ITSS'2007 – Pforzheim, July 7th-14th
Stability of Nonlinear Circuits
Giorgio Leuzzi
University of L'Aquila - Italy
ITSS'2007 – Pforzheim, July 7th-14th
Motivation
Definition of stability criteria and design rules
for the design of stable or intentionally unstable nonlinear circuits
(power amplifiers)
(frequency dividers)
under large-signal operations
Standard criteria are valid only under small-signal operations
ITSS'2007 – Pforzheim, July 7th-14th
Outline
Linear stability – a reminder:
Linearisation of a nonlinear (active) device
Stability criterion for N-port networks
Nonlinear stability – an introduction:
Dynamic linearisation of a nonlinear (active) device
The conversion matrix
Extension of the Stability criterion
Examples and perspectives
Frequency dividers
Chaos
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
A nonlinear device can be linearised around a static bias point
Example: a diode
Vo
I
I ( t )  I 0  g  V  t   V 0 
I0
Rs
v t 
v (t)
g
i( t ) 
Vth
1
Vo
g
 Rs
V
V0
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
The stability of the small-signal
circuit is easily assessed
Rs
s
d
s
d
Vs
v (t)
g
Vth
Vo
Oscillation condition:
s  d  1
d  1
stable
d  1
potentially unstable (negative resistance)
s  1
(passive)
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Oscillation condition
 s   d  a 
a
d
s
d a
Oscillation condition:
 s   d   a  a
s  d  1
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Example: tunnel diode
C
I
L
1
2
Rs
Vo
V
s
C
1
Rs
d
L
2
-g
Vth
Vo
Oscillation possible
d  1
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Example: tunnel diode oscillator
C
s
d
L
1
2
Rs
-g
Vth
Vo
Oscillation condition:
s 
s  d  1
f0 
1
2
LC
1
d
Rs 
1
g
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
2
L
Stability of a two-port network:
transistor amplifier
2
L
Vo
0
1
1
Rs
v (t)
RL
0
s
Vs
 in
 out
L
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Stability of a two-port network
s
 in
 out
L
Vs
Oscillation condition:
 s   in  1
 in  1
stable
 in  1
potentially unstable (negative resistance)
s  1
(passive)
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Stability of a two-port network
s
 in
 out
L
Vs
 in  1
Stability condition
 in  S 11 
S 12  S 21   L
1  S 22   L
K - stability factor:
k 
1  S 11
2
 S 22
2
 
2 S 12  S 21
  S 11  S 22  S 12  S 21
2
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Stability of a two-port network
s
 in
 out
L
Vs
 in  f  L 
k  1
k  1
(stability circle)
L
L
 in  1
stable
 in  1
 in  1
potentially
unstable
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
k  1
Intentional instability: oscillator
s
 in
 out
 in
 in  1
L
s 
1
 in
oscillation condition
L
ITSS'2007 – Pforzheim, July 7th-14th
Linear stability
Stability of an N-port network
 in  f   L 1 ,  L 2 ,  L 3 ,  L 4 ,  L 5   1
 in
s
 L1
 L2
 L3
 L4
 L5
No stability
factor
available!
ITSS'2007 – Pforzheim, July 7th-14th
Outline
Linear stability – a reminder:
Linearisation of a nonlinear (active) device
Stability criterion for N-port networks
Nonlinear stability – an introduction:
Dynamic linearisation of a nonlinear (active) device
The conversion matrix
Extension of the Stability criterion
Examples and perspectives
Frequency dividers
Chaos
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
A nonlinear device can be linearised around a dynamic bias point
Example: a diode driven by a large signal
V _ L S (t )
I
I ( t )  I LS  t   g  t   V  t   V LS  t 
I LS (t)
Rs
V
i( t ) 
v t 
Rs 
V LS (t)
v (t )
g (t )
1
g t 
Vth
V _ L S (t )
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
The large signal is usually periodic (example: Local Oscillator)
The time-varying conductance is also periodic
g(t)
I
g t  

n
Gn  e
jn  LS  t
Rs
v (t )
g (t )
Vth
I LS (t)
V _ L S (t )
V
t
V LS (t)
Small-signal linear timedependent (periodic) circuit
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Example: switched-diode mixer
V _ L S (t )
The diode is switched periodically on and off
by the large-signal Local Oscillator
Rs
v (t)
RL
g (t)
g (t)
Rs
v (t)
RL
t
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Spectrum of the signals in a mixer
Red lines: large-signal (Local Oscillator) circuit
Blue lines: small-signal linear
time-dependent circuit
fLS
Rs
2fLS
v (t)
fs
DC
fLS-fs
fLS+fs
V _ L S (t )
2fLS-fs
2fLS+fs
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Linear representation of a time-dependent linear network (mixer)
Input signal
Frequency-converting element
(diode)
fs
fLS+ fs
2 fLS+ fs
fLS-fs
Conversion
matrix
2 fLS-fs
3 fLS-fs
Passive loads at converted
frequencies
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Stability of the N-port linear time-dependent
frequency-converting network (linearised mixer)
fs
fLS+ fs
2 fLS+ fs
fLS-fs
Conversion
matrix
2 fLS-fs
3 fLS-fs
…can be treated as any linear N-port network!
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
One-port stability - the input
reflection coefficient can be:
 in  1
stable
 in  1
potentially unstable
(negative resistance)
 in
fs
fLS+ fs
2 fLS+ fs
fLS-fs
Conversion
matrix
Instability at fs frequency
2 fLS-fs
s
3 fLS-fs
 in
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Important remark: the conversion phenomenon, and therefore
the Conversion matrix, depend on the Large-Signal amplitude
g(t)
I
The stability depends
on the large-signal
amplitude (power)
I LS (t)
V
t
fs
V LS (t)
fLS+ fs
2 fLS+ fs
fLS-fs
Conversion
matrix
2 fLS-fs
3 fLS-fs
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Instability at small-signal and converted frequencies
A large signal is applied
fLS
2fLS
fs
DC
fLS-fs
|in| >1
A spurious signal appears at
a small-signal frequency and
all converted frequencies
fLS+fs
2fLS-fs
2fLS+fs
Nonlinear stability
Instability in a power amplifier
Bifurcation diagram
mathematical
Pout
Pout(fLS)
real
|in| < 
|in| > 
Pout(fs)
The amplifier is stable
in linear conditions
PI
Pin(fLS
)
Nonlinear stability
Design procedure – one port (1)
First step: Harmonic Balance analysis at n•f0
S(nf0)
f0
L(nf0)
Pin(f0)
2f0
DC
Z0
Second step: Conversion matrix at fs and converted frequencies
f0
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
2f0
DC
fs
Nonlinear stability
Design procedure – one port (2)
Third step: Conversion matrix reduction to a one-port
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
S(fs)
in(fs)
fs
stable
 in  1
potentially unstable
2f0
design choice
Fourth step: verification of the stability at fs
 in  1
f0
s 
1
 in
Oscillation condition
yes/no
Nonlinear stability
Design procedure – two port (1): same as for one port
First step: Harmonic Balance analysis at n•f0
S(nf0)
f0
L(nf0)
Pin(f0)
2f0
DC
Z0
Second step: Conversion matrix at fs and converted frequencies
f0
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
2f0
DC
fs
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Design procedure – two port (2)
f0
fs
2f0
Third step: Conversion matrix reduction to a two-port
f0+fs
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
S(fs)
L(f0+fs)
Fourth step: verification of the stability of the two-port
k  1
k  1
stable
potentially unstable
L
yes/no
s 
1
 in
Oscillation condition
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Design procedure – N port (1): same as for one and two port
First step: Harmonic Balance analysis at n•f0
S(nf0)
f0
L(nf0)
Pin(f0)
2f0
DC
Z0
Second step: Conversion matrix at fs and converted frequencies
f0
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
2f0
DC
fs
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Design procedure – N port (2)
Third step: Conversion matrix reduction to a one-port
fs
f0+fs
f0-fs
2f0+fs
2f0-fs
3f0+fs
S(fs)
in(fs)
…and simultaneous optimisation of all the loads at converted frequencies until:
 in  1
stable
 in  1
s 
1
 in
intentionally unstable
(maybe)
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Design procedure – important remark
Loads at small-signal and converted frequencies
are designed for stability/intentional instability
Loads at fundamental frequency and
harmonics must not be changed!
fLS
2fLS
fs
DC
fLS-fs
…otherwise the Conversion
matrix changes as well.
fLS+fs
2fLS-fs
2fLS+fs
This is not easy from
a network-synthesis
point of view
ITSS'2007 – Pforzheim, July 7th-14th
Nonlinear stability
Design problem: commercial software
Currently, no commercial CAD software allows
easy implementation of the design scheme
SUBCKT
ID=S1
NET="S_conv_matrix_real_imag_1_dBm"
A relatively straightforward procedure has
been set up in Microwave Office (AWR)
0.8
S(2 ,2 )
i n p ut_n e two rk _2 f0
1.0
Smith_loads
S(1 ,1 )
o utpu t_ n etwork
PORTF
P=13
Z=50 Ohm
Freq=fs GHz
Pwr=Ps dBm
Swp Max
1.728GHz
6
0.
S(2 ,2 )
i n p ut_n e two rk _3 f0
PORT
P=1
Z=50 Ohm
2.
0
S(1 ,1 )
o utpu t_ n etwork _ 2 f0
S(1 ,1 )
o utpu t_ n etwork _ 3 f0
S(2 ,2 )
d es i g n ed _ i n p ut_n e twork
0.
4
S(2 ,2 )
d es i g n ed _ i n p ut_n e twork _ 2 f0
PORT
P=14
Z=50 Ohm
PORT
P=15
Z=50 Ohm
PORT
P=18
Z=50 Ohm
PORT
P=16
Z=50 Ohm
PORT
P=17
Z=50 Ohm
PORT
P=10
Z=50 Ohm
PORT
P=11
Z=50 Ohm
PORT
P=12
Z=50 Ohm
S(2 ,2 )
i n p ut_n e two rk
0
3.
S(2 ,2 )
d es i g n ed _ i n p ut_n e twork _ 3 f0
0
4.
S(1 ,1 )
d es i g n ed _ o utpu t_ n etwo rk
12
11
10
1
9
2
8
3
7
PORT
P=9
Z=50 Ohm
5 .0
S(1 ,1 )
d es i g n ed _ o utpu t_ n etwo rk _ 2f0
0 .2
S(1 ,1 )
d es i g n ed _ o utpu t_ n etwo rk _ 3f0
PORT
P=3
Z=50 Ohm
PORT
P=2
Z=50 Ohm
2
PORT
P=5
Z=50 Ohm
2
f0<f<2*f0
0<f<f0
2*f0<f<3*f0
PORT
P=4
Z=50 Ohm
2
f0<f<2*f0
2
2
0<f<f0
4
4
SUBCKT
ID=S3
NET="output_couplers" 3
5
1
PORT1
P=19
Z=50 Ohm
Pwr=Pin dBm
2
1
P_METER3
ID=P1
2
W
1
2
SUBCKT
ID=S4
NET="input_network"
1
2
SUBCKT
ID=S2
NET="biased_transistor"
1
2
SUBCKT
ID=S5
NET="output_network"
I 2
1
V
3
RES
ID=R1
R=50 Ohm
PORT
P=3
Z=50 Ohm
.0
-2
-1.0
-0.8
-0
.6
.4
-0
SUBCKT
ID=S1
5 NET="input_couplers" 3
10.0
4.0
5.0
3.0
2.0
1.0
0.8
S(2 ,2 )
l i ne a r_ de s i g n ed _ am p l i fi e r
0.6
DPWRSMP
ID=U6
R=50
PORT
P=2
Z=50 Ohm
PORT
P=8
Z=50 Ohm
4
- 0.
2*f0<f<3*f0
10.0
S(1 ,1 )
l i ne a r_ de s i g n ed _ am p l i fi e r
S(2 ,2 )
l i ne a r_ am p l i fi er
DPWRSMP
ID=U5
R=50
4
PORT
P=6
Z=50 Ohm
S(1 ,1 )
l i ne a r_ am p l i fi er
1
0<f<f0
3
DPWRSMP
ID=U4
R=50
4
PORT
P=1
Z=50 Ohm
PORT
P=10
Z=50 Ohm
0.4
3
0.2
PORT
1
P=11
f0<f<2*f0
Z=50 Ohm
3
DPWRSMP
ID=U1
R=50
4
2
PORT
1
P=12
2*f0<f<3*f0
Z=50 Ohm
0
DPWRSMP
ID=U2
R=50
4
2
0<f<f0
1
3
DPWRSMP
ID=U3
R=50
4
PORT
P=7
Z=50 Ohm
4
5
6
PORT
P=7
Z=50 Ohm
-3
.0
3
-4
.0
- 5.
0
PORT
1
P=8
f0<f<2*f0
Z=50 Ohm
3
-10. 0
PORT
1
P=9
2*f0<f<3*f0
Z=50 Ohm
Swp Min
0.576GHz
PORT
P=4
Z=50 Ohm
PORT
P=5
Z=50 Ohm
PORT
P=6
Z=50 Ohm
It is advisable that commercial Companies
make the Conversion matrix and multifrequency design available to the user
ITSS'2007 – Pforzheim, July 7th-14th
Outline
Linear stability – a reminder:
Linearisation of a nonlinear (active) device
Stability criterion for N-port networks
Nonlinear stability – an introduction:
Dynamic linearisation of a nonlinear (active) device
The conversion matrix
Extension of the Stability criterion
Examples and perspectives
Frequency dividers
Chaos
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Frequency divider-by-three based on a 3 GHz FET amplifier
Harmonic Balance analysis
of a 3-GHz stable amplifier
Remark: a Harmonic Balance analysis will not
detect an instability at a spurious frequency,
not a priori included in the signal spectrum!
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Frequency divider-by-three based on a 3 GHz FET amplifier
Spectra for increasing input power
of the stable 3-GHz amplifier
Spectra from time-domain analysis
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Frequency divider-by-three based on a 3 GHz FET amplifier
Spectra for increasing input
power of the modified amplifier
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Frequency divider-by-two at 100-MHz
1
Rout @ 50 MHz
2
2
F RE Q = 10 0 M H z
1
0
Rout < 0
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Frequency divider-by-two at 100-MHz
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Chaotic behaviour
Bifurcation diagram
id
For increasing amplitude of
the input signal, many
different frequencies appear
Vs
ITSS'2007 – Pforzheim, July 7th-14th
Examples
Chaotic behaviour
The spectrum becomes dense
with spurious frequencies, and
the waveform becomes 'chaotic'
ITSS'2007 – Pforzheim, July 7th-14th
Conclusions
Nonlinear stability:
The approach based on the dynamic linearisation of a
nonlinear (active) device is a natural extension of the
linear stability approach
Can be studied by means of the well-known Conversion
matrix
Design criteria are available, even though not yet
implemented in commercial software
Stability criterion for N-port networks still missing