Date post: | 17-Jan-2016 |
Category: |
Documents |
Upload: | clifford-morgan |
View: | 216 times |
Download: | 0 times |
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
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
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
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)
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
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.
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
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.
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.
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
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
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
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
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
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
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
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
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.
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