muri
DESCRIPTION
MURI talk on December 10th, 2010TRANSCRIPT
Nonlinear Microwave Oscillators: Dynamics and Synchronization
Hien Dao (Chemical Physics Program)
John C. Rodgers (IREAP)
Thomas E. Murphy (ECE & IREAP)
OutlineOutline
• Motivation
• Dynamics of a nonlinear microwave system
• Synchronization of coupled microwave oscillators
• Conclusion
MotivationMotivation
Nonlinear time-delayed feedback loops can produce high dimensional chaos.
nonlinearity
H(s)20cos ( )
gain
delay
filter
Example: An optoelectronic chaotic oscillator
A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, Phys. Rev. Lett. 101, 154102 (2008)
Microwave transmission systems are used everywhere; and many of those rely on microwave carrier recovery with voltage controlled oscillator is key component.
MotivationMotivation
Our microwave chaotic system is based on time-delayed feedback loop architecture working in the frequency band, 2 GHz-4 GHz.
A chaotic signal in this band would potentially offer advantages such as lower probability of detection, less susceptible to noise and jamming, less likely to interfere with existing communication channels…
Phase- locked- loop using VCO could exhibit chaotic signal.
Sandia report, March 2004
VCOsplitter
d
mixer
Bias at operating point
H(s)
low pass filtergain
delay
Experimental setupExperimental setup
Voltage Controlled Oscillator (VCO) is a device that converts an input analog voltage into a signal whose frequency is linearly proportional to the magnitude of voltage
VCOTuning signal v(t) RF signal VCO(t) A cos( (t))
and 0
d (t)2 (t)
dt
with is named tuning sensitivity (VCO gain) and 0 is bias frequency.
d (t)2 (t)
dt
0.5 1 1.5 2 2.5 3 3.5 4 4.52.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4x 109
Tunning Voltage [V]
Mic
row
ave
freq
uen
cy [
Hz]
Slowly varying phase
0=2.65 GHz
=175 MHz/v
v VCO
d
Mixer0( ) 2 j te
dj ( t )e
Splitter
delay
0 0 dV cos( 2 )
Nonlinear function is created using delay-line frequency discriminator
outputdv 1/
VCOSplitter
d
Mixer
bidirectional coupler combiner
0.5 1 1.5 2 2.5 3 3.5 4 4.5
-0.1
-0.05
0
0.05
0.1
0.15
Vtune [V]
Vm
ixer
[V
]175MHz / V
d 5ns
VCOsplitter
d
mixer
Bias at operating point
H(s)
low pass filtergain
delay
Experimental setupExperimental setup
-60
-50
-40
-30
-20
-10
0
0.0 1.0 2.0 3.0 4.0 5.0
Frequency [MHz]
Po
we
r le
ve
l [d
B]
L/N L/N L/N
C/2N C/2N C/2N C/2N C/2N C/2N
N units
L=5 H
C=1nF
u=0.1 s/unit;
= 1.2 s
fcutoff ~ 3 MHz
Loop feedback delay is built in with transmission line design
Mathematical model for tuning signalMathematical model for tuning signal
H(s)0cos( ) nonlinearity
gain
delay
low pass filter
v(t)
system equation
0 0 d
duA.u B V cos( 2 Cu(t ))
dt
v(t) Cu(t)
fcutoff =3 MHzs
1.2 s
Varying from 0.5-9.5
5 ns
175 MHz/V
ValueParameters
0 / 2
0.5 1 1.5 2 2.5 3 3.5 4 4.5
-0.1
-0.05
0
0.05
0.1
0.15
Vtune [V]
Vm
ixer
[V
]
o
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-0.4
-0.2
0
0.2
0.4
0.6
Time [s]
Vtu
ne
[V]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time [s]
Vtu
ne
[V]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time [s]
Vtu
ne
[V]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time [s]
Vtun
e [V
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-0.4
-0.2
0
0.2
0.4
0.6
Time [s]
Vtu
ne
[V
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
-5
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time [s]
Vtu
ne
[V]
=1.6
=2.2
=6.5
Experiment Simulation
-40 -30 -20 -10 0 10 20 30 40-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
(t)
' (t
)
-100 -80 -60 -40 -20 0 20 40 60 80 100-0.6
-0.4
-0.2
0
0.2
0.4
0.6
(t)
' (t
)
-300 -200 -100 0 100 200 300-1.5
-1
-0.5
0
0.5
1
1.5
(t)
'(t
)
Phase portrait of system
d (t)(t) and 2 (t)
dt
plotting phase of envelope signal versus its derivative can tell us about dynamics of system
=1.6
=2.2 =6.5
Simulation
Experiment
Bifurcation diagram of system
-2
-1
2
1
0
-2
-1
2
1
0
1 2 3 4 65 7
Vtu
ne
[V
]V
tun
e [
V]
1 2 3 4 65 7
-2
-1
2
1
0
-2
-1
2
1
0
Vtu
ne
[V
]V
tun
e [
V]
increasing
decreasing
Historesis effect
-2
-1
2
1
0
Vtu
ne
[V
]
1 2 3 4 65 7
-0.2
0.4
0.2
-0.1
0.1
0
0.3
0.5
1emax imum [ms ]
Maximum Lyapunov exponent
Synchronization of coupled microwave oscillators Synchronization of coupled microwave oscillators
Chaotic synchronization had been achieved by coupling two optoelectronic systems.
How to couple two microwave systems and what kind of synchronization we should observe?
x1(t):
x1(t) – x2(t):
x2(t):
Two systems are coupled bi-directionally in microwave band,
VCOSplitter
d
Mixer
Bias
VCOSplitter
d
Mixer
Bias
H(s) H(s)
is coupling strength
v1 (t) v2 (t)
2 (t)1(t)
Behavior depends on whether the VCO difference frequency exceeds the filter bandwidth
• Phase synchronization (PS) is achieved when two RF signals has locking of phases.
• Envelope Synchronization (ES) happens when two tuning voltage signals synchronized while two microwave signals can stay uncorrelated
VCOTuning signal RF signal VCO(t) A cos( (t))
VCO(t) A cos( (t)) RF signal
1 2(t) (t)?
1 2(t) (t)?
(t)
RF signal VCO(t) A cos( (t))
Constant amplitude
Phase varies around a bias value 0
0
d (t)2 (t)
dt
RF signal collected from scope Analytic signalHilbert transform
Analytic signal j ( t )(t) (t) j (t) A(t)e
Where is Hilbert transform of
1 (t)(t) P.V. d
t
(t) (t)
Using Hilbert transform to estimate phase
VCOTuning signal RF signal
1 2(t) , (t)
=1.2 and =0.1
0 10050 200 250150
t (s)
1 2t t (rad)
0 0.5 1 1.5 2 2.5x 10
-4
-50
0
50
100
150
200
250
Time [s]
[Rad
]
0
100
200
0.88rad / st
(140KHz)
1 2(t) , (t)
VCOTuning signal RF signal
=2.1 and =0.1
0 0.5 1 1.5 2 2.5
x 10-4
-1000
0
1000
2000
3000
4000
5000
6000
0
3000
6000
1 2t t (rad)
0 10050 200 250150
t (s)
22.8rad / st(3.6MHz)
ConclusionConclusion
We designed and modeled a nonlinear microwave circuit which can exhibit
chaotic signal. The circuit is very applicable due to range of operating frequency,
small size and reasonable price.
We also coupled two microwave systems and achieved envelope synchronization
and some promising data indicated phase synchronization between RF signals.
To avoid delay loop in coupling part, we will try unidirectional coupling case and
increase coupling strength as well.
Improve modeling of coupled systems.