di
ffe
re
nt
m
odul
at
i
on st
rat
e
g
i
es, i
t
i
s
nece
ssary
t
o
st
udy
t
h
ei
r i
m
pact
i
n
co
n
v
ert
e
r
o
p
erat
i
o
n.
I
n
o
r
der t
o
achi
e
ve t
h
ei
r
goal
p
o
we
r c
o
nve
rt
ers
m
u
st
be a
p
p
r
op
ri
at
el
y
m
odel
l
e
d i
n
si
m
u
l
a
t
i
on
or
anal
y
t
i
cal
st
udi
es.
Hen
ce it is n
ecessary to
create si
m
p
le
m
o
d
e
ls. Th
e p
r
o
b
l
e
m
has been l
a
rgel
y
st
udi
e
d
and a wi
de var
i
et
y
of
m
o
d
e
ls h
a
v
e
been
propo
sed
[1
]-[6
]. However, th
e
u
s
e of
t
h
o
s
e m
o
d
e
ls an
d
t
h
eir sim
u
la
tio
n
in
a co
m
p
u
t
er still
req
u
i
r
es a l
a
r
g
e am
ount
o
f
r
e
so
urces a
nd
ci
rcui
t
sim
p
l
i
f
i
cat
i
on an
d m
a
t
h
em
at
i
cal
t
r
an
sfo
r
m
a
t
i
ons. I
n
t
h
i
s
pape
r, a
com
p
arat
i
v
e e
v
al
uat
i
on
of
t
w
o m
odel
l
i
ng a
p
pr
oa
ch
of c
o
m
p
l
e
x di
rect
AC
-
A
C
fre
q
u
ency
c
o
n
v
ert
e
r
s
i
s
prese
n
t
e
d. T
h
e
prese
n
t
e
d
m
odel
l
i
ng m
e
tho
d
s a
r
e
base
d
o
n
ci
rc
ui
t
ave
r
agi
ng t
e
c
h
ni
q
u
e [
1
]
,
[5]
a
n
d
st
at
e-
space a
v
era
g
e
d
m
odel [5].
The c
h
an
ge
of
fre
que
ncy
i
n
AC
v
o
l
t
a
ge i
s
no
w
one
o
f
t
h
e im
port
a
nt
fu
nct
i
o
n
s
o
f
s
o
l
i
d
st
at
e p
o
we
r
co
nv
erters. The
m
o
st d
e
sirable featu
r
es
o
f
freq
u
e
n
c
y
conv
erters in
clud
e th
e po
ssib
ility o
f
gen
e
rating
lo
ad
vol
t
a
ge
s wi
t
h
a
r
bi
t
r
a
r
y
am
pl
itude a
n
d ge
ne
ra
t
i
ng si
n
u
s
o
i
d
al
sou
r
ce a
nd l
o
a
d
cu
rre
nt
s a
nd
vol
t
a
ge
wa
vef
o
rm
s,
th
e po
ssi
b
ility
o
f
prov
id
i
n
g un
ity p
o
wer fact
o
r
fo
r an
y
lo
ad, and
fin
a
lly, their con
s
tru
c
tion
u
s
ing
a
sim
p
l
e
and
com
p
act powe
r
circuit. T
h
e
past few years
have witne
sse
d
rem
a
rkable p
r
o
g
ress
in re
search
into direct powe
r
AC–AC freq
uen
c
y conv
erters
witho
u
t
a DC en
erg
y
storag
e elem
en
t. Man
y
ex
citing
ap
p
lication
s
h
a
v
e
b
e
en
devel
ope
d
[7]
-
[
1
2
]
.
T
h
e m
o
st
com
m
on i
s
t
h
e
m
a
t
r
i
x
con
v
e
r
t
e
r (M
C
)
t
o
p
o
l
ogy
[
7
]
.
A
n
ot
her
gr
o
up
o
f
A
C
–A
C
freq
u
e
n
c
y conv
erters
with
a
b
u
c
k–
boo
st
v
o
ltag
e
tran
sfo
r
m
a
tio
n
po
ssib
ility and
with
ou
t
DC en
erg
y
st
orag
e is
propose
d
i
n
[10]-[12], a
n
d
are known as
m
a
trix-r
eacta
n
ce fre
que
ncy
conve
rters
(MRFC). T
h
e
expecte
d
bene
fi
t
o
f
t
h
es
e co
n
v
ert
e
rs
i
s
t
h
e
v
o
l
t
a
ge t
r
ans
f
er
rat
i
o
whi
c
h i
s
m
u
c
h
great
e
r
t
h
a
n
o
n
e a
n
d
dep
e
nds
on
reactive elem
e
n
ts [12].
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
A C
o
mp
ari
s
on
Bet
w
een Tw
o
Avera
g
e M
o
del
l
i
ng Tec
h
ni
q
u
e
s
of
AC
-AC
P
o
w
e
r C
o
nvert
e
r
s
(
P
aw
e
ł
Szcze
ś
ni
ak)
33
The a
n
al
y
s
i
s
a
n
d
m
odel
l
i
ng
of M
R
FC
p
r
es
ent
s
si
g
n
ificant ch
allen
g
es,
du
e to th
eir d
i
sco
n
tinuo
us
switching be
ha
vior and the increasing num
b
er of di
ffe
rent
m
odulation strategies [13]
. It is necessary to study
t
h
e m
odul
at
i
o
n
p
r
oces
s i
m
pact
i
n
co
n
v
ert
e
r
o
p
erat
i
o
n.
Th
e m
a
in
ai
m o
f
th
is
p
a
p
e
r
is to
presen
t two m
a
th
e
m
a
t
i
cal
m
o
d
e
ls
o
f
th
e selected
t
o
p
o
l
o
g
i
es
of
M
R
FC
s wi
t
h
s
i
m
p
l
e
Vent
u
r
i
n
i
m
odul
at
i
o
n
[
13]
.
The
res
u
l
t
s
o
f
t
h
e
st
u
d
y
a
n
d
m
a
t
h
em
ati
c
al
anal
y
s
i
s
p
r
e
s
ent
e
d
i
n
t
h
i
s
pape
r a
r
e base
d
on
t
h
e
pre
v
i
o
usl
y
p
r
e
s
ent
e
d
res
u
l
t
s
prese
n
t
e
d
i
n
[1
4]
-[
1
6
]
.
O
n
e
w
e
l
l
-
kn
o
w
n a
p
pr
oach
t
o
t
h
e
m
odel
l
i
ng
of
po
wer c
o
n
v
e
r
t
e
rs i
s
t
o
app
r
o
x
i
m
at
e thei
r o
p
erat
i
on
usi
n
g ave
r
agi
n
g t
echni
que
s [
1
]
.
The
gene
ral
i
zed av
eragi
ng m
e
t
hod i
s
based
on t
h
e fact
t
h
at
t
h
e wave
fo
rm
s can be ap
p
r
o
x
i
m
at
ed usi
n
g a d
e
fi
ne
d
ti
m
e
in
terv
al,
wh
ich
is
d
e
termin
ed
b
y
a
switch
i
ng
seq
u
en
ce
p
e
ri
o
d
T
Se
q
. In
itially, the av
erag
e m
e
t
h
od
was
wid
e
ly
u
s
ed for
DC–DC co
nverter m
o
d
e
lling
[1
]. Th
en it wa
s app
lied
t
o
o
t
h
e
r typ
e
s of
co
nv
erters:
[2
]
-
[6
].
As
a
m
a
in
ach
iev
e
men
t
o
f
th
e
p
a
p
e
r is to
sho
w
th
e d
i
fferen
ces in
th
e resu
lts o
f
two
an
alyzed
m
o
d
e
llin
g
meth
od
s,
whic
h a
r
e s
u
mmarized in t
h
e
section
4.
2.
DESC
RIPTI
O
N OF
THE
AN
ALYZ
ED MAT
R
I
X
-
R
E
ACT
AN
CE
F
R
EQUE
N
C
Y
CO
NVE
RTER
The fam
i
l
y
of M
C
R
F
C
s
cont
ai
ns 9 t
o
pol
og
i
e
s - t
w
o t
o
pol
ogi
es
based
on
buc
k-
b
oost
,
Ć
uk
, SEP
I
C
and Zet
a
t
o
p
o
l
ogi
es a
nd
o
n
e
base
d o
n
t
h
e
b
oost
t
o
p
o
l
o
gy
[1
0]
, [
12]
. Fi
rs
t
an M
R
FC
ba
sed o
n
buc
k
-
b
oost
to
po
log
y
(MRFC-I-bu
ck
-bo
o
st), sho
w
n
in
Fig
.
1
,
will b
e
an
alyzed
[1
0
]
.
Th
e d
e
scrip
tion
s
, in
g
e
n
e
ral fo
rm
, o
f
t
h
e co
nt
r
o
l
st
ra
t
e
gy
o
f
t
h
e
di
s
c
usse
d M
R
FC
s
i
s
sh
o
w
n
i
n
Fi
gu
re
2
[
12]
.
Ea
ch se
q
u
ence
pe
ri
o
d
i
s
di
vi
ded
i
n
t
o
t
w
o pa
rt
s
t
S
and
t
L
. Tim
e
t
S
is related
to m
a
tr
ix
conn
ected switch
sets
operations. In eac
h
switching cycl
e
T
Seq
,
in
th
e in
terv
al
t
S
, th
e m
a
trix
co
nn
ected
switch
sets are in th
e
process
of switch
i
ng
wi
th
selected
switch
i
n
g
m
odul
at
i
on,
w
h
i
l
e
t
h
e l
o
ad
s
y
nch
r
o
n
ous
co
nnect
e
d
s
w
i
t
c
h
set
s
are
t
u
r
n
e
d
-
o
f
f
.
The
v
o
l
t
ages
u
a
,
u
b
,
u
c
are
fo
rm
ed by
set
t
i
ng t
h
e re
q
u
est
e
d
out
put
fre
q
u
ency
f
L
,
with
sequ
en
tial p
i
ecewise section
s
o
f
th
e i
n
pu
t
vo
ltag
e
wave
f
o
rm
s
u
A
,
u
B
,
u
C
.
At the s
a
m
e
time the e
l
ectrical
energy is stored in t
h
e induct
o
r
L
S
1
,
L
S
2
,
L
S
3
. I
n
contrast,
in
th
e ti
m
e
p
e
rio
d
t
L
all o
f
th
e
m
a
trix
co
n
n
ected
switch
set
s
(MCS) are turn
ed
-off and
the lo
ad
syn
c
h
r
on
ou
s
connected s
w
itch sets a
r
e t
u
rned-on. T
h
e e
n
ergy store
d
i
n
s
o
urce i
n
duc
t
ors
L
S
1
,
L
S
2
,
L
S
3
is tran
sferred
to th
e
lo
ad
cap
acitor
C
L
1
,
C
L
2
,
C
L
3
. In
th
is way, we ob
tain
t
h
e
p
o
s
sib
ility o
f
in
creasin
g th
e
ou
tpu
t
vo
ltag
e
. Th
e du
ty
fact
or
s of
l
o
ad
swi
t
c
hes de
pen
d
on
t
h
e
e
x
pect
ed out
put
v
o
l
t
a
ges.
Figure 1.
Topology of
m
a
trix
-reactance
frequency converter ba
sed on buck–boost
topology
(M
RFC-I
-
b
u
c
k
-b
o
o
st) [1
0]
, [1
2]
Fi
gu
re
2.
Ge
ne
ral
f
o
rm
of t
h
e
cont
rol
st
rat
e
g
y
The state of the conve
r
ter s
w
itches can
be represe
n
ted
b
y
mean
s o
f
t
h
e so
-called
tran
sfer m
a
trix
T
(1
), (
2
). T
h
e m
a
trix
T
i
s
defi
n
e
d by
usi
n
g m
o
d
u
l
a
t
i
on
st
rat
e
gy
of m
a
t
r
i
x
con
n
ect
ed
swi
t
ch set
s
, a
nd
de
scri
bes
t
h
e l
o
w
fre
que
ncy
i
n
put
t
o
ou
t
put
c
u
r
r
ent
an
d
vol
t
a
ge
s rel
a
t
i
ons
hi
ps
[
1
2]
.
Ne
xt
Seq
uen
ce
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
IJPE
DS
V
o
l
.
6, N
o
. 1,
M
a
rc
h 20
1
5
:
3
2
– 44
34
S
C
B
A
cC
cB
cA
bC
bB
bA
aC
aB
aA
c
b
a
t
u
t
u
t
u
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
u
t
u
t
u
Tu
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
,
(1
)
L
T
c
b
a
cC
bC
aC
cB
bB
aB
cA
bA
aA
C
B
A
t
i
t
i
t
i
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
s
t
i
t
i
t
i
i
T
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
.
(2
)
Whe
r
e:
s
jK
-s
w
itch state f
u
nction,
j
= {
a
,
b
,
c
},
K
= {
A
,
B
,
C
}
–
n
am
es
of inp
u
t and o
u
tp
ut phase
s. The
M
C
S
wo
rk
wit
h
a
hi
gh
switc
hin
g
f
r
eq
ue
ncy
.
A lo
w f
r
e
que
nc
y
l
o
ad
v
o
ltage
o
f
va
riable am
plitude a
n
d
f
r
eq
uency
can be ge
ne
rated by
m
odulating the
duty
cy
cle of th
e switches usi
ng thei
r resp
ective sw
itching f
u
nctio
ns
s
jK
.
A m
odulatio
n
duty
cy
cle sh
o
u
ld
be
defi
ned
fo
r eac
h s
w
itch
in
or
der t
o
determ
ine the ave
r
age
be
ha
vio
u
r
of
the
M
C
S o
u
tp
ut v
o
ltage
wave
f
o
r
m
[12]
, [
1
3]
.
T
h
e m
odulatio
n
duty
cy
cle is
d
e
fine
d
by
:
Seq
jK
jK
T
t
t
d
(3
)
Whe
r
e
t
jK
represents the
time when switch S
jK
is turne
d
on a
n
d
T
Seq
represents the ti
me
of the com
p
le
te
sequence i
n
the PW
M
pattern, a
n
d 0 <
d
jK
<
1.
B
a
se
d on the
switch d
u
t
y
-ratios,
t
h
e
a
v
erage
d
out
put voltage
s
and the a
v
e
r
aged input curre
nts can
be
related to the i
n
put
voltages and the output
curre
nt
s, res
p
ectively, as:
S
L
t
t
u
M
u
)
(
,
L
T
S
t
t
i
M
i
)
(
,
(4
)
cC
cB
cA
bC
bB
bA
aC
aB
aA
d
d
d
d
d
d
d
d
d
t
M
.
(5
)
The classical Vent
uri
n
i cont
rol strategy
is
taking
int
o
consi
d
eration
with
low freque
ncy trans
f
er
m
a
trix descri
bed by (5)
[13]. Taking
int
o
account lim
ite
d switchi
ng ti
m
e
t
S
of M
C
S, the m
odulation
duty
cycles
d
jK
for
M
R
FC
-I-
b
u
ck
-
b
o
o
st are
defi
n
e
d by
(
6
) a
nd
(
7
) [
1
2]
. Exem
plary
tim
e
wavefo
rm
s of the cont
rol
signals, illustrating
ope
ration of
the discussed
MRFC
is shown in
Figure 3.
Figu
re
3.
Exe
m
plary
tim
e
wavef
o
rm
s of
th
e co
ntr
o
l si
gn
als in
MRFC-I
-b
u
c
k-
boo
st fo
r
switch
e
s in
o
n
e ph
ase
3
/
))
3
/
4
cos(
2
1
(
3
/
))
3
/
2
cos(
2
1
(
3
/
))
cos(
2
1
(
t
q
D
d
d
d
t
q
D
d
d
d
t
q
D
d
d
d
m
S
cB
bA
aC
m
S
bC
cA
aB
m
S
cC
bB
aA
,
(6
)
Seq
S
S
T
t
D
,
(7
)
t
t
t
t
t
S
jA
S
jB
S
jC
S
L
1
u
r
dj
A
dj
A
+
dj
B
t
L
t
S
T
Seq
dj
A
+
dj
B
+
dj
C
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
A C
o
mp
aris
on
Betw
een Tw
o
Avera
g
e M
o
delling Tec
h
niq
u
e
s
of AC
-AC
P
o
w
e
r C
o
nverter
s
(
P
aw
e
ł
Szcze
ś
niak)
35
Whe
r
e:
L
m
,
ω
,
ω
L
– pulsatio
n of the s
u
p
p
ly
a
nd load voltages respectivel
y,
D
S
–se
q
uen
ce
pulse
duty
factor
,
q
–
voltage
gai
n
(0
≤
q
≤
0.
5)
.
The m
a
trix
M
(
t
) is kn
o
w
n a
s
the
m
odulation m
a
trix or low
-
f
r
e
que
ncy
trans
f
er m
a
trix. B
a
sed o
n
these relationships in
(4) and (5), a
m
a
trix c
o
nverter on a sw
itching-cycle
avera
g
ed
basi
s can be
represented
by
nine id
eal trans
f
orm
e
rs with vary
in
g tu
r
n
-ratios
,
as sh
o
w
n i
n
Fig
u
re
4
(
a),
w
h
ereas
o
u
tp
ut switches
can be
represe
n
ted by ideal
transfor
m
e
rs
with turn-ratios equal
1-
D
S
,
as sh
own
in
Fig
u
r
e
4
(
b)
[12].
(a)
(b
)
Figure 4.
Averaged-switchi
ng-cycle represe
n
tation: a) m
a
trix c
o
nn
ected switches
(MCS), b) l
o
ad switc
hes
3.
AVE
RAGE MODELLING
TECHNIQUES
Ave
r
a
g
e m
odelling techni
que
s are base
d pr
incipally
on re
placing all cur
r
ents an
d v
o
ltages o
f
th
e
sy
stem
by
their m
ean value
o
v
er a
switchi
n
g
perio
d
a
n
d ig
no
rin
g
th
us t
h
e
i
r hig
h
fre
q
u
en
cy
com
pone
nts
.
The
local ave
r
age
of function
d
(
t
) i
s
de
fine
d as
f
o
llows
[
4
]
:
d
q
T
t
d
t
Seq
T
t
Seq
1
,
(8
)
Whe
r
e
d
(
t
) is the continuous duty factor.
F
o
r the next se
q
u
ence
peri
ods
T
Seq
becom
e
s
d
(
kT
Seq
)=
d
k
(
t
), whe
r
e
d
k
(
t
) is the actual duty factor i
n
the
k-th
cycl
e. If function
d
(
t
) is peri
odic
with pe
rio
d
T
Seq
, then
d
(
t
)=
D
w
h
er
e
D
is t
h
e steady
-
state duty ratio
[4].
There a
r
e am
ong
othe
rs tw
o
prese
n
ted i
n
th
is pape
r st
rateg
i
es used
fo
r the
deri
vation
of t
h
e ave
r
a
g
e
d
m
odel: circuit avera
g
e tec
hni
que
and
state-space a
v
era
g
ing techni
que
.
3.1.
Circuit Averaging Technique
Sinus
oi
dal tim
e
-va
r
y
i
ng
sy
stem
s can be cha
nge
d t
o
tim
e-inva
riant sy
ste
m
by
the
dq
0 t
r
ans
f
orm
a
tion
[
2
],
[1
4
]-[
18
].
Th
e
dq
0
tran
sf
orm
a
tion o
f
t
h
e va
ri
ables is
given as follows:
,
,
0
1
0
dq
abc
abc
dq
x
K
x
Kx
x
(9
)
Whe
r
e:
x
abc
=[
x
a
,
x
b
,
x
c
]
T
,
x
dq
0
=[
x
q
,
x
d
,
x
0
]T,
x
d
-
fo
rwar
d (r
otatin
g
)
ph
aso
r
,
x
q
- bac
k
wa
r
d
(r
otating
)
pha
s
o
r
,
x
0
-
zero
-
se
que
nce com
pone
nt
[
2
]
,
L
S
K
K
K
0
0
,
(1
0)
2
/
1
3
/
2
sin
3
/
2
cos
2
/
1
3
/
2
sin
3
/
2
cos
2
/
1
sin
cos
3
2
t
t
t
t
t
t
S
K
,
(1
1)
u
B
u
C
u
A
u
b
u
c
u
a
1:
d
aA
1:
d
bA
1:
d
cA
1:
d
aB
1:
d
bB
1:
d
cB
1:
d
aC
1:
d
bC
1:
d
cC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
IJPE
DS
V
o
l. 6, N
o
. 1,
M
a
rc
h 20
1
5
:
3
2
– 44
36
2
/
1
3
/
2
sin
3
/
2
cos
2
/
1
3
/
2
sin
3
/
2
cos
2
/
1
sin
cos
3
2
t
t
t
t
t
t
L
L
L
L
L
L
L
K
,
(
1
2
)
Whe
r
e:
K
S
an
d
K
L
are the
dq
0 tra
n
sf
orm
a
tion m
a
trices de
fine
d f
o
r
p
u
lsation o
f
the s
u
p
p
ly
and l
o
ad
v
o
ltages
,
ω
and
ω
L
res
p
e
c
tively
[1
4]
, [
1
6]
.
The circ
uit
dq
0 transform
a
tion is ob
tained by the
followi
n
g
p
r
oce
d
ures
[
14]
:
a)
Partition
of the averaged ci
rcu
it
m
odel into basic subcircuits.
b)
Transform
a
tion of each
of
the subcirc
u
its into
dq
0 e
qui
valent circ
uits base
d o
n
the
dq
0
trans
f
o
r
m
a
tion equatio
ns
.
3.
1.
1. Par
t
itio
n
o
f
the Circu
it into B
a
sic Subcircuits
We can di
vide the avera
g
e
d
circuit m
odel of
the presented MRFC into
seve
ral fundam
ental
subcircuits al
ong the
dotted li
nes i
ndi
cated in Fi
gure
5. After
partitioni
ng
, we
obtain ei
ght basic subci
r
cuits.
C
F
1
C
F
2
C
F
3
L
L
1
L
L
2
L
L
3
1:
d
aA
u
a
u
b
u
c
b
c
a
B
C
A
L
F
1
L
F
2
L
F
3
C
L
2
C
L
3
C
L
1
u
L
1
u
L
2
u
L
3
i
L
3
i
L
2
i
L
1
u
S
1
u
S
2
u
S
3
R
L
1
R
L
2
R
L
3
1:
d
aB
1:
d
aC
1:
d
bA
1:
d
bB
1:
d
bC
1:
d
cA
1:
d
cB
1:
d
cC
(1
-
D
S
):
1
u
A
u
B
u
C
Pa
r
t
1
Pa
r
t
2
Pa
r
t
3
Pa
r
t
4
Pa
r
t
5
Pa
r
t
6
Pa
r
t
7
Pa
r
t
8
Figure
5. Averaged circ
uit m
odel
of the considered MRFC
3.1.2. Tr
ans
f
ormati
on
of Basic Subc
ircuits into d
q
0 Eq
uivalen
t
Circu
its
For a t
h
ree
-
pha
s
e bala
nced voltage source
se
t (Par
t
1), t
h
e
proce
d
ure is as
fo
llo
ws
[2
]-[
3
]
,
[
1
4
]
:
0
cos
sin
)
3
/
2
sin(
)
3
/
2
sin(
)
sin(
1
1
1
1
1
0
S
S
S
S
S
Sdq
U
t
t
t
U
K
u
K
u
(1
3)
Whe
r
e
u
S
is the vector of the vo
ltage
so
urc
e
s. Th
us, t
h
e
dq
0 tr
ans
f
orm
e
d circuits o
f
the voltage
source set is
sho
w
n
in Fig. 6a. Usin
g basic
pri
n
ciples
f
r
o
m
circuit
theory, t
h
e source
inductor
s (
P
ar
t 2) are
m
odelled by
Equ
a
tio
n (1
4)
[2
],
[14
]
:
LFabc
LFabc
F
dt
d
L
u
i
(1
4)
Whe
r
e
L
F
1
=
L
F
2
=
L
F
3
=
L
F
. A
p
plication
of
(
9
)
to
(
1
4
)
y
i
elds:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
A C
o
mp
ari
s
on
Bet
w
een Tw
o
Avera
g
e M
o
del
l
i
ng Tec
h
ni
q
u
e
s
of
AC
-AC
P
o
w
e
r C
o
nvert
e
r
s
(
P
aw
e
ł
Szcze
ś
ni
ak)
37
LFabc
LFdq
S
LFdq
S
F
dt
d
dt
d
L
u
i
K
i
K
0
1
0
1
(1
5)
Finally, the
dq
0
tr
an
sf
or
m
o
f
so
ur
ce in
du
ct
ors
ca
n be form
ulated
as:
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
LFdq
LFdq
F
LFdq
S
LFdq
S
S
F
LFdq
F
L
dt
d
L
dt
d
L
u
i
u
K
i
K
K
i
(16)
An
d t
h
e ci
rc
ui
t
m
odel
s
are sh
ow
n i
n
Fi
gu
re
6(
b)
. The
dq
0
“i
nd
uct
o
r” i
s
r
e
prese
n
t
e
d
by
real
dy
nam
i
c ind
u
ct
o
r
L
F
in series
wi
th an im
aginary static reactor ±
j
ω
L
F
.
Since
the voltage a
n
d c
u
rrent
of t
h
e static reactor obeys
Ohm
’
s law, the
reactor is
repl
aced
by
a lossless resist
or
sym
bol [2].
Sim
i
l
a
r, equat
i
ons
a
n
d
ci
rcui
t
m
odel
s
a
p
p
l
y t
o
th
e lo
ad
i
n
ducto
r set
(Part 5):
LLabc
LLabc
L
dt
d
L
u
i
(17)
Whe
r
e
L
L
1
=
L
L
2
=
L
L
3
=
L
L
. Fr
om
ex
pres
si
o
n
(
9
)
,
(
1
2),
an
d
(
1
7
)
o
b
t
a
i
n
:
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
LLdq
LLdq
L
L
LLdq
L
LLdq
L
L
L
LLdq
L
L
dt
d
L
dt
d
L
u
i
u
K
i
K
K
i
(18)
Fig
u
re 6(c) illu
strates th
e
dq
0
co
m
p
on
en
ts
o
f
lo
ad
indu
cto
r
s. Fo
r th
e s
o
urce ca
pacitors
circuit (Part
3), the
d
i
fferen
tial eq
uatio
n
s
are in the fo
llo
wi
n
g
fo
rm
[2
], [14
]
:
CFabc
CFabc
F
dt
d
C
i
u
(19)
Whe
r
e
C
F
1
=
C
F
2
=
C
F
3
=
C
F
. T
a
k
i
ng
in
to
acco
un
t expr
ession
s (9
), (11
)
and
(19
)
, th
e
dq
0 t
r
a
n
sf
orm
of
sou
r
ce
cap
acito
rs is
defin
e
d
as
fo
llows:
CFabc
CFdq
S
CFdq
S
F
dt
d
dt
d
C
i
u
K
u
K
0
1
0
1
(20)
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
CFdq
CFdq
F
CFdq
S
CFdq
S
S
F
CFdq
F
C
dt
d
C
dt
d
C
i
u
i
K
u
K
K
u
(21)
Fo
r th
e lo
ad
cap
acito
rs circu
it (Part 7), th
e
dq
0
t
r
a
n
sf
orm
i
s
de
fi
ne
d as
f
o
l
l
o
ws:
CLabc
CLabc
L
dt
d
C
i
u
(22)
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
CLdq
CLdq
L
L
CLdq
L
CLdq
L
L
L
CLdq
L
C
dt
d
C
dt
d
C
i
u
i
K
u
K
K
u
(23)
The
dq
0 t
r
a
n
s
f
o
r
m
e
d ci
rcui
t
of
so
ur
ce an
d l
o
a
d
ca
p
acito
r
sets ar
e sho
w
n
in
Figure
6(
d)
and
Fig
u
re
6
(
e), resp
ectiv
ely. Similar as with
in
du
ctors, th
e
dq
0 “capacit
o
rs” are represe
n
ted by real dynam
ic
capacitors
C
F
and
C
L
in parallel with im
agin
ary static reactors ±1/(
j
ω
C
F
)
,
and ±
1
/
(
j
ω
L
C
L
)
[2]
.
If the swit
chin
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
: 2
088
-86
94
IJPE
DS
V
o
l
.
6, N
o
. 1,
M
a
rc
h 20
1
5
:
3
2
– 44
38
fun
c
tion
of th
e
m
a
trix
switches is d
e
fi
n
e
d by (4
)-(7) th
en
t
h
e
dq
0 t
r
a
n
sf
o
r
m
a
t
i
on of
t
h
e
M
C
S (Part
4) i
s
gi
ve
n
in
(24
)
[2
],
[14].
Th
e
dq
0 transform
e
d
circu
i
t
of m
a
trix
switc
h
e
s set is show
n in
Figu
r
e
6(
f)
.
0
0
0
0
1
0
1
0
0
0
0
0
0
Sdq
S
Sdq
dq
Sdq
S
L
SABC
L
Labc
L
Ldq
q
q
D
u
u
M
u
MK
K
Mu
K
u
K
u
(24)
(a)
(b
)
(c)
(d
)
(e)
(
f)
(g
)
(h
)
Fi
gu
re
6.
The
dq
0 t
r
ans
f
orm
a
t
i
on
of:
a
)
vol
t
a
ge s
o
urces
,
b)
so
urce
i
n
duct
o
rs, c
)
l
o
ad
i
n
du
ct
ors,
d
)
s
o
urce
capacitors
, e
)
l
o
ad capacit
o
rs
, f) m
a
trix switc
hes, g) loa
d
switches,
h) l
o
a
d
resistors
If t
h
e switch
i
ng
fun
c
tion
o
f
t
h
e lo
ad
switches (Part
6
)
is defin
e
d
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
A C
o
mp
ari
s
on
Bet
w
een Tw
o
Avera
g
e M
o
del
l
i
ng Tec
h
ni
q
u
e
s
of
AC
-AC
P
o
w
e
r C
o
nvert
e
r
s
(
P
aw
e
ł
Szcze
ś
ni
ak)
39
S
S
S
L
D
D
D
1
0
0
0
1
0
0
0
1
M
(25)
T
h
en
th
e
dq
0 t
r
ansf
o
r
m
i
s
descri
be
d as
f
o
l
l
o
ws
(Fi
g
ure
6(
g
)):
0
0
0
1
0
1
Ldq
L
Sdq
Ldq
L
L
Sdq
L
Labc
L
Sabc
u
M
u
u
K
M
u
K
u
M
u
(26)
Ass
u
m
i
ng t
h
at
,
R
L
1
=
R
L
2
=
R
L
3
=
R
L
,
th
e pr
o
c
ed
ur
e
o
f
dq
0 t
r
ansform
of the
resistor set
(P
art 8) is
as
follo
ws (Fi
g
u
r
e
6
(
h
)):
0
0
Ldq
L
Labc
L
L
Labc
L
Ldq
R
R
i
i
K
u
K
u
.
(27)
3.
1.
3. Ci
rcui
t
Reco
nstr
ucti
o
n
The e
q
ui
val
e
n
t
dq
0
ci
rcui
t
m
odel
s
of
t
h
e
p
r
esent
e
d M
R
FC
(Fi
g
u
r
e
1
)
are
obt
ai
ne
d
as s
h
o
w
n i
n
Fi
gu
re
7 by
re
j
o
i
n
i
n
g of
t
h
e
dq
0 t
r
ans
f
orm
e
d su
bci
r
cui
t
s
.
T
h
ere
f
o
r
e, t
h
e t
h
ree-
pha
se ci
rc
u
i
t
i
n
Fi
gu
re
1 c
a
n be
rep
r
ese
n
t
e
d
b
y
t
h
ree si
n
g
l
e
-p
hase s
u
bci
r
cui
t
s
f
o
r
fo
r
w
ar
d,
bac
k
wa
rd a
n
d zer
o
-
s
e
que
nce c
o
m
pone
nt
s.
Fu
rt
h
e
rm
o
r
e,
assu
m
i
n
g
th
at
th
e in
itial p
h
ase of inp
u
t
vo
ltag
e
s eq
uals zero
φ
1
=0
and
th
at t
h
e circu
it is
sym
m
et
ri
cal
and bal
a
nce
d
, we
o
b
t
a
i
n
[2]
:
0
1
0
0
S
S
S
Sdq
U
u
K
u
.
(28)
u
Sq
+
-
i
Sq
L
F
j
ω
L
F
C
F
1:
D
S
L
L
C
L
u
Sd
+
-
i
Sd
L
F
C
F
L
L
+
-
+
-
+
-
+
-
i
Lq
+
-
+
-
+
-
+
-
i
Ld
u
S
0
+
-
i
S
0
+
-
+
-
+
-
+
-
i
L
0
1:
D
S
1:
D
S
-j
ω
L
F
j
ω
L
F
j
ω
L
L
L
-j
ω
L
L
L
-j
ω
L
L
L
(1
-
D
S
):1
(1
-
D
S
):1
(1
-
D
S
):1
C
L
R
L
R
L
R
L
1/
j
ω
C
F
-1
/
j
ω
C
F
1/
j
ω
C
F
1/
j
ω
L
C
L
-1
/
j
ω
L
C
L
-1
/
j
ω
L
C
L
a)
b)
c)
Fi
gu
re
7.
The
dq
0 t
r
ans
f
orm
a
t
i
on
of
t
h
ree
ph
ase M
R
FC
-I
-
b
uck
-
bo
ost
(Fi
g
ure
1
)
:
a)
f
o
r
w
ard
seq
u
e
n
ce
com
pone
nt
,
b)
back
wa
rd
seq
u
e
nce c
o
m
pone
nt
, c
)
zer
o
-
seq
u
ence
com
p
o
n
e
nt
The e
q
ui
val
e
nt
ci
rcui
t
s
hav
e
b
een si
m
p
l
i
f
i
e
d
fr
om
t
h
ree ci
rcu
its to
on
e circu
it, wh
ich is sho
w
n
i
n
Fi
g
u
re
8
.
3.
1.
4. Ste
a
d
y
Sta
t
e An
al
ysi
s
There a
r
e seve
ral
anal
y
t
i
cal
m
e
t
hods
of an
al
y
s
i
s
average
d
m
odel
s
from
Fi
gu
re 8.
One
of t
h
em
i
s
a
fo
ur t
e
rm
i
n
al
net
w
or
k t
h
e
o
r
y
[17]
, t
o
st
eady
st
at
e ci
rc
uit analysis. The
n
, the stea
dy s
t
ate
m
odel is obtaine
d
si
m
p
ly b
y
eli
m
in
atin
g
th
e
reactiv
e ele
m
en
ts. Fig
.
u
r
e
9
sh
ows the stea
dy state
m
odel, where all inductors seem
to be
s
h
ort a
n
d all capacitors
ope
n. T
h
e stea
dy state cha
r
acteristics can
b
e
ob
tain
ed
b
y
co
n
s
i
d
eri
n
g th
e
circu
it
m
odel
of t
h
e
pr
esent
e
d
M
R
FC
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
: 2
088
-86
94
IJPE
DS
V
o
l
.
6, N
o
. 1,
M
a
rc
h 20
1
5
:
3
2
– 44
40
u
S
+
-
i
S
L
F
C
F
L
L
+
-
+
-
+
-
+
-
i
L
1:
D
S
-j
ω
L
F
-j
ω
L
L
L
(1-
D
S
):
1
C
L
R
L
-1/
j
ω
C
F
-1/
j
ω
L
C
L
Fi
gu
re 8.
The
dq
0 t
r
ans
f
orm
a
t
i
on
of
t
h
ree
ph
ase M
R
FC
-I
-
b
uck
-
bo
ost
M
R
C
(Fi
g
u
r
e
1)
f
o
r
φ
1
=0, a
n
d
b
a
lan
c
ed
-symmetrical circu
it con
d
ition
u
S
+
-
i
S
+
-
+
-
+
-
+
-
i
L
1:
D
S
-j
ω
L
F
-j
ω
L
L
L
(1
-
D
S
):
1
R
L
-1/
j
ω
C
F
-1/
j
ω
L
C
L
Fi
gu
re
9.
St
ead
y
st
at
e equi
val
e
nt
circ
uit of a
n
alysed circ
uit
3.
2.
St
ate
-
sp
ac
e A
v
era
gi
n
g
T
echni
que
The ge
ne
ral
fo
rm
of t
h
e aver
age st
at
e spac
e equat
i
o
ns i
s
descri
bed
by
f
o
l
l
o
wi
ng set
o
f
eq
uat
i
o
n
s
[1
2]
, [1
5]
, [1
6]
,
[
1
8]
:
d
d
dt
d
B
x
A
x
,
(29)
whe
r
e:
x
is the vector
of the
avera
g
e
d
state varia
b
les,
A
(
d
) an
d
B
(
d
) a
r
e the a
v
era
g
e
d
state m
a
trix
and
av
erag
ed
inp
u
t
m
a
trix
resp
ectiv
ely,
d
is the
continuous dut
y
factor
de
fi
ne
d
by (8). The s
t
ate-space
a
v
eraging
m
e
t
hod i
s
ba
s
e
d
on
anal
y
t
i
cal
m
a
ni
pul
at
i
o
ns
usi
n
g t
h
e
d
i
ffere
nt
c
o
n
v
e
r
t
e
r st
at
e re
pres
ent
a
t
i
ons
[
1
2]
, [
15]
,
[17].
T
h
e
ave
r
age state spa
ce
m
e
thod applied to the
m
a
trix-reactance freque
n
cy co
nve
r
ter in Fig.
1, is
illu
strated
in Fi
g
u
re
10
b
y
b
l
ock
d
i
agram
[1
2]:
Figure 10. Dia
g
ram
m
atic
representation
of
t
h
e state s
p
ace
avera
g
ing m
e
thod
for MRFC
s
Th
is m
o
d
e
llin
g
techn
i
qu
e con
s
ists in
d
e
termin
in
g
,
fi
rstly, the linear state
m
odel for ea
ch possi
ble
co
nfigu
r
ation
o
f
th
e ci
rcu
it an
d, then, to com
b
in
e all th
ese elem
en
tary m
o
d
e
ls in
to
a
sin
g
l
e and
un
ified
o
n
e
th
ro
ugh
a
d
k
du
ty facto
r
. Th
e in
pu
ts for th
e
m
o
d
e
llin
g
algo
r
ith
m
are all
ele
m
en
tary su
b
c
ircu
its for allowed
swi
t
c
h st
at
e co
m
b
i
n
at
i
ons (Fi
g
u
r
e 1
0
)
.
I
n
al
l
t
opol
ogi
es
of
M
R
FC
s, 28 s
w
i
t
c
h st
at
es can be use
d
. T
h
en
t
h
ere
are defi
ne
d di
f
f
ere
n
t
i
a
l
eq
uat
i
ons
f
o
r
eac
h of
t
h
e 28
s
w
i
t
c
h con
f
i
g
urat
i
o
ns
[1
2]
:
t
t
dt
d
k
k
B
x
A
x
,
(30)
Whe
r
e:
x
are the vect
ors
of t
h
e state varia
b
les;
A
k
(
t
) an
d
B
k
(
t
) are th
e st
ate
m
a
trix
an
d in
pu
t
m
a
trix
fo
r
k-
th
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
A C
o
mp
ari
s
on
Bet
w
een Tw
o
Avera
g
e M
o
del
l
i
ng Tec
h
ni
q
u
e
s
of
AC
-AC
P
o
w
e
r C
o
nvert
e
r
s
(
P
aw
e
ł
Szcze
ś
ni
ak)
41
switch
co
nfiguratio
n
resp
ectiv
ely. Th
e
ave
r
age state space
equati
ons
for
a MRFC can
be represe
n
ted
by the
fol
l
o
wi
n
g
e
q
ua
t
i
on [1
5]
, [1
8]
:
t
d
t
d
dt
d
,
,
B
x
A
x
,
(31)
and
28
1
1
k
k
d
,
28
1
,
,
k
k
k
t
d
d
t
d
A
A
,
28
1
,
,
k
k
k
t
d
d
t
d
B
B
.
The wei
ght c
o
efficient
d
k
i
s
t
h
e de
gre
e
o
f
o
ccur
r
ence
o
f
al
l
t
h
e pos
si
bl
e c
o
n
f
i
g
urat
i
o
ns
,
and
de
pen
d
s
on t
h
e s
w
itch
cont
rol strate
gy. Not all 28 s
w
itch c
onfigurations
occur in each s
w
itch s
e
que
nce
peri
od
T
Seq
.
Equation
(31)
defi
ne the
general form
of the
m
a
the
m
atica
l
avera
g
e state
space m
odel for MRFCs for
various
cont
rol
st
rat
e
gi
es [
18]
,
[
19]
.
The m
a
them
a
t
ical
m
odel,
of the a
n
alysed
MRFC
(Figure
1),
d
e
scrib
e
d
b
y
th
e m
a
trix
d
i
fferen
tial
eq
u
a
tion
(31
)
for Ven
t
urin
i co
n
t
ro
l strategy (4
)-(7
) is
de
fi
ne
d as (3
2
)
[
15]
. T
h
e m
odel
defi
ne
d by
e
quat
i
o
n
(32) is tim
e
-varying m
odel in state-sp
ace
form
, because
the pulse dut
y
factors
d
k
fo
r MRFCs is a ti
me
vari
a
b
l
e
[1
2]
.
A re
duce
d
t
i
m
e-i
n
vari
ant
m
odel
of t
h
e M
R
FC
can be
fo
u
nd
by
ex
pre
ssi
ng E
q
uat
i
o
n (
3
2) i
n
t
h
e
d-
q
r
o
t
a
t
i
ng
fr
am
e usi
ng t
h
e
t
w
o f
r
eq
ue
nc
y
t
r
ansf
orm
a
t
i
on m
a
t
r
i
x
(9
)-
(1
2)
[1
5]
, [
1
6
]
. Then
, we
o
b
t
a
i
n
statio
n
a
ry
tim
e
-
inv
a
rian
t set Equ
a
tio
n
(3
3). A d
e
tailed
m
a
t
r
ix
d
e
scrip
tion for equ
a
tio
ns
(33
)
are
presented
in
refe
rence
[
12]
.
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
2
2
1
3
2
1
3
2
1
3
2
1
3
2
1
3
3
3
2
2
2
1
1
1
3
3
3
3
2
2
2
2
1
1
1
1
3
3
3
3
2
2
2
2
1
1
1
1
3
2
1
3
2
1
3
2
1
3
2
1
3
2
1
S
S
S
L
L
L
CF
CF
CF
LL
LL
LL
S
S
S
L
L
L
S
L
L
L
S
L
L
L
S
F
cC
F
bC
F
aC
F
F
cB
F
bB
F
aB
F
F
cA
F
bA
F
aA
F
L
S
L
cC
L
cB
L
cA
L
S
L
bC
L
bB
L
bA
L
S
L
aC
L
aB
L
aA
F
F
F
L
L
L
CF
CF
CF
LS
LS
LS
S
S
S
u
u
u
u
u
u
u
u
u
i
i
i
i
i
i
C
R
C
D
C
R
C
D
C
R
C
D
C
d
C
d
C
d
C
C
d
C
d
C
d
C
C
d
C
d
C
d
C
L
D
L
d
L
d
L
d
L
D
L
d
L
d
L
d
L
D
L
d
L
d
L
d
L
L
L
dt
u
d
dt
u
d
dt
u
d
dt
u
d
dt
u
d
dt
u
d
dt
i
d
dt
i
d
dt
i
d
dt
i
d
dt
i
d
dt
i
d
(3
2)
B
AX
X
0
0
dq
dq
dt
d
,
(33)
The s
o
l
u
t
i
o
n
o
f
t
h
e E
q
uat
i
o
n
(
3
3
)
i
s
desc
ri
be
d
by
(
3
4)
[
12]
,
[1
5]
,
[1
6]
:
B
I
KA
Y
K
x
A
A
t
dq
t
e
0
e
1
0
,
(3
4)
Whe
r
e:
Y
qd
0
(0)–vect
or
of t
h
e initial values
of transform
e
d variables,
I
–unit
m
a
trix. The st
eady-state values of
the ave
r
a
g
ed
st
ate varia
b
les f
r
o
m
(34
)
a
r
e
de
scribe
d
by
(
3
5)
[
12]
,
[
15]
,
[
1
6
]
.
B
KA
x
1
.
(3
5)
The stea
dy state cha
r
acteristic of the
M
R
FC
top
o
lo
gy
gi
ven
in Fi
gu
re
1 ca
n
be a
n
aly
zed
with the
hel
p
o
f
th
e
solutio
n
of
ave
r
age
dif
f
ere
n
tia
l
Eq
uation
(
3
5)
[
12]
.
Evaluation Warning : The document was created with Spire.PDF for Python.