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IEICE TRANS.

LETTERFUNDAMENTALS,

VOL, E77rA, NO.

II NOVEMBER

1994

1887

Section of LettersSetected fromthe 1994 LEICE SpringConference
SpecialAnalysisfor a Second Order Phase-Locked
Loop in the Presence of a Weak CW tnterference･Melnikov

Hisa-Aki TANAKAt,Shin'ichiOISHIt

SUMMAR\

This letter
presentsthe results of an

the global,

orderanalyzes

wave

focusingthe geometric structure of the invariant manifolds. Te do so, we applied the MelnikoVsmethod tothe problem by utilizing the presenceof small parameters,As a result of the analysis, the transversal intersecis proven to occur
tions of the invariantmanifolds

equa-fromtion are focused and analyzed
as to how they are extended
Using the Melnikov integral
the hyperbolic periodic orbits.
and
which
evaluates
the distance
betwccnthe stable manifolds
theunstablemanifolds,
the transversal intersectionof theseis proven to occur under some conditions on themanifolds
the angularfrequency difTerence
andpower of the interference
NumericaLcomputations
between the signal and the interference,
wereperformed to confirm the transversal intersectionof thesystern-generated invariantmanifolds fbr a practicalset of parameters.kay
words:

phase-tbcked loop,CPV intederenee,Mbinikovana-ysisL

This situation led to the presentwork whichof these complex
behaviorsby
the mechanism

analysisdynamicalstructure of a second
ofthe
continuous
phase-lockedloop (PLL)inthe presenceof the PLL
The invariantmanifolds(CW) interference.
concerning

Members

Kazuo HORIUCHIt,andunderon theconditions

somepower ofthe interference

frequencydifferencebetweenthe sigand the angularIn addition, we numericallynal and the interference.
of the systemconfirmed
the transversal intersection
for a practicalset ofgenerated invariant manifolds
parameters.
2. PLL Equation in the Presence of CW

Introduction

Interfer-ence

In today's coherent
phaseforlocked loops (PLL's)are used extensively
demodulation of the signal. In such systems, manyfor
users always
coexist inthe sarne frequency channelofPLL'scommunication.
Consequently,
perfbrmancein the presenceof the continuous wave (CW) interferand has beenence is a problem of practicalinterest,
studiedby many authors.
communicationsystems,of nonlinearandKawakami
recently by Oie, Iritani,
Suzuki [3],earlier
workby Endo andand Takahashi [5],Theclarified
the tracking performance of PLL by
Iizuka[1]studying
the periodic solution via the harmonic balMoreover, they showed that in the initial
ance method,basin boundary is formed
phase plane, a complex
betweenthe basin of a periodicsolution for the signal
for theandthe basin of anotherperiodic solution
issuMcientlylarge,
when
theinterference
interference,
to
theto
this
longtransients
Addingphenomenon,
PLL)
areoftenlock-inof
stable fixed point (i,e.,numericallyobserved
undersome set of parameters andhave investigatedthe
initialstates. The works[1]-[5]intracking behavior to the signal or to the interference
behind these complex
detail.However, the mechanism
phenomena has not been fu11yelucidated,

Manuscript receiveds(t)the CWsin=Ss(t)

takes

the fbllowing
fbrm,

(1)

at,

interferenceI(t)can be describedby
NI(t)

==Z

Iisin(w+(2)yi) t,

ir-1

ui denotes the angularwherethe(3)v(t)=Vcos(cvt+es),

esdenotes the phase error between the desired

whereConsidered here areat VCO,and the outputa
voltage-controlled
oscillator
that
incorporate
PLL'ssignal(VCO),a phase detector(PD) having sinusoidal

loop filter(LF) comprised of afunction
F(S) =l/(1RC filterwith transfer
simple+ fS), which is known as a lagfilter.
Figure 1 shows a blockdiagramofthe system, anddifferential
equation
the fo11owing ordinary(ODE),of thedescribes
thedynamics
i,e.,the phase model
[1],
error
th:
phaseandcharacteristics,(i+T)
ddtaddatOKb(g==(- e,)+],R,g (y,t-a)),z".,

April 22, 1994.

t The authors are with the School of Science andneering,

desiredsignal

thefrequency difference
andthei-th mode of thedesiredsignalbetweenHence,
thereceived
signal isthe sum
CW interference,
outputv(t)at the voltagewhile thes(t) and I (t),[4], ofcontrolled
oscillator(VCO)can be describedby

dynamicalsysby Endo andtems, this problem has been investigatedEndo andIizuka [1],Endo, Matsubara,and Ohta [2],From the viewpoint

When

(4)

Engi-Tokyo, 169 Japan.
Waseda University,

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IEJCE

FUNDAMENTALS,TRANS.

l888s(t)+Kt)

PDVOL

E77-A, NO, 1r NOVEMBER

1994

X2LF
vco.

Fig.1 Block diagram ofpresenceofa

continuousawaye:=l,Ll:--::::::-:-:･elIi:::-:::::-::::!':

otoop (PLL) in thephase-locked(CW) interference.

e=2T

Kb denotesthe total loop gain, and Ri denotes
L/S. Let the PD characteristics besinusoidal,
i.e.,g(e)e, then the phase model isobtainedby applying the trigonometricidentity
to (4)
wheretheratio=sin

ase'==ocZ,+r-ia+

Kb T"i sin thFig.2field.
Heteroclinictrajectories in the three-dimensional

N+ Kbr-iZRi(cosvit sin 6b-sinvit cos ca)
i--1=O,yector(5)
By a scaling ofthe independent
variable:(5)gives.

t->t,(QeqriJ)--

Xlca+(1/Q(ii)G)+sinca
N-ZRi=2T(cosyit sin ca-sinyit cos ca)=O. (6)1=1''3.

The Melnikov Analysis for PLL Equation

Denotingth=xi,ca=x2,and
firstorder system of ODE

vit=0i,(6)isgivenasa(7a)Xi=x2,

L sin xi - (1/Vlki5T)x2

X2=N+ZRi(cosi--1ei sin xi-sin

ei cos xi),

ei=yi,here.

3. 1 Separationof the InvariantManifolds
In the limitof RnO

1/Jier-O, there exist a pairof heteroclinic
trajecteries as in Fig.2,
In this limit, the (7a)and(7b) component
becomes Hamiltoniansystem with Hamiltenian H=x22f 2-cos xi, which
givesthe analytic solution on themvariantmanifolds±

fo11ows,(PVS) and.

the unsta-±t),

2 sech t,

ei=yit+0o･(8)In addition,
the solution(xbx2, ei)=( ± rr,O, uit+ eo) represents a hyperbolicperiodic orbit respectively. When Ri=O(e) and lfV)?5i=O(e), such hyperbolic periodic orbits persistand the invariantmanifolds from the hyperbolic periodic orbits genericallyseparate
into the stable manifblds(PVS) and the unstable manifbldsor")
asinFig.
3.
(The distanced betweenthese manifolds depicted
in Fig.3 can be evaluated via the MelnikovintegralM
byd-M'(fp, 0o,vi, Kb r, Ri)

1D.H(x,±±,(-fo),x,andas± 2 arcsin(tanhih ==x2'h=×=:O(E),llV]?iE?=O(E)consideredx(7c)× Ri(xb)c2,ei)E(Rimanifolds(7b)Ti) with i--1,･･･,IV.
We concentratehere the case of N= 1,i.e,,a single tone
CW interference,The general case of Ar>l will be
discussedin the near future.Since the forcingterm in(7b)isperiodic with respect to ei,the Poincare section can be definedon the plane ei=const.,
wheretheMelnikov's method
canbe applied to evaluate thedistancebetween the system-generated
invariantmanifolds [7].
Te apply this method, the simplest case ofRiandyi=O(1)is primarily
whereFig.3 Separationof the stable
ble manifold(MU).

where,(-fp))11+O(E2),(9)'M'(fp, eo,yi, Kor,Ri)'==

DxH, g> (xih, x ih)dt,±./[:<NII-Electronic Mbrary
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1889

D.H=(sinxi,x2,O) and g=(O,-(1IV)}iJir)x2+Ri(cos ei sin xi-sin 0i cos xi), O) , <,> denotes thewithwhilelD.H(xii(-fo),xihC-fo))11

Since (7)isnot invariant
inner product ofthe vectors.
-x2), M'x2)-(-x"underthe transformation: (xi,does not necessarily hold. M can be expressed
asby the sum of the following integrals==2

sinh2(-fp)+cosh2(-fo)/cosh2(-th)==2

cosh2fo/cosh2fp>O

±

=M-M

holds.Fromeo,vi, KbT, Ri)(fo,±

II:vvkEirx(-=:+1:2dt±±

Rix 2±h Ct-fp)sin ei cos x 2 h (t-fo)dt±(11)-E+IS+1le,k, h, and h can be integratedanalyticallyin which

the residue

calculation(see[6]for a similarvia

tech-. , , (t-fp)2dt
k=IO.er (1/J)Gi)±(12)=-8/&.

Rix 2'h (t-fo)
cos 0i sin x ih (t-fo)dt

h=IZ±-4R,.[Z8.`g,h,(i-t,))･sin yi (t-fo)sin(yi fp+ em)
dt-4Ri{flvi2/(2sinh rrvi/2)}sin(vifpt==

uRix ih(t-fo) 0i
Y[i:lesin

==

-1:

==

± 2Ri sechcos x2i0o). (13)(t-fp)dt(t-fo)cosvi (t-fo)± 4R,± 4Ri{n(1 - yr)

must

weconsider,Since a second kind of periodic solution correspondappear
in the upper
ing to the interference
signal mayx2), we
focus our attention to thehalf plane of (xi,half plane here. Using the Melnikov integral
upper
we can consider
the geometric structure
given by (15),
PVS and PVU, It isrealized
ofthe
invariantmanifolds
andisthe sum ofthe constant term -8fwt
that (15)amplitude
is -Ri 2(1J y?)/the oscillating term whose-2sinh<rrvi/2) .
-2y;/sinh(nyV2)cosh(-yi/2)
holds.
Hence, the fo11owing proposition
Propositien1; For parameters satisfying Ri=O(e), 1/VIII=O(e)andyi=O(l),there existsacritical

PVU(c)For Ri>Rc,M(fo, ca) has the simplefo+ a))･
(vi/(2cosh nvi/2) }sin

8VIimr± 2(1-y,2)cosh(rrvi12)

3.3

2v,Z
sinh(rryi/2)F2sinh(nvi!2)]sin(yifp+es),i.e.,

is small but finite(i.e.,vi=
the interference'(i.e.
1/VIilJir=O(1)) and the PLL is low damping
dynamics such as longtransients toO(E)), complex

signalandandthe fractalbasin boundary mightlock-inof theintersection
due to the transversal
invariantmanifolds.
generated

is given by+nRi(zeros,ofPVS and MU occurs.
the transversal intersection
Proposition 1 leads the fbllowing conjecture:
frequencydifferencebetween theWhen the angular(I4)M

be O(E) respectively, as in (9).However, ifof yi, e,g. yi=O(11e),ha large value
in tsterm
4Ri{rr(1-yi2)/(2ceshrryi/2)}
andtheany
becomes O(e-iie) being smaller thanpower of E.
Ri=O(E) and 1/If we consider the case of yi=O(I),
Vjfi O(E),it is realized that thisdiMcultydoes notoccur
breakthroughon this diMculty has(Theoreticalbeen madeby Holmes, Marsden and Scheurle [8]).
Mtangency occurs.

=:F2Rinsinh(rrvi/2)sin(vifo+th)± =-comes(b)At Ri=R,, d=O (] fo,6h),i.e.the heteroclinictt--fofp))((: gionshh2,±

vi=O(1)

separate.

･cos yi (t-fp)sin(yi fo+ ca)
dt

Hence,M

assumption

parameter Rc such that the fbllowingholds:ca),i.e. WS and(Vfo,(a)For Ri<Rc,d<O･sin(vifo+ca)dt'ffrom thefo11owingreasons. To evaluate the distanced between
PVS and PV" via the Melnikov integral,L,h and k inTheoccurs.

of the InvariantManifolds
3.2 GeometricStructure- (11J)er)dt
4 sech2 (t-fo)

==1:

if

==

giye the fbllowingnique),tothat we check(PVS)

Rix 2 h (t-fo)cos
ei sin x i h (t-fo)dt-f:suMces(15)and (I6),itthe Melnikov integral
M has the simple zeros todetermine if the transversal intersectionof the stablemanifolds
manifolds
andthe unstable(MiU)t - a,)

±,,

(16)

To

(15)

read

Numerical
verifysystem-Results

the validity

problern,weappear,

ofthe abovein theintegration

conjecture

performed the numerical

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1'EICE TRANS.FUNDAMENTALS,VOL

E77-A,

1890

2.0

11 NOVEMBER

1994

2.5=2X2

e.2.0

NO.

o-2,5-T-T

XI

Fig,4 Numerically obtained invariant
manifolds
PVS and }V"
for a practicalset ofparameters:
l/VjE5ii=O.Ol,Ri=O.08 and vi=1

Fig.6 Numericallyobtained=[O.8and R]=O,02,PVS and M" fbr 1!Jiei7=O.Ol, vi

=O.8.

2.5=2

-xi, + 7rRi( ,,2,(hiinvYi)2)"-2
smh?f;'

sinh(rrui/2))==o,,f2)(l7)1/V)liff7=O,Ol
and vias R.=O.O06990.The
discrepancybetweenthe theoretical and experimental
valueof Rc isconsidered
to due the finite
value of lfwith

o=O,8,vfi,

4. Conclusions
In summary,-2.5-TXI

Fig.5 Numerically obtainedandRi==O.Ol.

MS andPVU for lfV)riii==O.Ol,
yi=O.8invariantmanifolds
PVS andPVU for a practicalset ofparameters,
We used the 4thQrder Runge-Kutta integration
schemewithintegration
step at O.Ol, 1400 initialpoints on the Poincare sectionofthe system-generated

ei==O werethe eigenvectors
the unstable manifblds
fl;oma hyperbolic
periodicQrbit. The result in Fig,4
confirms
the transversal intersectionof PVS andPV" onthe section ei=O,for a practicalset of parameters 1!
associatedletter
reports the analyticand numerithe geometric structure of the invariantmanifolds
in PLL in the presence of a weak CWinterference.The transversal intersectionof PVS andifU is proven to occur under some coditions on thesystem parameters
via the Melnikov integral,
while theexistenceof the transversal intersectionof these invarcal

set

with

respectively

along

the stable andwt=O.Ol,Ri=O,08, and vi=O,8,

The result in Fig.
5shows that WS and PVU are close butseparate, whilein Fig.6 the intersectionof ewS and MU can beidentified.Hence, the heteroclinictangency is suggested to occur
betweenRi=='O.Oland Ri=O.02 with 1/and vi==O.8.
The theoretical value ofthe
VIkiir==O,Ol
critical parameter Rc isobtained from the equation

study

thisoniant manifolds was confirmed
numericallyfor a practical set of parameters. These results
are considered
toof the complex
provide an understanding
phenomena
onsetof PLL
in the presence of a weak
CW interference. Although the Melnikov homoclinicitycondition
doesnot guarantee the existence of a chaotic attractor,
complexdynamicalpropertiessuch as the fractal
basinboundary or extremely long transientsis expected tooccur.
These will be discussedin the near future,

AcknowledgementWe would
like to thank ProfessorH.Kawakami,
TokushirnaUniversity,ProfessorY. Takahashi, Hime-of Technology, andjilnstitute

AssistantProfessorH,
Okazaki, Gifu National Collegeof Technology, fortheir valuable

discussions.

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LETTER

1891
1984.

T. and Kawakami, H.,[4]Oie,T. Iritani,

References[1]Endo, T. and Iizuka,N.,Interference
in
InteSecond-Order Phase-Locked Loops with Imperfbct"Influence

ofthegrator,"7}"ans.IECE, vol. j64-B,no. 3,pp. 191-198.Mar.
1981.
of[2]Endo, T,, Matsubara, T. and Ohta, T.,
Phase-Locked Loops in the Presence of LargeInterfering
Signal," 7)'ans, IECE, voLJ65-B, no.3, pp.294-301,"BefaviorMar. 1982.

[3] Endo, T, and"ResponseofPLL

Inputs,"Rroeeedings of
Demodulatorby Two Sinusoiclal

NOLT14
1993,
Takahashi, Y.,
on Locking Behaviorof PLL
Signal,"Trans,IECE,Caused by Frequency ofInterfering
vol. T77-A,no, r,pp, 84-87, l994.
Kinokuniya Publ. 1981.
[6]Niwa, T., llynamical S),stems,[7]Wiggins, S., Globat Bijltreationsand Chaos, SpringerVerlag, 1988.
[8]Holmes, P,,Marsden, J. and Scheurle,J.,
Small Splittingof Separatricswith Applicationsto KAM
Theery and Degenerate Bifurcations,"Contenrporary
Mathematies, vol. 81, pp. 213-244, 1988.
'SB,

[S]

"Infiuences

L`Exponentially

Suzuki, T,,
of Phase-Locked
Loops inthe PresenceofInterference by Galerkin'sProcedure,"7leans,
IECE, vol. J67-B,no. 3, pp, 241-248, Mar.
"`Analysis

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