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

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

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

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

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

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

Pole-zero identification Vs 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

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

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

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

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

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

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

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

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

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

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

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

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

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