1. Nonlinear Dynamics and Stability of Power Amplifiers
Sanggeun Jeon, Caltech
Almudena Suárez, Univ. of Cantabria
David Rutledge, Caltech
May 19th, 2006

2. Lee Center Workshop, May 19, 2006
Outline
Introduction
Bifurcation detection techniques
Stability analysis of power amplifiers
Oscillation, chaos, hysteresis
Noisy precursor, hysteresis in power-transfer curve
Conclusion
Lee Center Workshop, May 19, 2006

3. Lee Center Workshop, May 19, 2006
Introduction
Strong nonlinearity of power amplifiers
Instabilities
Performance degradation, interference, damage of circuit.
Bifurcations
Qualitative stability changes by varying a circuit parameter(s).
Oscillators are also based on bifurcation phenomenon.
Bifurcation detection
Solve nonlinear differential equations
difficult!
Must harness circuit simulator techniques like HB.
Lee Center Workshop, May 19, 2006

4. Types of instabilities and bifurcations - I
Spurious oscillation
Frequency division
Chaos
Hopf bifurcation
Flip bifurcation
Many routes lead to chaos
Quasi-periodic route
Period-doubling route
Torus-doubling route
Lee Center Workshop, May 19, 2006

5. Types of instabilities and bifurcations - II
Noisy precursors
Hysteresis
Reduced stability margin
D-type bifurcation
Lee Center Workshop, May 19, 2006

6. Lee Center Workshop, May 19, 2006
Auxiliary generator
VAG
fAG
IAG
Ideal BPF at fAG
nonlinear
circuit
Ain (large signal),
fin
(Non-perturbation condition)
Oscillating solution is obtained by solving:
Lee Center Workshop, May 19, 2006

7. Pole-zero identificationVs
Is(ε,ω)
nonlinear
circuit
Ain (large signal),
fin
Identify poles and zeros of the large-signal operated system.
Impedance function Zin(ω)=Vs/Is calculated thru the conversion-matrix approach in combination with HB.
Detect bifurcations and pole evolution with a circuit parameter varied.
Lee Center Workshop, May 19, 2006

8. 1.5kW, 29MHz Class-E/Fodd PA using a Distributed Active Transformer
choke M 4 3 2 1 V g – DD k C
res =
560 pF R 48 nH + 21 . nF 33
RF in :
Input -
power distribution network
Lee Center Workshop, May 19, 2006

9. Evolution of measured output spectrum in Pin
Pin = 5.5W
Pin = 13.0W
Pin = 13.0W
Pin = 5.0W
Pin = 5.3W
Self-oscillation at fa = 4 MHz
Chaos
Hysteresis in the lower Pin boundary of bifurcation.
Lee Center Workshop, May 19, 2006

10. Local stability analysis using pole-zero identification technique
Change input-drive power Pin (5W – 15W by 1W step).
Good agreement with the measurement in terms of bifurcation points.
Lee Center Workshop, May 19, 2006

11. Lee Center Workshop, May 19, 2006
Bifurcation locus
Auxiliary generator with the non-perturbation condition solved in combination with HB:
Delimit the stable and unstable operating regions.
Lee Center Workshop, May 19, 2006

12. Oscillating solution curve
Auxiliary generator with the non-perturbation condition (fixed VDD): O s c i l a t o n v g e V A G ( )
Input -
drive power P in W 4 6 8 10 12 14 20 30 40 50 60 70
Jump 1 2
Hopf bifurcations
Turning point
Lee Center Workshop, May 19, 2006

13. Lee Center Workshop, May 19, 2006
Chaos prediction
Two-tone based envelope-transient
Magnitude of fin harmonic component
Chaotic regime 2
nd Hopf birfurcation
Spectrum of harmonic component O s c i l a t o n v g e V A G ( )
Input -
drive power P in W 4 6 8 10 12 14 20 30 40 50 60 70
Self-oscillating regime with a single oscillation
Jump 2
Jump 1
3 non-commensurate frequencies
Quasi-periodic route to chaos
Hopf birfurcations
Lee Center Workshop, May 19, 2006

14. 7.4-MHz Class-E power amplifier
Pout = 360 W with 16 dB gain and 86 % drain efficiency
Lee Center Workshop, May 19, 2006

15. Measured output spectrum
Pin = 0.5W
Pin = 0.8W
Pin = 0.84W
Pin = 0.89W
Pin = 4.0W
Lee Center Workshop, May 19, 2006

16. Stability analysis over solution curve
Hysteresis in power-transfer curve.
Pole-zero identification performed along the power-transfer curve. F r e q u n c y ( M H z )
Real
poles - 8 6 4 2 1 . 5 π X 10 f j p s ζ1 ζ2 ζ4
Jump ζ2 ζ1
Lee Center Workshop, May 19, 2006

17. Simulated noisy precursor spectrum
Simulated by two different techniques
Envelope-transient
Conversion-matrix technique
Lee Center Workshop, May 19, 2006

18. Elimination of hysteresis in Pin-Pout curve
The cause of hysteresis: turning points in the curve.
Elimination of turning points by varying a circuit parameter.
Cusp bifurcation
Variation of a sensitive circuit parameter
At turning points, the Jacobian matrix for the non-perturbation equation YAG(|VAG|, φAG)=0 becomes singular.
Lee Center Workshop, May 19, 2006

19. Locus of turning points
Elimination of hysteresis
No hysteresis below 85pF.
Lee Center Workshop, May 19, 2006

20. Lee Center Workshop, May 19, 2006
Conclusion
Bifurcation detection techniques are introduced.
Linked to a commercial HB simulator.
Application to the stability analysis of power amplifiers.
Stabilization of power amplifiers by bifurcation control.
Versatility of techniques
General-purpose
Design of self-oscillating and synchronized circuits
Lee Center Workshop, May 19, 2006