Evolution of a nonlinear system- route to chaos
Gain= 2.13
Hanieh Nejadriahi (St. Olaf College), John Rodgers. Adviser (UMD)
Abstract
•
•
Phase locked loops (PLL) are widely important in modern electronic communication
and control systems. They are used in systems where it is necessary to estimate the
phase of a received (feedback) signal. The goal of this project was to investigate the
stability of a second order PLL and observe the route to chaos in a third order PLL as
well as a second order PLL with time delayed feedback by advanced usage of scientific
instruments and exploring and exploring new chaotic dynamics with different filter
topologies. To prove the validity, these experimental studies have been compared to
the theories of chaos along with computer simulations.
Main Components: Phase detector (mixer), Voltage
Controlled Oscillator (VCO), Low Pass filter.
•
To study detailed route to chaos with different filter topologies and the sensitivity of the
dynamics to filter orders.
Using advanced scientific instruments experimentally observe the chaotic behavior in 3rd
Order PLL/ Time delayed 2nd order PLL within the lock range by analyzing their
bifurcation diagrams and phase portraits.
•
•
Phase detector-Mixer
Ω=0
𝑓𝑐𝑎𝑟𝑟𝑖𝑒𝑟 = 910𝑀𝐻𝑧 − 𝑝𝑜𝑤𝑒𝑟 = 7𝑑𝐵𝑚
2nd order filter- 𝑓𝑐𝑢𝑡𝑜𝑓𝑓 = 100 𝐾𝐻𝑧
Integrator- 𝑓𝑐𝑢𝑡𝑜𝑓𝑓 = 30 KHz − 𝑔𝑎𝑖𝑛 = 20
Gain=749
Bifurcation Parameter= Gain
Gain range= 680-760
Gain=739
Applications in telecommunication, Frequency
synthesizers (found in modern devices: radio
receivers, mobile telephones, radiotelephones,
satellite receivers, GPS systems, etc.)
Gain=680
Gain=710
3rd order PLL- Exp
Voltage Controlled Oscillator (VCO)
compares the phase at each input and generates an error (phase difference). Mixer multiplies (mixes) the
amplitude of two input signals and compares their phase angles.
If the inputs are sinusoid, the ideal mixer output is the sum and the difference frequencies
𝑠1 = 𝑉𝐿𝑂 cos ω𝐿𝑂 𝑡 + φ𝐿𝑂
An electronic oscillator whose oscillation frequency is a function of the input tuning
voltage.
•
𝑠2 = 𝑉𝑅𝐹 cos(ω𝑅𝐹 𝑡 + φ𝑅𝐹 )
VCO is a linear time invariant system. Oscillates at an angular frequency ω. The
initial frequency for which the tuning voltage is 0 equals to ω0 . Frequency is linearly
𝑉𝐼𝐹 = 𝐾𝛾 𝑉𝐿𝑂 𝑉𝑅𝐹 sin (ω𝐿𝑂 −ω𝑅𝐹 𝑡 + (φ𝐿𝑂 − φ𝑅𝐹 ))
𝐾𝛾 (conversion gain= 1/V)
Typically, either the sum (up conversion), or the difference (down conversion) frequency is removed with a
filter.
An ideal mixer translates the modulation around one carrier to another, usually from
a higher frequency (RF) to a lower IF frequency.
proportional to the tuning voltage with a gain coefficient 𝐾𝑉𝐶𝑂 (
ω𝑜𝑢𝑡 = ω0 + 𝐾𝑉𝐶𝑂 𝑉𝑡𝑢𝑛𝑒
•
𝑟𝑎𝑑
𝑉
𝑠
)
Gain=760
The source of nonlinearity in the system
Mixer output frequency
(MHz)
Internal Circuitry
PLL loop filter usually a low pass filter has two main functions:
1) To determine the loop dynamics (stability); common
considerations such as the range over which the loop reaches its lock range
and how fast it achieves lock and whether or not there is a damping behavior.
2) To limit the reference frequency energy (ripples) appearing
at the output of the phase detector which then is applied to the VCO’s input
as the tuning voltage. This frequency modulates the VCO and produces
Frequency Modulation sidebands (harmonic and sub-harmonic frequencies).
The low pass filter is used to reduce these energies.
Gain=630
Frequency
(Hz)
Filter
•
Gain=760
A PLL is a control system that generates an output
signal whose phase is related to the phase of the
input and the feedback signal of the local oscillator.
The purpose of a low pass filter is to force the VCO
to replicate and synchronize the frequency and
phase at the input. When locked the frequencies
must match, but it is possible to have a phase
offset.
Objectives
•
This is a homodyne system with a sine
nonlinearity.
Phase Locked Loop (PLL) Basics
Gain=500
Ω = 0.05 MHz
𝑓𝑐𝑎𝑟𝑟𝑖𝑒𝑟 = 910𝑀𝐻𝑧 − 𝑝𝑜𝑤𝑒𝑟 = 7𝑑𝐵𝑚
2nd order filter- 𝑓𝑐𝑢𝑡𝑜𝑓𝑓 = 100 𝐾𝐻𝑧
Integrator- 𝑓𝑐𝑢𝑡𝑜𝑓𝑓 = 30 𝐾𝐻𝑧 − 𝑔𝑎𝑖𝑛 = 20
Bifurcation Parameter= Gain
Gain range= 500-700
Gain=605
Gain=508
Tuning Voltage (V)
Digital Signal Processor (DSP) Filter Topologies
𝐻
Frequency (Hz)
1
𝐻 𝑠 𝑃𝐿𝐿 =
𝑠 𝑠+𝑃
1
𝑠 𝑃𝐿𝐿 =
𝑠 𝑠+𝑃1 𝑠+𝑃2
Feedback system with time delay and sine nonlinearity
•
While ordinary differential equations must be at least of third order to produce chaos, first order nonlinear delayed
differential equations can produce chaotic dynamics.
The principle difference between this system and the previous one is that it is a homodyne, time delayed PLL that also has
a sine nonlinearity.
A digital system processor has been used to create time delay via different filter topologies.
•
•
Open Loop Transfer Function
Phase (rad)
Gain (dB)
Open Loop Phase
Gain (dB)
Frequency (Hz)
Phase (rad)
Gain (dB)
Phase (rad)
The transfer function H(s) or the gain of a
filter where s is the complex frequency
(𝑠 = 𝜎 + 𝑗𝜔)
𝑥 𝑡 = −𝑅𝑠𝑖𝑛 𝑥 𝑡 − 1
𝑅=
𝐹𝑐𝑢𝑡𝑜𝑓𝑓 = 800 𝐻𝑧
𝐿𝑖𝑛𝑒𝑎𝑟 𝐺𝑎𝑖𝑛 = 1
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
2𝜋K VCO K 𝛾 𝑉𝐿𝑂 VIF τ𝑑 𝜏𝑔
𝑇
𝐹𝑐𝑢𝑡𝑜𝑓𝑓 = 800 𝐻𝑧
𝐿𝑖𝑛𝑒𝑎𝑟 𝐺𝑎𝑖𝑛 = 1
Operating voltage = 6.27 Volts
Frequency (Hz)
PLL Stability Analysis
The order of the filter in a PLL is the key factor in terms of determining the behavior of the system (stable, or chaotic)
Other than the two main functions, PLL LPF also affects the loop response including parameters such as the loop filter time constant 𝜏𝑟 , loop bandwidth
𝜔𝐶 and the damping factor 𝜕 of the loop.
Linkwitz-Riley
An important aspect of PLL design is the steady state stability of the loop.
PLL’s are negative feedback loops-output of the system tries to oppose changes to the input of the system- to make the system self regulating.
Gain= 2.80
Gain= 2.13
−𝐾𝛾 𝐻 𝑠 𝐾𝑉𝐶𝑂
𝜃𝑂
G s
𝑇 𝑠 =
=
=
𝜃𝐼 𝑠 + 𝐾𝛾 𝐻 𝑠 𝐾𝑉𝐶𝑂 1 + G s
Gain= 2
T(s): Closed loop PLL transfer function in the frequency domain
G(s): Open loop transfer function of the PLL
…2nd Order Filter Stability Criteria
• In order to fully determine the stability of the loop with a single pole
low pass filter, finding the open loop transfer function is vital
1
2
𝜔𝑁 = 𝐾𝑡𝑜𝑡 𝜔𝐿
Natural frequency of the loop
𝜔𝑁
𝜕=
Damping factor in the loop
2𝐾𝑡𝑜𝑡
𝜔 ≫ 𝜔L
𝜔 = 𝜔L
𝜔 ≪ 𝜔L
Gain= 2.80
Gain= 2.13
𝐺 𝑠 =
𝜔 = 𝜔LO − 𝜔𝑅𝐹
Unstable
Critical Point (Marginally stable)
𝐾𝛾 𝐾𝑉𝐶𝑂 𝐻 𝑠
𝐾𝑡𝑜𝑡
=
𝑠
𝑠
𝑠
+1
𝜔𝐿
Bessel
Bessel
𝜙
Butterworth
• 𝐾𝑡𝑜𝑡 = 𝐾𝑉𝐶𝑂 𝐾𝛾
• 𝜔𝐿 = Cutoff frequency of the LPF
Stable
𝜙
Gain =20
𝑉
𝜔L = 1 𝑀𝐻𝑧
𝜔= 1 MHZ
Gain =100
Phase (arb)
𝑀𝐻𝑧
𝑀𝐻𝑧
𝑉
Gain= 1.7
Gain= 2.01
Gain= 2.43
Gain= 1.5
Gain= 1.73
Gain= 2.04
Conclusion
Tuning Frequency (arb)
PLL Equations-3rd order
𝑥 = Ω𝑛 − 𝑘𝑛 𝑧
𝑔−1
𝑦 = sin 𝑥 + 𝑔 − 2 𝑦 − 𝑔 −
𝑔
𝑧 = 𝑔𝑦 − 𝑧
Filter topology: 2nd order low pass Butterworth, cutoff
frequency = 1MHz
References
Where Ω𝑛 = ΩT, k n = kT
Normalized Ω𝑛 = .5, k n = 1
Y= Mixer Output
X= phase difference (ϕ)
Z= tuning voltage
Ω = frequecny detuning
K= Loop gain
In a 3rd order PLL with a 2nd order low pass filter, the
gain of the filter and its time constants are the
controlled parameters which by generating different
topologies of the filter could lead to generating chaos
Phase (rad)
• A 3rd order PLL is a heterodyne system that has a built-in sine
nonlinearity to it which comes from the mixer.
Phase (rad)
3rd Order PLL and chaos- simulation
Using different filter topologies chaotic dynamics of phase locked loops have been studied. The systems use both a digital signal
processor and microwave components to apply nonlinearity, filtering functions, and time delay. The order of the filter and time delay
are the determining factors in generating chaos. In a feedback system with time delay we demonstrated that a Butterworth is the most
unstable filter due to having the highest gain while Linkwitz-Riley is the most stable. Butterworth bifurcated at 3dB before Bessel.
𝜋
Period doubling route to chaos at a hopf bifurcation point has been also shown at which agrees with the theory. Chaotic dynamics
2
of the 3rd order PLL also come to an agreement with the theory since by increasing the gain of the loop PLL breaks into chaotic
oscillations as shown in the bifurcation diagrams.
Gain
G=.7
x
G=1.5
x
z
z
G=1.1
x
z
G=1.93
x
z
G=1.9
G=1.83 x
x
z
TREND 2013
Training and Research Experiences in Nonlinear Dynamics
Gain
Acknowledgments
• Dao, Hien, John Rodgers, and Thomas Murphy. "Chaotic dynamics of a frequency modulated microwave
oscillator with delayed feedback." American Institute of Physics. (2013): n. page. Web. 2 Aug. 2013.
<http://chaos.aip.org/resource/1/chaoeh/v23s/i1/p013101_s1?view=fulltext>.
• Schanz, Michael, and Axel Pelster. "Analytical and numerical investigations of the phase locked loop with
time delay." American Institute of Physics. (2003): n. page. Web. 2 Aug. 2013.
<http://pre.aps.org/abstract/PRE/v67/i5/e056205>.
• Charan Sarkar, Bishu, and Saumen Chakraborty. "Chaotic Dynamics of a Third Order PLL with Resonant Low
Pass Filter in Face of CW and FM Input Signals."ACEEE. (2012): n. page. Web. 2 Aug. 2013.
<http://www.slashdocs.com/knzrpr/chaotic-dynamics-of-a-third-order-pll-with-resonant-low-pass-filter-inface-of-cw-and-fm-input-signals.html>.
• Harb, Bassam A. "Chaos and Bifurcation in Third Order Phase Locked Loop." IEEE. (2003): n. page. Web. 4
Aug. 2013. <http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1194549>.
• Mansukhani, Arun. "Phase Lock Loop Stability Analysis."Applied Microwave & Wireless. (2000): n. page.
Web. 4 Aug. 2013.
<http://www.rfdh.com/ez/system/db/lib_jnl/upload/312/[AMW0002]_Phase_Lock_Loop_Stability_Analysis.p
df>.
• I would like to express my sincere gratitude to my advisor Dr. John Rodgers
for the continuous support with my research project, for his patience,
motivation, enthusiasm, and immense knowledge. His guidance helped me in
all the time of research. I could not have imagined having a better mentor
during this short period of time.
• My thanks also go to Bisrat Addissie for helping me with developing new
and advanced programming skills in MATLAB.
• At last but not least I would like to acknowledge all the TREND faculty
members who made this great research opportunity for students encouraged
in pursuing science and engineering.
z
REU program sponsored by the
National Science Foundation
Award Number: PHY1156454