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.