Chaotic I'liciioiiiciia
in a Swit,clicd Capacitor Pliase-Locltcd Lcop
N a \ v a i u Kiitsuzawh, Yosliifunii Nisliio and Sliinsaku Mori

IIcpartiiiciiL of 13lcclric.nl Eiiginccriiig, Keio Uiiiversily

3 - 1 4 1 Iliyoslii, l<oliolcu-~kii,Yoltolinnia, 223 Japnii

CIIICUIT MODELA B S T R A C T : Chaos in Phase-Locked Loops have beenstudied over the years[]], [2]. In those papaers, the existenceof horseshoe chaos 1ia.s been proved via the Melnikov methodor the perf urbation methods based on the idea of Melnikov. In
this papaer, we discuss the chaotic plienoniena i n a SwitichedCapacitpor Phase-Locked Loop proposed by D.Asta et al. in[3]. We prove the existence of the logistic chaos by tlie onedimensional map. And it has turned out to be cleared thatin the map, there exist parameter volues where attractors
coexist. Moreover 2-parameter space in which the chaotic
phenomena occur is calculated. The analytical estimate isverified by circuit experiments.

1"igiire 1 sliows the block diagratn of a Switclied Capacitor
Pliasc-Locked Loop. Tliis circuit contains two sample-holdcircuits as a sampler ancl one VCO. T h e loop input is modeled here as a continuos-time sinusoid with amplitude A andfrequency W . Tlie phase is an arbitrary time Iunction, 8.

The loop operates by tracking the zero crossings of the input waveform. The output of sampler is to be a sinusoidal
function of the phase diffrence between the input wavelorm
and waveform controlling tlie sampler. Tlie period of VCO,r k = t k - t k - 1 , is controlled by the output u ( t k ) of the sampler
as follows:

INTRODUCTION
A Switched-Capacitor Phase-Locked Loop proposed by D.
Asta et al. exhibits a fundamental behavior inherently diffrent
from any previosly studied. T h e diffrence between this systern and conventional PLL's is due to its structure as both
continuous- and discrete-time conponents. And this loop ischaracterized by both nonuniform sampling and a sinusoidal
iionlinearity. In this papaer, we will show that a SwitchedCapacitor Phase-Locked Loop can exhibit chaotic behavior.
Though t.liis system allow for a filter of arbitrary order, weristric our discussion only a first-order loop. The reason forabove limitation is t h a t a rigorous analysis of the chaotic phenomena can be achived by discribing the governing equationa s a one-diinensinal map.

where wo is VCO's free-ruiiniiig frequency

p is the VCO charxteiistic
The phase error is denoted by the diffrence between thepositive-going zero crossing of tlie input signal and the risingedge of the VCO oulput(see Pig. 2), as follows.

T h e scalor first-order equation describing the system can be
w ri t tenninI-+2 ~ ( l A)
/Isin ( 4 ( k - 1))- 1))+ sin ($(k(4)

The above equation exhibit a circular symmetry in d(lc).
We can therefore ristrict 4(k) t o
where we identify--x and T .

[-.,I91-645128/92$03.00 0 1992 IEEE

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1)eiiotc (lie mapping iri (5) as &(n+1) = F ( $ , ( n ) ) , which__.- . ..__.

I I ~ I I S ~ O llie
~ I Un-tli pliasc erior & ( n ) into the ( I % +l)-th phaseciior & ( U+ 1).

. .. .... . . -3i
II 1/Ii
3.6F ig . 4 1-par aine t e r bi f u r c ation di agr am
starting froin the right extram ( A = 0)

A n exainple of the one-dimensional map is shown in Fig.
3. For soine parameter volues this map has two extram andchaotic solution is considered to appear.

2.

-*1.1P

3.6

Fig. 5 I-parameter bifurcation diagram
starting from the left extram ( A = 0)_.

+..
Fig. 3 One-dimensional inap

Fig.4 and Fig.5 are examples of 1-parameter bifurcation diagram. Thc initial vdue are started from the right/left ex-

tram in Fig. 4/5, respectively. Period-doubling route chaos,periodic windows a i d crisis, are observed in these figures.
The diffrence between Fig.4 and Fig.5 exists from the factthat there is within tlie bounds of possibility of coexistingclilTrent type of attractors w!ien there exist two extram ina one-dimensional map. Phi$ically, this means that for theparameter rigion where 1-periodic solution and n-periodic solut.ioii coexisl., initial phase error decides the succesion of thelocking s t a k in the circuit. In logistic map there never exists a sta.ble 1-periodic solution after clinotic phenomena haveoccured, while i n the one-dimensional map which represents
tlie circuit. is not. tlie same case. T h a t is, even after crisis oneperiodic solubions appear. We show this fact in Fig.6.

The mechanism of the emergence of 1-periodic solution after crisis is as follows, With parameter p decreases, the left
extram increases and intersects 45’-line infinitely many times
in the parameter space(see Fig. 7).
To determin the relation between the real system and theone-dimensional map, that is, n-priodic solution in the onedimensional map corresponds t o what periodic state in thereal system, we should calculate rotation numbers. But themap usually don’t satisfy homeomorphism, so it is no use evaluting rotation numbers. Instead we executed circuit simulations, and the simulations led us the fact t h a t n-periodic solution in the one-dimensional m a p corresponds t o n/m-periodic
state in the system, where the ratio n/m denotes the outputof VCO oscillation times divided by the input signal oscillation times. We prsent Zparameter diagram and boundries ofthe basion of attraction of several periodic solution associatedthe switched-capacitor phase-looked loop circuit , moreover wewill show chaotic regions in the parameter space. With thisdiagram, we confirm routes to chaos. Now, let denote theLiapunov exponents X as equation (5).

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21

1.45

1245

1.55

P

1.255(c)Fig. G 1-paiaiiieler Iirtiicat,ioii diagiaiiis ( A = 0 )( a ) p ; 1.9

--

2.1
1.55
( c ) P ; 1.245
1.255( b ) /1; 1.45

The Liapunov exponents formalizes the concept of stretching rates and it represent the figures t h a t distiguish between
periodic-orbits ( A < 0), quasi-periodic orbit ( A = 0) a.nd chaos( A > 01. so, in this diagraA we evaluated chaotic plienotnena wit11 the Liapunov numbers. Fig. 8 is an example of 2parameter bifurcation diagram. In this figure, period-n-orbits
are i i i the shape of "Arnold's tongue", and we confirmed thatiiifinitly niniiy "Arnold's t.ongue" exist i n the para.meter space.

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CIRCUIT EXPERIMENTS
We confirmed the existence of logistic chaos and crisis bycircuit experiments. Fig. 9 show experimental results. In
these figures, parameter A was fixed to 0 ,that is, input signal
frequency coinsides with PLL’s free running frequency, andthe parameter p was changed with changing the input signal
ampulitutle A .

CONCLUTION
In this article, we have analyzed chaotic phenomena in aswitched-capacitor phase-locked loop by the one-dimensional
map. With analyzing the map, the existence of logistic chaos,coexistence of attractors, one-periodic solution after crisis andperiodic solution in the shape of ” Alnold’s tongue” in the 2parameter space were cleared. Moreover experimental results
justified the analytical ones.

REFERENCES(1) T. Endo and L. 0. Chua, ”Chaos from Phase-Locked
Loop,” IEEE Trans. Circuits Syst., Vol. CAS35, No.
8, pp. 987-1003, August 1988.
(2) Y.-H. Chu, J.-H. Chou and S. Chang, ”Chaos from Third-

Order Phase-Locked Loops with a Slowly Varying Parameter,” IEEE Trans. on Circuits Syst., Vol. CAS-37, No.
9, pp. 1104-1115, September 1990.
[3] D.

Asta and D. Green, ”Analysis of a Hybrid
Analog/Switched-Capacitor Phase-Locked Loop,” IEEETrans. Circuits Syst., Vol. CAS-37, No. 2, pp. 183-197,February 1990.

[4] P. Collet and J.-P. Eckrnann, Iterated Maps on the In-terval as Dynamical Systems, Boston, MA: Birkhauser,
1980.
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