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

1

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 2

Outline

Introduction

Bifurcation detection techniques

Stability analysis of power amplifiers

Oscillation, chaos, hysteresis

Noisy precursor, hysteresis in power-transfer curve

Conclusion


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.

00 X)(X ),,X(X

ttf


Types of instabilities and bifurcations - I

Real (poles)

imag (poles)

−fa

fa

Hopf bifurcation

Out

put s

pect

rum

Frequency

fin

2fin

3finfin/23fin/2

5fin/2

Frequency division

Out

put s

pect

rum

Frequency

fin

2fin

Chaos

Out

put s

pect

rum

Frequency

fin

2fin

3finfosc

Spurious oscillation

−fin/2

Real (poles)

imag (poles)fin/2

Flip bifurcation

Many routes lead to chaos

- Quasi-periodic route

- Period-doubling route

- Torus-doubling route


Types of instabilities and bifurcations - II

Noisy precursors

Out

put s

pect

rum

Frequency

fin

2fin

Reduced stability margin

Real (poles)

imag (poles)

−fa

fa

Hysteresis

Ou

tpu

t p

ow

er

Po

ut

Input-drive power Pin

T1

T2 J1

J2

D-type bifurcation

Real (poles)

imag (poles)


Auxiliary generator

nonlinearcircuitAin (large signal),

fin

0AG

AGAG V

IY (Non-perturbation condition)

0)(

0),( AGAGAG

XH

VfY

• Oscillating solution is obtained by solving:

VAG

fAG

IAG

Ideal BPF at fAG


Pole-zero identification

VsIs(ε,ω)nonlinear

circuitAin (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.


1.5kW, 29MHz Class-E/Fodd PA using a Distributed Active Transformer

LchokeLchoke

M4M3M2M1

Vg4–Vg2

–

V DD

k

VDD

k

Cres=560 pF

C res=560 pF

R L

48 nH 48nH

Vg1+ Vg3

+

Vg3+

21nH

2.2nF

33nF

Vg1+

RF in 3 : 1

21nHVg4

–Vg2

–

Input -power distribution network


Evolution of measured output spectrum in Pin

Out

put

sp

ectr

um

(d

BW

)

Frequency (MHz)

0 20 40 60 80 100 120 140-60

-40

-20

0

40

20 Low-power leakage

Out

put s

pect

rum

(dB

W)

Frequency (MHz)

0 20 40 60 80 100 120 140-60

-40

-20

0

40

20 Chaotic spectrum

Pin = 5.5W

Out

put s

pect

rum

(dB

W)

Frequency (MHz)

0 20 40 60 80 100 120 140-60

-40

-20

0

40

20

fin

2fin

3fin

4fin

5fin

Pin = 13.0W

Pin = 13.0W

Ou

tpu

t sp

ectr

um

(d

BW

)

Frequency (MHz)

0 20 40 60 80 100 120 140

-60

-40

-20

0

20

40

3fin

finfin+2fa

fin+fa

Pin = 5.3WPin = 5.0W

Self-oscillation at fa = 4 MHz

Chaos

Hysteresis in the lower Pin boundary of

bifurcation.


Local stability analysis using pole-zero identification technique

Change input-drive power Pin (5W – 15W by 1W step).

Fre

qu

enc

y (M

Hz)

Real (poles) / 2

0

5

10

x105-4 -2 0 2 4 6

5W

10W

15W

Hopf bifurcation(Pin = 6.1W)

Inverse Hopf bifurcation(Pin = 13.5W)

Good agreement with the measurement in terms of bifurcation points.


Bifurcation locus Auxiliary generator with the non-perturbation condition solved in

combination with HB:

Delimit the stable and unstable operating regions.

. 0),,( inDDaAG PVfY

Drain bias voltage VDD (V)

0 20 40 60 80 100 1200

5

10

15

20

25

Inpu

t-dr

ive

pow

er P

in (

W)

Stable

Unstable

Stable


Oscillating solution curve

Auxiliary generator with the non-perturbation condition (fixed VDD):

. 0),,( inAGaAG PVfY

Osc

illa

tion

volta

ge V

AG

(V)

Input-drive power Pin (W)

4 6 8 10 12 140

10

20

30

40

50

60

70

Jump1

Jump2

Hopf bifurcations

Turning point


Osc

illa

ti on

v ol ta

ge V

AG

( V)


4 6 8 10 12 140

10

20

30

40

50

60

70

Chaos prediction Two-tone based envelope-transient

lk

tlfkfjlk et

,

)(2,

AGin)( Xx

fin

Ha

rmon

ic v

alu

es (

dBV

)

Frequency (MHz)0 10 20 30 40 50 60

-100

-80

-60

-40

-20

0

20

40

60

1st oscillation

2nd oscillation

Spectrum of harmonic component

Self-oscillating regime with a single oscillation

Jump1

Jump2

Hopf birfurcations

Vol

tage

(V

)

Time (μs)

0 10 20 30 4063.80

63.81

63.82

63.83

63.84

63.85

Magnitude of fin harmonic component

Chaotic regime

2nd Hopf birfurcation

3 non-commensurate frequencies

Quasi-periodic route to chaos


7.4-MHz Class-E power amplifier

Llpf

Clpf

C2nd

L2nd

LresCres

Cout

Cin

Lin

Cbypass

RF in

Lchoke

VDD

6 : 1RL

Pout = 360 W with 16 dB gain and 86 % drain efficiency


Measured output spectrum

Out

put s

pect

rum

(dB

W)

Frequency (MHz)

0 2 4 6 8 10-80

-60

-40

-20

0

20

40

Noise bumps fin

fc

Pin = 0.5W

Out

put s

pect

rum

(dB

W)

Frequency (MHz)0 2 4 6 8 10

-80

-60

-40

-20

0

20

40

Noise bumps

fin

fc

Pin = 0.8W

Out

put s

pect

rum

(dB

W)


-80

-60

-40

-20

0

20

40

fin

fa

Self-oscillating mixer regime

Pin = 0.84W

Out

put s

pect

rum

(dB

W)


-80

-60

-40

-20

0

20

40

fin

fin / 7

Sub-harmonic oscillation

Pin = 0.89W

Out

put s

pect

rum

(dB

W)


-80

-60

-40

-20

0

20

40

finProper spectrum

Pin = 4.0W


Stability analysis over solution curve

Hysteresis in power-transfer curve.

Pole-zero identification performed along the power-transfer curve.

Out

put p

ower

Pou

t (dB

W)

0.70 0.75 0.80 0.85


14

16

18

20

T1

T2

ζ1

ζ2

Fre

qu

en c

y ( M

Hz)

Real (poles)

-8 -6 -4 -2 0 2

-1.0

-0.5

0.0

0.5

1.0

2π X 105

fj 2

22 2 fj

ζ1

ζ1

ζ2

ζ2

ζ2

ζ2

ζ1

ζ1

ζ4

Jump

ζ4

Jumpζ4


Simulated noisy precursor spectrum

Out

put s

pect

rum

(dB

W)


-150

-100

-50

0

50

by conversion-matrix

by envelope transient

Simulated by two different techniques Envelope-transient

Conversion-matrix technique


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.

0detdet

AG

iAG

AG

iAG

AG

rAG

AG

rAG

AG

Y

V

Y

Y

V

Y

JY

Ou

tput

po

wer

Pou

t


T1

T2 J1

J2

Out

put

pow

er

Pou

t


T1

T2 J1

J2

Out

put

pow

er

Pou

t


Out

put

pow

er

Pou

t



Locus of turning points

Inpu

t po

we

r a

t tu

rnin

g p

oin

t (W

)

Capacitance in LPF Clpf (pF)

40 60 80 100 120 1400.5

0.6

0.7

0.8

0.9

1.0

CP1

CP2

CP3

Turning points for the original PA

Llpf = 100nH

Llpf = 257nH

Llpf = 400nH

Locus of turning points

Ou

tput

pow

er P

out (

dBW

)


0.76 0.78 0.80 0.82 0.84 0.8615

16

17

18

19

20

Clpf = 100pFClpf = 90pFClpf = 85pFClpf = 80pF

Elimination of hysteresis

No hysteresis below 85pF.


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

Documents

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