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
. 1
~
9
I
S
SN
: 208
8-8
6
9
4
1
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
Modified Single Stage AC-AC Converter
Derick Mathe
w
,
Athira
P Ashok
,
Binc
y Math
ew
Departem
ent
of
Ele
c
tri
cal
Eng
i
n
eering
,
Karun
y
a
Univers
i
t
y
,
T
a
m
il Nadu
, Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
J
u
n 27, 2014
Rev
i
sed
O
c
t 22
, 20
14
Accepted Nov 10, 2014
The paper d
e
scr
i
bes the single s
t
age AC-AC converter
. This co
nverter is a
good alternative to quasi dir
e
ct back
to
back
converter
. Th
is single stage
converter is called Matrix Co
nverter
.
Matrix
converter is an array
of
controlled semiconductor sw
itch
es that conn
ects
th
ree phase source
to th
e
three phas
e
lo
ad. This
conv
erter
provid
e
s bidirectional power flow,
sinusoidal input and output
waveforms and th
ey
hav
e
no dc link storage
elem
ents. Sim
u
l
a
tion m
odel
and
results
presented showing Venturini contro
l
method of matr
ix conver
t
er
.
Keyword:
Ba
c
k
to b
a
ck
co
nv
er
te
r
B
i
di
rect
i
onal
p
o
we
r fl
o
w
Matrix
conv
ert
e
r
Stora
g
e elem
e
n
ts
Vent
uri
n
i
co
nt
r
o
l
m
e
t
hod
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
:
D
e
r
i
ck
Ma
th
ew
,
Depa
rtem
ent of Elect
ri
cal
E
n
gi
nee
r
i
n
g,
Karun
y
a Un
iv
ersity,
Kar
u
ny
a Na
gar
,
C
o
i
m
bat
o
re,
Tam
i
l
Nadu
6
4
1
1
1
4
,
I
ndi
a.
Em
a
il: d
e
rickmath
ew1@g
m
ail.co
m
1.
INTRODUCTION
The t
r
a
n
s
f
o
r
m
a
t
i
on a
nd c
o
nt
rol
of e
n
er
gy
i
s
one
o
f
t
h
e
m
o
st
im
port
a
nt
pr
ocesses i
n
el
ect
ri
cal
engi
neeri
n
g. In rece
nt years, this
work has
been done wi
th the use of
po
we
r sem
i
cond
uct
o
rs an
d
ener
gy
stora
g
e elem
e
n
ts suc
h
as c
a
pacitors a
n
d inductances
. S
e
veral
co
n
v
ert
e
r fam
i
li
es have bee
n
de
vel
ope
d:
rectifiers, i
n
v
e
rters, cho
p
p
e
rs, cyclo
c
onv
ert
e
rs, etc.
Each o
f
th
ese
families h
a
s its own
adv
a
n
t
ag
es and
li
mitatio
n
s
. The
m
a
in
ad
v
a
n
t
ag
e o
f
all
static converters over ot
her e
n
ergy
process
o
rs
is the high efficiency
that can be ac
hieve
d
.
One
of the m
o
st interesting
fa
m
i
l
i
e
s o
f
con
v
e
rters is th
at of th
e so-called
matrix
co
nv
erters
(M
Cs).
It is
h
oped
th
at t
h
e
AC
-AC m
a
trix
con
v
e
rter t
o
po
log
y
will rep
l
ace th
e
work of st
an
d
a
rd
AC-DC-AC con
v
e
rters sin
c
e
stan
d
a
rd
conv
erters are
bu
lk
y
and
co
stly. Th
is co
nv
erter t
o
po
log
y
will play
a
larg
e ro
le in
the ap
p
licatio
n
of an
indu
strial AC driv
es a
nd
wi
n
d
ene
r
gy
p
o
we
r ge
ne
rat
i
o
n. T
h
i
s
t
o
pol
og
y
can
for instance
be used in t
h
e following
area
s:
i
n
wi
nd e
n
e
r
gy
p
o
w
er
gen
e
ra
tio
n, in
an
in
du
strial AC m
o
to
r
d
r
i
v
es, in a marin
e
ap
p
licatio
n, in
a m
ili
ta
ry app
licatio
n
esp
ecially fo
r
mil
itary v
e
h
i
cles, in
an
aerosp
a
ce
ap
p
lication
.
2.
LITERATU
R
E
REVIE
W
The first stu
d
y
of di
rect AC/AC fre
q
u
ency
con
v
e
r
t
e
rs wa
s
prese
n
t
e
d i
n
19
7
6
by
[
1
]
.
I
n
a ge
neral
sense,
an
AC
/
A
C
p
o
w
er
fre
q
u
ency
c
o
nve
rsi
on i
s
t
h
e
pr
oce
sses o
f
t
r
a
n
sf
o
r
m
i
ng AC
p
o
w
er o
f
one
fr
eq
u
e
ncy
to
AC
p
o
wer
of ano
t
h
e
r frequen
c
y. In
ad
d
itio
n
t
o
th
e cap
a
bilit
y o
f
prov
i
d
in
g
con
tin
uou
s
co
n
t
ro
l
o
f
the
o
u
t
p
u
t
freq
u
e
n
c
y relativ
e to
th
e in
put frequ
en
cy th
e p
o
wer frequ
e
n
c
y co
nv
erter p
r
ov
id
e a con
tin
uou
s con
t
ro
l o
f
the
a
m
p
litu
d
e
o
f
t
h
e ou
tpu
t
v
o
l
t
a
g
e
. Th
eseconv
erters h
a
v
e
in
h
e
ren
t
b
i
d
i
rectio
n
a
l p
o
wer flow cap
ab
ilit
y. Static
po
we
r f
r
e
que
n
c
y
co
nve
rt
ers
can
be
di
vi
ded
i
n
t
o
t
w
o
m
a
in cat
e
g
o
r
i
e
s.
T
h
e
fi
rst
t
y
pe
i
s
a t
w
o
st
age
p
o
we
r
co
nv
er
ter
w
ith an
in
ter
m
ed
ia
te D
C
lin
k
cal
led
in
d
i
r
ect AC/D
C/A
C
p
o
wer
f
r
e
q
u
e
n
c
y co
nv
er
ter
.
Th
e second
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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088
-86
94
IJPE
DS
V
o
l
.
6, N
o
. 1,
M
a
rc
h 20
1
5
:
1 – 9
2
t
y
pe i
s
cal
l
e
d a di
rect
AC
/
A
C
po
wer
fre
q
u
e
ncy
co
nve
rt
er
. Th
is latter typ
e
is a on
e stag
e power conv
erter
whi
c
h c
onsi
s
t
s
basi
cal
l
y
of a
n
ar
ray
o
f
se
m
i
cond
uc
tor switch
e
s co
nn
ected
d
i
rectly between
t
h
e inpu
t and
out
put
t
e
rm
i
n
als.
2.
1.
Indirec
t
AC/
DC/
AC
P
o
w
er C
o
n
v
er
t
e
r
Th
e m
o
st tr
ad
i
tio
n
a
l top
o
l
ogy f
o
r
A
C
/A
C
p
o
w
e
r
co
nv
er
t
e
r
is a
d
i
od
e
rectif
ier
b
a
sed
p
u
l
se w
i
d
t
h
m
odul
at
ed v
o
l
t
a
ge so
urce i
n
vert
er
(P
W
M
-
V
SI
) w
h
i
c
h i
s
sho
w
n i
n
Fi
g
u
re
1. Thi
s
c
o
nsi
s
t
s
of t
w
o
po
we
r
stages and an interm
ediate
energy stora
g
e ele
m
ent.
In the first stage the AC power is conve
rted to
unc
o
n
t
r
ol
l
e
d D
C
pow
er by
t
h
e
m
eans of a d
i
ode rect
i
f
i
e
r
circu
it. Th
e co
nv
erted
DC power is th
en
st
ored
in
DC link ca
paci
tor. In t
h
e sec
o
nd stage
a
high
fre
quen
cy
s
w
i
t
c
hi
n
g
o
p
era
t
ed
P
W
M
-
VS
I gene
rat
e
s AC
s
i
gnal
s
wi
t
h
a
r
bi
t
r
a
r
y
am
pl
i
t
ude
a
n
d
fre
que
ncy
[2]
.
Fig
u
re
1
.
Di
o
d
e rectifier-PWM VSI co
nv
erter
2.
2.
Direc
t A
C
/A
C Co
nver
ter
The direct AC
/AC converter prov
id
es a d
i
rect co
n
n
ection b
e
tween
th
e in
pu
t and
o
u
t
pu
t ter
m
in
als
with
ou
t an
i
n
term
ed
iate en
erg
y
storage elemen
t th
ro
u
g
h
an a
rray
of
se
m
i
cond
uct
o
r s
w
i
t
c
hes a
s
s
h
o
w
n
i
n
Fi
gu
re 2.
Figu
re
2.
Direc
t
AC/AC c
o
n
v
e
rter
The m
a
in features
of m
a
trix co
nv
erter are the fo
llo
wi
n
g
[3
]:
a)
Si
nus
oi
dal
i
n
p
u
t
cu
rre
nt
a
n
d
si
nus
oi
dal
o
u
t
p
ut
v
o
l
t
a
ge.
b)
It
em
pl
oy
s bi
di
rect
i
onal
s
w
i
t
c
hes,
w
h
i
c
h
ena
b
l
e
s re
ge
nerat
i
ng
ene
r
gy
bac
k
t
o
t
h
e
s
o
u
r
ce.
c)
It ab
les to
adju
st th
e in
pu
t po
wer fact
o
r
of th
e co
nv
erter d
e
sp
ite th
e typ
e
o
f
th
e lo
ad
co
nn
ected. Un
ity
powe
r fact
or is easily achieva
ble.
d)
Th
ere is
n
o
i
n
term
ed
iary DC-lin
k
en
er
gy storage
.
Since t
h
e
conve
r
ter is
D
C
-l
i
nk l
e
ss, t
h
e
si
ze and c
o
st
of
th
e conv
erter i
s
relativ
ely redu
ced.
In add
itio
n,
th
e power
at th
e inp
u
t
is seen
at the
o
u
t
pu
t.
e)
It h
a
s fo
und
u
tility in
h
i
g
h
te
m
p
eratu
r
e,
h
i
gh
v
i
br
atio
n and
low
vo
lume/weig
h
t
applicatio
n
s
su
ch as
aeros
pace.
3
.
MATRI
X
CONVERTER
Th
e stud
y o
f
th
e m
a
trix
co
nv
erter
h
a
s b
e
en
go
in
g on for the last 25 years. T
h
e progress in the
d
e
v
e
l
o
p
m
en
t of
p
o
wer d
e
v
i
ce (silico
n
) tech
no
log
y
and larg
e po
wer i
n
tegrated
circu
its enco
urag
ed th
e i
n
terest
of rese
arc
h
to expl
ore a
n
AC
-AC m
a
trix conve
rter as an
el
egant silicon-i
ntensi
ve an
d efficient way to conve
r
t
electric p
o
w
er fo
r th
e
fo
llowing
: AC m
o
to
r driv
es, un
in
terrup
tib
le po
wer supp
lies, v
a
riab
le frequ
e
n
c
y
g
e
n
e
rators, and
reactiv
e en
erg
y
con
t
ro
ls.
Howev
e
r,
th
e
p
o
wer con
v
e
rt
er is still n
o
t
u
tilized
in
ind
u
s
t
r
y
because of the
diffic
u
lties involve
d
in
the practical i
m
p
l
e
m
entation relate
d
to bidi
rectional switch real
ization,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mod
ified
S
i
n
g
l
e S
t
ag
e
AC – AC Co
n
v
erter (Derick Ma
th
ew
)
3
P
W
M
c
o
nt
rol
m
e
t
hod
, t
h
e
sy
nch
r
o
n
i
zat
i
o
n a
n
d
t
h
e
p
r
ot
ect
i
on
p
r
o
b
l
e
m
s
[3]
.
T
h
i
s
sect
i
o
n
desc
r
i
bes a
m
a
t
h
em
at
i
c
al
tool
s an
d m
ode
l
s
whi
c
h are
us
ed t
o
anal
y
s
e t
h
e di
f
f
ere
n
t
co
nt
r
o
l
m
e
t
hods
com
m
onl
y
empl
oy
ed
for m
a
trix
conv
erter im
p
l
e
m
e
n
tatio
n
.
3.
1. Wor
k
i
n
g
Pri
n
ci
pl
e
In g
e
n
e
ral, th
e m
a
trix
co
nv
erter is a
sing
le-stag
e
c
o
n
v
e
r
t
e
r
wi
t
h
m
×
n bi
di
rection
a
l power switches,
desi
g
n
e
d
t
o
co
nnect
a
n
m
-
ph
ase vol
t
a
ge s
o
urce t
o
a
n
n-ph
ase lo
ad
. Th
e
m
a
trix
co
n
v
er
t
e
r of
3×3 s
w
i
t
ches,
shown in Fi
gure.3, is t
h
e m
o
st im
por
tant c
o
nverter
from
a practical point
of vie
w
,
becaus
e
it conne
cts a three
-
pha
se source to a three phase
load.
In
th
e
basic to
po
log
y
o
f
t
h
e MC show
n
i
n
Figure 3, Vi,i={a,b,c}
are the
sou
r
ce vol
t
a
ge
s,
i ={a,b,c},
are the s
o
urce
currents
, V
jn
,
j ={A,B,C}, are th
e lo
ad
vo
ltag
e
s
with
respect to
t
h
e neut
ral
poi
nt
of t
h
e l
o
a
d
n
,
and i
j
, j
=
{A,B,C} are th
e lo
ad
cu
rren
ts. Ad
d
ition
a
lly, o
t
her au
x
iliary v
a
riab
les
have
been
defi
ned t
o
be use
d
as a basi
s of t
h
e
m
odul
a
t
i
o
n and c
ont
rol
st
rat
e
gi
es an
d V
j
N
,
j = {A,B
,C
} a
r
e t
h
e
lo
ad
vo
ltag
e
s
with
resp
ect t
o
th
e n
e
u
t
ral
po
in
t N of t
h
e so
urce.
Each switch S
ji
, i ={a,b,c}, j =
{
A,B,C}, ca
n connect or disc
onnect phase I
of the
in
pu
t stage to
p
h
a
se
j of the load a
n
d, the prope
r com
b
ina
tio
n
of th
e co
ndu
ctio
n stat
es o
f
th
ese switch
e
s, arb
itrary ou
tpu
t
v
o
l
tag
e
s
V
jN
czn be syntheized. Each switch is characterized
b
y
a switch
i
ng
fu
n
c
tion
called ex
isten
ce functio
n
,
pr
o
pose
d
by
Wo
o
d
[
4
]
, pr
ovi
des a m
a
t
h
em
at
i
cal expr
e
ssi
on
fo
r desc
r
i
bi
ng s
w
i
t
c
hi
n
g
pat
t
e
r
n
s,
def
i
ned a
s
fo
llows:
S
ji
(t) =
0ifs
w
it
ch
S
iso
p
en
1ifs
w
ichS
is
clo
s
ed
(1)
Fig
u
re
3
.
Basic Power Circu
it o
f
th
e Matri
x
Co
nv
erter
3.
2. M
a
t
h
ema
tical Mo
del
o
f
Ma
trix
C
o
n
v
e
r
ter
A sim
p
lified
th
ree-ph
ase m
a
trix
conv
erter
m
o
d
e
l
i
s
sho
w
n i
n
Fi
g
u
r
e
2 an
d c
o
nsi
s
t
s
of
9 i
d
eal
bidirectional s
w
itches which allows each
of the three output lines to
be
connected t
o
any of
t
h
e thre
e input
lines. The t
h
ree converte
r
inputs are c
o
nnected to a
3-pha
se syst
em
, Va, V
b
,
Vc.
The output lines ar
e
connected to a
three-phase c
u
rrent s
o
urce
, iA, iB and iC,
which acts as t
h
e l
o
ad.
In
p
u
t
v
o
l
t
a
ges
and
o
u
t
p
ut
c
u
r
r
e
nt
s are
gi
ven
by
E
quat
i
o
n
(2
) a
n
d
(
3
)
,
re
spe
c
t
i
v
el
y
.
Vip
h
Va
Vb
Vc
V
i
n
cos
ω
t
cos
ω
t
cos
ω
t
(
2
)
I
oph
iA
iB
iC
i
cosω
t
φ
cos
ω
tφ
cos
ω
tφ
(
3
)
Whe
r
e Va,
Vb
and
Vcare t
h
r
e
e-p
h
ase i
n
put
s
i
nus
oi
dal
v
o
l
t
a
ges an
d Vi
ni
s t
h
e pea
k
val
u
e
of t
h
e i
n
put
v
o
l
t
a
ges.
Ass
u
m
i
ng that the output vol
t
age wa
veforms are sinu
s
o
i
d
a
l
and as
sum
i
ng a l
i
n
ear
l
o
a
d
, t
h
e
out
put
c
u
rre
nt
s
iA, iBand iC
are also
si
n
u
s
o
i
d
a
l. i
out
is th
e p
e
ak
v
a
lue
o
f
th
e
ou
tpu
t
curren
t
s an
d
is
the phase between
out
put
vol
t
a
ge
s and c
u
rre
nt
s
and
are th
e inpu
t and
ou
tpu
t
an
gu
lar
fre
quencies respectiv
e
l
y. The colum
n
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
:
1 – 9
4
matrices V
iph
and i
oph
pr
ovi
de
a com
p
act
m
a
t
h
em
ati
cal
form
of ex
p
r
ess
i
ng t
h
e i
n
p
u
t
vol
t
a
ge
s a
n
d
out
put
currents
, re
spe
c
tively.
4.
VENT
URI
NI
METHO
D
The M
a
t
r
i
x
C
o
n
v
e
r
t
e
r c
ont
r
o
l
st
rat
e
gi
es
were
fi
rst
m
e
nt
i
one
d
by
Al
esi
n
a [
1
]
an
d
Vent
uri
n
i
[5]
.
Vari
ous
m
odul
at
i
on t
e
c
hni
qu
es can
be
ap
pl
i
e
d t
o
t
h
e
AC
-
A
C
m
a
t
r
i
x
C
o
nve
rt
er t
o
ac
hi
eve si
nus
oi
dal
out
put
vol
t
a
ge
s an
d i
n
put
c
u
r
r
ent
s
. A
n
o
p
t
i
m
al
m
o
d
u
l
a
t
i
on st
rat
e
g
y
sho
u
l
d
m
i
ni
m
i
ze t
h
e i
nput
cur
r
ent
a
n
d t
h
e
out
pu
t
vol
t
a
ge
harm
oni
c di
st
ort
i
on
and
devi
ce
po
wer l
o
s
s
. The
fi
rst
m
odul
at
or
pr
op
ose
d
f
o
r M
a
t
r
i
x
C
o
n
v
ert
e
r
s
,
Kn
o
w
n a
s
t
h
e
Vent
uri
n
i
m
odul
at
i
o
n, em
pl
oy
ed a sc
al
ar
m
odel
[6]
.
T
h
i
s
m
odel
gi
ves
a m
a
xim
u
m
vol
t
a
ge
trans
f
er
ratio
of 0.5. T
h
e c
onc
ept of
switch
i
ng
fun
c
tio
ns is
u
s
ed
to
d
e
ri
v
e
a m
a
th
e
m
atical
m
o
d
e
l o
f
t
h
e
Matrix
C
o
n
v
ert
e
r
whi
c
h i
s
d
one
i
n
t
h
e p
r
evi
ous
cha
p
t
e
r.
In this analysis
, a three-pha
se
input, three
-
phase ou
tput conve
rter is cons
idere
d
. Becaus
e
the Matrix
Co
nv
erter is sy
mmetrical, th
e d
e
sign
atio
n
o
f
inp
u
t
and
ou
tpu
t
p
o
rts is arb
itrary. Ho
wev
e
r, for an
y sen
s
ib
le
m
ode of ope
ra
t
i
on, o
n
e p
o
rt
sho
u
l
d
be co
ns
i
d
ere
d
t
o
hav
e
a vol
t
a
ge st
i
f
f
charact
eri
s
t
i
c
and t
h
e
ot
her
po
rt
a
cu
rren
t stiff ch
aracteristic.
In
th
is
case stiff m
ean
s th
at
th
e vo
ltag
e
o
r
cu
rren
t m
u
st b
e
co
n
s
tan
t
with
no
in
ter
r
u
p
tion
s
or
su
dd
en
var
i
atio
n
s
. Fo
r
t
h
e
fo
llo
w
i
n
g
an
aly
s
is it is assu
m
e
d
th
at t
h
e input p
o
r
t
is vo
ltage stif
f
and t
h
e output
port is curre
nt
stiff. In a
pract
ical Matrix Conve
rter a
n
input filter is include
d to ci
rcula
t
e the
hi
g
h
f
r
eq
ue
nc
y
swi
t
c
hi
n
g
ha
rm
oni
cs and
p
r
o
v
i
d
e t
h
e v
o
l
t
a
ge st
i
ff c
h
ar
a
c
t
e
ri
st
i
c
. The
out
put
i
n
d
u
ct
a
n
ce i
s
usu
a
l
l
y
part
of
t
h
e l
o
a
d
gi
vi
ng
a c
u
r
r
ent
st
i
ff c
h
aract
erist
i
c. This st
udy
consider
s that
uppe
r case s
u
ffixes
al
way
s
de
n
o
t
e
t
h
e o
u
t
p
ut
ph
as
es an
d l
o
we
r c
a
se su
ffi
xes
de
not
e t
h
e i
n
p
u
t
pha
ses as
sh
o
w
n i
n
Fi
g
u
r
e
3.
If c
o
nve
nt
i
o
na
l
P
W
M
i
s
em
pl
oy
ed t
h
e
swi
t
chi
n
g se
que
nc
e Tshas
a fi
xe
d
peri
o
d
.
A
m
odul
at
i
o
n
dut
y
cy
cl
e sho
u
l
d
b
e
de
fi
ne
d
fo
r e
ach s
w
i
t
c
h
i
n
o
r
de
r t
o
de
term
ine t
h
e a
v
era
g
e
beha
viour
of the Matri
x
C
o
nverte
r
out
put
v
o
l
t
a
ge
wave
f
o
rm
. The
m
odul
at
i
o
n
d
u
t
y
cy
cl
e i
s
defi
ned
by
:
(
4
)
Whe
r
e tA are
prese
n
ts the time when s
w
itch Aa is ON an
d
Ts rep
r
esen
ts th
e ti
m
e
o
f
th
e co
m
p
lete seq
u
e
n
ce i
n
t
h
e P
W
M
pat
t
ern
.
The m
odu
l
a
t
i
on st
rat
e
gi
e
s
are defi
ned
b
y
usi
ng t
h
ese c
ont
i
n
u
o
u
s
t
i
m
e
fu
nct
i
ons
. Eq
uat
i
o
n
(
4
)
sh
ow
s th
e
use of
th
ese functio
n
s
f
o
r
t
h
e t
h
r
e
e-
ph
ase Matr
ix
Conv
er
ter
.
∙
(
5
)
∙
(6)
Vo
ltag
e
s VA,
VB &VCand
cu
rren
ts ia, ib&icin
(5) and
(6) are now val
u
es avera
g
e
d
over the sam
p
ling time
Ts. In
(7) and
(8), wh
ich
is a rep
r
esen
tation in
a
m
o
re co
m
p
act
n
o
t
atio
n o
f
(5
) and
(6
), th
e
m
a
trix
M
(
t) i
s
kn
o
w
n
as t
h
e
m
odul
at
i
on m
a
t
r
i
x
.
∙
(
7
)
∙
(
8
)
In this section, the basi
c Venturini
m
odula
tion strategy for m
a
trix conve
rte
r
will be
prese
n
ted.Modulation is the proce
d
ure use
d
to ge
nera
te the appropriate firing
pul
ses
to each of the nine
b
i
d
i
rection
a
l switch
e
s (Sj
i
) i
n
o
r
d
e
r to
g
e
n
e
rate th
e d
e
sired
ou
tpu
t
v
o
ltage. In
th
is case,
th
e p
r
im
ary o
b
j
ectiv
e
of the m
odulat
ion is to ge
nerate va
riable-fre
que
ncy and va
ri
able-am
p
litude sinusoi
d
al output volta
ges (VjN)
fr
om
t
h
e fi
xe
d-
fre
q
u
ency
a
nd
fi
xe
d
-
am
pli
t
ude i
n
p
u
t
v
o
l
t
a
ges (
V
i
)
. T
h
e easi
e
st
wa
y
of
doi
ng
t
h
i
s
i
s
t
o
co
nsid
er ti
m
e
wind
ows i
n
wh
ich
t
h
e instantan
e
ou
s
v
a
lu
es o
f
th
e d
e
si
red
o
u
t
p
u
t
vo
ltag
e
s are sam
p
led
an
d th
e
in
stan
tan
e
ou
s i
n
pu
t
v
o
ltag
e
s
are
u
s
ed to syn
t
h
e
size a signal who
s
e low
frequ
e
n
c
y co
m
p
on
en
t is th
e desired
o
u
t
p
u
t
vo
ltag
e
. If tj
i is
d
e
fin
e
d
as th
e tim
e d
u
ring
wh
ich
switch
sj
iis
o
n
an
d Tsas th
e
sa
m
p
lin
g
i
n
terval, th
e
sy
nt
hesi
s pri
n
c
i
pl
e
desc
ri
be
d abo
v
e
ca
n be e
x
p
r
esse
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
Mod
ified
S
i
n
g
l
e S
t
ag
e
AC – AC Co
n
v
erter (Derick Ma
th
ew
)
5
∙
∙
∙
∙
∀
j= {
A
,
B, C}
(9)
Obv
i
ou
sly,
Ts = tj
a + tjb
+ tj
c
∀
j
a
n
d
the
r
efore duty
cycles can be defi
ned as:
,
,
(
1
0
)
In
pu
t
Vo
ltag
e
an
d ou
tpu
t
cu
rren
t
s ar
e
sinus
o
idal and ca
n e
x
press
e
d as:
Vip
h
Va
Vb
Vc
V
i
m
cos
ω
t
cos
ω
t
cos
ω
t
(11)
io
ph
iA
iB
iC
i
cos
ω
tφ
cos
ω
t
φ
cos
ω
tφ
(12)
Su
pp
ose t
h
at
t
h
e
desi
re
d
out
p
u
t
v
o
l
t
a
ge
an
d
i
n
p
u
t
cu
rre
nt
v
ect
ors i
s
gi
ven
by
:
V
oph
VA
VB
VC
q
V
i
m
cos
ω
t
cosω
t
cos
ω
t
(13)
iip
h
ia
ib
ic
i
cos
ω
t
cos
ω
t
cos
ω
t
(14)
And
t
h
at
th
e follo
wing
activ
e p
o
wer b
a
lan
ce eq
u
a
tion
m
u
st b
e
satisfied
with
:
The e
xpl
i
c
i
t
f
o
rm
of m
a
t
r
i
x
M
(t
) ca
n be
obt
ai
ne
d
fr
om
(Al
b
ert
o
et
al
.
1
9
8
1
)a
n
d
i
t
c
a
n
be re
d
u
ced
t
o
t
h
e
fol
l
o
wi
n
g
e
x
pr
essi
on
[
7
]
:
1
2
(
1
5
)
Whe
r
e i= {a
,
b, c} a
n
d j=
{A
,
B, C}
Not
e
t
h
at
,
beca
use o
f
t
h
e ave
r
agi
n
g w
o
r
k
i
n
g
pri
n
ci
pl
e, t
h
e
o
u
t
p
ut
vol
t
a
ge l
o
w
fre
que
ncy
com
pone
nt
cannot e
x
ceed
the m
a
xim
u
m
available am
plitudes
for all instants. T
h
e refere
nce c
a
n atta
in its m
a
xim
u
m
a
t
an
arb
itrary ti
m
e
, th
erefore th
e
w
o
rst
-
case m
a
xi
m
u
m
avai
l
a
bl
e am
pl
it
udes a
r
e eq
ual
t
o
0.
5
V
i
as i
n
Fi
g
u
r
e
4 a
n
d,
th
erefore,
t
h
e v
o
ltag
e
g
a
in
of
th
e
m
a
tr
ix
co
nv
erter is
restricted
to
b
e
less th
an
0
.
5
.
It
m
u
st b
e
clarified
,
h
o
wev
e
r, th
at th
is li
m
i
t is s
m
all sin
ce th
e
mo
du
latio
n
u
n
d
e
r con
s
id
eratio
n u
s
es th
e ph
ase-to
-n
eu
tral voltag
e
s
t
o
sy
nt
hesi
ze t
h
e
out
put
v
o
l
t
a
ges,
i
.
e. t
h
i
s
i
s
a l
i
m
i
t
a
t
i
on ari
s
i
n
g
fr
om
t
h
e m
odul
at
i
o
n
use
d
,
n
o
t
f
r
o
m
t
h
e
matrix
con
v
e
rt
er [7
].
Fi
gu
re
4.
O
u
t
p
ut
v
o
l
t
a
ges
,
Vo
= 0
.
5
Vi
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
:
1 – 9
6
5.
SIMULATION MOD
EL AND RESULT
S
In
o
r
de
r to
ve
rify
the e
ffecti
v
ene
ss
of t
h
is
p
o
wer sup
p
l
y
and
th
e ou
tpu
t
vo
ltag
e
contro
l strateg
y
p
r
op
o
s
ed
, th
e
Matrix
Co
nv
erter syste
m
is s
i
m
u
la
ted
c
onsi
d
eri
n
g pract
i
cal
val
u
es
for the
electronic com
p
one
n
t
a
n
d lo
ad
co
nd
itio
n
s
. T
h
e s
y
s
t
e
m
h
a
s
ma
in
technical s
p
ecifi
cations as
follow [8]:
a)
Rated
inp
u
t
voltag
e
: Three-p
h
ase 415
V (rm
s
,
lin
e vo
ltag
e
),
5
0
Hz m
a
in
s sup
p
l
y.
b)
Rated
ou
tpu
t
po
wer: Th
e Mat
r
ix
C
o
nv
erter syste
m
is d
e
sign
ed fo
r a to
tal
p
o
wer
o
f
3k
VA.
c)
Rated
ou
tpu
t
vo
ltag
e
: Th
r
e
e-
ph
ase
11
7V
(
r
m
s, ph
ase vo
ltage)
.
Th
e t
o
tal ou
tput p
o
wer is
7
.
5
k
VA. Th
e powe
r per phase
is
c
a
lculated
as:
1000
(
1
6
)
An
eq
ui
val
e
nt
l
o
ad i
m
peda
nc
e can
be
o
b
t
a
i
n
ed as:
∙
∙
(
1
7
)
∴
1
3
.
689
C
onsi
d
eri
n
g
a val
u
e o
f
0.
8
f
o
r
t
h
e
p
o
w
er fac
t
or (
pf
),
Pf
0
.
8
c
o
s
∴
a
cos
0.8
36.87
(
1
8
)
|
|
√
(
1
9
)
|
|
cos
1
0
.
95
(20)
Xl
|
|
sin
8
.
2
Ω
(
2
1
)
2
300
(
2
2
)
8
.
7
(
2
3
)
Th
e
fin
a
l
v
a
lu
es fo
r t
h
e equ
i
valen
t
ph
ase
resistiv
e-in
du
ctive lo
ad, co
nsid
ering
a
pf
=
0
.8, are:
R = 10
Ω
L =
8.
7m
H
5.
1. Si
mul
a
ti
o
n
M
o
del
Det
a
i
l
s
of
B
a
s
i
c Vent
uri
n
i
c
ont
rol
M
e
t
h
o
d
was
ex
pl
ai
ne
d i
n
det
a
i
l
e
d i
n
sect
i
o
n
V
I
.
Thi
s
m
odel
desi
g
n
e
d
an
d
s
i
m
u
l
a
t
e
d usi
n
g
t
h
e M
a
t
l
a
b -
S
i
m
u
l
i
nk pac
k
a
g
e to
dem
onstrate the basic
prin
cip
l
e of th
e
matrix
co
nv
er
ter
.
Equaatio
n
(
15)
is used
t
o
obtain t
h
e elem
ents of the l
o
w-freque
n
c
y tran
sfer
matrix
M (t) an
d tim
es
t
j
i
.
Fi
g
u
re
5 s
h
o
w
s t
h
e
ge
ne
ral
st
ruct
ure
o
f
t
h
e m
odul
e t
h
at
ge
nerat
e
s t
h
e com
p
o
n
ent
s
m
j
i
o
f m
a
t
r
i
x
M
(
t
)
,
tak
i
n
g
as i
n
pu
t
s
th
e cu
rren
t sam
p
les o
f
th
e M
C
in
pu
t
v
o
ltages (Vi) and
o
f
t
h
e
d
e
sired
o
u
t
pu
t vo
ltag
e
s.
(V
jN=
V
j
ref
)
The m
o
st
im
p
o
rt
a
n
t
part
of
t
h
e si
m
u
l
a
ti
on i
s
t
h
e
gene
r
a
t
i
on o
f
t
h
e s
w
i
t
c
hi
n
g
f
u
nct
i
ons
of t
h
e
bi
di
rect
i
o
nal
s
w
i
t
c
hes
(S
ji
(t
)
)
.
T
h
ese
fu
nct
i
o
ns c
o
r
r
es
po
n
d
to
th
e g
a
te
driv
e si
g
n
a
ls of t
h
e
p
o
wer switch
e
s i
n
the real converter. Figure
6 presents
th
e
b
l
o
c
k
d
i
ag
ram
u
s
ed
to
g
e
n
e
rate th
ese fun
c
tio
ns in
th
e case of th
e jt
h
out
put
p
h
ase.
If we co
nsi
d
e
r
t
h
e vari
abl
e
s and wa
vef
o
r
m
s show
n i
n
Fi
gu
re
7 t
h
e "Int
r
o
d
u
ct
i
o
n" chapt
e
r can
u
lti
m
a
tel
y
result in
"Resu
lts
an
d Discu
ssi
on
" ch
ap
ter, so th
ere is co
m
p
atib
ility. Mo
reo
v
e
r, it can
also
b
e
adde
d
t
h
e
p
r
os
pect
of
t
h
e
de
v
e
l
opm
ent
of
r
e
search
re
sul
t
s
a
n
d
a
ppl
i
cat
i
o
n
pr
os
pect
s
of
f
u
rt
her
st
u
d
i
e
s i
n
t
o
t
h
e
next
(
b
ased
o
n
resul
t
a
n
d
di
sc
ussi
o
n
)
.
Fre
que
ncy
f
s=
2
k
H
z i
e
sam
p
l
i
n
g
t
i
m
e
Ts=0.
5
m
s
and c
o
nd
uct
i
o
n
t
i
m
e’s tja =
0.
2
3
m
s
, t
j
b =
0
.
1m
s
and tjc=
0.17ms.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
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:
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8-8
6
9
4
Mod
ified
S
i
n
g
l
e S
t
ag
e
AC – AC Co
n
v
erter (Derick Ma
th
ew
)
7
Fi
gu
re
5.
Ge
ne
rat
i
o
n
o
f
dut
y
c
y
cl
e
m
j
i
i
n
M
a
t
l
a
b –
S
i
m
ul
i
nk
soft
ware
pac
k
a
g
e
Fi
gu
re
6.
P
u
l
s
e ge
nerat
i
o
n sc
h
e
m
e
for
o
n
e
o
u
t
put
pha
se
The ram
p
f
unc
t
i
on st
art
i
n
g f
r
o
m
zero wi
t
h
s
l
ope
1 at
t
h
e
b
e
gi
n
n
i
n
g o
f
ea
ch sam
p
l
i
ng i
n
t
e
rval
. T
h
i
s
ram
p
sig
n
a
l is
co
m
p
ared
with ti
mes tj
aan
d
tja+tj
b
,
u
s
ing
com
p
arato
r
s Com
p
A an
d
Co
m
p
B
resp
ectively. Th
e
out
put
o
f
com
p
arat
or C
o
m
p
A i
s
t
h
e req
u
i
r
e
d
swi
t
c
hi
n
g
f
unct
i
o
n S
j
a
,
whi
c
h co
rres
p
o
n
d
s t
o
a p
u
l
se of
am
pl
i
t
ude
1
wi
t
h
a
d
u
rat
i
o
n e
qual
t
o
t
j
a.
The
f
o
l
l
o
wi
ng
l
o
gi
c deci
si
o
n
i
s
u
s
ed t
o
ge
nerat
e
swi
t
c
hi
ng
f
u
n
c
t
i
ons
fo
r S
j
b,
Sjc
.
.
B
(2
3)
5.
2. Resul
t
s
Som
e
st
udi
es have
bee
n
do
n
e
usi
n
g t
h
e
f
o
l
l
o
wi
n
g
param
e
t
e
rs:
so
urce
v
o
l
t
a
ge am
pl
it
ud
e 23
0
V
,
50
Hz, loa
d
resistance R =10
Ω
,
lo
ad
indu
ctan
ce L=8
m
H
,
vo
ltag
e
g
a
i
n
q
=
0.45
,
o
u
t
p
u
t
f
r
e
quen
c
y f0
=50H
z th
is
means that the refere
nce has
an am
plitude equal to
0
.
45
×23
0
=
10
3.5V
and
a f
r
e
qu
en
cy o
f
50H
z, sw
itch
i
ng
f
r
e
q
u
e
n
c
y
f
s
=1
/Ts=2kH
z. Fo
r th
e r
e
so
lu
ti
o
n
of
t
h
e equa
tio
n
s
a fi
v
e
-o
rd
er fi
x
e
d-step
so
lv
er, in
clud
ed in
M
a
t
l
a
b_–
Si
m
u
l
i
nk
(O
DE
5 (
D
o
r
m
a
nd-P
r
i
c
e)),
has
bee
n
use
d
. Fi
gu
re
8
sh
ow
s t
h
e
o
u
t
put
v
o
l
t
a
ge
V
A
N
an
d
Fi
gu
re
9 s
h
o
w
s t
h
e l
o
a
d
c
u
r
r
ent
i
A
f
o
r t
h
e
ab
ove c
o
ndi
t
i
ons
. T
h
e
wo
rk
i
ng
pri
n
ci
pl
e
of t
h
e M
C
i
s
cl
earl
y
d
e
m
o
n
s
trated. Th
e lo
w-p
a
ss ch
aracteristic o
f
th
e lo
ad
p
r
od
uces an
al
m
o
st s
i
n
u
s
o
i
d
a
l curren
t
ia. In
add
itio
n, it
can be
ob
ser
v
ed t
h
at
t
h
e M
C
can gene
rat
e
out
put
freque
ncies that
are not re
stricted by the sourc
e
freq
u
e
n
c
y,wh
i
c
h
typ
i
cally is th
e case in
phas
e-controlled
cy
cloconve
rters
.
Fi
gu
re
7.
Va
ri
abl
e
s
used
f
o
r
t
h
e p
u
l
s
e
gene
rat
o
r
o
f
one
o
u
t
p
u
t
pha
se
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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088
-86
94
IJPE
DS
V
o
l
.
6, N
o
. 1,
M
a
rc
h 20
1
5
:
1 – 9
8
Fi
gu
re
8.
O
u
t
p
ut
V
o
l
t
a
ge
V
A
N
, i
t
s
refe
rence
(
bol
d
lin
e)
Fi
gu
re
9.
O
u
t
p
ut
cu
rre
nt
i
A
Figure 10. Output fre
quency changes
as the
refe
rence
f
r
eq
u
e
ncy
cha
n
ges
It
i
s
im
port
a
nt
t
o
n
o
t
e
t
h
at
t
h
e
pr
op
ose
d
s
i
m
u
l
a
t
i
on st
rat
e
gy
Fi
g
u
re
1
0
sh
ows
t
h
e
o
u
t
put
phas
e
vol
t
a
ge i
n
t
h
e
sam
e
condi
t
i
o
n
s
of Fi
g
u
re
8,
but
co
nsi
d
eri
n
g a vol
t
a
ge
gai
n
of
q=0
.
9 a
n
d
an out
put
f
r
eq
uenc
y
o
f
f0
=20Hz. No
te th
at t
h
e
v
o
l
tag
e
gain
is greater th
an
the max
i
m
u
m
allo
wab
l
e (q
=
0
.5) and the
r
efore the
low-
fre
que
ncy
com
p
o
n
e
n
t
o
f
t
h
e
gene
rat
e
d
v
o
l
t
a
ge i
s
hea
v
i
l
y
d
i
sto
r
ted
.
Th
ere are in
terv
al
s in
wh
ich
t
h
e in
pu
t
vol
t
a
ge
l
e
vel
i
s
n
o
t
en
o
u
g
h
t
o
sy
nt
hesi
ze t
h
e
desi
re
d
out
put
vol
t
a
ge
.
6.
CO
NCL
USI
O
N
The
pape
r
pres
ent
s
Ve
nt
u
r
i
n
i
m
odul
at
i
on t
e
c
hni
que
s f
o
r
t
h
r
ee p
h
ase-t
o
-t
h
r
ee p
h
ase m
a
t
r
ix c
o
n
v
ert
e
r
.
In t
h
is technique we will ass
u
m
e
the desire
d out
put
volta
ge a
nd
from
that we
will de
rive the
m
odul
ation
matrix
fro
m
th
is we
will find th
e
d
u
t
y
ratio of each
sw
itch
e
s, so will g
e
t th
e sinu
so
id
al in
pu
t curren
t
and
out
put
voltage
. The feasi
b
ility and validity of the m
e
thod
were
verifie
d
by MATL
AB sim
u
lation. It can be
concl
ude
t
h
at
t
h
i
s
m
e
t
hod i
s
a
c
t
o
ac c
o
nv
ers
i
on i
n
m
a
t
r
i
x
con
v
e
r
t
e
r.
REFERE
NC
ES
[1]
Alberto Alesina,
Marco GB
Venturini, Solid-state conv
ersion: A fourier
analy
s
is
approach to
gen
e
ralized
tra
n
sforme
r s
y
nthe
sis,
IEEE Transactions on
C
i
rcuits and
System
, 1981; 28(4): 31
9.
[2]
Abdelm
alek
, et al,
Direct Power Control for AC/DC/AC Converters
in Doubley
f
e
d induction gen
e
rator b
a
sed wind
Turbine
,
IJ
E
C
E
, 2012; 2(3)
: 425-
432.
[3]
Gebregergis
,
et al.
, A thr
e
e phas
e
AC/AC matrix
conver
t
er
s
y
stem,
Thesis Ste
llen
bosch Universit
y
.
[4]
Wood P. Switching Power Converter,
Van
Nostrand Reinhold Co
mpany
, New
y
or
k, N.Y.10020
, 1
981.
[5]
Venturini M.
A
New s
i
ne wave o
u
t, con
ver
s
i
on te
chni
que which eliminates
r
e
ac
tiv
e elem
ents
, Proceeding Powerco
n
7, 1980; E3
: 1
.
[6]
Alberto Alesina
et al.
, Analy
s
is and design
of
optimum amplitude
nin
e
– switch dir
ect AC-AC conver
t
er
,
IEEE
Transactions on
Power Electronics
, 1989; 4(1): 1
01.
[7]
Rodriguez j
et
al
. Mat
r
ix
conver
t
er con
t
rol
l
ed wi
th the direct
tr
ansfer funct
i
on approach: An
al
y
s
is, m
odelling
,
and
sim
u
lation,
International journal of
electronics
, 2
005.
[8]
Saul Lop
e
z Arev
alo. Matr
ix conv
erter
for fr
equen
c
y
ch
anging
po
we
r supply
applications,
University o
f
Nottingham
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mod
ified
S
i
n
g
l
e S
t
ag
e
AC – AC Co
n
v
erter (Derick Ma
th
ew
)
9
BIOGRAP
HI
ES
OF AUTH
ORS
Mr. De
ri
ck
Ma
th
e
w
receiv
e
d B
achelor’s degree in Electr
ical an
d Electronics En
gineer
ing fro
m
Kannur University
in 2011 and
is now pursuing
his Masters degree in Power Electronics and
Drives at Karuny
a University
.
M
s
.
Athir
a
P
A
s
hok
received
t
h
e Ba
chelo
r
‟
s
d
e
gree
in
El
ec
tric
al
and
Ele
c
troni
cs
Engin
eerin
g
from MG University
in 2012
an
d is now pursuing
her Masters
degree in Power
Electron
ics and
Drives at Karuny
a University
.
Ms
. Bin
c
y M
Math
e
w
was bor
n in Kerala
. She rec
e
ived th
e
B.Te
ch Degre
e
in Ele
c
tri
c
a
l
an
d
Electronics Engineering from Mar BaseliosColle
ge,Ker
ala Univ
ersity
in 2012 an
d now pursuing
M
.
Tech
in
P
o
wer El
ectron
i
cs
and
Drives
from
Ka
run
y
a
Univers
i
t
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.