thermal noise in nonlinear devices and circuits wolfgang mathis and jan bremer institute of...

Thermal Noise in Nonlinear Devices and Circuits

Wolfgang Mathis and Jan Bremer

Institute of Theoretical Electrical Engineering (TET)

Faculty of Electrical Engineering und Computer ScienceUniversity 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




4. Steps in Noise Analysis in Design Automation



7. Examples

8. Noise Analysis of Phase Locked Loops (PLL)

9. Conclusions


A. Meissner, 1913

Bob Pease, National Semiconductors

Real Circuit

Modelling

Circuit-Model

Models for Electronic Circuits

Partitioning b

b

Electrical and Electronic Circuits: The Ohm-Kirchhoff-Approach

b

b

Electrical Circuits are defined in

1 1,..., ; ,.( , ..) : ,( )bb

bbi ii uu u Z := R R

Space of all currents and Voltages

O

( , ) ( , ) 0, : ;b b b b kR Ri u F i u F k b O := R R R R R

Description of resistive NW elements:

Ohm Space

K

Description of connections:

( , ) 0 0, ( , ) exaktb bi u Ai B u A B K := R R

Kirchhoff Space

S :=K OState-Space

of Electrical Circuits

S

0 ( , )g x y

( , )dx

f x ydt

DAE System:

Differential-

algebraic System

DAE Systems consist of manye.g. (100-) thousand Equations

numerical solutions necessary!

Dynamics electronic Circuits (Networks)

Special Cases: State-Space Equations (ODEs) ( )dx

F xdt

Capacitors Inductors

D e t e r m i n i s t i c D e s c r i p t i o n : O D E s

d x

d tF xi

i ( )

w h e r e

: ( 1 , , )n niF i n a n d x R R R

00 )(, xtx

00 )( xtx

Are initial value problems suitable for studying the qualitative behavior?

Deterministic Description: ODEs

Initial value problems suitablefor studying

the quantitative behavior!

Reformulation of the deterministic Dynamics

Qualitative behavior: Considering a whole family of systems

IRStxxxFdt

dxi

i )(),( 00

( )p x

where

generalized Liouville equation

( , ) ( ( ))tp x t p xP

1

ni

i i

p Fp

t x

Dynamics of a Density Function p (Frobenius-Perron-Operator ):tP

Set of Initial Values

Density Function p

Thermal Noise in linear and nonlinear electrical Circuitswith noise sources

Noise Model


Microscopic

Approach

DeviceModelling

Circuits

Network Thermodynamics

Deterministic Approach

Circuit Equations

GeneralizedLiouville Equation

Generalization: (Non)linear Circuits including noise

DeterministicCircuits

MacroscopicApproach

NoisyCircuits

MesoscopicApproaches

Microscopic Approach: Statistical Physics

drift movemente.g.Recombination-

Generation-Noise

Multi-Body System (approx. 1023 particles)

C. Jungemann (see his talk this morning)

Noisesources

as inputs

D y n a m ic s o f t h e D e n s i t y F u n c t io n p : ( m a th e m a t ic a l e q u iv a le n t to S O D E )

2 2

1 , 1

( ) 1 ( )

2

n ni

i i ji i j

p Fp p

t x x x

F o k k e r - P la n c k e q u a t io n


Remarks: Fokker-Planck equation as modified generalized Liouville equation

Stochastic ODE (SODE):

( ) ( )ii i

dxF x x

dt

: ( 1, , ) andn niF i n x R R R

white noise and ( ) coupling coefficienti x

The Langevin Approach:

Deterministic Circuit

(without inputs)

Langevin’s Approach:

Noisy input output

Applications: e.g in Communication Systems

Transmission of noisy signals through a deterministic channel

(Mathematics: Transformation of stochastic processes)

iij j

ji

dxA x

dt

Physical Interpretation of SODE (Langevin, 1908)

a) Linear Case

iij j

j

dxA x

dt

( white noise, konst.)i

Average:ii

ij j ij jj

ij

i

d xdxA x A x

dt dt

=0

stoch.

Conclusion: First Moment satisfies a determinstic differential equation

( )ii i

dxF x

dt

b) Nonlinear Case ( white noise, konst.)i

( )ii

dxF x

dt stoch.

Average: ( ) ( )iij i j i

d xdxF x F x

dt dt

=0

( )jF x Coupling ofMoments

(van Kampen, 1961)

Compare: In nonlinear systems

Deterministic nonlinear System:

Energetic Coupling of Frequencies

Sinusoidal input

Coupling of

Moments

of the probility density

Stochastic nonlinear System:

Alternative: Analyzing nonlinear circuits including noise

Numerous papers

Methods:

• Calculation of desired spectra

• Numerical Methods in Stochastic Differential Equations

• Geometric Analysis of Stochastic Differential Equations

Extraction of Noise Sources (then using the Langevin approach)

1. Motivation 5

Stochastic Equation

dw)II2(qdt1kT

quexpIduC ss

Average: (first moment)

2

2

s ukT

q

2

1u

kT

qIu

dt

dC

Thermodynamic Equilibrium:

Fluctuations-Voltage-Converter

However: Brillouin‘s Paradoxon of nonlinear electrical Circuits

Contradiction against the second law of thermodynamics

(white noise sources in device models are forbidden: …., Weiss, Mathis, Coram, Wyatt (MIT))

PN-Diode

White noise

:

„A diode can rectifyits own noise“

White noise

Electrical current is related to noisy electron transport

internal noise (cannot switched off)

in nonlinear systems (electrical circuits) Drift Movement

phys.

No systematic extraction of deterministic equations

The entire behavior has to be described as a stochastic process

Assumption: description as a Markov process

Nonlinear Electronic Circuits

Mesoscopic Approach based on statistical thermodynamics

“First Principle” Mesoscopic Approach for Circuits with Internal Noise

Starting Point: Markovian Stochastic Processes are defined by the Chapman-Kolmogorov Equation (Integral equation for the transition probability density)

),( tx

Types of Markov Processes

time domain probability density domain

Stochastic differential Equation (SODE)

Fokker-Planck equation

mathematical equivalent!

more general partial differentialequations 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

( )

the equilibrium density function is known:

However: Restricted to reciprocal circuits (no transistors!)

„Irreversible Statistical Thermodynamics of Circuits“

Weiss; Mathis (1995-2001), Dissertation (Weiss) 1999

kT

W

eq

X

ep

(stable)

Perturbation analysis for calculating coefficients

Kramers-Moyal Equation

)t,X(p)X(KXX!m

)1(

t

)t,X(pm1

m1 m11m

mm

?

Circuit Equations )kT(o)X(KXdt

d

Thermodynamic Equilibrium )kT/Wexp()X(p Xeq

Detailed Balance (Reciprocity)

Nonlinear „distributed“ Generalization

of the Nyquist’s Formula

Statistical Thermodynamics of Thermal Noise in Nonlinear Circuit Theory

Using Stratonovich‘s Approach: Basic is the Markov Assumption

determination

3.1 Anwendung: Vollständige nichtlineare Netzwerke

Network Equations

U

)U,I(M

dt

dU)U(C

I

)U,I(M

dt

dI)I(L

- Reciprocal

- DAE Description (Index 1)

Mixed Potential M: Dissipation

M(I,U) = F(I) - G(U) + (I,U)

Complete Reciprocal Circuits: Brayton-Moser Description

Nonlinear Circuits (Weiss und Mathis (1995-1999))

Starting Point:

Y)0(X ,

F o k k e r -P la n c k E q u a t io n : „ D is t r ib u te d N o is e “

)U,I(P)U,I(

t

)U,I(p1

21 1

21

L

)0(F ,

21

2

1

II 21

2111

1

IL

U

12

2111

1

UC

I

1

21

C

)0(G ,

2

1

UU

S ig n a l

21

kT21

21

LL

)0(F ,

21211

kTII

2

21

21

CC

)0(G ,

21UU

2

)U,I(p N o is e

S D E „D if fe re n t ia l“ N y q u is t F o rm u la

,dU

)0(dIkT2S I dI

)0(dUkT2S U

N o is e S o u r c e

Linear Approximation: , (0)X Y

, ,(0) 1/ 2 (0)X Y Y Y Quadratic Approximation:Quadratische Approximation:

YY)0(2/1Y)0(X ,,

3. Anwendung auf Schaltungen und Bauelemente 13

Kramers-Moyal Equation: „Distributed Noise“

)U,I(t

)U,I(p1

321 1

321

L

)0(F

2

1 ,

1I

321 1

321

32 C

)0(GII ,

1U

32UU

+ kT

21 21

221

LL

)0(F,

1I

+ kT

21 21

221

CC

)0(G,

1U

321 21

321

LL

)0(FkT3

,

21II

2

321 21

321

3 CC

)0(GkTI ,

21UU

2

3U

321 3121

321

LLL

)0(F)kT(2

,2

1 2 3

3

I I I

)t,U(p

there is no equivalent Noise Source Model

3

22not of

Fokker-Planck type

Cubic Approximation: , , ,

1 1(0) (0) (0)

2 6X Y Y Y Y Y Y

)U,I(P)U,I()U,I(

t

)U,I(p21

2

1321 31

3321

432

14321 4321 1

4321

432

11

4321 IILL

)0(FkT3UUU

UC

)0(GIII

IL

)0(F

6

1 ,,,

4321 21

2

1321 31

3321

LL

kT

12

1U

UCC

)0(GkT3 ,

4321 ,c

4

43

214321 L

kTII

II)0(F2

2

,

4321 21CC

kT

12

143,21

c

4

43

21

3321 C

kTUU

UU)0(G2

2

,

)0(F4LLL

)kT(4321

4321 321

,

2

413242314321 ,,, ccc

44

4

321IL2

kTI

III

3

)0(G4CCC

)kT(3321

4321 321

,

2

413242314321 ,,, ccc )U,I(p

UC2

kTU

UUU44

4

321

3

R a u s c h e n t h e r m o d y n a m i s c h n i c h t v o l l s t ä n d i g b e s t i m m b a rNoise cannot be determined thermodynamical!

Our Approach of Noise Spectra Calculations

StratonovichMachine

Current-VoltageRelation

Circuit Topology

PhysicalAssumptions

Correct NoiseSpectra

(if the physicalassumptions valid)

Note: Assumptions are not satisfied if non-thermal effects are included (hot electron effects)

The “Thermodynamic Window”of a Circuit

Currents and Voltages

Microscopic Behavior

Stochastic Diff.Equ. ( Noise Source )

duC )u(K dt IS dw

Signal Noise

Fokker-Planck Equation ( distributed Noise )

t

)t,U(pC

)U(KU

21

)t,U(p 2I

2

2

C

S

U

)t,U(p

Network Equation K(U) = - U / R

Thermodynamic Equilibrium )kT/Wexp()U(p Ceq

Linear RC Networks: Classical Result

Nyquist‘s Formular (linear approximation)

)/1(2 RGkTGSI

our approach

equivalent SODE

Dissipation Fluctuation

3 . 2 N i c h t l i n e a r e s R C N e t z w e r k : L i n e a r - q u a d r a t i s c h e N ä h e r u n g

N e t w o r k O D E 22

2

UdU

)0(gd

2

1U

dU

)0(dg

dt

dUC

S O D E dwudu

)0(gd

2

1

du

)0(dgkT2dt

C

kTu

du

)0(gd

2

1u

du

)0(dgduC 2

22

2

2

C/kTu 2 t h a t i s S h o r t C i r c u i t

G e n e r a l i z e d N y q u i s t F o r m u l a

U

dU

)0(gd

2

1

dU

)0(dgkT2S

2

2

I

Some Results: Nonlinear RC Network (Linear-quadratic Approximation)

our approach(equivalent SODE)

22

T

s

T

s

Ts U

U

I

2

1U

U

I1

U

UexpI)U(g

U

U

I

2

1

U

IkT2S 2

T

s

T

sI )U(IqIq2 s )U(O 2

Results about Thermal +Noise in Semiconductor Devices (Weiss, Mathis: IEEE Electron Devices Letters 1999)

1. Schottky Diode

Shot Noise!

Note: Shot noise has a thermal background (see Schottky (1918))

our approach

MOSFET:

IS

DStGS U

2

1UUkT2

tGS

DS

2tGS

2DS

tGS

DS

tGStherm,I

UU

U

2

11

UU

U

3

1

UU

U1

UUkT2S

DStGS U

2

1UUkT2 2

DSUO

2. MOSFET

2( )DSO UthermIS

known from microscopic analysis (see textbooks):our approach

2DSDStGSD U

2

1UUUI

known from

(simple model)

JFET:

IS

0GSdiff

DS

0

GSdiff0

UUU4

U

U

UU1gkT2

,yx

3

2yx

yx2

1yx

3

4yx

gkT2S2/32/3

222/32/3

0therm,I

0

GSdiff

0

DSGSdiff

U

UUy,

U

UUUx

therm,IS

0GSdiff

DS

0

GSdiff0

UUU4

U

U

UU1gkT2 )U(O 2

DS

3. JFET

2( )DSO UthermIS

known from microscopic analysis (e.g. van der Ziel (1962):

2/1

0

2/3GSdiff

2/3DSGSdiff

DS0DU

UUUUU

3

2UgI

our approach

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 - …)

Small-signal noise models do not work if e.g. bias changes occur,oscillators, more general nonlinear circuits

Idea: „Linearization with respect to an operational point (constant solution)“

State Space

• Second Generation: LPTV Models Variational Linear Noise Analysis of Periodical Systems (Hull, Meyer (1993), Hajimiri, Lee (1998))

Useful for periodic driven systems, however heuristic assumptionsand concept will be needed for oscillators (Leeson‘s formula)

Idea: „Linearization with respect to a periodic solution“

State Space

• Third Generation: SDAE Models Noise Analysis by Stochastic Differential Algebraic Equations (Kärtner (1990), Demir, Roychowdhury (2000))

0 ( , )g x y

( , )dx

f x ydt

DAE System:

Differential-

algebraic System

+ Noise (stochastic processes)

Systematic Results in Phase Jitter of Oscillators as well asother nonlinear systems (e.g. PLL),however the onset of oscillations cannot described

)()()()( txfxgdt

dxxB

Given: , Cgs, Cds, RL, y22; Choice: CG, CL (influence of Cgs and Cds „small“)

fosz

L

Cgs//CG Cds//CL

RLgm

Linearization?

What is happened if the circuit is non-hyperbolic?

x

x

x

CI0 2 4 6 8 10 12 14 16 18 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

The dynamical behavior of state space equations is related to the dynamics of the „linearized“ equationsin hyperbolic cases.

Theorem of Hartman-Grobman:

FET Colpitts Oscillator


Non-reciprocal

Barkhausen or Nyquist Criteria

In certain cases Limit Cycles can be observed

damping term

Example: Sinusoidal Oscillators

( ) x x x x x 2 2 1 0

Obvious solution: x t t or x t t( ) cos ( ( ) sin )

positive

State space interpretation:

x

x

Type of damping

2 2( 1) 0x x

Periodic Solution

negative

Idea: (Poincaré; Mandelstam, Papalexi - 1931)

Embedding of an oscillator (equation) into a parametrized family of oscillator (equations)

( ) x x x x x 2 2 0embedding

with

( ) x x x x x 2 2 1 0

Example: Van der Pol equation

Analysis of Systems with Limit Cycles

Cut

planeCut

plane

BifurcationPoint

Andronov-Hopf Bifurcation

Stable equilibriumpoint

Limit Cycle

State Space

x

x

Poincaré-Andronov-Hopf Theorem (1934,1944)

( , ), : n nx f x f R R RLet

with f ( , )0 0 for all in a neighborhood of 0. If

• the Jacobi matrix includes a pair of

imaginary eigenvalues • the other eigenvalues have a negative real part •

D fx ( , )0 0 1 2, j

d

d 1 0 0( )

• the equilibrium point for asymptotic stable 0

Then there exist

• an asymptotic stable equilibrium point for • a stable limit cycle for

( , ) 1 0( , )0 2

Transient Behavior of a Sinusoidal Oscillator(Center Manifold Mc)

Concept for Analysis of Practical Oscillators

• Transformation of the linear part: Jordan Normal Form

• Transformation of the Equations: Center Manifold

• Transformation of the reduced Equations: Poincaré-Normal Form

• Averaging

( , ), : n nx f x f IR IR IR

Symbolic Analysis(MATHEMATICA, MAPLE)


Different Concepts:

I) The physical (phenomenological) approach (e.g. van Kampen)

Behavior of P.D.F. p neara stable equilibrium point

initialP.D.F.

initialP.D.F.

Behavior of P.D.F. p neara unstable equilibrium point?

Special Case: Dynamics in a Potential U(x)

U(x) U(x)

?

Dynamical Equation:'( )

dxU x

dt

Fokker-Planck

2

2'( )

p pU x p

t x x

42

2 0 uni

0 bi

moda

moda( )

l

lx

x

statp x C e

stationary

( )2

2

!)0 '( ) (

U x

stat

pU x p p x C e

x x

42 0 single we

e.g. l

( )2 >0 double we

l

ll

xU x x

3t t t tdx x x dt dW

SODE

For the equilibrium P.D.F. p(x) changes its type0

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”*

* Arnold, p. 473 ** Arnold, p. 473

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 of a family of stochastic dynamics with a ergodic I.M. ( . . . )w r t R

D

“Near” we have another I.M. with D

D

Question: Is there any relationship between these types of bifurcation

In general, there is not!

Our example: (above)

The corresponding invariant measure to

is unique*

42

2 0 uni

0 bi

moda

moda( )

l

lx

x

statp x C e

There is noD-Bifurcation

Remark: There are cases with D-bifurcation but no P-bifurcation*

* Arnold, p. 476

Case 1: Pitchfork Bifurcation

3t t t t tdx x x dt x dW

D P

0D 2

2P

Invariant Measures

0

q

q

q

q

Case 2: Andronov Bifurcation

3 21 1

0 0 1 0 0

1 0t t t tdx dt x dt x dWx x x

D1 P

1D 0P

D2

2D

0 stable

invariantmeasures

0

1

saddle

stable

0

1

2

unstable

stable

saddle

Main Questions: There are stochastic generalizations of geometric theorems

• Hartman-Grobman

• Poincaré-Andronov-Hopf

• Center Manifold

• Poincarè Normal Form

Global H-G: Wanner, 1995 (local H-G: still open question) Arnold, Schenk-Hoppe Namachchivaya 1996

Boxler 1989, Arnold, Kadei 1993

Elphick et al. 1985, C.+G. Nicolis 1986 (physics)Namachchivaya et al. 1991 Arnold, Kedai 1993, 1995Arnold, Imkeller 1997

Remark: Until now a research program (Arnold, ...)

Meissner oscillator and van der Pol’s equation

Meissner‘s Tube Oscillator

BEu

2

2 2 20 1 2 3 02

0

2 3 0BE BEBE BE BE

d u duRM k k u k u u

dt L dt

1 2 3, ,k k k

R

C

0 ,L M

20

0

1

L C

7. Examples

Normalization and Scaling:

1 12 2 2 2 20 0 1 0 2 1 2 0 3 1 22 2

0

0 10

2 3

x xdR

Mk k Mx x k Mx xx xdtL

1 2( , )BE BEx u x u

1 1

22 2 1 2

000 1 0

1 0 t

x xd dt

x x xx

xdW

2 20 1 0 3

0

where

:= , : Wiener p3 1, rocesst

RMk k M

LW

Noisy frequency:20 2:white noise( ) and 0k

(using results from Arnold, Imkeller 1997)

Linearization:

1 1

2 2

0 1 0 0

1 0 t

x xd dt x dW

x x

Center Manifold: 2 20If 0 ! , < 4 : / 4d 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)

BEu1 2 3, ,k k k

RC

0 ,L M

20

0 0

1

L C

2

2 3 20 1

32 3 02

0

2 3 0BBE BE

BE BE BE E

d u R duM k k u k u u

dt L dtu

Meissner oscillator with nonlinear capacitor

Duffing-van der Pol equation

q

Normal Form in Resonant Case: (polar coordinates)

Solution of stochastic cohomology equation results(results: Arnold, Imkeller, 1997)

2 23 2 2

2

5 212 4 sin 2 sin 4 cos 2

2 2 8t t t

t t t t d t t t t td d d

r r rd r r r dt r r dW

2 2 2 22

2 2

3 5 36 3 11 1 cos 2 cos 4 sin 2

4 2 2 2 2t t t t

t d t t t t td d d d d d

r r r rd r dt dW

1 1

3 22 2 1 1 2

00 1 0 0

1 0 t

x xd dt x dW

x x x x x

noisycase

Pardoux-Wihstutzformula

1988

2

41,2 2

,2 8 d

O

determinsticAndronov-Hopf

bifurcation0

0

0

2 1

0

0, 0 0,

first stochasticD-Andronov-Hopf

bifurcation

Stability is lost

Ljapunov-Exponents:

0 stochastic

P- Andronov-Hopfbifurcation

0

8. Noise Analysis of Phase Locked Loops (PLL)

Equivalent Base-band Modell:

F(s)+ KA sin(.)

0

t

d

0

( )sintd d

K A f t ddt dt

sin ,d d

K A tdt dt

State Space Equation

( )t

ˆ t

t( )x t

( )e tF(s)=1

F(s)P

VCO

Phase Detector Lowpass Filter

Voltage Controlled Oscillator

( )s t

( )r t

2nd order PLL with ideal Integrator

1

1

2 sin

sin

n

dx

d

d x

d

Equivalent Base-band Modell

System State Space Equation:

Noisy state equation

Formally, one obtains:

But:

How do we interpret this equation, if n’(t) is not exactly known?

- need generic results

noise

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 well-defined 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

• 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), 161- 164, 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-von-Guericke-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. 635- 648, 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

TET References:

• 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. 407-410, 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

thermal noise in nonlinear devices and circuits wolfgang mathis and jan bremer institute of...

Documents