Thermal Noise in Nonlinear
Devices and Circuits
Wolfgang Mathis and Jan Bremer
Institute of Theoretical Electrical Engineering (TET)
Faculty of Electrical Engineering und Computer Science
University of Hannover
Germany
Content
1.
Deterministic Circuit Descriptions
2.
Stochastic Circuit Descriptions
3.
Mesoscopic Approaches
4.
Steps in Noise Analysis in Design Automation
5. Bifurcation in Deterministic Circuits
6. Bifurcation in Noisy Circuits and Systems
7. Examples
8. Conclusions
1.
Deterministic Circuit Descriptions
2.
Stochastic Circuit Descriptions
3.
Mesoscopic Approaches
4.
Steps in Noise Analysis in Design Automation
5.
Bifurcation in Deterministic Circuits
6.
Bifurcation in Noisy Circuits and Systems
7.
Examples
8.
Noise Analysis of Phase Locked Loops (PLL)
9.
Conclusions
1. Deterministic Circuit Descriptions
A. Meissner, 1913
Bob Pease, National Semiconductors
Electrical and Electronic Circuits: The Ohm-Kirchhoff-Approach
Real Circuit
Circuit-Model
Modelling
b
Partitioning
b
Models
for Electronic Circuits
O
Electrical Circuits are defined in
Z := (i, u ) : (i1 ,..., ib ; u1 ,..., ub )  Rb  R b 
Space of all currents and Voltages
Description of resistive NW elements:


O := (i, u )  Rb  Rb FR (i, u)  0, FR : Rb  Rb  R k ; k  b
Ohm Space
b
Description of connections:
b
K
K := (i, u )  Rb  Rb Ai  0 B u  0, ( A, B) exakt
Kirchhoff Space
S := K O
State-Space
of Electrical Circuits
Dynamics electronic Circuits (Networks)
S
DAE System:
dx
 f ( x, y )
dt
0  g ( x, y )
Capacitors
Inductors
Differentialalgebraic System
DAE Systems consist of many
e.g. (100-) thousand Equations
numerical solutions necessary!
Special Cases: State-Space Equations (ODEs)
dx
 F ( x)
dt
Deterministic Description: ODEs
Deterministic Description: ODEs
d xi
 Fi ( x ) ,
dt
x(t0 )  x0
where
Fi : R n  R (i  1,
, n) and
x  Rn
Initial value problems suitable
for studying
the quantitative behavior!
x(t0 )  x0
Are initial value problems suitable for studying the qualitative behavior?
Reformulation of the deterministic Dynamics
Qualitative behavior:
Considering a whole family of systems
 dxi

 Fi ( x), x0  x(t0 )  S  IR

 dt

Density Function p
Dynamics of a Density Function
p (Frobenius-Perron-Operator Pt ):
p( x, t )  Pt ( p( x))
where
n
  p Fi 
p
 
t
xi
i 1
generalized Liouville equation
p( x)
Set of Initial Values
2. Stochastic Circuit Descriptions
Thermal Noise in linear and
nonlinear electrical Circuits
with noise sources
Noise Model
Deterministic Approach
Circuits
Network Thermodynamics
Circuit
Equations
Generalized
Liouville
Equation
Generalization: (Non)linear Circuits including noise
Deterministic
Circuits
Noisy
Circuits
Device
Modelling
Macroscopic
Approach
Mesoscopic
Approaches
Microscopic
Approach
Microscopic Approach: Statistical Physics
e.g.
RecombinationGenerationNoise
drift movement
Multi-Body System (approx. 1023 particles)
C. Jungemann (see his talk this morning)
3. Mesoscopic Approaches
The Langevin Approach:
Stochastic ODE
(SODE):
dxi
 Fi ( x)   ( x) i
dt
Fi : R n  R (i  1,
, n) and
Noise
sources
as inputs
x  Rn
i white noise and  ( x) coupling coefficient
Dynamics of the Density Function p: (mathematical equivalent to SODE )
n
 ( pFi )
p
 
t
xi
i 1

1 n  2 ( 2 p)

2 i , j 1 xi x j
Fokker-Planck equation
Remarks: Fokker-Planck equation as modified generalized
Liouville equation
Langevin’s Approach:
Noisy input
Deterministic
Circuit
output
(without inputs)
Applications:
e.g in Communication Systems
Transmission of noisy signals through a deterministic channel
(Mathematics: Transformation of stochastic processes)
Physical Interpretation of SODE (Langevin, 1908)
a) Linear Case (i  white noise,   konst.)
dxi
  Aij x j
dt
j
Average:
dxi

dt
dxi
  Aij x j   i
dt
j
stoch.
d xi
dt

A x
ij
j
j
 i   Aij x j   i
=0
j
Conclusion: First Moment satisfies a determinstic differential equation
b) Nonlinear Case (i  white noise,   konst.)
dxi
 Fi ( x)
dt
Average:
dxi
dt
stoch.
(van Kampen, 1961)
dxi
 Fi ( x)   i
dt
=0
d xi

 Fj ( x)  i  Fj ( x)   i
dt
 Fj ( x )
Coupling of
Moments
Compare: In nonlinear systems
Deterministic nonlinear System:
Energetic Coupling of
Frequencies
Stochastic nonlinear System:
Coupling of
Moments
of the probility density
Sinusoidal input
Alternative: Analyzing nonlinear circuits including noise
Extraction of Noise Sources (then using the Langevin approach)
Methods:
• Calculation of desired spectra
• Numerical Methods in Stochastic Differential Equations
• Geometric Analysis of Stochastic Differential Equations
Numerous papers
However: Brillouin‘s Paradoxon of nonlinear electrical Circuits
White noise
White noise
Stochastic Equation

 qu  
C du   I s  exp    1 dt  q(2I s  I) dw
 kT  

PN-Diode
Average: (first moment)
2
 q

d
1
q


2

C  u    Is
 u      u 


dt
kT
2
kT




: u  q
Thermodynamic Equilibrium:
eq
2C
Fluctuations-Voltage-Converter
„A diode can rectify
its own noise“
Contradiction against the second law of thermodynamics
1. Motivation
5
(white noise sources in device models are forbidden: …., Weiss, Mathis, Coram, Wyatt (MIT))
Nonlinear Electronic Circuits
Electrical current is related to noisy electron transport
 internal noise (cannot switched off)
 in nonlinear systems (electrical circuits)
Drift Movement
No systematic extraction of deterministic equations
The entire behavior has to be described as a stochastic process
phys.
Assumption: description as a Markov process
Mesoscopic Approach based on statistical thermodynamics
“First Principle” Mesoscopic Approach for Circuits
with Internal Noise
Starting Point: Markovian Stochastic Processes x(t ,  ) are defined
by the Chapman-Kolmogorov Equation
(Integral equation for the transition probability density)
Types of Markov Processes
time domain
Stochastic differential
Equation (SODE)
probability density domain
mathematical
equivalent!
Fokker-Planck
equation
more general partial differential
equations for the
probability density domain
General solutions of the Chapman-Kolmogorov equation by the Kramers-Moyal series
Derivation of the Kramers-Moyal Coefficients for nonlinear systems
by Nonlinear Nonequilibrium Statistical Thermodynamics (Stratonovich):
„Thermal noise“:
In thermodynamical equilibrium
(
(stable)
)
the equilibrium density function is known:
peq  e
W X
kT
Perturbation analysis for calculating coefficients
„Irreversible Statistical Thermodynamics of Circuits“
Weiss; Mathis (1995-2001), Dissertation (Weiss) 1999
However: Restricted to reciprocal circuits (no transistors!)
Statistical Thermodynamics of Thermal Noise
in Nonlinear Circuit Theory
Using Stratonovich‘s Approach: Basic is the Markov Assumption
Kramers-Moyal Equation
p(X, t )  (1) m

t
m1 m!
m
K 1m (X) p(X, t )

1 m X 1  X  m
determination
?
 Circuit Equations
d
X   K  (X)  o(kT)
dt
 Thermodynamic Equilibrium
p eq (X)  exp(  WX / kT)
 Detailed Balance (Reciprocity)

Nonlinear „distributed“ Generalization
of the Nyquist’s Formula
Nonlinear Circuits (Weiss und Mathis (1995-1999))
Starting Point:
3.1 Anwendung: Vollständige nichtlineare Netzwerke
Complete Reciprocal Circuits: Brayton-Moser Description
Network Equations
L  (I  )
dI 
dt

M(I, U)
I 
dU 
M(I, U)

dt
U 
Mixed Potential M: Dissipation
C  (U  )
- Reciprocal
- DAE Description (Index 1)
M(I,U) = F(I) - G(U) + (I,U)
Linear Approximation:
X 
   (0) Y
,
Quadratic Approximation:
X 
   (0) Y
,
2
2
3
 1/ 2
  (0) Y Y


,
not of
Fokker-Planck
type
Cubic Approximation:
X 
   (0) Y
,

1
2
1
  (0) Y Y     (0) Y Y Y

6 

,
Noise cannot be determined thermodynamical!
,
Our Approach of Noise Spectra Calculations
Physical
Assumptions
Current-Voltage
Relation
Circuit Topology
Stratonovich
Machine
Correct Noise
Spectra
(if the physical
assumptions valid)
Note: Assumptions are not satisfied if non-thermal effects are included
(hot electron effects)
The “Thermodynamic Window”
of a Circuit
Microscopic
Behavior
Currents and Voltages
Linear RC Networks: Classical Result
Stochastic Diff.Equ. (NoiseSource)
C du   K(u ) dt 
Signal
S I dw
Noise
equivalent SODE
 Fokker-PlanckEquation (distributed Noise)
 K ( U)
2
 p( U, t )
SI
1
p( U, t )


p( U, t )
2
2
t
U C
2 U C
 Network Equation
our approach
K(U) = - U / R
 Thermodynamic Equilibrium
p eq ( U)  exp( WC / kT)
Nyquist‘s Formular (linear approximation)
S I  2kTG
Dissipation
(G  1 / R)
Fluctuation
Some Results: Nonlinear RC Network (Linear-quadratic Approximation)
3.2 Nichtlineares RC Netzwerk: Linear-quadratische Näherung
Network ODE
dU
dg(0)
1 d 2 g ( 0) 2
C

U
U
2
dt
dU
2 dU
our approach
(equivalent SODE)
 dg(0)
1 d 2g (0)
SODE C du   
u
du
2 du 2

 dg(0) 1 d 2g (0)
 2 kT  
u  C   dt  2kT  du  2 du 2



u  dw

u 2  kT / C that is Short Circuit
Generalized Nyquist Formula
 dg(0) 1 d 2 g (0) 
 SI  2kT 

U 
2
2 dU
 dU

Results about Thermal +Noise in Semiconductor Devices
(Weiss, Mathis: IEEE Electron Devices Letters 1999)
1. Schottky Diode
our
approach
Shot Noise!
Note: Shot noise has a thermal background
(see Schottky (1918))
2. MOSFET (simple model)
MOSFET:
1
2
I D  U GS  U t  U DS   U DS
2
our
approach
known from microscopic analysis (see textbooks):
 SI  2 kT  U GS  U t   1  U DS 

known from
S I, therm  2 kT U GS
2

2
U DS
U DS
1
1

U GS  U t 3 U GS  U t  2
 Ut 
1 U DS
1
2 U GS  U t

1


 2 kT  U GS  U t    U DS   O U DS2
2

 O (U
S I therm 

DS
2
)
FET:
3. JFET
3/ 2
3/ 2

2 U diff  U GS  U DS   U diff  U GS  

I D  g 0  U DS 
1/ 2

3
U0


our
approach
known from microscopic analysis (e.g. van der Ziel (1962):



 SI  2 kT g 0 1 
SI, therm  2 kT g 0
x
U diff  U GS
U DS

U0
4 U diff  U GS U 0

 





4 3/ 2
1
x  y3 / 2  x 2  y 2
3
2
,
2 3/ 2
3/ 2
xy x y
3
xy


U diff  U GS  U DS
U  U GS
, y  diff
U0
U0

U DS
1  U diff  U GS 
2
kT
g

S I Stherm

0
I , therm

U0
4 U diff  U GS U 0


  O (U 2 )
DS

 O (U
2
DS
)
4. Steps of Noise Analysis in Design Automation
• First Generation: LTI-Noise Models
Linear Noise Analysis based on Schottky-Johnson-Nyquist
(Rohrer, Meyer, Nagel: 1971 - …)
Idea: „Linearization with respect to an operational point (constant solution)“
State Space
Small-signal noise models do not work if e.g. bias changes occur,
oscillators, more general nonlinear circuits
• Second Generation: LPTV Models
Variational Linear Noise Analysis of Periodical Systems
(Hull, Meyer (1993), Hajimiri, Lee (1998))
Idea: „Linearization with respect to a periodic solution“
State Space
Useful for periodic driven systems, however heuristic assumptions
and concept will be needed for oscillators (Leeson‘s formula)
• Third Generation: SDAE Models
Noise Analysis by Stochastic Differential Algebraic Equations
(Kärtner (1990), Demir, Roychowdhury (2000))
DAE System:
dx
 f ( x, y )
dt
0  g ( x, y )
Differential-
algebraic System
+ Noise (stochastic processes)
dx
B( x)  g ( x)  f ( x)   (t )
dt
Systematic Results in Phase Jitter of Oscillators as well as
other nonlinear systems (e.g. PLL),
however the onset of oscillations cannot described
5. Bifurcation in Deterministic Circuits
Given:
Choice:
f osz , Cgs, Cds, RL, y22;
CG, CL (influence of Cgs and Cds „small“)
L
gm
RL
Linearization?
Cgs//CG
Cds//CL
FET Colpitts Oscillator
Theorem of Hartman-Grobman:
What is happened if the circuit is
non-hyperbolic?
1
0.8
0.6
The dynamical behavior of state space equations is
related to the dynamics of the „linearized“ equations
in hyperbolic cases.
Barkhausen or Nyquist Criteria
x
I
C
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
x
x
2
4
6
8
10
12
14
16
18
20
In certain cases Limit Cycles can be observed
Example: Sinusoidal Oscillators
damping term
x  ( x 2  x 2  1) x  x  0
Obvious solution:
x (t )  cos t
(or
x (t )  sin t )
State space interpretation:
x
Type of damping
positive
Periodic Solution
( x 2  x 2  1)  0
x
negative
Analysis of Systems with Limit Cycles
Idea: (Poincaré; Mandelstam, Papalexi - 1931)
Embedding of an oscillator (equation) into
a parametrized family of oscillator (equations)
Example: Van der Pol equation
  ( x 2  x 2  1) x  x  0
x
  ( x 2  x 2   ) x  x  0
x
embedding
with

Andronov-Hopf Bifurcation
x
Stable equilibrium
point
x
Limit Cycle
State Space
Bifurcation
Point

Poincaré-Andronov-Hopf Theorem (1934,1944)
Let
x  f ( x,  ),
with f
(0,  )  0
f : Rn  R  Rn
for all  in a neighborhood of 0. If
• the Jacobi matrix Dx f (0,0) includes a pair of
imaginary eigenvalues  1, 2   j
• the other eigenvalues have a negative real part
d
• d   1 ( )  0  0
• the equilibrium point for   0 asymptotic stable
Then there exist
• an asymptotic stable equilibrium point for (  1 ,0)
• a stable limit cycle for (0,  2 )
Transient Behavior of a Sinusoidal Oscillator
(Center Manifold Mc)
Concept for Analysis of Practical Oscillators
x  f ( x,  ),
f : IR n  IR  IR n
• Transformation of the linear part: Jordan Normal Form
• Transformation of the Equations: Center Manifold
• Transformation of the reduced Equations: Poincaré-Normal Form
• Averaging
Symbolic Analysis
(MATHEMATICA, MAPLE)
6. Bifurcation in Noisy Circuits and Systems
Different Concepts:
I) The physical (phenomenological) approach (e.g. van Kampen)
Special Case: Dynamics in a Potential U(x)
U(x)
U(x)
initial
P.D.F.
Behavior of P.D.F. p near
a stable equilibrium point
initial
P.D.F.
?
Behavior of P.D.F. p near
a unstable equilibrium point?
Dynamical Equation:
Fokker-Planck
stationary
dx
 U '( x)
dt
dxt   xt  xt3  dt   dWt
SODE
p 
2 p
 U '( x) p   2
t x
x
! 
2 p
0  U '( x) p   2
x
x
4
x
e.g. U ( x)   x 2 
2

pstat ( x)  C  e

U ( x)
  0 single well

 >0 double well
x4
x 
2
2
pstat ( x)  C  e

  0 unimodal

  0 bimodal

For
 0
the equilibrium P.D.F. p(x) changes its type
It is called P-bifurcation (e.g. L. Arnold)
Obvious disadvantage: (Zeeman, 1988)
“It seems a pity to have to represent a dynamical system by y static picture”*
II) The mathematical approach (e.g. L. Arnold)
Observation:**
One-one correspondence: stationary P.D.F. and invariant measures (I.M.)
Consider more general invariant measures (if exist)
D(ynamical)-Bifurcation Point  D
of a family of stochastic dynamics ( w.r.t. 
“Near”
D
we have another I.M. 
* Arnold, p. 473 ** Arnold, p. 473
   
 R)
   
D
with a ergodic I.M. 
with
Question: Is there any relationship between these types of bifurcation
In general, there is not!
Our example: (above)
The corresponding invariant measure to
 x2 
pstat ( x)  C  e

x4
2
  0 unimodal

  0 bimodal
There is no
D-Bifurcation
is unique*
Remark: There are cases with D-bifurcation but no P-bifurcation*
* Arnold, p. 476
Case 1: Pitchfork Bifurcation
dxt   xt  x  dt   xt dWt
3
t
Invariant Measures
q
q
  0
D
P
D  0
P 
q

2
2
q
Case 2: Andronov Bifurcation
0

dxt   3 2
  x1  x1 x

0
 dt  


1
 0 0
 xt dt   
 xt dWt

 1 0
invariant
measures
D1
D2
P
P  0
D
1
D
2
 0 unstable
 0 stable
 0 saddle
 1 stable
 1 stable
 2 saddle

Main Questions:
There are stochastic generalizations of geometric theorems
• Hartman-Grobman
Global H-G: Wanner, 1995
(local H-G: still open question)
• Poincaré-Andronov-Hopf
Arnold, Schenk-Hoppe
Namachchivaya 1996
• Center Manifold
Boxler 1989, Arnold, Kadei 1993
• Poincarè Normal Form
Elphick et al. 1985, C.+G. Nicolis
1986 (physics)
Namachchivaya et al. 1991
Arnold, Kedai 1993, 1995
Arnold, Imkeller 1997
Remark: Until now a research program (Arnold, ...)
7. Examples
Meissner oscillator and
van der Pol’s equation
Meissner‘s Tube Oscillator
k1 , k2 , k3
L0 , M
uBE
C
1
 
L0C
2
0
R
 duBE
d 2uBE  R
2
2
2



M
k

2
k
u

3
k
u





0
1
2 BE
3 BE 
0 u BE  0
2
dt
 L0
 dt
Normalization and Scaling:
( x1  uBE , x2  uBE )
1
 0

0

 x1  
d  x1  

     2  2 Mk  R      2
2
2
2

k
Mx
x

3

k
Mx
x
x
dt  x2   0
0
1
0 3
1 2 
 2   0 2 1 2
L
0 

Noisy frequency:
  02   
( :white noise) and k2  0
(using results from Arnold, Imkeller 1997)
 x1   0
d    
 x2   
1   x1   0  
 0 0
  x     x x 2   dt   
 x dWt
   2   1 2 
 1 0
where
 := 02 Mk1 
R
, 3 02 k3 M  1, Wt : Wiener process
L0
Linearization:
 x1   0
d 
 x2   
1   x1 
 0 0
  x  dt   
 x dWt
  2 
 1 0
Center Manifold: If   0   02 ! ,  < 4

d
:    2 / 4  R 
2-dim. Center space and no stable space
Normal Form in Resonant Case: (polar coordinates)
Solution of stochastic cohomology equation results
(see approach: Arnold, Imkeller, 1997)

Meissner oscillator with nonlinear capacitor
k1 , k2 , k3
L0 , M
uBE
C
q
 02 
1
L0C0
R
 duBE
d 2uBE  R
2
3
2
3



M
k

2
k
u

3
k
u


u


u



0
1
2 BE
3 BE 
0 BE
BE  0
2
dt
 L0
 dt
Duffing-van der Pol equation
noisy
case
 x1   0
d    
 x2   
0
1   x1  

 0 0
  x     x3  x x 2   dt   
 x dWt
   2   1 1 2 
 1 0
Normal Form in Resonant Case: (polar coordinates)
Solution of stochastic cohomology equation results
(results: Arnold, Imkeller, 1997)

 5rt 2

rt 
2rt 2
1 3

2
d rt   rt  rt  dt  2  2rt  
 4 d  sin 2t 
sin 4t  rt 2 cos 2t   dWt
2 
8 d 
d
2
 d



6  3 2 
1
d t    d 
rt  dt 
4 d
2 d




3rt 2 
5rt 2 
rt 2
3rt 2

1



1

cos
2


cos
4


sin
2



t
t
t 
2
2 
  dWt

2

2

2

d
d 
d
d



Ljapunov-Exponents:

PardouxWihstutz
formula
1988
2
1,2   ,     2  O  4 
2 8 d
 0
 0
 0
 0
stochastic
P- Andronov-Hopf
bifurcation
first stochastic
D-Andronov-Hopf
bifurcation
 0
Stability is lost
determinstic
Andronov-Hopf
bifurcation
2  0,    0  1  0,  
 0
8. Noise Analysis of Phase Locked Loops (PLL)
Phase Detector
s (t )
Lowpass Filter
F(s)
P
r (t )
VCO
Voltage Controlled Oscillator
Equivalent Base-band Modell:
 t 
 (t )
+
x(t )
e (t )
A sin(.)
K
ˆ t 

F(s)
F(s)=1
t
 d
0
d d 

 K  A  f (t   ) sin     d
dt
dt
0
t
d
d
  K  A sin   t   
,
dt
dt
State Space Equation
2nd order PLL with ideal Integrator
Equivalent Base-band Modell
System State Space Equation:
d  

 2 sin     x1   
d
n
d x1  
 sin   
d
Noisy state equation
Formally, one obtains:
noise
But:
How do we interpret this equation, if n’(t) is not
exactly known?
- need generic results
Noisy state equation
Necessary Assumption:
PLL-bandwidth is small compared with BW of the noise-process
- sufficient to model n’(t) as white noise again
- rewrite state equation as SDE
Normalized SDE
After time-normalization and introducing parameters
from linear PLL noise theory, one obtains:
Interpretation:
• SDE in the Stratonovich sense
• dw() = (t)d: increment of a normalized Wiener process
• BL: loop noise bandwidth,  : frequency offset between input
and VCO output, : SNR in the loop
The Euler-Maruyama scheme (1)
• based on Ito-Taylor expansion ) consistent with Ito-calculus
• Ito stochastic integrals ) evaluate Riemann sum approximation at
lower endpoint
Consider the scalar Ito-SDE
And the corresponding Euler-Maruyama scheme
The Euler-Maruyama scheme
Consistency with Ito-calculus
Noise term in the EM scheme approximates the Ito
stochastic integral over interval [tn, tn+1] by evaluating its
integrand at the lower end point of this interval, that is
Phase-acquisition time
• Time to reach locked state from an initial state
Transient PDF – Lock-in
Meantime between cycle slips
Simulation approach
• numerically solve the SDE using the Euler-Maruyama
scheme
• estimate probabilities using relative frequencies
• verify the accuracy with the results from the Fokker-Planck
method
• a relative tolerance level of 5% was allowed
Still no simulation required more than 5
minutes on a standard PC
9. Conclusions
•
Determinstic and stochastic behavior are related
in time domain and density function domain
•
Physical description of noise with a nonlinear Langevin
equation fails with respect to its physical interpretation
•
For thermal noise in nonlinear reciprocal circuits a welldefined theory is available (L.E. as approx.)
•
For nonhyperbolic circuits (e.g. oscillators) first concepts
for a geometric theory is available
There is a difference between P- and D-Bifurcation
•
•
Stochastic D-Andronov-Hopf theorem is illustrated by
means of versions of a Meissner oscillator circuit
10. References
TET References:
•
•
•
•
•
•
•
•
•
•
•
•
B. Beute, W. Mathis, V. Markovic: Noise Simulation of Linear Active Circuits by Numerical Solution of Stochastic
Differential Equations. Proceedings of the 12th International Symposium on Theoretical Electrical
Engineering (ISTET), 6 - 9 July 2003, Warsaw, Poland
Mathis, W.; M. Prochaska: Deterministic and Stochastic Andronov-Hopf Bifurcation in Nonlinear Electronic
Oscillations, Proceedings of the 11th workshop on Nonlinear Dynamics of Electronic Systems (EDES), 161164, 18-22 May 2003, Scuols, Schweiz
W. Mathis: Nonlinear Stochastic Circuits and Systems – A Geometric Approach. Proc. 4th MATHMOD, 5-7 Februar
2003, Wien (Österreich)
L. Weiss: Rauschen in nichtlinearen elektronischen Schaltungen und Bauelementen - ein thermodynamischer
Zugang. Berlin; Offenbach: VDE Verlag, 1999. Also: Ph.D. thesis, Fakultät Elektrotechnik, Otto-vonGuericke-Universität Magdeburg, 1999.
L. Weiss, W. Mathis: A thermodynamic noise model for nonlinear resistors, IEEE Electron Device Letters, vol. 20,
no. 8, pp. 402-404, Aug. 1999.
L. Weiss, W. Mathis: A unified description of thermal noise and shot noise in nonlinear resistors (invited paper),
UPoN'99, Adelaide, Australia, July 11-15, 1999.
L. Weiss, D. Abbott, B. R. Davis: 2-stage RC ladder: solution of a noise paradox, UPoN'99, Adelaide, Australia,
July 11-15, 1999.
W. Mathis, L. Weiss: Physical aspects of the theory of noise of nonlinear networks, IMACS/CSCC'99, Athens,
Greece, July 4-8, 1999.
W. Mathis, L. Weiss: Noise equivalent circuit for nonlinear resistors, Proc. ISCAS'99, vol. V of VI, pp. 314-317,
Orlando, Florida, USA, May 30 - June 2, 1999.
L. Weiss, W. Mathis: Thermal noise in nonlinear electrical networks with applications to nonlinear device models,
Proc. IC-SPETO'99, pp. 221-224, Gliwice, Poland, May 19-22, 1999.
L. Weiss, W. Mathis: Irreversible Thermodynamics and Thermal Noise of Nonlinear Networks, Int. J. for
Computation and Mathematics in Electrical and Electronic Engineering COMPEL, vol. 17, no. 5/6, pp. 635648, 1998.
W. Mathis, L. Weiss: Noise Analysis of Nonlinear Electrical Circuits and Devices. K. Antreich, R. Bulirsch, A. Gilg,
P. Rentrop (Eds.): Modling, Simulation and Optimization of Integrated Circuits. International Series of
Numerical Mathematics, Vol. 146, pp. 269-282, Birkhäuser Verlag, Basel, 2003
• L. Weiss, M.H.L. Kouwenhoven, A.H.M van Roermund, W. Mathis: On the Noise Behavior of a Diode, Proc.
Nolta'98, vol. 1 of 3, pp. 347-350, Crans-Montana, Switzerland, Sept. 14-17, 1998.
• L. Weiss, W. Mathis: N-Port Reciprocity and Irreversible Thermodynamics, Proc. ISCAS'98, vol. 3 of 6, pp. 407410, Monterey, California, USA, May 31 - June 03, 1998. *
• L. Weiss, W. Mathis, L. Trajkovic: A Generalization of Brayton-Moser's Mixed Potential Function, IEEE CAS I, vol.
45, no. 4, pp. 423-427, April 1998.
• L. Weiss, W. Mathis: A Thermodynamical Approach to Noise in Nonlinear Networks, International Journal of
Circuit Theory and Applications, vol. 26, no. 2, pp. 147-165, March/April 1998.
Further references:
• Langevin, P., Comptes Rendus Acad. Sci. (Paris) 146, 1908, 530
• W. Schottky, W.: Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern, Ann. d. Phys. 57, 1918,
541-567
• J. Guckenheimer; P. Holmes: Nonlinear oscillations, dynamical systems, and bifurcation of vector fields.
Springer-Verlag, Berlin-Heidelberg 1983
• N.G. van Kampen: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam 1992
• R.L. Stratonovich.: Nonlinear Thermodynamics I. Springer-Verlag, Berlin-Heidelberg, 1992
• L. Arnold: The unfoldings of dynamics in stochastic analysis. Comput. Appl. Math. 16, 1997, 3-25
• W. Mathis: Historical remarks to the history of electrical oscillators (invited). In: Proc. MTNS-98 Symposium, July
1998, IL POLIGRAFO, Padova 1998, 309-312.
• L. Arnold; P. Imkeller: Normal forms for stochastic differential equations. Probab. Theory Relat. Fields 110, 1998,
559-588
• L. Arnold: Random dynamical systems. Berlin-Heidelberg-New York 1998
• W. Mathis: Transformation and Equivalence. In: W.-K. Chen (Ed.): The Circuits and Filters Handbook. CRC Press &
IEEE Press, Boca Raton 2003