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![Page 1: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/1.jpg)
1
Nonlinear Dynamics and Stability of Power Amplifiers
Sanggeun Jeon, Caltech
Almudena Suárez, Univ. of Cantabria
David Rutledge, Caltech
May 19th, 2006
![Page 2: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/2.jpg)
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
![Page 3: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/3.jpg)
Lee Center Workshop, May 19, 2006 3
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
![Page 4: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/4.jpg)
Lee Center Workshop, May 19, 2006 4
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
![Page 5: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/5.jpg)
Lee Center Workshop, May 19, 2006 5
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)
![Page 6: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/6.jpg)
Lee Center Workshop, May 19, 2006 6
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
![Page 7: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/7.jpg)
Lee Center Workshop, May 19, 2006 7
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.
![Page 8: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/8.jpg)
Lee Center Workshop, May 19, 2006 8
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
![Page 9: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/9.jpg)
Lee Center Workshop, May 19, 2006 9
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.
![Page 10: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/10.jpg)
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).
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.
![Page 11: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/11.jpg)
Lee Center Workshop, May 19, 2006 11
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
![Page 12: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/12.jpg)
Lee Center Workshop, May 19, 2006 12
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
![Page 13: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/13.jpg)
Lee Center Workshop, May 19, 2006 13
Osc
illa
ti on
v ol ta
ge V
AG
( V)
Input-drive power Pin (W)
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
![Page 14: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/14.jpg)
Lee Center Workshop, May 19, 2006 14
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
![Page 15: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/15.jpg)
Lee Center Workshop, May 19, 2006 15
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)
Frequency (MHz)0 2 4 6 8 10
-80
-60
-40
-20
0
20
40
fin
fa
Self-oscillating mixer regime
Pin = 0.84W
Out
put s
pect
rum
(dB
W)
Frequency (MHz)0 2 4 6 8 10
-80
-60
-40
-20
0
20
40
fin
fin / 7
Sub-harmonic oscillation
Pin = 0.89W
Out
put s
pect
rum
(dB
W)
Frequency (MHz)0 2 4 6 8 10
-80
-60
-40
-20
0
20
40
finProper spectrum
Pin = 4.0W
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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.
Out
put p
ower
Pou
t (dB
W)
0.70 0.75 0.80 0.85
Input-drive power Pin (W)
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
![Page 17: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/17.jpg)
Lee Center Workshop, May 19, 2006 17
Simulated noisy precursor spectrum
Out
put s
pect
rum
(dB
W)
Frequency (MHz)0 1 6 7 8 9
-150
-100
-50
0
50
by conversion-matrix
by envelope transient
Simulated by two different techniques Envelope-transient
Conversion-matrix technique
![Page 18: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/18.jpg)
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.
0detdet
AG
iAG
AG
iAG
AG
rAG
AG
rAG
AG
Y
V
Y
Y
V
Y
JY
Ou
tput
po
wer
Pou
t
Input-drive power Pin
T1
T2 J1
J2
Out
put
pow
er
Pou
t
Input-drive power Pin
T1
T2 J1
J2
Out
put
pow
er
Pou
t
Input-drive power Pin
Out
put
pow
er
Pou
t
Input-drive power Pin
![Page 19: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/19.jpg)
Lee Center Workshop, May 19, 2006 19
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
)
Input-drive power Pin (W)
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.
![Page 20: 1 Nonlinear Dynamics and Stability of Power Amplifiers Sanggeun Jeon, Caltech Almudena Suárez, Univ. of Cantabria David Rutledge, Caltech May 19th, 2006.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bf771a28abf838c81761/html5/thumbnails/20.jpg)
Lee Center Workshop, May 19, 2006 20
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