I
nte
rna
t
io
na
l J
o
urna
l o
f
Appl
ied P
o
wer
E
ng
i
neer
ing
(
I
J
AP
E
)
Vo
l.
13
,
No
.
4
,
Dec
em
b
er
20
24
,
p
p
.
9
3
4
~
9
4
3
I
SS
N:
2252
-
8
7
9
2
,
DOI
:
1
0
.
1
1
5
9
1
/ijap
e
.
v
1
3
.
i
4
.
pp
934
-
943
934
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
a
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Phas
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nchro
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th
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f
re
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e
wa
b
le
e
n
e
rg
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so
u
rc
e
s
(RES
)
b
a
se
d
p
o
we
r
g
e
n
e
ra
ti
o
n
,
t
h
e
d
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RES
with
re
sp
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iss
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e
s
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f
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R
ES
c
o
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tr
o
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ig
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iza
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e
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g
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t
o
d
e
tec
t
th
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ti
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t
y
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d
v
a
riab
les
in
term
s
o
f
a
m
p
li
tu
d
e
,
p
h
a
se
,
a
n
d
fre
q
u
e
n
c
y
.
M
o
st
ly
,
th
e
p
h
a
se
-
lo
c
k
e
d
l
o
o
p
(P
LL
)
s
y
n
c
h
r
o
n
iza
ti
o
n
sc
h
e
m
e
s
a
re
u
ti
li
z
e
d
fo
r
c
o
n
tro
l
o
f
RES
a
n
d
m
o
n
it
o
rin
g
o
f
g
ri
d
v
o
lt
a
g
e
.
T
h
e
d
y
n
a
m
ics
o
f
th
e
g
rid
sid
e
c
o
n
v
e
rter
(G
S
C)
is
d
irec
tl
y
in
flu
e
n
c
e
d
b
y
t
h
e
p
e
rfo
rm
a
n
c
e
a
n
d
d
e
si
g
n
c
rit
e
ria
o
f
th
e
P
LL
s.
T
h
is
p
a
p
e
r
p
r
o
p
o
se
s
a
n
o
v
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ra
ll
re
v
iew
o
f
th
e
p
e
rfo
rm
a
n
c
e
s
o
f
th
re
e
p
h
a
se
P
LL
b
a
se
d
g
ri
d
sy
n
c
h
r
o
n
iza
ti
o
n
m
e
th
o
d
s
u
n
d
e
r
d
iv
e
rse
we
a
k
g
ri
d
c
o
n
d
i
ti
o
n
s.
K
ey
w
o
r
d
s
:
Mo
v
in
g
a
v
er
ag
e
f
ilter
Ph
ase
-
lo
ck
ed
lo
o
p
R
en
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le
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g
y
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s
Seco
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d
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er
alize
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teg
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ato
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Sy
n
ch
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o
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s
r
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r
en
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f
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T
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s
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p
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n
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c
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ss
a
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u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Po
o
n
am
T
r
i
p
ath
y
Sch
o
o
l o
f
E
lectr
ical
E
n
g
in
ee
r
in
g
,
KI
I
T
Un
iv
er
s
ity
B
h
u
b
an
eswar
,
I
n
d
ia
E
m
ail:
p
o
o
n
am
t
r
ip
ath
y
.
lali@
g
m
ail.
co
m
1.
I
NT
RO
D
UCT
I
O
N
R
ec
en
tly
,
p
h
ase
-
lo
ck
ed
lo
o
p
s
(
PLL
s
)
ar
e
m
o
s
tly
u
s
ed
f
o
r
th
e
d
is
tr
ib
u
ted
elec
tr
ical
en
er
g
y
p
o
wer
g
en
er
atio
n
s
co
n
n
ec
ted
with
g
r
id
f
o
r
s
y
n
c
h
r
o
n
izatio
n
p
u
r
p
o
s
e.
PLL
s
ar
e
r
eq
u
ir
ed
f
o
r
ex
tr
ac
tin
g
th
e
g
r
id
’
s
v
o
ltag
e
p
r
o
f
ile
in
f
o
r
m
atio
n
lik
e
am
p
litu
d
e,
p
h
ase,
an
d
f
r
e
q
u
en
cy
to
co
o
r
d
in
ate
an
d
co
n
t
r
o
l
th
e
p
o
wer
f
lo
w
b
etwe
en
r
en
ewa
b
le
en
e
r
g
y
s
o
u
r
ce
s
(
R
E
S
)
p
o
wer
g
e
n
er
atio
n
co
n
n
ec
ted
to
m
ain
u
tili
ty
g
r
id
an
d
lo
ca
l
lo
ad
th
o
u
g
h
p
o
wer
co
n
v
er
ter
s
as a
n
in
ter
f
ac
e
[
1
]
-
[
5
]
.
T
h
e
ch
allen
g
e
f
o
r
all
th
e
p
o
w
er
elec
tr
o
n
ics
-
b
ased
elec
tr
ical
s
y
s
tem
with
s
y
n
ch
r
o
n
izatio
n
m
eth
o
d
s
is
to
h
av
e
a
s
tab
le
o
p
er
atio
n
u
n
d
er
v
ar
io
u
s
g
r
id
d
is
tu
r
b
a
n
ce
s
s
u
ch
as
v
o
ltag
e
u
n
b
alan
ce
s
,
v
o
l
tag
e
s
ag
s
o
r
s
wells,
h
ar
m
o
n
ics
an
d
d
i
r
ec
t
cu
r
r
en
t
(
DC
)
o
f
f
s
ets
p
r
esen
t
in
th
e
u
tili
ty
v
o
ltag
e.
T
h
ese
k
in
d
s
o
f
d
is
tu
r
b
a
n
ce
s
ar
e
m
ajo
r
ly
ca
u
s
ed
d
u
e
to
t
h
e
in
cr
ea
s
in
g
d
is
p
er
s
io
n
o
f
d
is
tr
ib
u
te
d
g
en
er
atio
n
s
o
r
r
e
n
ewa
b
le
en
er
g
y
s
y
s
tem
s
alo
n
g
with
p
o
wer
elec
tr
o
n
ics
d
e
v
ices
o
r
n
o
n
lin
ea
r
p
o
wer
elec
tr
o
n
ic
lo
ad
s
.
So
,
th
e
co
n
t
r
o
l
o
f
t
h
e
p
o
wer
co
n
v
er
ter
s
f
o
r
s
y
n
c
h
r
o
n
izatio
n
b
etwe
en
R
E
S
co
n
v
er
ter
an
d
th
e
u
tili
ty
g
r
id
is
h
ig
h
ly
ess
en
tial
to
en
s
u
r
e
s
af
e
an
d
ef
f
ec
tiv
e
o
p
er
atio
n
.
T
h
e
s
y
n
ch
r
o
n
izatio
n
m
eth
o
d
s
in
p
o
wer
co
n
v
er
ter
s
ar
e
class
if
ied
as c
lo
s
ed
lo
o
p
-
b
ased
an
d
o
p
en
lo
o
p
-
b
ased
s
tr
ateg
ies
[
6
]
,
[
7
]
.
T
h
e
clo
s
ed
lo
o
p
-
b
ased
s
y
n
c
h
r
o
n
i
za
tio
n
n
ee
d
s
a
f
ee
d
b
ac
k
lo
o
p
f
o
r
o
n
e
o
r
m
o
r
e
s
ig
n
als
in
its
s
tr
u
ctu
r
e
an
d
is
o
f
two
g
r
o
u
p
s
(
PLL
s
[
8
]
-
[
1
4
]
an
d
f
r
eq
u
en
cy
-
lo
ck
ed
lo
o
p
s
(
FLL
s
)
[
1
5
]
-
[
1
8
]
)
.
T
h
is
p
ap
er
em
p
h
asizes
th
e
o
v
e
r
v
iew
o
f
p
r
o
m
in
en
t
clo
s
ed
-
lo
o
p
PLL
s
with
th
eir
d
esig
n
s
an
d
ap
p
licatio
n
s
f
o
r
s
o
lv
in
g
p
o
wer
q
u
ality
p
r
o
b
lem
s
b
y
im
p
r
o
v
in
g
th
eir
p
e
r
f
o
r
m
an
ce
u
n
d
er
d
if
f
er
en
t
g
r
i
d
cir
cu
m
s
tan
ce
s
[
1
9
]
-
[
2
4
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
P
h
a
s
e
-
lo
ck
ed
lo
o
p
b
a
s
ed
s
yn
c
h
r
o
n
iz
a
tio
n
s
ch
eme
s
fo
r
th
r
ee
-
p
h
a
s
e
u
n
b
a
la
n
ce
d
…
(
P
o
o
n
a
m
Tr
ip
a
th
y
)
935
2.
DE
S
I
G
N
AND
CO
N
T
RO
L
O
F
A
F
UNDA
M
E
NT
A
L
P
H
ASE
L
O
CK
E
D
L
O
O
P
SYS
T
E
M
T
h
e
r
ep
r
esen
tatio
n
o
f
th
e
b
a
s
ic
PLL
s
y
s
tem
i
s
d
ef
in
ed
b
y
th
r
ee
n
u
m
b
e
r
s
o
f
b
asic
el
em
en
ts
as
p
r
esen
ted
in
Fig
u
r
e
1
,
wh
er
e
th
e
p
h
ase
d
etec
to
r
(
PD)
d
etec
ts
th
e
p
h
ase
er
r
o
r
b
y
m
ea
s
u
r
in
g
th
e
d
if
f
er
e
n
ce
b
etwe
en
th
e
p
h
ase
a
n
g
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o
f
t
h
e
in
p
u
t
an
d
o
u
tp
u
t
s
ig
n
al
an
d
p
ass
es
it
to
th
e
lo
o
p
f
ilter
(
L
F
)
f
o
r
th
e
elim
in
atio
n
o
f
p
h
ase
er
r
o
r
.
T
h
e
lo
o
p
f
ilte
r
also
ca
lled
as
a
p
r
o
p
o
r
tio
n
a
l
in
teg
r
al
co
n
tr
o
ller
s
en
d
s
an
o
u
tp
u
t
s
ig
n
al
th
at
en
ab
les
th
e
v
o
ltag
e
-
c
o
n
tr
o
lle
d
o
s
cillato
r
(
VC
O)
to
p
r
o
d
u
c
e
th
e
esti
m
ated
p
h
ase
as
o
u
t
p
u
t
th
at
f
o
llo
ws
th
e
in
p
u
t
f
o
r
tr
ac
k
in
g
th
e
m
ain
g
r
id
p
h
ase.
PD
is
ac
co
u
n
tab
le
f
o
r
g
en
er
atin
g
p
h
ase
er
r
o
r
in
f
o
r
m
atio
n
as
o
u
tp
u
t
wh
er
ea
s
th
e
L
F
d
r
iv
es
p
h
ase
er
r
o
r
s
ig
n
al
to
b
ec
o
m
e
ze
r
o
.
T
h
en
,
VC
O
g
en
er
ates
an
esti
m
ated
s
y
n
ch
r
o
n
ized
s
ig
n
al
in
ter
m
s
o
f
p
h
ase
as
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o
u
tp
u
t.
T
h
e
d
q
PLL
/s
y
n
c
h
r
o
n
o
u
s
r
ef
er
en
ce
f
r
am
e
PL
L
(
SR
F
PLL
)
an
d
α
βP
L
L
/f
ix
ed
r
ef
er
en
ce
f
r
am
e
PLL
(
FR
F
PLL
)
ar
e
th
e
tr
ad
itio
n
al
PLL
s
with
s
im
p
lest
im
p
lem
en
tatio
n
.
PLL
s
ar
e
ca
teg
o
r
ized
in
to
t
h
r
ee
p
h
ase
an
d
s
in
g
le
-
p
h
ase
ty
p
e
o
n
th
e
b
asis
o
f
ap
p
licatio
n
s
.
A
n
ef
f
icie
n
t
PLL
is
ch
ar
ac
ter
ized
b
y
lo
wer
co
m
p
u
tatio
n
al
b
u
r
d
e
n
,
r
o
b
u
s
tn
ess
to
g
r
i
d
in
s
tab
ilit
ies
(
i.e
.
,
i
m
b
alan
ce
d
v
o
ltag
es,
h
ar
m
o
n
ics,
DC
o
f
f
s
ets,
in
ter
h
ar
m
o
n
ics
,
an
d
f
r
e
q
u
en
c
y
v
ar
iatio
n
s
)
,
an
d
f
aster
d
y
n
a
m
ic
r
esp
o
n
s
e
with
im
p
r
o
v
e
d
s
tab
ilit
y
.
I
n
r
ec
en
t
y
ea
r
s
,
d
esig
n
i
n
g
o
f
m
o
r
e
ad
v
an
ce
d
t
h
r
ee
-
p
h
ase
PLL
s
with
en
h
an
ce
d
d
is
tu
r
b
an
c
e
r
ejec
tio
n
ab
ilit
y
co
m
p
a
r
ed
to
class
ical
SR
F
PLL
an
d
m
o
r
e
m
o
d
if
ied
v
er
s
io
n
s
o
f
class
ical
SR
F
PL
L
ar
e
e
n
co
u
r
ag
ed
to
tack
le
th
e
g
r
o
win
g
p
o
wer
q
u
ality
p
r
o
b
lem
s
in
elec
tr
ical
p
o
wer
s
y
s
tem
s
[
2
5
]
-
[
2
7
]
.
T
h
e
f
u
n
d
am
e
n
tal
SR
F
PLL
[
2
8
]
is
a
s
tan
d
ar
d
ized
PL
L
wh
ich
is
th
e
f
o
u
n
d
atio
n
f
o
r
all
th
e
ex
ten
d
ed
PLL
s
in
th
r
ee
-
p
h
ase
ap
p
licatio
n
s
.
T
h
e
u
p
c
o
m
in
g
s
ec
tio
n
p
r
esen
t
s
s
o
m
e
o
f
th
e
ad
v
an
ce
d
v
er
s
io
n
s
o
f
th
r
ee
-
p
h
ase
PLL
b
ased
g
r
id
s
y
n
ch
r
o
n
izatio
n
tech
n
iq
u
es
wh
ich
ar
e
th
e
ad
v
an
ce
d
v
er
s
io
n
s
o
f
SR
F
PLL
s
with
ce
r
tain
m
o
d
if
icatio
n
s
f
o
r
b
etter
p
er
f
o
r
m
an
ce
u
n
d
er
v
ar
io
u
s
g
r
id
d
is
tu
r
b
an
ce
s
.
Fig
u
r
e
1
.
B
asic PL
L
s
tr
u
ctu
r
e
3.
RE
VI
E
W
O
F
ADVA
NCE
D
T
H
RE
E
-
P
H
ASE
P
L
L
S WI
T
H
E
NH
ANCE
D
F
I
L
T
E
RIN
G
AB
I
L
I
T
Y
T
h
e
in
cr
ea
s
in
g
r
e
n
ewa
b
le
en
er
g
y
r
eso
u
r
ce
s
in
jectio
n
in
to
th
e
elec
tr
ical
g
r
id
with
th
e
g
r
o
win
g
in
ter
co
n
n
ec
tio
n
o
f
th
e
n
o
n
lin
e
ar
lo
ad
s
h
av
e
led
to
s
ev
er
e
p
o
wer
q
u
ality
p
r
o
b
lem
s
m
ak
i
n
g
th
e
s
y
n
ch
r
o
n
izatio
n
p
r
o
ce
s
s
m
o
r
e
ch
allen
g
i
n
g
.
T
o
tack
le
th
is
is
s
u
e,
ad
v
an
ce
d
PL
L
s
with
en
h
an
ce
d
d
is
tu
r
b
an
ce
ca
n
ce
llatio
n
ab
ilit
y
h
av
e
b
ee
n
p
r
o
p
o
s
ed
.
All
th
ese
PLL
s
ar
e
n
o
th
in
g
b
u
t
b
asi
c
SR
F
PLL
with
ad
d
itio
n
al
f
ilter
s
im
p
lem
en
ted
in
s
id
e
its
co
n
tr
o
l
lo
o
p
s
tr
u
ct
u
r
e
o
r
p
r
io
r
to
its
in
p
u
t
[
2
9
]
,
[
3
0
]
.
T
h
e
ty
p
ical
th
r
ee
-
p
h
ase
PLL
s
with
th
eir
o
v
er
all
class
if
icatio
n
ar
e
s
u
m
m
ar
ized
in
Fig
u
r
e
2
.
T
h
is
s
ec
tio
n
p
r
e
s
en
ts
a
co
m
p
r
eh
e
n
s
iv
e
r
ev
iew
o
f
th
ese
a
d
v
an
ce
d
th
r
ee
-
p
h
ase
PLL
alg
o
r
ith
m
s
.
Fig
u
r
e
2
.
Gen
e
r
al
class
if
icatio
n
s
o
f
th
r
ee
-
p
h
ase
p
h
ase
-
lo
ck
e
d
lo
o
p
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
13
,
No
.
4
,
Dec
em
b
er
20
24
:
9
3
4
-
943
936
3
.
1
.
Cla
s
s
ica
l
s
y
nchro
no
us
re
f
er
ence
f
ra
m
e
P
L
L
T
h
e
co
n
v
e
n
tio
n
al
s
y
n
ch
r
o
n
o
u
s
r
ef
er
en
ce
f
r
a
m
e
PLL
(
SR
F
PLL
)
is
th
e
b
asic
PLL
f
r
o
m
wh
ich
v
ar
io
u
s
m
o
d
if
ie
d
PLL
s
ar
e
d
ev
elo
p
e
d
u
s
in
g
C
lar
k
e
an
d
Par
k
tr
an
s
f
o
r
m
atio
n
m
o
d
u
les
f
o
r
co
n
v
e
r
s
io
n
o
f
d
q
/s
y
n
ch
r
o
n
o
u
s
r
ef
er
en
ce
f
r
a
m
e
f
r
o
m
n
atu
r
al/ab
c
r
e
f
er
en
ce
f
r
am
e.
T
h
en
,
p
r
o
p
o
r
tio
n
al
-
in
t
eg
r
al
(
PI)
co
n
tr
o
lle
r
is
u
s
ed
to
r
eg
u
late
th
e
q
v
ar
ia
b
le
with
g
r
id
f
r
eq
u
en
cy
as
an
o
u
tp
u
t.
Af
ter
th
at,
th
e
u
tili
ty
g
r
id
p
h
ase
an
g
le
is
g
en
er
ated
as
an
o
u
tp
u
t
b
y
in
te
g
r
atin
g
th
e
g
r
id
f
r
eq
u
en
cy
as
a
n
in
p
u
t
t
o
VC
O/in
teg
r
ato
r
/p
h
a
s
e
an
g
le
g
en
er
at
o
r
.
T
h
e
o
b
tai
n
ed
o
u
tp
u
t
p
h
ase
a
n
g
le
is
f
ed
a
g
ain
b
ac
k
in
t
o
th
e
PD
(
α
β
-
d
q
tr
a
n
s
f
o
r
m
atio
n
b
l
o
ck
)
f
o
r
clo
s
ed
-
lo
o
p
co
n
tr
o
l.
T
h
e
SR
F
PLL
is
m
ain
ly
u
tili
ze
d
in
t
h
r
ee
-
p
h
ase
g
r
i
d
in
teg
r
ated
p
o
wer
co
n
v
er
ter
s
d
u
e
to
its
s
im
p
lest
o
p
er
atio
n
with
ex
ac
t
an
d
f
ast
p
h
ase
o
r
f
r
eq
u
en
cy
ass
ess
m
en
t
in
m
o
s
tly
id
ea
l
g
r
id
co
n
d
itio
n
s
.
T
h
e
class
ical
SR
F
PLL
is
r
ep
r
esen
ted
in
Fi
g
u
r
e
3
.
T
h
e
SR
F
r
o
tatio
n
wit
h
r
esp
ec
t
to
its
p
o
s
itiv
e
an
g
u
l
ar
s
p
ee
d
allo
ws
th
e
class
ical
S
R
F
PLL
in
f
aster
a
n
d
p
r
ec
is
e
d
eter
m
i
n
atio
n
o
f
th
e
g
r
id
f
r
e
q
u
en
c
y
a
n
d
p
h
ase
a
n
g
le
in
n
o
r
m
al
g
r
id
co
n
d
itio
n
s
th
at
is
u
n
d
er
b
alan
ce
d
f
au
lt
cir
cu
m
s
tan
ce
s
.
W
h
er
ea
s
,
it
f
ail
s
in
tr
ac
k
in
g
th
e
p
h
ase
an
g
le
in
ca
s
e
o
f
u
n
b
alan
ce
d
f
au
lt
co
n
d
itio
n
a
s
a
r
esu
lt
o
f
th
e
d
o
u
b
le
g
r
id
f
r
eq
u
e
n
cy
(
2
ω
)
o
s
cillatio
n
s
in
tr
o
d
u
ce
d
b
y
th
e
f
u
n
d
am
e
n
tal
n
eg
ativ
e
s
eq
u
en
ce
co
m
p
o
n
e
n
ts
wh
ich
lead
s
t
o
m
is
m
atch
o
f
/d
ir
ec
t
ax
is
v
o
ltag
e
co
m
p
o
n
e
n
t
f
r
o
m
th
e
p
o
s
itiv
e
s
eq
u
en
ce
m
ag
n
itu
d
e
[
3
1
]
o
f
d
q
-
co
m
p
o
n
e
n
ts
.
T
h
e
S
R
F
-
PLL
i
s
al
s
o
u
n
ab
le
to
wo
r
k
p
r
o
p
er
l
y
u
n
d
er
h
ar
m
o
n
ically
p
o
llu
ted
g
r
id
v
o
ltag
es.
T
h
e
m
u
lt
ip
le
SR
F
f
ilter
-
b
ased
PLL
(
MSR
F
PLL
)
is
o
n
e
o
f
th
e
f
am
o
u
s
ex
ten
d
ed
v
er
s
io
n
s
o
f
SR
F
-
PLL
s
.
T
h
e
d
u
al
s
y
n
ch
r
o
n
o
u
s
r
ef
er
en
ce
f
r
am
e
p
h
ase
-
lo
ck
e
d
lo
o
p
(
DSR
F
PLL
)
[
2
]
co
n
s
is
t
s
o
f
two
SR
F
m
o
d
u
les r
o
tatin
g
with
s
am
e
an
g
u
lar
s
p
ee
d
th
o
u
g
h
in
r
e
v
er
s
e
d
ir
ec
tio
n
s
with
a
cr
o
s
s
-
f
ee
d
b
a
ck
s
y
s
tem
in
ten
d
ed
to
ex
tr
icate
an
d
s
eg
r
e
g
ate
th
e
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
n
eg
at
iv
e
an
d
p
o
s
itiv
e
s
eq
u
e
n
ce
co
m
p
o
n
e
n
ts
f
r
o
m
th
e
u
tili
ty
in
p
u
t
v
o
ltag
e.
T
h
e
im
b
alan
ce
d
g
r
id
in
p
u
t
v
o
ltag
e
h
as
n
o
n
eg
ativ
e
co
n
s
eq
u
en
ce
o
n
th
e
p
er
f
o
r
m
a
n
ce
o
f
th
e
DSR
F
PLL
b
u
t
th
e
h
ar
m
o
n
ics
m
ix
ed
v
o
ltag
e
as
in
p
u
t
o
f
DSR
F
PLL
m
ig
h
t
o
r
i
g
in
ate
f
l
u
ctu
atin
g
e
r
r
o
r
s
i
n
th
e
ass
es
s
ed
q
u
an
titi
es.
T
o
o
v
er
co
m
e
th
is
d
if
f
icu
lty
,
s
o
m
e
SR
F
b
lo
ck
s
wh
ich
r
o
tate
o
n
th
e
b
asis
o
f
tar
g
eted
h
ar
m
o
n
ic
f
r
eq
u
en
cies
with
r
es
p
ec
t
to
th
e
f
u
n
d
am
en
tal
SR
F
PLL
ca
lled
as
MSR
F
PLL
is
u
s
ed
[
3
2
]
b
u
t
m
o
r
e
SR
F
s
lead
to
in
cr
ea
s
ed
PLL
co
m
p
u
tatio
n
al
b
u
r
d
e
n
.
T
h
e
DSR
F
PLL
i
s
th
e
m
ath
em
atica
l
eq
u
iv
alen
ce
o
f
th
e
d
ec
o
u
p
le
d
d
o
u
b
le
SR
F PLL
(
DDSR
F PLL
)
wh
en
th
e
PI
co
n
tr
o
ller
’
s
in
p
u
t v
o
ltag
e
s
ig
n
al
is
ad
ju
s
ted
to
,
1
+
.
3
.
2
.
M
o
v
ing
a
v
er
a
g
e
f
ilte
r
b
a
s
ed
P
L
L
Fig
u
r
e
4
illu
s
tr
ates
th
e
m
o
v
in
g
av
er
ag
e
f
ilter
b
ased
PLL
(
MA
F
PLL
)
[
3
3
]
wh
ich
is
a
c
o
n
v
en
tio
n
al
SR
F
PLL
with
a
m
o
v
in
g
av
e
r
ag
e
f
ilter
(
MA
F)
th
at
is
a
f
ilter
o
f
lin
ea
r
p
h
ase
ty
p
e
wh
ic
h
is
d
ef
in
ed
in
th
e
L
ap
lace
d
o
m
ain
s
h
o
wn
as (
1
)
.
W
h
er
e,
r
ep
r
esen
ts
th
e
win
d
o
w
len
g
th
o
f
MA
F.
(
)
=
1
−
−
(
1
)
I
t
is
n
o
ted
th
at
in
clu
s
io
n
o
f
MA
F
b
lo
ck
s
in
s
id
e
th
e
SR
F
PLL
’
s
co
n
tr
o
l
lo
o
p
/in
lo
o
p
MA
F
SR
F
PLL
r
em
ar
k
ab
l
y
im
p
r
o
v
es
its
f
ilter
i
n
g
ab
ilit
y
b
u
t
with
a
co
n
s
id
er
a
b
le
r
ed
u
ctio
n
o
f
its
d
y
n
am
ic
r
esp
o
n
s
e
d
u
e
to
th
e
MA
F’s
in
-
lo
o
p
p
h
ase
d
elay
wh
en
MA
F’s
win
d
o
w
len
g
th
(
)
is
s
am
e
as
th
e
MA
F
PL
L
in
p
u
t
s
ig
n
al’
s
n
o
m
in
al
p
er
io
d
(
)
th
at
is
=
.
I
t
is
r
ec
o
m
m
en
d
ed
to
s
elec
t
th
e
MA
F
win
d
o
w
len
g
th
s
a
m
e
as
th
e
f
u
n
d
am
e
n
tal
tim
e
p
er
io
d
o
f
in
p
u
t
v
o
ltag
e
o
f
th
e
g
r
id
(
=
)
d
u
r
i
n
g
u
n
k
n
o
w
n
id
ea
ab
o
u
t
th
e
D
C
o
f
f
s
et
an
d
g
r
id
h
ar
m
o
n
ic
p
r
esen
t
in
th
e
PLL
’
s
in
p
u
t.
Dif
f
er
en
t
s
elec
tio
n
s
o
f
th
e
MA
F’s
win
d
o
w
le
n
g
th
s
u
c
h
as
=
2
an
d
=
6
ar
e
a
p
p
r
o
p
r
iate
f
o
r
th
e
ap
p
licatio
n
s
with
p
o
s
s
ib
le
ex
is
ten
ce
o
f
o
d
d
h
ar
m
o
n
ics
ex
i
s
tin
g
in
th
e
PLL
’
s
in
p
u
t.
T
h
e
MA
F
p
er
m
its
th
e
DC
co
m
p
o
n
en
t
to
p
ass
b
y
co
m
p
letely
b
lo
c
k
in
g
th
e
in
teg
r
al
m
u
ltip
le
f
r
eq
u
e
n
cy
c
o
m
p
o
n
en
ts
o
f
1
.
Fo
r
f
u
r
th
er
im
p
r
o
v
em
e
n
t
o
f
th
e
MA
F
PLL
d
y
n
a
m
ics
with
im
p
r
o
v
ed
h
ar
m
o
n
ics
f
ilter
in
g
o
p
e
r
atio
n
,
d
if
f
er
e
n
t a
p
p
r
o
ac
h
es h
av
e
b
ee
n
r
ec
o
m
m
en
d
ed
in
th
e
liter
atu
r
e
s
tu
d
ies.
Fig
u
r
e
3
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
a
class
ical
S
R
F PLL
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
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n
g
I
SS
N:
2252
-
8
7
9
2
P
h
a
s
e
-
lo
ck
ed
lo
o
p
b
a
s
ed
s
yn
c
h
r
o
n
iz
a
tio
n
s
ch
eme
s
fo
r
th
r
ee
-
p
h
a
s
e
u
n
b
a
la
n
ce
d
…
(
P
o
o
n
a
m
Tr
ip
a
th
y
)
937
Fig
u
r
e
4
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
t
h
e
MA
F PLL
T
h
e
co
n
v
e
n
tio
n
al
PI
in
-
lo
o
p
c
o
n
tr
o
ller
r
e
p
lace
d
b
y
a
n
in
-
lo
o
p
p
r
o
p
o
r
tio
n
al
in
teg
r
al
d
e
r
iv
ativ
e
(
PID
)
co
n
tr
o
ller
in
th
e
MA
F
PLL
’
s
L
F
p
r
o
v
id
es
ad
d
itio
n
al
d
eg
r
e
e
o
f
f
r
ee
d
o
m
.
T
h
e
p
h
ase
d
ela
y
p
r
o
d
u
ce
d
b
y
th
e
MA
F
ca
n
also
b
e
ef
f
icien
tl
y
ca
n
ce
led
o
u
t
b
y
a
p
o
le
-
z
er
o
ca
n
ce
llatio
n
ar
r
an
g
em
en
t
in
th
e
MA
F
PLL
s
tr
ateg
y
[
1
9
]
.
A
d
d
itio
n
ally
,
p
l
ac
in
g
a
s
p
ec
ial
lead
c
o
m
p
en
s
ato
r
p
r
i
o
r
t
o
th
e
MA
F
PLL
’
s
PI
co
n
tr
o
ller
[
3
4
]
,
wh
er
e
th
e
co
m
p
en
s
ato
r
’
s
tr
an
s
f
er
f
u
n
ctio
n
th
at
is
r
ev
er
s
e
o
f
th
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
MA
F
en
ab
les
in
th
e
r
ed
u
ctio
n
o
f
th
e
p
h
ase
d
elay
i
n
th
e
MA
F
b
ased
PLL
’
s
co
n
tr
o
l
lo
o
p
.
T
h
e
MA
F
PL
L
h
a
v
in
g
a
win
d
o
w
len
g
th
s
am
e
as
th
e
f
u
n
d
am
en
tal
tim
e
p
er
io
d
o
f
g
r
id
in
p
u
t
ca
n
eli
m
in
ate
co
m
p
letely
th
e
h
a
r
m
o
n
ics
with
f
r
e
q
u
en
c
y
er
r
o
r
s
an
d
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
d
is
to
r
ted
c
o
m
p
o
n
en
ts
.
Pre
v
iew
s
tu
d
y
[
3
5
]
,
r
em
o
v
al
o
f
in
l
o
o
p
MA
Fs
an
d
b
y
in
s
er
tin
g
th
em
b
ef
o
r
e
th
e
in
p
u
t
o
f
a
s
ep
ar
ate
SR
F
PLL
is
p
r
o
p
o
s
ed
.
T
h
is
MA
Fs
b
ased
S
R
F b
eh
av
es lik
e
a
p
r
ep
r
o
ce
s
s
in
g
f
ilter
th
at
ca
n
e
f
f
icien
tly
elim
in
ate
th
e
d
is
tu
r
b
in
g
c
o
m
p
o
n
en
ts
th
e
r
e
b
y
en
h
an
cin
g
th
e
PLL
’
s
d
y
n
am
ic
b
e
h
av
io
r
.
T
h
e
p
r
o
p
o
s
ed
p
r
e
f
ilter
in
g
p
h
ase
co
m
p
r
is
es
o
f
an
ex
tr
a
f
r
e
q
u
en
cy
d
etec
to
r
.
T
h
e
r
e
m
o
v
al
o
f
th
e
ad
d
e
d
f
r
eq
u
e
n
cy
d
etec
to
r
is
also
p
o
s
s
ib
le
b
y
th
e
am
p
litu
d
e
s
ca
lin
g
an
d
p
h
ase
s
h
if
t
co
r
r
ec
tio
n
p
e
r
f
o
r
m
ed
b
y
th
e
f
r
eq
u
en
cy
n
o
n
a
d
ap
tiv
e
MA
F
p
r
ef
ilter
in
g
p
h
ase
in
s
id
e
th
e
SR
F PLL
[
3
6
]
.
3
.
3
.
No
t
ch
f
ilte
r
ba
s
ed
P
L
L
Gen
er
ally
,
a
b
an
d
r
ejec
tio
n
f
ilter
ca
lled
as
n
o
tch
f
ilter
(
NF)
is
k
n
o
win
g
ly
ca
p
ab
le
o
f
a
tten
u
atin
g
f
r
eq
u
e
n
cy
o
f
s
p
ec
if
ic
r
an
g
e
an
d
p
ass
in
g
all
o
th
er
f
r
eq
u
e
n
cy
s
ig
n
als to
tally
u
n
alter
ed
.
T
h
is
ch
ar
ac
ter
is
tic
o
f
NF
en
ab
les
it
to
ca
n
ce
l
o
u
t
th
e
s
el
ec
tiv
e
r
an
g
e
o
f
h
ar
m
o
n
ic
co
m
p
o
n
en
ts
p
r
esen
t
in
th
e
PLL
in
p
u
t
s
ig
n
al
[
1
9
]
.
T
h
e
NFs
ar
e
class
if
ied
as
ad
ap
tiv
e
an
d
n
o
n
-
ad
ap
tiv
e
ty
p
e
f
ilter
s
.
T
h
e
a
d
ap
tiv
e
NF
is
h
i
g
h
ly
c
h
o
s
en
o
v
e
r
th
e
o
th
er
d
u
e
to
ea
s
ier
s
elec
tio
n
o
f
n
ar
r
o
w
b
an
d
wid
t
h
f
o
r
NF
th
er
e
b
y
m
in
im
izin
g
t
h
e
PLL
co
n
tr
o
l
lo
o
p
’
s
p
h
ase
d
ela
y
b
u
t in
cr
ea
s
es th
e
co
m
p
u
tatio
n
al
b
u
r
d
en
.
T
h
e
NF PL
L
’
s
co
n
s
tr
u
ctio
n
is
s
am
e
as th
e
ty
p
ical
MA
F PLL
wi
th
th
e
MA
F
b
lo
ck
r
e
p
lace
d
b
y
th
e
N
Fs
as
illu
s
tr
ated
in
Fig
u
r
e
5
.
I
n
clu
s
io
n
o
f
m
u
ltip
le
NFs
in
s
id
e
th
e
PLL
co
n
tr
o
l
lo
o
p
to
p
o
lo
g
y
is
also
co
n
s
id
er
ed
as
ca
s
ca
d
ed
NF
PLL
an
d
p
ar
allel
NF
PLL
d
if
f
er
en
tiated
o
n
th
e
b
asis
o
f
its
f
r
eq
u
e
n
cy
esti
m
atio
n
m
eth
o
d
[
3
7
]
,
[
3
8
]
.
All
th
e
NFs
in
p
ar
allel
co
n
f
ig
u
r
atio
n
u
tili
ze
th
e
s
am
e
f
r
eq
u
en
cy
esti
m
ato
r
wh
er
ea
s
,
ea
ch
NF
h
as
its
o
wn
in
d
iv
id
u
al
f
r
eq
u
e
n
cy
esti
m
ato
r
in
ca
s
ca
d
ed
co
n
f
ig
u
r
atio
n
th
er
eb
y
s
ettin
g
a
tr
ad
e
-
o
f
f
b
etwe
en
th
eir
f
ilter
in
g
ca
p
ac
ity
an
d
co
m
p
u
tatio
n
al
p
r
o
b
lem
.
Fo
r
s
atis
f
ac
to
r
y
p
er
f
o
r
m
an
ce
with
a
r
o
b
u
s
t
PLL
,
u
s
u
ally
th
r
ee
n
u
m
b
e
r
s
o
f
NFs
with
n
o
tch
f
r
eq
u
en
cies
at
2
,
6
,
an
d
1
2
ar
e
r
ec
o
m
m
en
d
ed
[
1
9
]
.
3
.
4
.
Sin
us
o
ida
l sig
na
l int
eg
ra
t
o
r
ba
s
ed
P
L
L
T
h
e
r
ep
r
esen
tatio
n
o
f
th
e
s
in
u
s
o
id
al
s
ig
n
al
in
teg
r
ato
r
PLL
(
SS
I
PLL
)
is
s
h
o
wn
in
Fig
u
r
e
6
wh
er
e
th
e
g
r
id
’
s
v
o
ltag
e
is
ex
tr
ac
ted
b
y
t
h
e
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
p
o
s
itiv
e
s
eq
u
en
ce
co
m
p
o
n
e
n
t (
FF
PS
C
)
with
th
e
S
R
F
PLL
d
esig
n
f
o
r
an
e
n
h
an
ce
d
o
p
er
atio
n
u
n
d
e
r
u
n
b
alan
ce
d
a
n
d
h
ar
m
o
n
ically
p
o
llu
ted
g
r
i
d
s
itu
atio
n
s
.
T
h
e
SS
I
PLL
lo
o
p
r
esp
o
n
s
e
s
p
ee
d
an
d
lo
o
p
b
an
d
wid
th
ar
e
co
n
tr
o
lle
d
b
y
th
e
p
ar
am
eter
‘
K’
.
An
e
q
u
iv
alen
t
s
tr
u
ctu
r
e
u
s
in
g
a
s
in
g
le
SS
I
is
p
r
esen
te
d
in
[
3
9
]
th
at
e
x
tr
ac
ts
p
o
s
itiv
e
s
eq
u
en
ce
co
m
p
o
n
e
n
t
th
r
o
u
g
h
th
e
s
in
g
le
SS
I
f
ilter
f
o
r
in
p
u
t
g
r
id
v
o
ltag
e
th
at
is
ca
lcu
lated
th
r
o
u
g
h
a
90
d
eg
r
ee
d
elay
in
th
e
s
ig
n
al.
T
h
e
m
ajo
r
m
er
it
o
f
SS
I
-
PLL
is
its
ab
ilit
y
to
o
p
e
r
ate
in
u
n
b
alan
ce
d
g
r
id
s
itu
atio
n
s
with
r
esis
tan
ce
to
th
e
g
r
id
v
o
ltag
e
d
is
to
r
tio
n
.
Fu
r
th
er
m
o
r
e
,
its
s
im
p
lifie
d
v
er
s
io
n
with
ce
r
tain
m
o
d
if
icatio
n
s
ca
n
b
e
ap
p
lied
to
s
in
g
le
-
p
h
ase
g
r
id
ap
p
licatio
n
.
3
.
5
.
Co
m
plex
co
e
f
f
icient
f
ilt
er
-
ba
s
ed
P
L
L
T
h
e
co
m
p
lex
c
o
ef
f
icien
t
f
ilte
r
PLL
(
C
C
F
PLL
)
f
ea
tu
r
es
asy
m
m
etr
ical
f
r
eq
u
en
c
y
r
esp
o
n
s
e
o
f
th
e
C
C
Fs
ar
o
u
n
d
ze
r
o
f
r
eq
u
en
cy
i
m
p
ly
in
g
th
eir
a
b
ilit
y
to
d
is
tin
g
u
is
h
b
etwe
en
th
e
n
eg
ativ
e
an
d
p
o
s
itiv
e
s
eq
u
en
ce
co
m
p
o
n
en
ts
at
d
if
f
er
en
t
f
r
eq
u
en
cy
[
4
0
]
.
T
h
is
tr
ait
m
ak
es
t
h
e
C
C
F
s
s
u
itab
le
f
o
r
th
e
p
r
ec
is
e
ex
tr
ac
tio
n
an
d
n
u
llifica
tio
n
o
f
h
ar
m
o
n
ics/
d
is
tu
r
b
an
ce
co
m
p
o
n
e
n
ts
f
r
o
m
p
o
l
lu
ted
g
r
id
in
p
u
t
v
o
ltag
e
b
ef
o
r
e
s
en
d
in
g
as
in
p
u
t
to
th
e
SR
F
PLL
.
T
h
e
d
iag
r
a
m
o
f
a
p
o
p
u
lar
ize
d
C
C
F
PLL
u
s
in
g
two
n
u
m
b
er
s
o
f
c
o
m
p
l
ex
co
ef
f
icien
t
-
b
ased
b
an
d
p
ass
f
ilter
s
(
C
C
B
Fs
)
a
s
p
r
ef
ilter
s
in
s
id
e
SR
F
PLL
lo
o
p
is
s
h
o
wn
in
Fig
u
r
e
7
wh
ich
is
r
ef
er
r
ed
as
d
u
al
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
13
,
No
.
4
,
Dec
em
b
er
20
24
:
9
3
4
-
943
938
co
m
p
lex
c
o
ef
f
icien
t
f
ilter
PLL
(
DC
C
F
PLL
)
.
T
h
e
C
C
Fs
in
th
e
SR
F
PLL
in
p
u
t
wo
r
k
co
llab
o
r
ativ
ely
to
e
x
tr
ac
t
a
s
p
ec
if
ic
co
m
p
o
n
en
t
f
r
o
m
th
e
PLL
in
p
u
t.
T
h
e
DC
C
F
PLL
i
s
th
e
m
ath
em
atica
l
alik
e
o
f
th
e
DSR
F
PLL
wh
en
th
e
DSR
F PLL
h
as f
ir
s
t
-
o
r
d
er
L
PF
s
with
cu
t
-
o
f
f
f
r
eq
u
en
cy
[
4
1
]
.
Pre
v
iew
s
tu
d
y
[
4
1
]
,
th
e
s
tab
ilit
y
an
aly
s
is
,
s
m
all
-
s
ig
n
al
m
o
d
ellin
g
a
n
d
a
s
y
s
tem
atic
p
r
o
ce
s
s
f
o
r
p
ar
am
eter
s
tu
n
in
g
o
f
th
e
DC
C
F
PLL
is
also
m
en
tio
n
ed
.
I
n
ca
s
e
o
f
d
o
m
in
a
n
t
h
a
r
m
o
n
i
c
co
m
p
o
n
en
ts
,
th
e
DC
C
F
PLL
ca
n
b
e
m
o
d
if
ied
b
y
th
e
u
s
e
o
f
a
d
d
itio
n
al
c
o
m
p
le
x
b
an
d
p
ass
f
ilter
s
(
C
B
F
s
)
ce
n
ter
ed
at
th
e
ch
o
s
en
h
ar
m
o
n
ic
f
r
eq
u
e
n
cies.
T
h
e
u
s
e
o
f
an
in
-
lo
o
p
PID
b
ased
L
F
SR
F
PLL
wi
th
a
p
o
le
-
ze
r
o
ca
n
ce
llatio
n
ar
r
an
g
em
e
n
t
m
in
im
izes
th
e
in
ter
co
n
n
ec
tio
n
b
etwe
en
th
e
SR
F
PLL
an
d
C
C
Fs
th
er
eb
y
im
p
r
o
v
in
g
th
e
DC
C
F
PLL
’
s
d
y
n
am
ic
p
er
f
o
r
m
an
ce
an
d
its
ex
ten
d
ed
f
o
r
m
as
[
4
2
]
.
T
h
e
C
C
F
s
ar
e
also
em
p
lo
y
ed
as
in
-
lo
o
p
f
ilter
s
with
in
th
e
SR
F PLL
’
s
lo
o
p
co
n
tr
o
l b
u
t t
h
is
to
p
o
lo
g
y
h
as n
o
t
y
et
g
ain
ed
m
u
ch
atten
tio
n
.
Fig
u
r
e
5
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
t
h
e
NF PL
L
Fig
u
r
e
6
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
t
h
e
SS
I
PLL
Fig
u
r
e
7
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
t
h
e
DC
C
F PLL
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
P
h
a
s
e
-
lo
ck
ed
lo
o
p
b
a
s
ed
s
yn
c
h
r
o
n
iz
a
tio
n
s
ch
eme
s
fo
r
th
r
ee
-
p
h
a
s
e
u
n
b
a
la
n
ce
d
…
(
P
o
o
n
a
m
Tr
ip
a
th
y
)
939
3
.
6
.
Dela
y
ed
s
ig
na
l
ca
ncella
t
io
n
-
ba
s
ed
P
L
L
T
h
e
d
elay
e
d
s
ig
n
al
ca
n
ce
llati
o
n
PLL
(
DSC
PLL
)
is
u
s
u
al
ly
in
ten
d
ed
f
o
r
en
h
an
cin
g
th
e
f
ilter
in
g
ab
ilit
y
o
f
th
e
s
tan
d
a
r
d
SR
F
P
L
L
s
b
ec
au
s
e
o
f
its
s
im
p
le
cu
s
to
m
f
ea
tu
r
es
u
n
d
er
v
ar
io
u
s
g
r
i
d
v
o
ltag
e
c
o
n
d
itio
n
s
[
4
3
]
.
I
t
is
an
i
n
-
lo
o
p
f
ilter
o
f
co
n
v
en
tio
n
al
SR
F
PLL
th
a
t
wo
r
k
s
lik
e
a
p
r
ep
r
o
ce
s
s
in
g
f
ilter
p
r
io
r
to
t
h
e
class
ical
S
R
F
PLL
in
p
u
t.
As
in
cr
ea
s
ed
p
h
ase
d
elay
an
d
s
lo
wer
d
y
n
am
ic
r
esp
o
n
s
e
m
ig
h
t
b
e
ca
u
s
ed
b
y
th
e
DSC
o
p
er
ato
r
s
th
er
eb
y
ca
u
s
in
g
th
e
i
n
s
tab
ilit
y
o
f
t
h
e
PLL
s
,
t
h
ey
ar
e
m
o
s
tly
em
p
lo
y
e
d
as
a
p
r
ep
r
o
ce
s
s
in
g
f
ilter
f
o
r
im
p
r
o
v
in
g
th
e
f
ilter
in
g
a
b
ilit
y
o
f
th
e
b
asic
SR
F
PLL
[
1
9
]
.
T
h
e
ex
p
ec
ted
h
ar
m
o
n
ic
el
em
en
ts
o
f
p
o
llu
ted
th
r
ee
-
p
h
ase
g
r
id
v
o
ltag
es
d
ete
r
m
in
e
th
e
DSC
b
lo
ck
s
n
u
m
b
e
r
in
th
e
PLL
’
s
co
n
tr
o
l
l
o
o
p
.
I
n
co
n
tr
ast,
th
e
SR
F
PLL
’
s
esti
m
ated
f
r
eq
u
en
cy
i
s
f
r
eq
u
en
tly
f
e
d
b
ac
k
f
o
r
a
d
ap
tab
ilit
y
to
th
e
v
ar
iatio
n
s
in
f
r
eq
u
en
cy
o
f
th
e
s
y
s
tem
.
Ad
d
itio
n
al
DS
C
s
f
u
n
ctio
n
in
g
in
th
e
SR
F
PLL
’
s
co
n
tr
o
l
lo
o
p
in
cr
ea
s
es
th
e
o
p
er
at
io
n
al
d
if
f
icu
lty
an
d
co
m
p
u
tatio
n
al
lo
ad
.
T
h
e
f
r
eq
u
en
cy
f
ee
d
b
ac
k
lo
o
p
lead
s
t
o
an
ex
tr
em
ely
n
o
n
lin
ea
r
an
d
to
u
g
h
-
to
-
p
er
f
o
r
m
s
tab
ilit
y
an
aly
s
is
o
f
th
e
PLL
s
y
s
tem
.
T
h
o
u
g
h
f
ew
d
if
f
er
e
n
t
m
eth
o
d
s
ar
e
p
r
esen
ted
i
n
[
1
9
]
,
th
ey
s
till
n
ee
d
f
u
r
th
er
c
o
m
p
u
tatio
n
al
ef
f
o
r
ts
.
Ad
d
itio
n
ally
,
th
e
p
h
ase
an
d
am
p
litu
d
e
er
r
o
r
co
r
r
ec
tio
n
m
eth
o
d
s
in
PLL
s
ar
e
r
ec
o
m
m
en
d
ed
f
o
r
co
m
p
u
tatio
n
al
b
u
r
d
en
r
e
d
u
ctio
n
d
u
e
to
th
e
f
ix
e
d
DSC
d
elay
len
g
th
s
lead
in
g
to
ea
s
ier
s
tab
ilit
y
an
aly
s
is
[
4
4
]
.
T
h
is
m
eth
o
d
’
s
d
em
e
r
it
is
th
e
in
ab
ilit
y
to
en
h
an
ce
d
is
tu
r
b
an
ce
elim
in
atio
n
ca
p
a
b
ilit
y
o
f
th
e
f
r
eq
u
e
n
cy
n
o
n
ad
a
p
tiv
e
DSC
s
d
u
r
in
g
th
e
u
tili
ty
g
r
id
f
r
eq
u
en
cy
d
ev
iatio
n
f
r
o
m
its
ac
tu
al
v
alu
e
wh
ich
lim
its
its
ad
d
itio
n
al
u
s
ag
e
d
u
r
in
g
lar
g
er
f
r
e
q
u
en
c
y
d
r
if
ts
an
d
s
er
io
u
s
asy
m
m
etr
ical
g
r
id
f
au
lts
.
3
.
7
.
Seco
nd
-
o
rder
g
ener
a
lize
d
inte
g
ra
t
o
r
-
ba
s
ed
P
L
L
A
s
ec
o
n
d
-
o
r
d
er
g
en
er
alize
d
in
teg
r
ato
r
PLL
(
SOGI
PLL
)
wo
r
k
s
lik
e
an
SS
I
PLL
as
well
a
s
b
eh
av
es
lik
e
a
q
u
ad
r
atu
r
e
s
ig
n
al
g
en
er
ato
r
(
QSG)
an
d
b
an
d
p
ass
f
ilter
(
B
PF
)
,
as
r
ep
r
esen
ted
in
Fig
u
r
e
8
[
4
5
]
.
Fig
u
r
e
8
s
h
o
ws
its
u
s
ag
e
f
o
r
t
h
e
FF
PS
C
ex
tr
ac
tio
n
p
r
i
o
r
t
o
th
e
in
p
u
t
to
SR
F
PLL
.
T
h
e
Q
SG
an
d
B
PF
o
f
th
e
SOGI
also
k
n
o
wn
as
QSG
-
SOGI
is
an
ap
p
r
o
p
r
iate
m
eth
o
d
f
o
r
ex
tr
ac
tin
g
an
d
s
ep
ar
atin
g
th
e
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
p
o
s
itiv
e
an
d
n
eg
ati
v
e
s
eq
u
en
ce
co
m
p
o
n
e
n
ts
o
f
th
e
th
r
ee
-
p
h
ase
g
r
i
d
v
o
ltag
e
s
ig
n
al.
T
wo
n
u
m
b
er
o
f
QSG
-
SOGI
s
ar
e
u
tili
ze
d
f
o
r
e
x
tr
ac
tio
n
o
f
th
e
d
ir
ec
t
(
d
)
an
d
q
u
ad
r
atu
r
e
(
q
)
ax
is
c
o
m
p
o
n
en
ts
o
f
an
d
an
d
th
en
in
s
tan
tan
e
o
u
s
s
y
m
m
etr
ic
al
co
m
p
o
n
en
t
m
eth
o
d
is
u
s
ed
f
o
r
t
h
e
f
u
n
d
a
m
en
tal
p
o
s
itiv
e
s
eq
u
en
ce
co
m
p
o
n
en
t
ca
lcu
latio
n
.
I
t
h
as
a
m
at
h
em
atica
l
eq
u
iv
alen
t
s
tr
u
ctu
r
e
s
im
ilar
ity
with
th
e
d
u
al
SOGI
-
b
ased
PLL
(
DSOGI
PLL
)
,
DSR
F
PLL
,
an
d
DC
C
F
PLL
[
1
9
]
.
Fo
r
en
h
an
ce
d
h
ar
m
o
n
ic
f
iltra
tio
n
ab
ilit
y
o
f
DSOGI
PLL
,
ex
tr
a
QSG
-
SOGI
s
ca
n
b
e
co
m
b
in
ed
with
th
e
s
tan
d
ar
d
s
tr
u
ctu
r
e
t
u
n
ed
at
th
e
h
ar
m
o
n
ic
f
r
eq
u
en
cies
[
1
9
]
.
An
o
th
er
m
eth
o
d
th
at
is
a
B
PF
b
ased
o
n
a
t
h
ir
d
-
o
r
d
er
g
e
n
er
alize
d
in
te
g
r
ato
r
(
T
OGI
)
an
d
a
QSG
(
QSG
-
T
O
GI
)
in
p
lace
o
f
th
e
QSG
-
SOGI
ca
n
b
e
u
s
ed
in
D
SOGI
PLL
ca
lled
T
OGI
PLL
is
u
s
ed
f
o
r
im
p
r
o
v
in
g
th
e
d
y
n
am
ic
r
esp
o
n
s
e
[
3
]
.
On
e
m
o
r
e
s
im
ilar
to
th
e
DSO
G
PLL
is
f
o
u
n
d
in
[
4
6
]
th
at
p
r
ac
tice
u
s
in
g
an
ad
a
p
tiv
e
n
o
tch
f
ilter
(
ANF)
b
u
ilt
o
n
th
e
least
m
ea
n
s
q
u
a
r
e
m
eth
o
d
(
L
MS)
with
two
ad
a
p
tiv
e
weig
h
ts
as
a
r
ep
lace
m
en
t
f
o
r
t
h
e
QSG
-
SOGI
in
its
s
tr
u
ctu
r
e.
T
h
e
QSG
-
SOGI
an
d
ANF
ar
e
m
ath
e
m
atica
lly
e
q
u
al
im
p
ly
i
n
g
th
at
th
e
ANF
-
b
ased
PLL
[
4
6
]
a
n
d
DSOGI
PLL
ar
e
m
ath
em
atica
lly
eq
u
iv
alen
t stru
ct
u
r
es.
Fig
u
r
e
8
.
B
asic stru
ctu
r
al
d
iag
r
am
o
f
t
h
e
DSOGI
PLL
3
.
8
.
O
t
her
t
hree
-
ph
a
s
e
P
L
L
s
T
h
er
e
ar
e
v
ar
i
o
u
s
ty
p
es
o
f
th
r
ee
p
h
ase
PLL
s
b
ased
o
n
d
if
f
er
en
t
ap
p
licatio
n
s
.
Pre
v
iew
s
t
u
d
y
[
4
7
]
,
SR
F
PLL
’
s
co
n
tr
o
l
lo
o
p
h
as
a
r
e
p
etitiv
e
r
eg
u
lato
r
(
R
R
)
b
ased
co
n
t
r
o
ller
in
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f
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r
id
h
ar
m
o
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n
ee
d
s
th
e
s
am
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c
o
m
p
u
tatio
n
al
ef
f
o
r
ts
b
y
th
e
r
eg
u
lato
r
m
a
k
i
n
g
R
R
-
PLL
m
o
r
e
ad
v
an
tag
e
o
u
s
.
T
h
e
h
ig
h
er
d
e
p
en
d
en
cy
o
f
R
R
PLL
r
eg
u
lato
r
with
r
esp
ec
t
to
th
e
s
am
p
lin
g
f
r
eq
u
en
c
y
in
cr
ea
s
es
th
e
co
m
p
u
tatio
n
al
co
s
t
d
u
r
in
g
in
cr
ea
s
ed
s
am
p
lin
g
f
r
e
q
u
en
c
y
m
ak
in
g
R
R
PLL
u
n
s
u
itab
le
f
o
r
ap
p
licatio
n
s
with
h
ig
h
er
s
am
p
lin
g
f
r
e
q
u
en
c
y
.
A
ze
r
o
-
cr
o
s
s
in
g
d
etec
tio
n
PLL
(
Z
C
D
PLL
)
[
4
8
]
is
r
ec
o
m
m
en
d
ed
f
o
r
r
em
o
v
in
g
th
e
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
n
eg
ativ
e
s
eq
u
en
ce
co
m
p
o
n
en
t
f
o
r
g
en
er
atin
g
th
e
b
alan
ce
d
g
r
id
s
ig
n
als
f
r
o
m
th
e
im
b
alan
ce
d
in
p
u
t
s
ig
n
als.
T
h
e
Z
C
D
PLL
h
as
im
p
lem
en
tatio
n
s
im
p
licity
with
ef
f
ec
tiv
e
o
p
e
r
atio
n
ev
e
n
wh
en
th
e
PLL
in
p
u
t
h
as
m
u
ltip
le
lo
ca
tio
n
s
o
f
ze
r
o
cr
o
s
s
in
g
s
.
T
h
e
Z
C
D
PLL
ac
co
u
n
ts
ex
clu
s
iv
ely
f
o
r
th
e
g
r
id
v
o
ltag
e
am
p
litu
d
e
im
b
alan
ce
s
as
th
e
PLL
in
p
u
t
b
u
t
with
in
a
b
ilit
y
to
wo
r
k
d
u
r
in
g
th
e
g
r
id
v
o
ltag
e
p
h
ase
im
b
alan
ce
s
.
Mo
r
eo
v
er
,
th
e
Z
C
D
PLL
h
as a
v
er
y
lim
it
ed
h
ar
m
o
n
ic
f
ilter
in
g
ca
p
ac
ity
.
T
h
e
s
p
ac
e
v
ec
to
r
Fo
u
r
ier
tr
a
n
s
f
o
r
m
s
PLL
(
SVFT
PLL
)
[
4
9
]
is
a
s
o
r
t
o
f
th
r
ee
-
p
h
ase
PLL
wh
er
e
th
e
SVFT
wo
r
k
s
lik
e
a
p
r
ef
ilter
in
g
p
h
ase
in
th
e
SR
F
PLL
.
T
h
e
SVFT
s
u
cc
e
s
s
f
u
lly
r
ejec
ts
n
ea
r
ly
all
th
e
h
ar
m
o
n
ic
elem
en
ts
with
v
er
y
litt
le
co
m
p
u
tatio
n
al
s
tr
u
g
g
le.
T
h
e
r
ec
u
r
s
iv
e
ap
p
licatio
n
o
f
SVFT
f
ilter
s
in
v
o
lv
es
p
o
s
s
ib
le
s
tab
ilit
y
is
s
u
es
in
th
e
elec
tr
i
ca
l
s
y
s
tem
wh
ich
ca
n
b
e
p
r
o
h
ib
ited
b
y
th
e
n
o
n
-
r
ec
u
r
s
iv
e
f
o
r
m
o
f
SVFT
im
p
lem
en
tatio
n
th
at
m
ig
h
t
in
cr
ea
s
e
th
e
co
m
p
u
tatio
n
al
c
o
s
t.
A
s
ec
o
n
d
-
o
r
d
e
r
lead
co
m
p
e
n
s
ato
r
s
PLL
(
SOLC
PLL
)
[
5
0
]
is
p
r
o
p
o
s
ed
b
y
th
e
im
p
lem
e
n
tatio
n
o
f
a
S
OL
C
s
ch
em
e
in
s
id
e
th
e
SR
F
PLL
wh
er
e
th
e
co
m
p
en
s
ato
r
s
co
n
tain
s
ets
o
f
p
u
r
e
im
ag
in
a
r
y
p
o
les
an
d
ze
r
o
es
p
r
o
v
id
in
g
a
s
elec
tiv
e
h
ar
m
o
n
ic
elim
in
atio
n
as
th
e
NFs
d
o
n
’
t
p
r
o
d
u
ce
p
h
ase
d
elay
s
in
th
e
SR
F
PLL
.
T
h
u
s
,
th
e
co
m
p
en
s
ato
r
s
o
f
SOLC
PLL
s
h
av
e
im
p
r
o
v
e
d
f
ilter
in
g
ab
ilit
y
with
o
u
t lim
itin
g
th
eir
b
a
n
d
wid
th
b
u
t w
ith
r
e
d
u
ce
d
n
o
is
e
im
m
u
n
ity
.
4.
P
E
RF
O
RM
A
NCE CO
M
P
A
RIS
O
N
O
F
T
H
R
E
E
-
P
H
AS
E
P
L
L
S
A
p
er
f
o
r
m
a
n
ce
co
m
p
ar
is
o
n
o
f
v
ar
io
u
s
th
r
ee
-
p
h
ase
PLL
s
wit
h
th
eir
ty
p
ical
s
tr
u
ctu
r
es
is
an
aly
ze
d
an
d
s
u
m
m
ar
ized
in
T
ab
le
1
.
I
n
an
id
ea
l
g
r
id
co
n
d
itio
n
ca
s
e,
th
e
m
o
s
t
co
m
m
o
n
ly
u
s
ed
PLL
is
SR
F
PLL
b
ec
au
s
e
o
f
its
s
im
p
le
s
tr
u
ctu
r
e,
lo
wer
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m
p
u
tatio
n
al
b
u
r
d
e
n
,
an
d
f
aster
d
y
n
am
ic
r
esp
o
n
s
e.
Du
e
to
t
h
e
h
ig
h
s
en
s
itiv
ity
o
f
th
e
SR
F
PLL
to
th
e
g
r
id
v
o
lta
g
e
im
b
alan
ce
s
a
n
d
h
ar
m
o
n
ics,
s
o
m
e
o
t
h
er
PLL
s
lik
e
SS
I
PL
L
,
DC
C
F
PLL
,
an
d
SOGI
PLL
ar
e
s
u
g
g
ested
to
in
cr
ea
s
e
th
e
s
y
s
tem
r
eliab
ilit
y
to
war
d
s
th
e
g
r
id
d
is
tu
r
b
an
ce
s
an
d
u
n
b
ala
n
ce
d
g
r
id
f
au
lts
.
W
h
er
ea
s
,
th
e
s
tab
ilit
y
p
er
f
o
r
m
an
ce
an
d
th
e
d
y
n
am
i
c
r
esp
o
n
s
e
o
f
th
e
PLL
s
ar
e
t
h
e
m
ajo
r
co
n
ce
r
n
s
.
Nea
r
ly
all
th
e
PLL
s
in
T
ab
le
1
ex
h
ib
it
d
ec
en
t
d
y
n
am
ic
r
esp
o
n
s
e
an
d
r
o
b
u
s
tn
ess
b
u
t
th
e
DS
C
PLL
,
Z
C
D
PLL
,
an
d
MA
F PLL
g
en
er
ally
r
eq
u
i
r
e
PID
-
b
ased
p
r
ef
ilter
b
lo
ck
o
r
ad
d
itio
n
al
p
h
ase
co
m
p
e
n
s
ato
r
f
o
r
im
p
r
o
v
in
g
th
e
s
y
s
tem
’
s
d
y
n
am
ic
r
esp
o
n
s
e
th
at
m
ig
h
t
r
aise
th
e
co
m
p
u
ta
tio
n
al
p
r
o
b
lem
.
Ad
d
itio
n
ally
,
th
e
Z
C
D
PLL
i
s
in
ad
v
is
ab
le
f
o
r
d
ir
ec
t
ap
p
lica
tio
n
in
th
e
g
r
id
-
in
te
g
r
ated
p
o
wer
s
y
s
tem
d
u
e
to
its
n
o
n
-
ad
ap
tiv
e
f
r
e
q
u
en
c
y
n
atu
r
e
an
d
lim
ited
h
ar
m
o
n
ic
f
iltra
tio
n
ab
ilit
y
.
A
l
l
o
f
t
h
e
t
h
r
e
e
-
p
h
as
e
P
L
L
s
i
n
T
a
b
l
e
1
b
e
n
e
f
i
t
e
d
f
r
o
m
d
i
f
f
e
r
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n
t
d
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s
t
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r
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a
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r
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t
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n
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Per
f
o
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m
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a
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th
r
ee
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p
h
ase
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Ty
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e
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o
f
t
h
r
e
e
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p
h
a
se
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LLs
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e
a
t
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r
e
s
D
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g
n
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l
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c
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t
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F
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e
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a
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e
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k
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st
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h
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h
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A
F
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e
d
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m
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e
d
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u
m
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e
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k
Le
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M
e
d
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m
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e
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m
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e
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Evaluation Warning : The document was created with Spire.PDF for Python.