This document summarizes research on nonlinear microwave oscillators and their synchronization. Key points include: 1) A nonlinear microwave system was designed using a voltage controlled oscillator (VCO), splitter, mixer, and delay line to produce chaos. 2) Coupling two such systems bidirectionally with strength κ led to envelope synchronization between the tuning voltages when the filter bandwidth exceeded the VCO frequency difference. 3) Increasing the coupling strength κ showed initial evidence of phase synchronization between the RF signals, as measured by the decreasing difference between their phases φ1(t) and φ2(t). 4) Future work aims to improve the modeling of coupled systems and try unidirectional coupling with stronger

1. Nonlinear Microwave Oscillators:
Dynamics and Synchronization
Hien Dao (Chemical Physics Program)
John C. Rodgers (IREAP)
Thomas E. Murphy (ECE & IREAP)

2. Outline
• Motivation
• Dynamics of a nonlinear microwave system
• Synchronization of coupled microwave oscillators
• Conclusion

3. Motivation
Nonlinear time-delayed feedback loops can produce high
dimensional chaos.
Example: An optoelectronic chaotic oscillator
nonlinearity
delay
2
cos (• + φ0 ) b t H(s)
gain filter
A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, Phys. Rev. Lett. 101, 154102 (2008)

4. Motivation
Microwave transmission systems are used everywhere; and many of
those rely on microwave carrier recovery with voltage controlled
oscillator is key component.
Phase- locked- loop
using VCO could
exhibit chaotic signal.
Sandia report, March 2004
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…

5. Experimental setup
τd
VCO
splitter
mixer
Bias at operating point
gain low pass filter
β τ H(s)
delay

6. 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
Tuning signal v(t) VCO RF signal ψ (t) = A VCO cos(ϕ (t))
dϕ (t) with g is named tuning sensitivity (VCO gain)
and =ϖ 0 + 2πγν (t)
dt and w0 is bias frequency.
dφ (t)
Slowly varying phase = 2πγν (t)
dt
9
x 10
3.4
3.3
Microwave frequency [Hz]
3.2
3.1 w0=2.65 GHz
3
2.9
g=175 MHz/v
2.8
2.7
2.6
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Tunning Voltage [V]

7. Nonlinear function is created using
delay-line frequency discriminator
τ d
e jω (t −τ d )
delay
v VCO V0 cos(ω0τ d + 2πγυ )
jω t
ω (υ ) = ω0 + 2πγυ Splitter e
Mixer
output
∆v =1 / γτ d

8. bidirectional coupler combiner
τ d = 5ns
τd
0.15
VCO 0.1
Splitter Mixer
γ = 175MHz / V 0.05
Vmixer [V]
0
-0.05
-0.1
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Vtune [V]

9. Experimental setup
τd
VCO
splitter
mixer
Bias at operating point
gain low pass filter
β τ H(s)
delay

10. Loop feedback delay t is built in with transmission
line design
L/N L/N L/N
C/2N C/2N C/2N C/2N C/2N C/2N
N units
L=5 µH
0
C=1nF
-10
Power level [dB]
-20 τu=0.1 µs/unit;
-30
-40
-50
τ= 1.2 µs
-60
0.0 1.0 2.0 3.0 4.0 5.0
fcutoff ~ 3 MHz
Frequency [MHz]

11. Mathematical model for tuning signal
nonlinearity delay
v(t)
cos(• + φ0 ) b t H(s)
gain low pass filter
φo
0.15
Parameters Value
0.1
g 175 MHz/V 0.05
Vmixer [V]
td 5 ns 0
t 1.2 ms -0.05
-0.1
b Varying
from 0.5-9.5 0.5 1 1.5 2 2.5
Vtune [V]
3 3.5 4 4.5
θ0 −π / 2
system equation
H(s) fcutoff =3 MHz
du
= A.u + Bβ V0 cos(θ 0 + 2πγτ d Cu(t − τ ))
dt
v(t) = Cu(t)

12. Experiment Simulation
0.4
0.4
0.3
0.3
0.2
0.2
0.1
Vtune [V]
0.1
b=1.6
Vtune [V]
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [s] -5
Time [s] -5
x 10 x 10
0.6
0.6
0.4
0.4
0.2 0.2
b=2.2
Vtune [V]
0 0
Vtune [V]
-0.2
-0.2
-0.4
-0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s] -5
x 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s] -5
x 10
2
2
1.5
1.5
1
1
0.5
Vtune [V]
0.5
Vtune [V]
0 b=6.5 0
-0.5
-0.5
-1
-1
-1.5
-1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [s] -5
Time [s] -5 x 10
x 10

13. Phase portrait of system
0.4
0.3
plotting phase of envelope signal versus
0.2
0.1
its derivative can tell us about dynamics
0
of system
φ ' (t)
-0.1
-0.2
dφ (t)
-0.3 φ (t) and = 2πγν (t)
-0.4
dt
-40 -30 -20 -10 0 10 20 30 40
φ(t)
b=1.6
0.6 1.5
0.4 1
0.2 0.5
φ ' (t)
0
φ '(t)
0
-0.2
-0.5
-0.4
-1
-0.6
-100 -80 -60 -40 -20 0 20 40 60 80 100 -1.5
φ(t) -300 -200 -100 0 100 200 300
φ(t)
b=2.2 b=6.5

14. Bifurcation diagram of system
2
1
Vtune [V]
0
Experiment
-1
-2
2
1
Vtune [V]
Simulation
0
-1
-2
1 2 3 4 5 6 7
b

15. Historesis effect
2
1
Vtune [V]
increasing b
0
-1
-2
2
1
Vtune [V]
0
decreasing b
-1
-2
1 2 3 4 5 6 7
b

16. Maximum Lyapunov exponent
2
1
Vtune [V]
0
-1
-2
0.5
0.4
0.3
0.2
−1
max imum λe [ms ] 0.1
0
-0.1
-0.2
1 2 3 4 5 6 7
b

17. Synchronization of coupled microwave oscillators

18. Chaotic synchronization had been achieved by coupling two optoelectronic
systems.
x1(t):
x2(t):
x1(t) – x2(t):
How to couple two microwave systems and what kind of synchronization we
should observe?

19. Two systems are coupled bi-directionally in microwave band,
κ
τd τd
VCO ω1 (t) VCO ω2 (t)
Splitter Splitter
Mixer Mixer
Bias v1 (t) Bias v2 (t)
β τ H(s) β τ H(s)
κ is coupling strength
Behavior depends on whether the VCO difference frequency exceeds
the filter bandwidth

20. Tuning signal υ (t) RF signal ψ (t) = A VCO cos(ϕ (t))
VCO
• Envelope Synchronization (ES) happens when two tuning voltage signals
synchronized while two microwave signals can stay uncorrelated
υ1 (t) =υ2 (t)?
• Phase synchronization (PS) is achieved when two RF signals has locking of phases.
RF signal ψ (t) = A VCO cos(ϕ (t))
ϕ1 (t) = ϕ2 (t)?

21. Using Hilbert transform to estimate phase
Hilbert transform
RF signal collected from scope Analytic signal
Constant amplitude
RF signal ψ (t) = A VCO cos(φ (t))
Phase varies around a bias value w0
dφ (t)
=ϖ 0 + 2πγν (t)
dt
Analytic signal Ω(t) =ψ (t) + jψ (t) = A(t)e jθ ( t )
ɶ
Where ψ (t) is Hilbert transform of ψ (t)
ɶ
∞
ψ (t)
ψ (t) = π P.V. ∫
ɶ −1
dτ
−∞
t −τ

22. Tuning signal RF signal
VCO
b=1.2 and k=0.1
υ1 (t), υ2 (t)

23. 200
ϕ1 ( t ) − ϕ2 ( t ) (rad)
100 ∆ϕ
= 0.88rad / µs
t
(140KHz)
0
0 0.5 1 1.5 2 2.5
0 50 100 150 200 250
t (µs)

24. Tuning signal RF signal
VCO
b=2.1 and k=0.1
υ1 (t), υ2 (t)

25. 6000
3000
ϕ1 ( t ) − ϕ 2 ( t ) (rad)
∆ϕ
= 22.8rad / µ s
t
(3.6MHz)
0
0 0.5 1 1.5 2 2.5
0 50 100 150 200 250
t (µs)

26. Conclusion
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.