Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 2
,
A
p
r
il
201
4, p
p
.
24
3
~
25
6
I
S
SN
: 208
8-8
7
0
8
2
43
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
/
IJECE
Achieving Pull-in Avoiding Cycl
e Slip using Second-order PLLs
A
b
u-
Sa
y
e
ed
A.
H
u
q
u
e
Department o
f
Electrical Engin
e
ering, Univ
ersity
of Tabuk
, KSA
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 3, 2013
Rev
i
sed
Feb
26
, 20
14
Accepted
Mar 10, 2014
S
y
nchronization
is an essential process and o
n
e of the first tasks of the
receiver
in case
of coher
e
nt com
m
uni
cations as
well s
y
nchronou
s digital data
trans
f
er.
The p
h
as
e lock loop
(P
LL), which em
plo
y
s
the err
o
r trackin
g
techn
i
que, has b
een a ver
y
popular way
to
implement this s
y
n
c
hr
onizer sin
c
e
the early
1930s.
A phenomenon
called cy
cle slip
occurs when th
e number of
c
y
cl
es
pres
ent i
n
the trans
m
itt
e
d
carri
er (clo
ck
) differs
from
that of the
recover
e
d
carr
i
er (c
lock)
at
t
h
e re
ce
iver
. Th
e c
y
c
l
e s
lip
c
a
n
be v
e
r
y
detrimental to
some applicati
ons such
as frequen
c
y modulated
com
m
unications
sy
st
em
s (FSK,
m
u
lti-carr
i
er et
c.
), burst digital d
a
ta tr
ansfer,
training pulse retriev
a
l, and so on. This
paper presents a remed
y
to avoid th
e
cy
cle slip
b
y
using properly
de
signedsecond-ord
e
r Ty
p
e
II
PLL.
Keyword:
Bifurcatio
n
False lo
ck
L
i
m
i
t
c
y
cl
e
Phase
p
o
rt
rait
Pu
ll-o
u
t
frequ
en
cy
Copyright ©
201
4 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
:
Abu-Syaeed A. Huque
,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
U
n
i
v
er
sity of
Tabu
k,
Tabu
k,
K
i
ng
dom
o
f
Saud
i
A
r
ab
ia.
Em
a
il: ah
u
q
u
e
@u
t.edu
.
sa
1.
INTRODUCTION
Whe
n
i
n
f
o
rm
ati
on i
s
se
nt
fr
o
m
one p
o
i
n
t
t
o
anot
her
,
re
gar
d
l
e
ss o
f
t
h
e t
y
pe o
f
p
r
ocessi
ng
(anal
o
g o
r
d
i
g
ital), ex
tractio
n
o
f
th
e time b
a
se fro
m
th
e tran
sm
i
tter i
s
i
mmen
s
ely i
m
p
o
r
tan
t
. Th
e p
r
o
c
ess of th
is ti
me
base e
x
traction, at the
receive
r, is
known as
sync
hro
n
i
z
at
i
o
n
. M
o
st co
mmo
n
l
y
f
oun
d synch
r
on
izatio
n
can
be
divide
d i
n
thre
e m
a
in categories. In the ca
se of
ca
rrier syn
c
hron
iza
tion
, fo
und
bo
th in
d
i
g
ital an
d an
alog
comm
unications syste
m
s, a l
o
cal oscillator in the receiver
is locked to the carrier
of the
transm
i
tted
message.
I
n
t
h
e
c
a
s
e
o
f
cl
ock sy
nch
r
oni
z
a
t
i
o
n
, fou
n
d
p
a
rticu
l
arly in
d
i
g
ital co
m
m
u
n
i
catio
n
s
, th
e rising
o
r
t
h
e fallin
g
edge
of t
h
e re
cei
ver cl
ock i
s
li
ned up wi
t
h
t
h
at
of t
h
e t
r
ansm
i
t
t
er, so t
h
at
t
h
e dat
a
can be ca
pt
u
r
e
d
correctly. In t
h
e third cate
g
ory
of
sy
nc
hr
o
n
i
zat
i
on,
k
n
o
w
n
as
w
o
r
d
sy
nc
h
r
oni
z
a
t
i
o
n
foun
d in d
i
g
ital pack
et
data tra
n
sfe
r
, t
h
e
begi
nni
ng a
n
d the e
n
d
of a
data
packet are ascertaine
d
by the recei
ver.
Th
e
p
r
o
cess
of syn
c
hron
izatio
n
is equ
a
lly essen
tial for any
cohere
nt
c
o
mm
u
n
i
c
at
i
o
n
i
nde
pe
nde
nt
of
t
h
e m
e
di
u
m
used, e.
g.
, wi
re o
r
wi
rel
e
ss
, cab
l
e
or fi
ber;
ho
weve
r, t
h
e i
m
p
l
em
ent
a
t
i
on de
t
a
i
l
s
m
a
y
vary
. Thi
s
fact rem
a
in
s tru
e
for an
y tran
sm
it
ter(s)-
rec
e
i
v
er(
s
) c
o
nne
ct
i
on t
o
p
o
l
o
gy
, suc
h
a
s
poi
n
t
-t
o-
poi
nt
,
mu
l
tica
s
t
,
b
r
oad
ca
st
,
shared
,
mes
h
, etc
.
For insta
n
ce, in the case
of a
sha
r
e
d
ch
annel
c
o
m
m
uni
cat
i
ons,
whe
r
e t
h
e
dedicate
d
m
e
s
s
ages for each user a
r
e c
o
m
b
ined
(m
ultiple
xed) in
differe
n
t ti
m
e
slots (TDMA) or fre
que
ncy
slots (FDM
A), or, a c
o
m
b
ina
tion
of
both, t
h
e indivi
dual
receiver
has t
o
perform
the synchronization
before
extracting
(dec
odi
ng) t
h
e m
e
s
s
age i
n
tended for it.
There a
r
e at least three ways
to im
ple
m
ent
a sync
hr
o
n
i
zer
, nam
e
l
y
, 1) erro
r t
r
acki
n
g
,
2
)
m
a
xim
u
m
seek
ing
fo
ll
o
w
ed
b
y
selection
filterin
g
and
3) n
o
n
-
li
n
ear o
p
e
ration
fo
llowed
b
y
p
a
ssi
ve
filtering
. The
m
o
st
co
mm
o
n
syn
c
hr
on
izer
, kno
wn as
phase l
o
ck
loop that
uses
the error
t
r
ac
ki
ng
t
e
c
h
ni
q
u
e, has been
use
d
si
nce
t
h
e co
he
rent
c
o
m
m
uni
cat
i
ons
has
bee
n
i
n
ve
n
t
ed i
n
t
h
e ea
rl
y
1
9
3
0
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
24
3 – 2
5
6
24
4
2.
PLL MODEL
In esse
nce, a
mo
del
is an
abstractio
n
o
f
someth
in
g
real.
Th
at th
ing
can b
e
an
id
ea, a
co
nd
itio
n, an
ex
istin
g
system o
r
a
p
o
t
en
tial syste
m
.
Ma
th
ema
tica
l
mo
d
e
lin
g
, i
n
p
a
rticular, is a techn
i
q
u
e
of tran
slatin
g
on
e
o
f
th
e ab
ov
e ite
m
s
fro
m
an
ap
p
lication
aren
a in
to
tractable
m
a
th
em
a
tic
al fo
rm
u
l
atio
ns who
s
e th
eo
retical
and/
or n
u
m
e
rical
anal
y
s
i
s
pr
ovi
des usef
ul
i
n
si
g
h
t
s
, det
a
i
l
s
an
d gui
danc
e.
As
f
o
r
a
n
exi
s
t
i
ng o
r
po
t
e
nt
i
a
l
syste
m
, th
e mo
d
e
l
rep
r
esen
t
s
its k
e
y ch
aracteristics
an
d attrib
u
t
es in
a brok
en
-do
w
n
,
co
m
p
on
en
t
lev
e
l
fashi
o
n.
Ofte
n, m
odels are
a
n
alyzed i
n
a
n
effort t
o
b
u
i
l
d
a sy
st
em
or t
o
c
ont
ro
l, rem
e
d
i
ate or
op
timize it
s
p
e
rf
or
m
a
n
ce [
1
],
[
2
].
T
h
e
r
e a
r
e thr
e
e k
e
y s
t
ep
s
in for
m
u
l
a
tin
g
a
ma
th
em
a
tical
m
o
del as
de
picted in Fi
gure
1.
Fi
gu
re 1.
Key
st
eps of
m
a
t
h
em
at
i
cal
m
odel
i
ng
2.
1. Wh
y
We
Need a M
o
del
There
are
m
a
ny practical reas
ons
to use m
a
them
ati
cal
m
odel
s
. I
n
t
h
e e
n
gi
neeri
n
g
de
si
g
n
p
r
oce
ss, i
n
p
a
rticu
l
ar, th
e
u
s
ag
e o
f
m
a
th
e
m
atical
m
o
d
e
ls are p
r
im
aril
y
of gr
eat
bene
fi
t
due t
o
t
h
e fact that they allow us
t
o
det
e
rm
i
n
e t
h
e po
ssi
bl
e
beha
vi
o
r
o
f
a sy
st
e
m
wi
t
hout
ha
vi
ng t
o
b
u
i
l
d
i
t
,
whi
c
h i
n
m
a
ny
cases, c
oul
d b
e
hu
ge
and/
or c
o
st
l
y
. Sur
p
ri
si
n
g
l
y
, som
e
tim
es
t
h
e m
odel
s
hel
p
cl
ari
f
y
un
de
rl
y
i
ng assum
p
t
i
ons
i
n
t
h
e desi
g
n
p
r
oces
s.
Th
ey m
a
y a
l
so
su
gg
est/id
en
tify th
e cru
c
ial d
a
ta th
at shou
ld
b
e
co
llected
wh
ile cond
u
c
ti
n
g
m
easu
r
em
en
ts or
may generate
data that cannot be c
o
llected from
real
-life
m
easurem
ents. Anot
her im
porta
nt feature
of t
h
e
m
odel
s
i
s
t
h
at
t
h
ey
m
a
y
pred
i
c
t
t
h
e out
c
o
m
e
un
de
r va
ry
i
n
g co
n
d
i
t
i
ons
b
y
m
odi
fy
i
ng t
h
e sy
st
em
param
e
t
e
rs
[
3
],
[4
].
2.
2.
B
a
si
c Arc
h
i
t
ecture
o
f
a PL
L
As m
e
n
tio
n
e
d earlier, a
p
h
a
se lo
ck
l
o
op
(PLL) is
basic
a
lly an error t
r
acki
ng
fee
d
back control
syste
m
. It tracks the
pha
se
of
a refe
re
nce signal (m
ost comm
only
b
a
ndp
ass
) an
d tries to
lo
ck
on
to it. Fi
g
u
re
2
shows the
bas
i
c architecture
of a PLL syste
m
,
cont
ai
ni
ng t
h
ree com
p
o
n
e
n
t
bl
oc
ks,
nam
e
l
y
, t
h
e
ph
ase
co
m
p
arato
r
(detecto
r), th
e loo
p
filter and
th
e vo
ltag
e
contro
lled
o
s
cilla
tor (VCO
)
.
Wh
ile b
u
ild
ing
a fu
ll-up
syste
m
, each of these
com
p
onent bl
ocks ca
n be realized
in many
diffe
re
nt
ways.
The e
x
t
e
r
n
al
r
e
fere
nce i
n
p
u
t
si
gnal
i
s
c
o
m
m
onl
y
m
odel
e
d as t
h
e s
u
m
of a
desi
re
d f
r
eq
ue
ncy
com
pone
nt, the undesi
red fre
que
ncy
c
o
m
ponents and
the additive noise
com
pone
nt. A voltage
control
l
ed
o
s
cillato
r with a
cen
ter frequ
en
cy
at
i
s
c
h
os
en t
o
t
r
ack
t
h
e
pha
se
of
t
h
e
re
fere
nce i
n
p
u
t
.
Su
pp
ose
and
are the
in
st
an
tan
e
ou
s ph
ase
of
t
h
e
reference input si
gnal
an
d th
at
o
f
th
e
VCO
o
u
t
p
ut
si
g
nal
,
respectively. T
h
e phase c
o
m
p
arat
or
produce
s a signal that
contai
ns a co
m
p
on
en
t qu
an
tifyin
g
th
e
ph
ase
error
∅
, al
on
g wi
t
h
t
h
e
ra
ndo
m
no
ise
. Th
is sign
al is fed
in
t
o
th
e
l
o
op
filter
(u
sually
l
o
w
pass
),
an
d th
e
filtered ou
tpu
t
is app
lied
to
t
h
e
VCO
as th
e
co
n
t
ro
l in
pu
t
with
th
e
ho
p
e
th
at it adjusts th
e ph
ase of the
VC
O
out
put
si
gnal
s
o
t
h
at
t
h
e phase e
r
r
o
r
∅
di
m
i
ni
shes m
onot
oni
cal
l
y
. If
t
h
e l
o
o
p
i
s
des
i
gne
d an
d
ope
r
a
t
e
d
properly, the
phase e
r
ror
∅
(ab
s
o
l
u
t
e
v
a
lu
e)
b
e
co
m
e
s v
e
ry small u
n
d
e
r th
e
ph
ase-l
o
ck
ed
con
d
ition
.
In
th
e PLL lite
ratu
re, th
e d
e
fau
lt arch
itectu
r
e sh
own
in
Fig
u
re 2
is called
a
sho
r
t-loop
o
r
b
a
seband
m
odel
,
t
o
be s
p
eci
fi
c. T
h
e
r
e
i
s
anot
her
ve
rs
i
on,
cal
l
e
d
long
-
l
o
o
p
,
found in s
o
m
e
receivers
that
deals with a
n
IF signal as the re
fere
nce.
In th
e long-loop arc
h
itecture, a stage of
do
wn c
o
nve
rsi
o
n
t
o
IF f
o
l
l
o
we
d by
b
a
ndp
ass filterin
g
is fo
llowed
b
y
ano
t
h
e
r stag
e o
f
do
wn
co
nv
ersion
(t
o b
a
seb
a
nd
)
fo
l
l
o
w
ed
b
y
a lowp
ass
filterin
g
t
h
at is d
o
n
e
p
r
i
o
r to
th
e VC
O feed
. Our work
will b
e
restricted
to
th
e baseb
a
nd
loo
p
(sh
o
rt-l
o
op)
m
odel that ope
r
ates in
a no
is
e-
fr
e
e
en
v
i
ron
m
e
n
t.
2.
3. C
o
mp
one
n
t M
o
del
s
There a
r
e t
h
ree
m
a
i
n
com
pon
ent
bl
oc
ks i
n
a
basi
c (s
h
o
rt
- l
o
op
) PL
L co
nst
r
uct
i
on a
s
sh
o
w
n i
n
Fi
gu
re
2. A
n
y
o
n
e o
f
t
h
ese t
h
ree bl
ock
s
can be i
m
pl
em
ent
e
d i
n
vari
o
u
s way
s
usi
n
g vari
ou
s
di
ffe
rent
t
ech
ni
q
u
es
and/
or t
ech
n
o
l
ogi
es;
he
nce,
t
h
ei
r cor
r
es
p
o
n
d
i
n
g m
odel
s
chan
ge acc
or
di
n
g
l
y
. Eac
h
f
o
rm
has i
t
s
ow
n
adva
nt
age
s
an
d di
sa
dva
nt
age
s
. C
o
m
m
onl
y
,
t
h
e t
echni
cal
req
u
i
r
em
ent
s
of a gi
ve
n ap
pl
i
cat
i
on an
d t
h
e cost
associated
with a pa
rticular implem
en
tatio
n
determin
e th
e
rig
h
t
cho
i
ce.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Achi
evi
n
g
P
u
l
l
-
i
n
Av
oi
di
n
g
C
ycl
e Sl
i
p
usi
n
g
Seco
n
d
-
o
r
d
er
PLLs (
A
bu
-
Say
eed A
.
Hu
q
u
e)
24
5
As m
e
nt
i
oned
i
n
t
h
e p
r
e
v
i
o
u
s
sect
i
on, t
h
e
pha
se com
p
ara
t
or,
herea
f
t
e
r
kn
o
w
n a
s
t
h
e
pha
se det
ect
o
r
(PD
)
,
pr
o
duces a si
g
n
al
t
h
at
cont
ai
ns a com
pone
n
t
quant
i
f
y
i
n
g
t
h
e i
n
st
ant
a
neo
u
s p
h
ase di
f
f
er
ence bet
w
ee
n t
h
e i
n
p
u
t
refe
rence a
n
d t
h
e VC
O
out
pu
t
si
gnal
.
T
h
us,
a sim
p
l
e
m
ode
l
for
t
h
e
phas
e
det
ect
or
, s
h
o
w
n i
n
Fi
g
u
re
2,
coul
d
be e
x
p
r
esse
d a
s
a f
u
nct
i
o
n
o
f
t
h
e i
n
st
a
n
t
a
ne
o
u
s
ph
ase
di
ffe
r
e
nce
Fi
gu
re
2.
B
a
si
c arc
h
i
t
ecture
of a PLL
system
∅
(1
)
Exact f
u
nction
fo
r
∅
will b
e
determin
ed
b
y
t
h
e natu
re of the p
h
ase detecto
r
u
s
ed
in
t
h
e
syste
m
an
d
ou
r
wo
rk
will b
e
restrict
ed
to
two
m
o
st co
mm
o
n
l
y u
s
ed
PDs,
on
e kn
own
as sinu
so
id
al PD and
t
h
e o
t
h
e
r
one
as t
r
i
a
ng
ul
ar P
D
.
Math
em
a
tical
l
y
, an
n
t
h
-
o
r
d
e
r filter, sho
w
n
as lo
op
filter in
Fig
u
re 2, can
be
m
o
d
e
led
in
t
h
e Lap
l
ace
dom
ai
n as
⋯
⋯
,
(2
)
Bo
th
activ
e and
p
a
ssiv
e loop
filters are co
m
m
o
n
in
th
e PLLs, tho
ugh
activ
e filters im
p
l
e
m
en
ted
with
OP-
A
M
P
s
are
m
o
re com
m
on i
n
m
oder
n
a
ppl
i
cat
i
ons.
Th
e vo
ltag
e
con
t
ro
lled
o
s
cillato
r (VCO) tak
e
s in
a con
t
ro
l
vo
ltag
e
et
and p
r
o
duce
s
a si
n
u
soi
d
wi
t
h
instantane
ous pha
se
, as
de
picted in Figure
2.
A popular
way
to m
a
th
e
m
atical
ly
m
o
d
e
l su
ch
a VC
O is
b
y
relating
th
ese two
q
u
a
n
tities as
(3
)
whe
r
e c
onst
a
nt
s
and
are known as t
h
e ce
nter or quiesce
n
t
freq
u
e
n
cy
an
d gai
n
param
e
ter o
f
the
VC
O,
resp
ectiv
ely. Th
u
s
, t
h
e freq
u
e
n
c
y of t
h
e
VCO is
p
r
op
ortion
a
l to
t
h
e con
t
ro
l
vo
ltag
e
around the
ce
nte
r
fre
que
ncy
.
R
e
defi
ni
n
g
t
h
e i
n
st
ant
a
ne
ous
p
h
a
ses of t
h
e ref
e
rence i
n
p
u
t
t
o
t
h
e PD a
nd t
h
e out
put
o
f
t
h
e
VC
O
by
s
ubt
ract
i
n
g
a
pha
se
ram
p
t
e
rm
, so
m
e
times known as
a
quiesce
nt
phase
,
sim
p
lif
ies th
e an
alysis.
Thu
s
, t
h
e
n
e
w
in
stan
tan
e
ou
s
relativ
e ph
ase
qu
an
tities in
to th
e PD and
o
u
t
o
f
th
e
VCO b
e
co
m
e
s resp
ectiv
ely
(4
)
and
(5
)
whe
r
e
is th
e i
n
stan
tan
e
ou
s
ph
ase
o
f
t
h
e
referen
c
e inp
u
t
t
o
th
e PD, as
defin
e
d
earlier.
W
i
t
h
th
is, th
e
pha
se er
r
o
r at
t
h
e
out
put
o
f
t
h
e PD
i
s
al
so
re
defi
ned
as
∅
(6
)
Furt
her
ass
u
m
i
ng
0
0
lead
s to th
e VCO m
o
d
e
l in th
eti
m
e d
o
m
ai
n
as
(7
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
24
3 – 2
5
6
24
6
whi
c
h i
s
rep
r
e
s
ent
e
d i
n
t
h
e VC
O
bl
oc
k di
agram
sho
w
n
i
n
Fi
g
u
re 3
.
A
f
t
e
r ap
pl
y
i
ng t
h
e p
r
o
p
e
r
b
o
u
nda
ry
condition, t
h
e
VCO m
odel i
n
the La
place domain becom
e
s
(8
)
Fi
gu
re
3.
M
a
t
h
em
at
i
cal
m
odel o
f
a
VC
O
Thu
s
, it im
p
lie
s th
at th
e respon
se
of
t
h
e
V
C
O
is i
n
v
e
r
s
ely
p
r
op
or
tio
n
a
l
to th
e
f
r
e
qu
en
cy
o
f
th
e
cont
rol
vol
t
a
g
e
. Su
ch
a
first-o
r
d
e
r lowp
ass filter is so
meti
m
e
s k
n
o
wn
as an
in
teg
r
at
or. Th
erefore, t
h
e
VCO add
s
an
o
r
d
e
r to
th
e ord
e
r of th
e lo
op filter (wh
i
ch
i
s
u
s
u
a
lly an
o
t
her lo
wp
ass filter) wh
en
it co
mes to
th
e ov
er
all or
der
o
f
t
h
e PLL syste
m
.
2.
4. Order an
d
T
y
pe
of a P
L
L
Sys
t
em
The order
of a
PLL is the (hi
g
hest) degree of the de
n
o
m
i
nator
pol
y
n
o
m
i
al
of t
h
e cl
osed l
o
o
p
t
r
a
n
sfe
r
fun
c
tion
of th
e syste
m
. Th
erefore, th
e
ord
e
r
o
f
t
h
e loo
p
filter is on
e less th
an
t
h
at of th
e
PLL, si
n
ce th
e VCO
itself is a first-o
r
d
e
r l
o
wp
ass
filter, co
mm
o
n
l
y ter
m
ed
as
an
in
tegrator.
On
th
e
o
t
h
e
r
h
a
nd
, th
e Ty
p
e
of
a PLL
is d
e
term
in
ed
b
y
th
e nu
m
b
er
o
f
in
tegrators
presen
t
in
t
h
e sy
st
em
. Ty
pe
I P
LL, t
h
ere
f
o
r
e,
doe
s
not
co
nt
ai
n a
n
y
in
teg
r
at
o
r
i
n
th
e loo
p
filter; th
e so
le in
teg
r
ato
r
is con
t
ri
bu
ted
b
y
th
e
VCO. Lik
e
wise, Typ
e
II con
t
ain
s
on
e
in
teg
r
at
o
r
in
t
h
e lo
op
filter,
Typ
e
III con
t
ain
s
two
a
n
d
so o
n
.
For in
stance, a th
ird-order Typ
e
II PLL will
h
a
v
e
on
e i
n
tegrato
r
in
t
h
e
VC
O an
d ano
t
h
e
r
in
teg
r
at
o
r
i
n
its second
-o
rd
er lo
op
filter.
2.
5.
Wh
y Sec
o
nd-O
r
der PL
L
The c
o
m
p
lexity of be
ha
vioral analysis as
we
ll as
th
e co
nstru
c
tio
n of a PLL system
g
r
o
w
s drastically
as the
order
of
the PLL i
n
crea
ses. Fr
o
m
th
at p
e
rsp
ectiv
e, the first-o
r
d
e
r loo
p
app
ears to
be th
e m
o
st attractiv
e
choi
ce.
H
o
we
v
e
r, t
h
ey
are
n
o
t
oft
e
n
use
d
bec
a
use na
rr
o
w
ba
nd
wi
dt
h an
d
g
o
o
d
t
r
ac
ki
n
g
, c
o
m
m
onl
y
achieve
d
b
y
larg
e DC gain
, are in
co
m
p
atib
le in
th
e first-o
r
d
e
r loop
s [5
]. Nex
t
co
m
e
s
th
e secon
d
-o
rd
er PLL, wh
ich
of
fers sat
i
s
fact
ory
pe
rf
orm
a
nce for m
o
st
appl
i
cat
i
ons des
p
i
t
e
it
s sim
p
li
ci
ty
. As hi
nt
ed ea
rl
i
e
r, w
h
en t
h
e
ord
e
r
o
f
t
h
e PLL grows to th
ree, the co
m
p
lex
ity o
f
an
alysis
g
r
ows sign
ifican
tly. Th
erefo
r
e, it is rarely used
ex
cept
for sp
ecific app
licatio
n
s
where ex
t
r
em
el
y t
i
g
h
t
j
itter to
le
ran
ce is requ
ired
. Besi
d
e
s,
du
e to
t
h
e fact th
at a
second-order s
y
ste
m
can be
decom
pose
d
i
n
to a set
of
t
w
o
si
m
u
l
t
a
neous
fi
rst
-
o
r
de
r
sy
st
em
s, enabl
i
ng
t
h
e
vi
sual
i
zat
i
on
o
f
t
h
e sol
u
t
i
o
ns
i
n
a 2-
D pl
ane
,
kn
ow
n as p
h
a
s
e po
rt
rai
t
,
a secon
d
-
o
r
de
r l
o
o
p
has a di
st
i
n
ct
edge
in term
s of
dyna
m
i
cal syste
m
analysis
o
v
er
i
t
s
hi
ghe
r
or
der
cou
n
t
e
r
p
art
s
.
2.
6. C
o
mm
on T
y
pes of
Ph
as
e
De
tect
ors
As m
e
nt
i
oned
earl
i
e
r, eve
r
y
b
l
ock i
n
t
h
e PL
L sy
st
em
can be realized in more tha
n
one
way, thus, t
h
e
fi
rst
bl
ock i
n
t
h
e sy
st
em
, t
h
e pha
se det
ect
or
, can
be i
m
pl
em
ent
e
d i
n
m
a
ny
way
s
. T
h
e
Si
nu
soi
dal
pha
se
det
ect
or a
nd t
h
e
Tri
a
n
gul
ar
phase detectors are the
m
o
st comm
on one
s fo
und
in
th
e PLL
architectures
. The
sin
u
s
o
i
d
a
l
d
e
tecto
r
s are
u
s
ually i
m
p
l
e
m
en
ted
b
y
an
al
o
g
m
u
ltip
liers an
d
are v
e
ry co
mm
o
n
in
th
e leg
acy
syste
m
s. The
m
odel for it
c
a
n
be
expresse
d a
s
∅
sin
∅
, w
h
ere
is k
now
n
as
th
e p
h
a
se d
e
tector
gai
n
.
O
n
t
h
e ot
her
ha
nd
, t
h
e
t
r
i
a
ng
ul
ar
p
h
ase
det
ect
or
s,
wh
ich
are im
p
l
e
m
e
n
ted
relativ
ely easily with
an
XOR
gat
e
an
d
har
d
l
i
m
i
t
e
rs, are
ver
y
com
m
on t
h
ese day
s
.
Wh
en
the refere
nce i
n
put and
t
h
e output of the
VCO are
already in
digit
a
l form
at, even the ne
e
d
for hard lim
iters goes away and th
e PD ca
n
be re
alized with
onl
y a
n
XOR
gat
e
.
T
h
e m
odel
f
o
r
a
t
r
i
a
ng
ul
ar
PD, likewise
,
ca
n be
e
x
presse
d
as
∅
tri
∅
,
wh
er
e ’
t
ri’
rep
r
ese
n
t
s
a
(bi
pol
a
r) t
r
i
a
n
gul
ar f
u
nct
i
o
n
o
f
∅
havi
ng
u
n
i
t
y
a
m
pli
t
ude.
3.
RESEARCH
METHOD: BEHAVIORAL ANAY
LISIS OF SECOND-ORDER PLL
No
w t
h
at
we
h
a
ve t
o
uc
hed
u
p
o
n
t
h
e i
n
di
vi
dual
m
odel
s
fo
r t
h
e c
o
m
pone
nt
bl
ocks
i
n
t
h
e sy
st
em
, we
are read
y to
con
s
tru
c
t th
e m
o
d
e
l for a fu
ll-up
second
-o
rd
er PLL system
.
As su
gg
ested
i
n
th
e
p
r
ev
iou
s
sectio
n
,
o
n
l
y a first-o
r
der lo
op
filter (lo
w
p
a
ss) is requ
ired
to
b
u
i
l
d
a seco
nd
-o
rd
er PLL sin
ce th
e VCO itself also
acts
as a fi
rst-o
r
d
e
r
lo
wp
ass
filter.
Th
e tran
sfer fun
c
tio
n of a first
-
ord
e
r l
o
wp
as
s filter can
b
e
written
as
(9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Achi
evi
n
g
P
u
l
l
-
i
n
Av
oi
di
n
g
C
ycl
e Sl
i
p
usi
n
g
Seco
n
d
-
o
r
d
er
PLLs (
A
bu
-
Say
eed A
.
Hu
q
u
e)
24
7
whe
r
e
K
is k
nown
as t
h
e loo
p
fi
lter g
a
in
and
and
are resp
ect
iv
ely th
e zero
an
d th
e
p
o
l
e
of th
e filter.
To
en
sure th
e
stab
ility o
f
th
e lo
op
, it is also
requ
ired
th
at
0
. The close
d
loop gai
n
of the s
y
ste
m
can
be defi
ned
as
≡
by lu
m
p
in
g
all th
e ind
i
v
i
du
al gain
param
e
ters to
g
e
t
h
er.
It is
worth no
ting
t
h
at
G
can be placed anywhere
in
the
loop
without affecting the
an
alysis. T
h
us, in t
h
e
bloc
k
diagram
for a
s
econd-
o
r
d
e
r PLL sh
ow
n in
Figu
r
e
4, th
e closed loop
g
a
in
G
is arbitrarily in
clu
d
e
d
in th
e
VC
O
blo
c
k
.
Fi
gu
re 4.
B
l
oc
k di
ag
ram
of
a seco
nd
-o
r
d
er P
LL
sy
st
em
Fin
a
lly, b
y
d
e
fin
i
n
g
an
o
t
h
e
r lo
op p
a
ram
e
ter, kno
wn
as th
e
d
e
tun
i
ng
p
a
rameter in
th
e PLL literatu
re,
as
∆
, th
e m
a
th
e
m
atical
m
o
d
e
l for a g
a
i
n
no
rmali
zed second-orde
r
PLL sy
st
e
m
can be express
e
d
as
∅
∅
∅
∅
∅
∆
(1
0)
whe
r
e
∅
∅
is th
e d
e
ri
v
a
tiv
e of
∅
, th
e PD
ou
tput, w
.
r.t.
∅
,
and the gain
normalized syste
m
param
e
ters are
/
/
∆
∆
/
(1
1)
For
no
n-z
e
r
o
b
, t
h
e
ab
o
v
e sy
s
t
em
has
onl
y
o
n
e i
n
t
e
grat
or
, c
o
m
i
ng f
r
om
t
h
e VC
O, a
n
d s
u
ch a
sy
st
em
is term
ed as second- order
Type I. In c
o
ntra
st, when
b
b
e
co
m
e
s zero
,
t
h
e lo
op
filter b
e
co
m
e
s a p
r
opo
rt
io
n
a
l
pl
us i
n
t
e
g
r
at
or
havi
n
g
a p
o
l
e
at
t
h
e o
r
i
g
i
n
a
nd
he
nce t
h
e
s
y
st
em
cont
ai
ns
t
w
o i
n
t
e
gr
at
or
s. S
u
ch a
sy
st
em
i
s
then te
rm
ed as
second-order
T
y
pe II.
In t
h
e t
h
e
o
ry
o
f
di
ffe
rent
i
a
l
e
quat
i
o
n,
a sec
o
nd
-
o
r
d
er e
q
uat
i
on ca
n
be
dec
o
m
posed i
n
t
o
a set
of
t
w
o
sim
u
l
t
a
neous
f
i
rst
-
o
r
de
r e
q
u
a
t
i
ons a
nd t
h
e
sol
u
t
i
o
ns ca
n b
e
po
rt
ray
e
d i
n
a 2-
D
pl
ane,
k
n
o
w
n as
phase
pl
ane
.
Thu
s
, t
h
e system
d
e
scrib
e
d
b
y
Eqn
.
(1
1) can
b
e
rewritten
as
∅
∅
∅
∅
∅
∅
∆
′
∅
(1
2)
Thi
s
at
t
r
act
i
v
e
t
echni
que
, cal
l
e
d p
h
ase
p
o
rt
r
a
y
e
d, ca
n
be a
ppl
i
e
d
t
o
sec
o
nd
-
o
r
d
er l
o
o
p
s
i
n
o
r
der t
o
vi
sual
i
ze t
h
e d
y
n
am
i
cal
behavi
o
r
of t
h
e sy
s
t
em
on t
h
e ph
ase pl
ane,
whi
c
h i
s
not
p
o
ssi
bl
e fo
r hi
g
h
e
r
or
de
r
l
o
o
p
s. M
a
ny
a
t
i
m
e
s, a no
n-l
i
near
seco
n
d
-
o
r
d
er
sy
st
em
doe
s n
o
t
ha
ve cl
os
ed f
o
rm
sol
u
t
i
ons
.
In
pa
rt
i
c
ul
ar f
o
r
t
h
e
sy
st
em
des
c
ri
be
d by
Eq
n
.
(1
0)
, th
e so
urce
o
f
no
n-linearity is in
∅
a
nd/
or
∅
∅
,
a
n
d
b
o
t
h of
ou
r
candi
dat
e
PD
s
pr
od
uce
no
n
-
l
i
n
ear
∅
. Furtherm
ore, due to t
h
e prese
n
ce of the peri
odicit
y
(of
2
π
pe
ri
o
d
)
in
∅
, as
well as in
∅
∅
, resu
lts in
a si
m
ilar p
e
riod
icity in
th
e ph
ase
p
o
rtrait of th
e system
.
Thu
s
, the
pha
se
po
rt
rai
t
of
t
h
e a
b
ove
sy
st
em
can be
com
p
l
e
t
e
l
y
vi
sual
i
zed
by
w
r
ap
pi
n
g
i
t
up
on
t
h
e
su
rface
o
f
a
cy
l
i
nder o
f
u
n
i
t
radi
us [
6
]
.
There
f
o
r
e, t
h
e d
y
n
am
i
c
beha
vi
oral
st
u
d
y
of t
h
e sy
st
em
can be restricted in one
su
ch
p
e
ri
o
d
, someti
mes also
k
n
o
wn
as cell, with
ou
t lo
ss of g
e
n
e
rality. Lastly, th
ere is a
th
eorem
in
th
e th
eory
of
di
ffe
re
nt
i
a
l
equat
i
o
n t
h
at
st
at
es t
h
at
t
h
e l
o
cal
beha
vi
o
r
of a
no
n
-
l
i
n
ear sy
st
em
can be
deri
ved
f
r
om
i
t
s
co
rresp
ond
ing
lin
earized
syst
e
m
arou
nd
an
eq
u
ilibriu
m
p
o
in
t, as l
o
ng
as th
e eq
u
ilibrium
p
o
i
n
t
is no
t
no
n-
hy
pe
rb
ol
i
c
. F
u
rt
herm
ore, at
t
i
m
es, t
h
i
s
l
o
cal
beha
vi
o
r
ca
n be ext
e
nde
d t
o
obt
ai
n a
reas
o
n
abl
e
i
d
ea a
b
o
u
t
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
24
3 – 2
5
6
24
8
gl
o
b
al
b
e
ha
vi
o
r
of t
h
e sy
st
e
m
. Al
l
of the
s
e features a
r
e
cap
italized
in
an
alyzin
g th
e
syste
m
at h
a
nd
and
f
o
llow
i
ng
two
su
bsectio
ns br
i
e
f
l
y to
u
c
h
u
pon
th
e
r
e
su
lts of th
e an
alysis f
o
r
two
p
opu
lar
k
i
nd
s
o
f
PDs used
i
n
man
y
PLLs up
to
th
is d
a
te.
3.
1. D
y
n
a
mi
c B
e
ha
vi
or of
a
Secon
d
-
o
rder
PL
L
Usi
n
g
Si
nusoi
d
a
l
P
D
The
analysis i
s
rest
ricted t
o
one
peri
od
∅
of
∅
(also
∅
∅
)
witho
u
t
th
e lo
ss
o
f
g
e
n
e
rality, sin
c
e th
e
p
h
a
se
p
o
rtrait rep
eats aft
e
r th
e sam
e
p
e
rio
d
.
Th
e two
equ
ilib
ri
u
m
p
o
i
n
t
s i
n
th
e p
e
riod
of in
terest are
∅,
∅
∅
,0
and
∅,
∅
∅
,0
,
whe
r
e
∅
s
i
n
∆
and
0
∅
/
2
.
Th
e
first equ
ilib
riu
m
p
o
i
n
t
tu
rn
s
ou
t to
b
e
a
fo
cu
s
and
th
e
second one is a
sa
ddl
e
n
o
d
e
[5],
[
6
], [7
].
It is im
p
o
r
tan
t
to
n
o
t
e
h
e
re that th
e lo
cations
o
f
th
e eq
u
ilibria are
d
e
p
e
nden
t
on
t
h
e loo
p
p
a
ram
e
ters
′,
′
and
∆
. Moreov
er, th
e eq
u
ilibriu
m
p
o
i
n
t
ex
ist
s
if
|
|
Ω
and
Ω
≡
. T
h
u
s
, t
h
e be
ha
vi
o
r
o
f
t
h
e
sy
st
em
change
s drast
i
cal
l
y
i
f
any
one
or a
n
y
co
m
b
i
n
at
i
o
n
of t
h
ose pa
ra
m
e
t
e
rs keeps
vary
i
n
g an
d cr
oss t
h
at
li
mit. Th
is ph
en
o
m
en
on
is
k
n
o
w
n
as
b
ifurcatio
n
.
Again, the phase
portrai
t
can al
s
o
be a
ffected by t
h
e
close
d
lo
op
g
a
in
G
o
f
t
h
e sy
st
em
. H
o
we
ve
r,
we
wi
l
l
bri
e
fl
y
di
scu
ss t
h
e
be
ha
vi
o
r
of
t
h
e
sy
st
em
for
onl
y
hi
g
h
gai
n
cases, si
nce the
low gai
n
l
oops
are
rare
in
pra
c
tice.
The
bi
f
u
rc
at
i
o
n par
a
m
et
er
, c
o
m
m
onl
y
deno
t
e
d by
, f
o
r
a T
y
pe I l
o
o
p
ca
n
be
defi
n
e
d
by
any
o
f
t
h
e
lo
op
param
e
ters o
r
an
y co
m
b
in
atio
n
t
h
ereo
f. Th
e m
o
st
commo
n
and
practical way to
defin
e
it is to
use th
e
det
u
ni
n
g
pa
ra
m
e
t
e
r as
∆
. Two
p
a
rticu
l
ar
v
a
lu
es of th
is p
a
ram
e
ter g
i
v
e
rise to
a drastic ch
ang
e
in
t
h
e
gl
o
b
al
be
ha
vi
o
r
of a
sec
o
n
d
-
o
rde
r
Ty
pe I
l
o
o
p
s em
pl
oy
i
n
g
a
si
n
u
soi
d
al
P
D
.
The first one
i
s
known as
pu
ll-in
ran
g
e
(
Ω
),
whe
r
e a
s
p
eci
a
l
peri
odi
c
o
r
bi
t
,
c
o
m
m
onl
y
cal
l
e
d
limit
cycle
, appea
r
s
on t
h
e
ph
ase p
o
rt
rai
t
.
T
h
i
s
p
h
e
nom
eno
n
i
s
com
m
onl
y
kno
wn as
s
a
d
d
l
e
no
de bi
f
u
rcat
i
o
n
. A
n
app
r
oxi
m
a
t
i
ng
f
o
rm
ul
a
t
o
ca
l
c
ul
at
e
t
h
e gai
n
no
rm
al
i
zed
pul
l
-
i
n
ran
g
e f
o
r
a hi
g
h
gai
n
Ty
pe I PLL
wi
t
h
a
si
nus
oi
dal
PD
i
s
gi
ven
as
[7]
Ω
Ω
/
2
(1
3)
A
num
erical procedure to calculate
it accurat
e
ly can be
found in [8].
The se
cond
one is known
as
h
a
lf-p
l
an
e pu
ll-in
frequ
e
n
c
y
(
Ω
),
where
a
peri
odic lim
it cycle connects
all th
e sadd
le no
d
e
s on
t
h
e ph
ase
po
r
t
r
a
i
t
. Th
is
ph
enomen
o
n
is co
mm
o
n
l
y k
n
o
wn as
se
par
atrix
cycle
b
ifu
rca
tio
n
, w
h
i
c
h
onl
y
ap
pl
i
e
s t
o
hi
g
h
gai
n
cases. T
h
ere
ha
s neithe
r
been an e
x
act
no
r
an
app
r
ox
imatin
g
fo
rm
ul
a t
o
cal
cul
a
t
e
t
h
e
hal
f
-p
l
a
ne p
u
l
l
-
i
n
f
r
e
que
ncy
fo
r a
T
y
pe I
l
o
op
em
pl
oy
i
ng a
si
n
u
s
o
i
d
al
PD
.
Figu
re
5.
Regi
on
I
p
h
ase
p
o
rt
rait fo
r a
Ty
pe
I PL
L
with Si
n
u
soi
d
al P
D
3.
1.
1.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
Thi
s
regi
o
n
i
s
defi
ned
by
t
h
e
ran
g
e
|
|
Ω
.
Figu
r
e
5
is a
r
e
pr
esen
t
a
tiv
e ph
ase portr
ait o
f
a second
-
or
der
Ty
pe I
P
LL em
pl
oy
i
ng a si
nus
oi
dal
P
D
i
n
t
h
i
s
regi
o
n
,
whi
c
h i
s
d
r
a
w
n
fo
r
0
.
5
,
′
0
.
1
,
∆
2
.
5
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Achi
evi
n
g
P
u
l
l
-
i
n
Av
oi
di
n
g
C
ycl
e Sl
i
p
usi
n
g
Seco
n
d
-
o
r
d
er
PLLs (
A
bu
-
Say
eed A
.
Hu
q
u
e)
24
9
usi
n
g a
n
o
p
en
sou
r
ce M
a
t
l
a
b
pr
o
g
ram
,
cal
l
e
d
ppl
a
n
e8
. T
h
e
f
o
ci
are t
h
e
gl
obal
at
t
r
act
i
n
g
poi
nt
s f
o
r t
h
e
ent
i
r
e
p
h
a
se
p
l
an
e excep
t
for th
e
po
in
ts on th
e sep
a
ratrices.
For
∆
0
, th
e
p
h
a
se
po
r
t
r
a
it
sw
ap
s t
h
e
upp
er half
plane with
t
h
e lowe
r half plane.
3.
1.
2.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
I
Thi
s
regi
on
i
s
defi
ned
by
t
h
e
ran
g
e
Ω
|
|
Ω
. Fi
gure
6 is a
re
present
a
tive phase
portraitwhi
c
h
is d
r
awn fo
r
0
.
5
,
′0
.
1
,
∆
3
.
1
. In
t
h
is reg
i
on
, th
ere ex
ist two
limit cycles.
Th
e
st
able li
mit cycle
move
s
hi
g
h
era
n
d
hi
gh
er u
p
o
n
t
h
e phas
e
po
rt
rai
t
an
d
t
h
e
un
stab
le limit cycle
m
oves closer a
n
d clos
er
to the
separat
r
ix as
in
creases.
All th
e traj
ectories
above t
h
e st
able lim
it c
y
c
l
e
and
betwee
n the lim
it cycles
asy
m
p
t
o
tically
ap
pro
a
ch th
e stab
le limit cycle.On th
e
o
t
h
e
r
h
a
nd
, all t
h
e traj
ect
o
r
ies below t
h
e unstab
l
e
li
mitcycle ap
p
r
o
ach
o
n
e
of the fo
ci.
For
∆
0
, t
h
e phase
port
rait swa
p
s t
h
e
upper hal
f
plane
with t
h
e l
o
we
r
h
a
lf
p
l
an
e.
Thu
s
, t
h
e stab
le an
d th
e
u
n
stab
l
e
li
m
i
t cycl
es
will sh
ow
up
i
n
th
el
o
w
er
h
a
lf p
l
an
e instead o
f
th
e
up
pe
r hal
f
pl
an
e.
Fi
gu
re
6.
R
e
gi
on
I
I
pha
se
po
r
t
rai
t
fo
r a
Ty
pe
I P
LL
wi
t
h
Si
nus
oi
dal
P
D
3.
1.
3.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
I
I
Thi
s
re
gi
on i
s
defi
ned
by
t
h
e
ran
g
e
|
|
Ω
′
. Fi
g
u
re 7
is a rep
r
esen
tativ
e ph
ase
p
o
rtrait wh
ich
is
dra
w
n fo
r
0
.
5
,
′
0
.
1
,
∆
6
.
I
n
t
h
i
s
r
e
gi
on
, t
h
e
be
havi
or
o
f
t
h
eTy
p
e
I P
L
L d
r
ast
i
cal
l
y
d
i
ffers
f
r
o
m
th
at o
f
th
e
p
r
ev
iou
s
two
reg
i
o
n
s
. All th
e equ
ilib
riu
m
p
o
i
n
t
s o
n
th
e
p
h
a
se p
l
an
e d
i
sapp
ear, th
erefo
r
e
n
o
p
u
ll-i
n
is p
o
ssib
l
e.
Howev
e
r, th
ere ex
ists a stab
le limit cyc
l
e
wh
ich
asym
p
t
o
tical
ly attracts all t
h
e traj
ectories
o
n
th
e
pha
se pl
ane
.
F
o
r
∆
0
,
t
h
e ph
ase p
o
rtrait
swap
s th
e up
p
e
r
h
a
lf p
l
an
e with
t
h
e lo
wer h
a
lf p
l
an
e.
Th
us,
th
e
stab
le li
m
it cyc
l
e will sho
w
up in
th
e lower half p
l
an
e in
stead
o
f
th
e
up
p
e
r
h
a
lf
p
l
an
e.
Fi
gu
re
7.
R
e
gi
on
I
I
I
p
h
ase
p
o
r
t
r
ai
t
f
o
r a
Ty
p
e
I
PLL
wi
t
h
S
i
nus
oi
dal
PD
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
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08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
24
3 – 2
5
6
25
0
Now, setting
b
= 0 t
u
rns the syste
m
into a second-order
but Type
II
PLL. In that c
a
se, the two
eq
u
ilibriu
m
p
o
in
ts in
th
e same cell b
eco
me
∅,
∅
∅
,0
and
∅,
∅
,
0
.Th
e
y still re
m
a
in
to
b
e
a
foc
u
s a
n
d a
sa
ddle
node
res
p
ectively, just like in Type
I ca
se.
Howe
ver, they always
ex
i
s
t and
th
ei
r lo
catio
n
s
are fix
e
d
reg
a
rd
less of th
e v
a
l
u
e of th
e loo
p
p
a
ram
e
ters
, u
n
lik
e th
e Typ
e
II case. Moreover, th
e
p
o
s
sib
ility o
f
b
i
fu
rcation
b
y
v
a
rying
an
y
o
f
th
e lo
op
p
a
rameters is eli
m
i
n
ated
.
Figu
re
8 is a rep
r
esen
tativ
e p
h
a
se
po
rt
rait o
f
a second
-o
rd
er Typ
e
II PLL em
p
l
o
y
in
g
a sinu
so
i
d
al PD, span
n
e
d
little over a
p
e
ri
o
d
,
d
r
awn fo
r
a
′
= 0
.
5.
3.
2. D
y
n
a
mi
c B
e
ha
vi
or of
a
Secon
d
-
o
rder
PL
L
Usi
n
g
T
r
i
a
ng
ul
ar PD
The t
w
o e
qui
l
i
b
ri
um
poi
nt
s
i
n
t
h
e
peri
o
d
o
f
i
n
t
e
rest
are
∅,
∅
∆
,0
and
∅,
∅
∆
,0
. T
h
e fi
rst
eq
ui
l
i
b
ri
um
poi
nt
t
u
r
n
s
o
u
t
t
o
be
a f
o
cus
an
d t
h
e
seco
n
d
one
i
s
a sad
d
l
e
no
de
[
5
]
,
[
6
]
,
[7]
.
Ju
st lik
e i
n
th
e
case of a
Typ
e
I PLL
u
s
ing
Si
n
u
s
o
i
d
a
l PD, t
h
e ex
isten
ce as well as t
h
e lo
catio
n
o
f
th
e
eq
u
ilibria o
f
a
Typ
e
I loop
usin
g
Triangu
lar
PD
d
e
p
e
nd
s on
a
′
,
b
′
, a
nd
∆
. T
h
erefore
,
the
phase portrait has
to
be st
udie
d
se
pa
rately
fo
r
diffe
re
nt
ranges
of t
h
ese
param
e
ters.
Fi
gu
re
8.
Ty
pi
cal
pha
se
po
rt
r
a
i
t
for
a Ty
pe
I
I
PL
L
wi
t
h
Si
n
u
soi
d
al
P
D
Unl
i
k
e t
h
e l
o
o
p
s em
pl
oy
i
ng
a si
nus
oi
dal
P
D
, t
h
e
r
e
has
n
e
i
t
h
er bee
n
a
n
exact
n
o
r a
n
a
p
p
r
oxi
m
a
t
i
ng
form
u
l
a to
calcu
late th
e pu
ll-i
n
rang
e
f
o
r a
T
y
pe I l
o
o
p
em
pl
oy
i
n
g a t
r
i
a
n
gul
a
r
P
D
.
Ho
w
e
ver
,
t
h
e
r
e exi
s
t
s
an
exact form
ula
de
duced by
J. L.
Stens
b
y
to
calcu
late the h
a
lf-p
lan
e
pu
ll-
in
f
r
e
qu
ency f
o
r
a Typ
e
I
l
oop
em
pl
oy
i
ng
a t
r
i
a
ng
ul
ar
P
D
[
9
]
.
3.
2.
1.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
This region is defined by the range
|
|
Ω
. Figure 9 is a representativ
e p
h
ase portraitof a second-
order
T
y
pe I
PLL
emplo
y
i
ng a triangul
ar
PD inthis
region, w
h
ich is drawn
fo
r
0
.5,
′
0.1
,
∆
2
.
0
.
Thisphase portrait res
e
mbles
that
of a
sinus
oidal PD
T
ype I PLLi
n the same region as
depicted in Figure 5.
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I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Achi
evi
n
g
P
u
l
l
-
i
n
Av
oi
di
n
g
C
ycl
e Sl
i
p
usi
n
g
Seco
n
d
-
o
r
d
er
PLLs (
A
bu
-
Say
eed A
.
Hu
q
u
e)
25
1
Fi
gu
re
9.
R
e
gi
on
I
p
h
ase
p
o
rt
rai
t
fo
r a
Ty
pe
I PL
L
wi
t
h
T
r
i
a
ng
ul
ar
P
D
3.
2.
2.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
I
Thi
s
regi
on
i
s
defi
ne
d
by
t
h
e ra
nge
Ω
|
|
Ω
. Figur
e 10
is a r
e
pr
esen
tativ
e
phase po
r
t
r
a
it
whi
c
h
i
s
dra
w
n f
o
r
0
.
5
,
′0
.
1
,
∆
4
.
4
and
it resem
b
les th
e
ph
ase portrait of a
T
y
p
e
I PLL
em
pl
oy
i
ng
si
n
u
soi
d
al
P
D
i
n
t
h
e sam
e
regi
o
n
as
depi
ct
ed
i
n
Fi
gu
re
6.
Fig
u
r
e
10
. Regio
n
I
I
ph
ase por
tr
ait fo
r
a Type I
PLL
w
ith
Tr
iang
u
l
ar PD
3.
2.
3.
Ph
ase P
o
rtr
a
i
t
i
n
Re
gi
on I
I
I
Thi
s
re
gi
o
n
i
s
defi
ned
by
t
h
e
ran
g
e
|
μ
|
Ω
′
. Figure
1
1
is a re
pres
entative phase
portraitwhich i
s
dra
w
n fo
r
0
.
5
,
′0
.
1
,
∆
8
an
d
i
t
resem
b
l
e
s t
h
ep
hase
p
o
r
t
r
a
i
t
of
a
T
y
pe
I P
L
L
em
pl
oy
i
n
g
si
nus
oi
da
l
PD i
n
t
h
e
sam
e
regi
on
as
depi
c
t
ed i
n
Fi
g
u
re
7
.
Now
,
settin
g
b =
0
t
u
rn
s th
e system
in
to
a second
-o
rd
er
b
u
t
T
y
p
e
II PLL.
In th
at
case, the two
eq
u
ilibriu
m
p
o
in
ts in
th
esame cell b
eco
m
e
∅,
∅
∅
,0
and
∅,
∅
,
0
.
Th
ey still re
m
a
in
to
be a
foc
u
s a
n
d a
s
a
ddle
node
re
spectively
,
jus
t
like in
T
y
pe
I case
.
However
,
they always exist and thei
r
l
o
cat
i
onsa
r
e fi
xed
re
gar
d
l
e
ss
of t
h
e
val
u
e
of t
h
e l
o
o
p
pa
ram
e
t
e
rs, unl
i
k
e t
h
e
T
y
pe I
I
case. M
o
reo
v
e
r
,
t
h
e
p
o
s
sib
ility o
f
b
i
fu
rcation
v
a
n
i
sh
es wh
en
an
y of t
h
e loop
p
a
ram
e
ters is v
a
ried
.
Figu
re
1
2
is a
represen
tativ
e
p
h
a
se po
rtrait
o
f
a seco
nd-ord
e
r
T
y
p
e
II PLL
em
p
l
o
y
in
g
a sin
u
s
o
i
d
a
l PD, sp
an
n
e
d
little
o
v
e
r a p
e
riod
, d
r
awn
fo
r
0
.
5
.
3.
3.
Wh
y T
y
p
e
II
B
a
sed o
n
ou
r
di
scussi
o
n
i
n
t
h
e p
r
e
v
i
o
us se
ct
i
on, i
t
i
s
clear at this point t
h
at
the a
n
alysis as well as
th
e b
e
h
a
v
i
or of a Typ
e
I lo
op
, co
m
p
ared
to
its Typ
e
II co
un
terp
art, is
m
u
ch
m
o
re co
m
p
lex
.
W
e
wil
l
to
u
c
h
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
24
3 – 2
5
6
25
2
u
pon
a
few reason
s, b
a
sed
on
thei
r
resp
ectiv
e
p
h
a
se
p
o
rtraits, in wh
at
fo
llows. Th
e eq
u
ilibria fo
r Typ
e
II
lo
op
s are i
n
d
e
pen
d
e
n
t
of an
y o
f
t
h
e loo
p
p
a
ra
m
e
ters. In
contrast, th
e eq
u
ilib
ria for th
e Ty
p
e
I l
o
op
s
d
e
p
e
n
d
on
t
h
e l
o
op
para
m
e
t
e
r/
s, and t
h
ere
f
o
r
e, t
h
ey
m
ove as a
n
y
one
of the
param
e
ters is varied.
They may eve
n
di
sap
p
ear
f
o
r
c
e
rt
ai
n
val
u
es
o
f
t
h
ose
pa
ram
e
t
e
rs.
Figu
re 1
1
.
Re
g
i
on II
I p
h
ase p
o
rtrait fo
r
a Ty
pe I
PL
L with Trian
gula
r
P
D
Figure 1
2
.
T
ypical phase portrait for
a
T
ype II PLL
with
T
r
iangular PD
Th
ere is a
symmetry in
th
e
ph
ase
po
rtrait
fo
r Typ
e
II
loops, m
eaning t
h
at a
tra
j
ectory rem
a
ins a
trajectory if
both
the a
x
es
are ne
gated. Howeve
r,
in
case
o
f
Typ
e
I l
o
op
s, no
su
ch
symmetry ex
ists.
In
t
h
e case
o
f
Typ
e
II loop
s,
th
e en
tire
ph
ase p
l
an
e con
s
titu
tes th
e
reg
i
o
n
o
f
con
v
e
rg
en
ce for th
e
fo
ci,
ex
cep
t
f
o
r
th
e sep
a
r
a
t
r
ices.
H
o
w
e
v
e
r, th
is
may b
e
tr
u
e
fo
r
Ty
p
e
I
l
o
op
s
u
p
t
o
a certain
r
a
ng
e
o
f
t
h
e loop
param
e
ters’ va
lues. F
o
r a
not
h
e
r ra
n
g
e
of
the
loo
p
pa
ram
e
ters (
o
r s
o
m
e
com
b
ination ther
eof
)
,
p
h
ase
p
o
r
t
rait
may h
a
v
e
m
o
re th
an on
e cell, exh
i
b
iting
d
i
stin
ctly d
i
fferen
t
b
e
h
a
v
i
ors.Fo
r
ano
t
h
e
r rang
e,
t
h
e who
l
e syste
m
may fail to
prod
u
c
e pu
ll-in all to
g
e
t
h
er and
so
o
n
.
In th
e case
o
f
Typ
e
II l
o
op
s,
th
ere is no
drastic
change
in
the phase
portra
it as th
e l
o
op
p
a
ram
e
ters
are v
a
ried
. There is n
o
app
e
aran
ce/
d
i
sapp
earan
ce
o
f
th
e
eq
u
ilibriu
m
p
o
i
n
t
an
d
/
or th
e
li
mit c
y
cle. Wh
ereas,
Typ
e
I loo
p
s ex
h
i
b
it two
in
st
an
ces of su
ch
b
i
fu
rcatio
n
.
In th
e PLL literatu
re,
o
n
e
is
k
n
o
w
n
as sad
d
l
e n
ode
bi
f
u
rcat
i
o
n a
n
d
t
h
e
ot
he
r
one
i
s
k
n
o
w
n as
sep
a
rat
r
i
x
cy
cl
e
bi
fu
rcat
i
o
n
.
The p
h
ase
-
l
o
c
k
ed c
o
n
d
i
t
i
on
of a Ty
pe I
I
P
LL im
pl
i
e
s bot
h zero
phase
and ze
ro f
r
e
q
u
e
ncy
err
o
r
.
Ho
we
ver
,
i
n
ca
se o
f
Ty
pe I
P
LL, i
t
o
n
l
y
i
m
pl
i
e
s zer
o
fre
q
u
ency
e
r
r
o
r.
F
o
r
n
o
n
zer
o
det
uni
ng
pa
ram
e
ter,
p
h
ase
er
ro
r is alw
a
ys
no
nzer
o
.
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