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ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
Chaotic Dynamics of a Third Order PLL with
Resonant Low Pass Filter in Face of CW and FM Input
Signals
Bishnu Charan Sarkar 1 , Saumen Chakraborty 2
1 Physics Department, Burdwan University, Burdwan, India
Email: [email protected]
2 Physics Department, B C College, Asansol-4, Burdwan, India.
Email: [email protected]
Abstract — Nonlinear dynamics of a third order phase locked
loop (PLL) using a resonant low pass filter in the face of
continuous wave (CW) and frequency modulated (FM) input
signals is examined. The role of design parameters of the
loop resonant filter and the modulation index of the input FM
signal on the system dynamics is studied numerically as well
as experimentally. The occurrence of chaotic oscillations in
the PLL is verified by evaluating some well-known chaos
quantifiers like Lyapunov Exponents from the numerical time
series data.
Index Terms — Third Order PLL, Resonant Filter, Stability
Analysis, Chaotic dynamics, Lyapunov Exponent.
I. Introduction
In recent years, chaotic oscillations in PLLs have been
reported by several authors in the literature [1-4]. PLLs are
used by communication system designers as both modula-
tors and demodulators and chaotic oscillations in them could
be a matter of both problem and benefit from the application
point of view. The self generation of chaotic oscillations in
third order PLLs and the inherent synchronization property
of PLLs could be easily exploited in the design of coherent
receivers of chaos-based communication systems. In the work
of Endo et al., chaos from PLL based frequency modulators
was reported in the condition when the carrier frequency of
the FM signal was outside the locked state of the PLL. In this
paper we report about the chaotic oscillations that are ob-
served in an FM demodulator based on a third order PLL
with the carrier signal within the lock range of the PLL. JRC
Piqueira analyzed the stability and determined the lock in
range of a similar type PLL form the view point of nonlinear
dynamics in [5]. The present work shows that the PLL breaks
into chaotic oscillations with the increase of the gain of the
loop filter beyond a limit with CW input signal. The gain of
the filter and its time constant can be the control design
parameters of the PLL system to generate the chaos. Chaos
is also observed with an FM input signal, the modulation
index of the FM signal can be an additional control parameter
of the generated chaos for both in tune and off tune carrier
conditions. The proposed system can provide both baseband
chaos and chaos-modulated RF signal obtainable at the con-
trol input terminal and the output terminal of the loop volt-
age-control oscillator (VCO) respectively.
©2012 ACEEE
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We organize the paper in the following way. In section-II, the
mathematical model of the system is formulated with its
stability analysis. Section-Ill describes the results of
numerical simulation study by solving the system equations
using fourth order Runge-Kutta (RK) method. Chaotic
dynamics of the system is examined by phase-plane portraits
of the state-variables as well as by calculating the Lyapunov
Exponents for different values of design parameters. The
details of experimental studies on a prototype hardware
system are given in section-IV. In the last section, some
discussions on the importance of the study are presented.
n. Formulation Of System Equations And Stability
Condition
Phase locked loop consists of three basic components:
phase detector (PD), Low pass filter and Voltage control
oscillator (VCO) as depicted in Fig. 1 . Schematic diagram of
the system is given in Fig. 2. A multiplier type phase detector
and a resonant low pass filter are used in the loop. A FM
signal of the form
s in (t) = sm{co c t-y/(t)} (1)
is applied to the input of the PD. Where co c is the carrier
frequency and the phase variable 0(f) denotes the
frequency modulation. For single tone modulation
y/ = mcosa> m t (2)
Where, co is the frequency of the modulating signal with m
modulating index. To study the dynamics of PLL with CW
input simply one can put £ r =0. Let the VCO output signal be
denoted by 2 COS [iu,. t — (f)} ; Where co v the free
running frequency of VCO and Q r (t ) is the phase.
The PD produces a output proportional to the sine of phase
difference , i.e.,
sin {_x(t) -#0)1
where, x(f) = ^ _ ^ )f _ ^ (f)
This PD output is the input of the resonant filter circuit.
We take two state variable y and z just before the feedback
capacitor of the filter and at the output of the filter
respectively. Using OP- AMP circuit equation, one can obtain
the following two equations,
vc ACEEE
ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
^= a sin(x - W) + a(g - 2) v - a(^)z
dt & ( 3 )
dz
di
agy — az
(4)
F.-
Where p is the gain of the filter defined as Q ~ 1 — IT and
a = — = — where, C, R and T are the capacitance,
T CR
resistance and time constant that are used in the filter circuit.
VCO's oscillating frequency changes about the free running
frequency (co ) in accordance with the controlling voltage z.
From the basic PLL theory, one can write the phase part of
the reference signal Q (?) as the time integrated version of
the control input to the VCO (with sensitivity k ). Thus,
dt
(5)
dt
(6)
Where Q = ai, r — cu c is the frequency detunning and k is
loop gain. For calculation one can define another variable
which define as,
4> = ^ t (7)
Equation (7) and (2) are first differentiated and then normalised
whereas (3), (4), (6) are directly normalised with the factor
r = at ; a dimentinless quantity. Finally one get the
following five equations(8a-8e) that describes the dynamics
of the system with FM input.
dtp [Uj,, £
dv
-sin (jp)
dx
d ~
dy =
dr
ds _
Where |j.
sinfx - W) - ig - 2) y
gy — z
(8a)
(8b)
(8c)
(8d)
(8e)
fiT, k„ = kT,f n = ^ = « m 7"are the
normalised detuning, normalized loop gain and nomalized
frequency of modulating signal. The dynamics of third order
PLL with CW input is described by the last three equations
with *y=0.
Using these generalized system-differential equations
with =0, the stability criterion can be determined by
techniques of nonlinear control theory. It is quite apparent
that the steady state values of the variables x, y and z are
sin 1 (
respectively. Evaluating the
determinant of the system Jacobian at these steady state
points, one can derive the characteristic equation for the
system as given in (9).
A 3 + (3 - g~)A 2 + X - ^klg 2 — fl 2 , =
(9)
By applying the Routh's Array technique, we obtain the
conditions for stability as:
g < 3 (10a)
(10b)
(10c)
©2012 ACEEE
DOr.01.IJCOM.3.1.6
Input >(|wl
hu Film
Figure 1. Block diagram of a conventional PLL
r
Figure2. State-space model of the third order PLL with FM signal
showing the state variables (p^x, y and z
III. Numerical Simulation And Results
The dynamics of the system would be completely under-
stood by obtaining the solutions of the system equations
(8a) to (8e). However they are nonlinear differential equa-
tions and hence closed form solutions are difficult to obtain
for all parameter values, if not impossible. Hence numerical
solutions of these equations are obtained by Fourth order
Runge-Kutta technique and the system dynamics is exam-
ined for a range of values of the design parameters like m, g,
O and k . Varying the parameters , ( O and k being fixed)
respectively the values of the state variables x,y and z are
calculated. z(x) being the VCO control voltage, the nature of
VCO-output signal could be easily obtained from the solu-
tions.
First we simulate the equations for simple CW input with
just increasing the gain. With the increase of gain parameter,
the system undergoes into the chaotic regime at j gr=2.5
through a period doubling sequence. Fig. 3 compares the
63
ACE EE
ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
phase-plane diagram of simulation results with that of the
experimental results in this regard. Fig. 4 summarized the dy-
namics of PLL with CW input through a bifurcation diagram
with gain as control parameter.
In the second part we study the dynamics of the system
with FM input. We plot the phase-plane diagram with state
variables y(x) and z(t) for increasing values of in, modulation
index of FM signal. Observation shows the PLL transits from
a steady state condition to a limit cycle state and then goes
to chaotic state. (Fig. 5,6 and 7).
Here we verified the occurrence of chaos for suitable
system-parameter values, using standard nonlinear dynamical
measures like Lyapunov Exponents (k.) in both cases. The
methodology adopted by Wolf et al. 1985 [6] has been used
to calculate the Lyapunov exponents from the system
differential equations.
Table- 1 gives the value of Lyapunov exponent for third
order PLL with CW input and Table 2 is that of second case
with FM input for different values of modulation index m and
in tune condition.
In Table- 1 above 2.5 we get high value of positive
lyapunov exponent. For FM input signal the results as shown
in Table -2 indicate that when m > 1.99, we have high value of
X + , which is indicative of the chaotic dynamics occurring in
the system. We also calculate the maximum lyapunov
exponent [MLE] from the time series data using users friendly
software of time series analysis [7-9] and get h = 2.39 for CW
input at ,£p2.5, 1. = -0. 17 and 0.28 for m= 1 .9 and 1 .99 when the
system is in tune. In off tune carrier condition X. = -0.01 and
0.91 form=1.26 and 1.34 which supports the chaotic oscillation.
Table I. Lyapunov Exponents for different values of gain (for k = 1 and
n „ = o.5 )
Gain paramf tsr g
7i£
13
-0.073115
-0.073462
-1.5534
1.5
-0.013655
-0.013736
-1.4726
1.7
:>.yy.:<:4~<
-0.16503
-1.136
2.1
0.0040214
-0.027036
-0.87699
■7 <z
0.095223
-0.00011108
-0.59511
2.9
0.33 87
0.0012514
-0.43995
Table II: Lyapunov Exponents (a^s) for different values of m (for
S = = 0.904, rt„ = 0,fr„ = 0.6511)
m
1
A t
J- 2
h
1
A +
*i
0.0001
-0.042S
-0.0423
-1.7432
0.1
0.0022
-0.0021
-0.0777
-0.7733
-1.6734
1.39
0.002D
-0.0021
-0.0367
-0.360}
-1.1369
1.97
0.0024
-0.0016
-0.0933
-06146
-11264
199
00147
0.0000
-0.O074
-0.6724
-1.1637
2.1
0.1060
0.0024
-0.0019
-0.7322.
-1.2031
(a) (b)
Figure 3. Comparison of chaos in Third order PLL with CW input
(a) Numerical result (b) Experimental result.
-
■i'
1 * •
2 22 2.4 2.6 2 5
Figure 4. Bifurcation diagram of z with gain as the control
parameter and k n = 0.42,D n = 0.22.
In tune
■•■i : if a US 1 :S -if -1 -us 0.5 I V5
I >
m=1.89 m=2.1
(c) (d)
Figure 5. Phase plane plot of PLL operation, with ,
g — J-72S,/„ = 0-9O4./7 K = Q.k n = 0-6511,
and the modulation index m for (a) 0, (b) 0.1, (c) 1.89, (d) 2.1.
©2012 ACEEE
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vcACEEE
ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
it
E
i-i
0.5
Mcdjfatio index (m)
Figure 6. Bifurcation diagram of z with modulation index as the
control parameter and k n = 0.65,£2 n = 0, g =1.728
OffTune
■11 I 4 5 o ai 1 IS
m=0
(a)
t
-J -I 1 2
f
m=1.26 m=1.34
(c) (d)
Figure 7. Phase plane plot of PLL operation, with ,
g = 1.72S. f n = 0.769, A M = 0.2261. fc„ =
and the modulation index m for (a) 0, (b) 0.1, (c) 1.26, (d) 1.34
IV. Experimental Studies
A prototype hardware circuit for a third order PLL with a
resonant second order filter has been designed using off-
the-shelf ICs and passive circuit components. The integrated
circuit chip AD532 has been used to realize the multiplier-
type phase detector used in the PLL structures. The active
low-pass filters have been constructed using uA741 OP-
AMPs and necessary passive circuit components. Aplab-
make signal generator with FM modulation facility using ex-
ternal modulating signal is used as the VCO of the PLL. In the
first case, the central frequency of the VCO is taken as 125
kHz and 127 kHz is that of CW input with 2 volt amplitude.
The carrier frequency of the FM signal is also taken as 125
kHz for in tune. The spectrum of the VCO output is observed
using Agilent-make spectrum analyzer. We observe a broad-
©2012 ACEEE
DOL01JJCOM.3.1.6
band output spectrum in case of chaotic oscillation. The gain
parameter of the loop filter is varied by changing the feed-
back resistance at the filter circuit. Filter gain is 1 .728 in the
second case, i.e. within the stable region. The frequency of
the modulating signal is 8 kHz in case of in tune and that is
6.8 kHz in case of off tune.
Fig. 3 shows the chaos obtained in third order PLL with
CW input. The experimental results with FM input of some
specific cases are shown in Fig. 8 to 10. Fig. 8 represents the
steady state, as is evident from single component at the
output spectrum of the VCO. In Fig. 8, the value of m is 0. At
this condition the PLL is operated by simple sinusoidal signal
or CW signal. For m=11.94, the loop is in chaotic state of
oscillation which is clear from Fig. 9 as its spectrum is a broad
one. Fig. 10 shows the chaos when the carrier signal frequency
is 127 kHz i.e. in off tune. Experiments have been carried out
with several other values of the loop-design parameters and
obtained results are in close agreement with the analytically
predicted results obtained via numerical solutions of system-
equations.
V. Discussions
In the present paper, we have described the response of
a class of third-order PLLs with a CW input as well as an FM
input signal. The loop filter is a resonant type second order
one. The chaotic oscillations are produced in the loop for a
suitable set of loop-design parameters like loop filter-gain
and time-constant, amplitudes of the input reference signal
and the VCO signal, the phase detector gain, VCO sensitivity
etc. Chaos has been observed for both CW and FM inputs. It
is important to note that the dynamics of an FM demodulator
may occasionally become chaotic if the signal parameters
(like amplitude, modulation index etc) cross a finite limit
depending on the values of loop design parameters. The
results of numerical simulation are in close agreement with
those of the hardware experiment.
The present study is useful in several respects. It shows
that a properly designed third order PLL can be used as a RF
chaos generator. Thus a chaotic modulator can be designed
with it. Secondly, it indicates that design imperfections can
make the output of a PLL demodulator can become chaotic.
Hence a new degree of freedom ( namely, modulation index)
can be had for a controlled chaos generator with FM input
signal.
::::::!
"mm
(a) (b)
Figure 8. Amplitude of modulating signal V =0; (a) Phase plane plot
and (b) Spectrum of VCO output voltage with Modulating signal
Frequency=8kHz, Frequency of carrier signal=125kHz, Gain of PLL-
1.728.
65
ACE EE
ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
(a) (b)
Figure 9. Amplitude of modulating signal V =3.98; (a) Phase plane
plot and (b) Spectrum of VCO output voltage with Modulating signal
Frequency=8kHz, Frequency of carrier signal=125kHz, Gain of PLL-
1.728.
(a) (b)
Figure 10. Amplitude of modulating signal V =2.18; (a) Phase plane
plot and (b) Spectrum of VCO output voltage with Modulating signal
Frequency=6. 8kHz, Frequency of carrier signal= 127kHz, Gain of PLL-
1.728.
References
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loops", IEEE Transactions on Circuits and Systems, 35, 987-
1003.
[2] Harb, B.A. and Harb, M.A. (2004), "Chaos and bifurcation in
third-order phase-locked loop", Chaos, Solitons and Fractals,
19,667-672.
[3] Matrosov, V.V. (2006), "Self-modulation regimes of a phase
locked loop with the second order filter", Radio physics and
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[4] Banerjee, T. and Sarkar, B.C. (2008), "Chaos and bifurcation
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[5] Piqueira, I.R.C., "Using bifurcations in the determination of
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[6] Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A. (1985),
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[8] Stane Kodba, Matjaz Perc and Marko Marhl, Detecting chaos
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[9] Holger Kantz and Thomas Schreiber, Nonlinear time series
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