# Techniques for the global stability analysis of microwave circuits

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```					Techniques for the global stability analysis
of microwave circuits.
Application to novel circuit design.
Almudena Suárez

Dpto. Ingeniería de Comunicaciones
OUTLINE
1 - Introduction

2 - Nonlinear dynamics of analog circuits
• Geometrical approach: the phase space
• Local stability
• Global stability. Bifurcations

3 - Simulation techniques
• Comparison
• Stability analysis
• Bifurcation analysis. Use of auxiliary generators

4 - Novel circuit design

5 - Conclusions
1 - INTRODUCTION

Nonlinear circuits: nonlinear differential equations

• It can include autonomous and subharmonic frequencies
• DC, periodic, quasi-peridic, chaotic

Coexistence of solutions:    Same or different type
Stable (or physical) and unstable
When using frequency-domain analysis (intrinsecally forced):

Measured solution may be very different from the simulated one

1
Mag VGS (V)

0.8

0.6

0.4

0.2

0
0   10      20   30   40   50

Frequency (GHz)
Objective:

• Techniques for the global stability analysis of nonlinear circuits

Stability analysis under the variation of the circuit parameters

Applications:

• Accurate determination of stable-operation borders

• Investigation of complex behavior

• Advanced and novel circuit design. Practical examples
2 - NONLINEAR DYNAMICS OF ANALOG CIRCUITS
2.1- GEOMETRICAL APPROACH: THE PHASE SPACE
Current (A)                                                                                                  o
0.8

0.6      Limit cycle                                                                                                  in m
in                                                           
o
0.4
n
0.2

0
EP
-0.2

Voltage Drain-Source2
-0.4

-0.6                                    Transient
-0.8
-2   -1.5   -1   -0.5    0   0.5     1   1.5    2

Voltage (v)
DC solution EP                                                                      Voltage Gate-Source2   Voltage Gate-Source1

Periodic solution LC                                     Quasi-periodic solution: Limit torus
Chaotic solution                                                                                         3

2

• Sensitivity to initial conditions                                                                      1

Voltage (v)
0

• Continuous spectrum                                                                                    -1

-2

• Fractal dimension                                                                                      -3
0   0.5     1   1.5      2
Time (s)
2.5       3   3.5     4     4.5
-3
x 10

6

4                                                                 -10

-20
Time derivative (v/s)

2

Power (dBm)
-30
0

-40

-2
-50

-4
-60

-6
-4   -3   -2   -1        0        1   2   3   4                               0       0.5       1            1.5         2         2.5          3
Voltage (v)                                                                       Frequency (Hz)                          x 10
4
2.2 - LOCAL STABILITY OF STEADY-STATE SOLUTIONS

Through small amplitude perturbations:                      (t)
Transversal surface
Current (A)
0.8
0.2

0.15
0.6                                                 
0.4
0.1
0.2
0.05
0
x o x
0
EP                                                          o              

xo
-0.05
xo           -0.2

-0.4
-0.1

-0.6
-0.15

-0.2                                           -0.8
-0.4   -0.2   0    0.2   0.4    0.6         -2   -1.5   -1   -0.5   0    0.5      1   1.5   2     2.5

Voltage (v)

Stable EP                                             Stable LC
Stability types:          Equilibrium point or
intersection point with a transversal surface 
(limit cycle in phase space)

U
S

Attractor                                               Repellor
Very dangerous non-observable point
Coexistence of stable and unstable steady-state solutions :

Stable and unstable solutions
alternate

Different basins of attraction
2.3 - GLOBAL STABILITY
Input generator
Variations in one or more circuit parameters  :
linear element value
Bifurcation
• Qualitative variation of the solution stability under continuous
modification of 

• Creation / destruction of solutions                           stable LC

Example Onset of a free-running oscillation
unstable EP
Vbif                        Vbias
EP          Limit cycle
stable EP
3 - SIMULATION TECHNIQUES
3.1 - COMPARISON
• Time-domain integration                                                • Harmonic balance
Natural circuit evolution (Physical) Transient is avoided
Continuation techniques

x 10
-3                                            Drawbacks:
4

2
Difficulty in autonomous and
I3 ( A )

0
subharmonic regimes
-2

-4
5
4
Convergence to unstable solutions
0

-2
0
2
(stability analysis required)
-5   -4
V2 (v)                  V1 (v)

Limited to periodic and
Small basin of attraction                                              quasi-periodic solutions
3.2 - STABILITY ANALYSIS

External Generator
H(X) 0
Zgen
Nonlinear

f in
microwave
x p (t ) =  Xm e   j( m in ) t
circuit                       m

• Perturbation:          e ( + j  t)

• Linearization:        JHK in    j  X  0
• Determinant:        det JHK in    j   0
3.3 - BIFURCATION ANALYSIS
b
Local bifurcation conditions                 x                    

det  JH ( jkin  jb )  0                                     
d                                                - b
 0                               x                    
d 
b                                         b

b 
m
in    Hopf                            Det complex,         b, b
n
                                          Det real,   b
b    in         I   or   Flip
2
b    0          Turning point / Pitchfork       Det real,   b
Bifurcation diagrams
Pout                      N
• Turning points b= 0                                              SN2
S
Jump and hysteresis                                 SN1

• Branching - type bifurcations
N

Pout
Hopf
in
Flip                                       U
S                         i
S             n
in
o ou     2

b                         
Global bifurcations
• Mode-locking at saddle-node : Synchronization / synchronization loss

Bifurcation diagram                   Poincaré map

N                  S

Pout           N

SN2              SN1

S                                        SN


Autonomous and subharmonic solutions:

X  Ag  G
o                                Leads to trivial mathematical solutions

Auxiliary Generator                 • 2 more unknowns:
f AG filter
A AG , f AG Autonomous
Nonlinear                                         

microwave                 (A AG , f AG , AG )    A , 
circuit                                            AG AG Synchronized

Y=0                                   Yr  0

• 2 more equations:         
Y  0
 i
• Stability analysis of steady-state regimes

External Generator                   Auxiliary Generator

Zgen
Nonlinear
microwave                         ,o 
f in        circuit            o filter

Y(o )
Re Y (o )  0
Im Y (o )  0 Combined with zero-pole identification techniques
• Continuation techniques                             Auxiliary Generator

Nonlinear
microwave                                A, o 
o filter
circuit

A n 1  A n                                           YA, o   0
1 Y
 n 1    n   JY n       
 o   o                n

 Yr   Yr 
 A    o                     HB
AG jacobian matrix:
JY    Y        
Yi                  Inner tier
 i
 A
       o 


Comparison of increments      ,     A, o           Parameter switching
Continuation technique                                              Application to VCO

Y=0                   Optimization goal

20
YT (A,)=0
15
Output Power (dBm)

10
5
Simulation                                                                         15,5
0
15,3
Measure
-5
Simulation                      YT( v)=0
,

Frequency (GHz)
15,1
-10                                              Measure
14,9
-15                                                                   14,7
YT ( v)=0
,                           YT ( v)=0
,
-20                                                                   14,5
YT (,v)=0
-25                                                                   14,3
-2   -1,5        -1         -0,5         0    0,5         1                                              14,1
Negative Varactor Bias (V)                                                                   13,9
YT (A, )=0
13,7
13,5
-2    -1,5        -1   -0,5          0      0,5          1
Negative Varactor Bias (V)
• Simple bifurcation conditions
 Yr   Yr 
 A    o 
AG jacobian matrix:    JY    Y        
Yi 
 i
 A
       o 

A n 1  A n            Y
 n 1    n   JY 1      
 o   o 
n
 n

Sadde-node bifurcation locus            det [JY]  0
2
 det (A, o , 1 , 2 )  0

 Yr (A, o , 1 , 2 )  0
 Y ( A,  ,  ,  )  0
 i       o    1    2

1
Pout                                    Hopf-type bifurcation locus

in                A  , 0
U
S                     in          Yr (o , 1 , 2 )  0

Yi (o , 1 , 2 )  0
S
in
o ou 2

b                      Flip- type bifurcation locus

New branch starts at zero amplitude                          g
A ,        o 
2
Yr (, 1 , 2 )  0

 Yi (, 1 , 2 )  0
Bifurcation loci   Frequency divider 28-14 GHz
Flip A o   
 
 2 
15                fin
Divider                fin
Input    10
power                                            f in
(dBm)         5                                  2
Synchro
0            (Arnold Tongue)                                          Hopf
Divider
detJY  0                                              Aa   
-5                                      f in
2
-10
fin, fo                               fin, fo

-15

23        25           27             29   31            33

Input frequency            (GHz)
S : Synchronization
Bifurcation diagram
T : Turning point (jump)

fin/2            fo
fo
Output power (dBm)

fin
fin

Input frequency (GHz)
4 - NOVEL CIRCUIT DESIGN
4.1 - SELF-OSCILLATING MIXERS

Input network
T1
Optimization
L       AG                              IF filter
l/4
l/4                                                        Zo
Ideal filter
Zo                            s.c. at f o
o.c. at f f o
lfb
fin                                 (Vo,fo)   Series Feedback

Fixed self-oscillation frequency and amplitude
Optimization of line length and width
Shift of inverse-Hopf bifurcation

YT(1, 2) = 0 with Vo = 10-2 V, fo = 5 GHz, Pin = PkinH       (Inverse Hopf)

YT(1', 2') = 0 with Vo = 0.8 V, fo = 5 GHz, Pin = 0 W    (Free-running oscillator)
4.2 - ANALOG FREQUENCY DIVIDER BY VARIABLE ORDER 6 TO 9

Arnold tongues of very nonlinear oscillators

Colpitts-type oscillator in very nonlinear regime
R1
Vcc

Vbb                          L1

Rdc
Q1
Lf         Rb               C1

Out
Cdc
Re    C2

Vin
Input Generator Voltage (V)   0.8
Rb=62 Ohm
Data              Simulation
Rb=10 Ohm
0.7

0.6

0.5

0.4

0.3
Arnold tongues
0.2

0.1
:7                :8                :9           :10
0
1.5   1.6     1.7   1.8        1.9    2      2.1   2.2   2.3   2.4   2.5
Input Frequency (GHz)

sequence
Experimental results

Division by 7, fin= 1.534 GHz (C1=2.9 pF)

Near-synchronization spectrum

Division by 8, fin= 1.534 GHz (C1=4.7 pF)
Conclusions
Nonlinear microwave circuits          Coexistence of solutions
Bifurcations
Bifurcations delimit operation borders. They precede chaotic behavior

Continuation techniques         Easily combined with harmonic balance

Difficulty with autonomous and synchronized regimes:
solved through auxiliary generators
Auxiliary generators enable simple techniques for bifurcation detection
Novel circuit design:
- Fixing of oscillation point
- Fixing of bifurcation point (SOM)
- Use of Arnold tongues and period-adding routes to chaos
Variable-order frequency dividers

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