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Title: Kein Folientitel

1
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
2
Content
• Deterministic Circuit Descriptions
• Stochastic Circuit Descriptions
• Mesoscopic Approaches
• Steps in Noise Analysis in Design Automation
• Bifurcation in Deterministic Circuits
• Bifurcation in Noisy Circuits and Systems
• Examples
• Conclusions

3
• Deterministic Circuit Descriptions
• Stochastic Circuit Descriptions
• Mesoscopic Approaches
• Steps in Noise Analysis in Design Automation
• Bifurcation in Deterministic Circuits
• Bifurcation in Noisy Circuits and Systems
• Examples
• Noise Analysis of Phase Locked Loops (PLL)
• Conclusions

4
1. Deterministic Circuit Descriptions
A. Meissner, 1913
Bob Pease, National Semiconductors
5
Electrical and Electronic Circuits The
Ohm-Kirchhoff-Approach
Models for Electronic Circuits
6
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7
Dynamics electronic Circuits (Networks)
DAE System Differential- algebraic System
8
Deterministic Description ODEs
Initial value problems suitable for studying the
quantitative behavior!
9
Reformulation of the deterministic Dynamics
Qualitative behavior
Considering a whole family of systems
Density Function p
10
2. Stochastic Circuit Descriptions
Thermal Noise in linear and nonlinear electrical
Circuits with noise sources
11
Deterministic Approach
Network Thermodynamics
Generalized Liouville Equation
Circuit Equations
12
Microscopic Approach Statistical Physics
13
3. Mesoscopic Approaches
The Langevin Approach
Noise sources as inputs
Remarks Fokker-Planck equation as modified
generalized Liouville equation
14
Langevins Approach
Deterministic Circuit (without inputs)
Noisy input
output
Applications e.g in Communication
Systems
Transmission of noisy signals through a
deterministic channel (Mathematics
Transformation of stochastic processes)
15
Physical Interpretation of SODE (Langevin, 1908)
a) Linear Case
stoch.
Conclusion First Moment satisfies a determinstic
differential equation
16
b) Nonlinear Case
(van Kampen, 1961)
stoch.
0
Average
17
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
18
electrical Circuits
White noise
White noise
PN-Diode

A diode can rectify its own noise
Contradiction against the second law of
thermodynamics (white noise sources in device
models are forbidden ., Weiss, Mathis, Coram,
Wyatt (MIT))
19
Nonlinear Electronic Circuits

Electrical current is related to noisy electron
transport

internal
noise
(
cannot
switched
off)

in
nonlinear
systems
(
electrical
circuits)
20
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)
General solutions of the Chapman-Kolmogorov
equation by the Kramers-Moyal series
21
Derivation of the Kramers-Moyal Coefficients for
nonlinear systems by Nonlinear
Nonequilibrium Statistical Thermodynamics
(Stratonovich)
(stable)
Perturbation analysis for calculating coefficients
22
Statistical Thermodynamics of Thermal Noise in
Nonlinear Circuit Theory
Using Stratonovichs Approach Basic is the
Markov Assumption
23
Nonlinear Circuits (Weiss und Mathis (1995-1999))
Starting Point
Complete Reciprocal Circuits Brayton-Moser
Description
24
Linear Approximation
25
2
2
3
26
Cubic Approximation
Noise cannot be determined thermodynamical!
27
Our Approach of Noise Spectra Calculations
Physical Assumptions
Stratonovich Machine
Correct Noise Spectra (if the physical assumption
s valid)
Current-Voltage Relation Circuit Topology
Note Assumptions are not satisfied if
non-thermal effects are included (hot
electron effects)
28
The Thermodynamic Window of a Circuit
29
Linear RC Networks Classical Result
Stochastic
Diff.Equ. (
Noise
Source
)
dw
Signal

Noise
equivalent SODE

Û

Fokker-Planck
Equation (
distributed
Noise
)
our approach

Network
Equation
K(U) - U / R

Thermodynamic
Equilibrium
30
our approach (equivalent SODE)
31
our approach
Shot Noise!
Note Shot noise has a thermal background
(see Schottky (1918))
32
(simple model)
our approach
known from microscopic analysis (see textbooks)
known from
33
known from microscopic analysis (e.g. van der
Ziel (1962)
our approach
34
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
35
• 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 (Leesons formula)
36
• Third Generation SDAE Models
• Noise Analysis by Stochastic Differential
Algebraic Equations
• (Kärtner (1990), Demir, Roychowdhury (2000))

DAE System Differential- algebraic System
Noise (stochastic processes)
37
5. Bifurcation in Deterministic Circuits
Given , Cgs, Cds, RL, y22
Choice CG, CL (influence of Cgs and Cds
small)
Non-reciprocal
FET Colpitts Oscillator
Theorem of Hartman-Grobman
The dynamical behavior of state space equations
is related to the dynamics of the linearized
equations in hyperbolic cases.
38
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39
Analysis of Systems with Limit Cycles
Idea (Poincaré Mandelstam, Papalexi - 1931)
Embedding of an oscillator (equation) into a
parametrized family of oscillator (equations)
40
Andronov-Hopf Bifurcation
State Space
Cut plane
Cut plane
41
Poincaré-Andronov-Hopf Theorem (1934,1944)
Let
for all e in a neighborhood of 0. If
with
• the Jacobi matrix includes a
pair of imaginary eigenvalues
• the other eigenvalues have a negative real part
• the equilibrium point for asymptotic
stable

42
Transient Behavior of a Sinusoidal
Oscillator (Center Manifold Mc)
43
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

Symbolic Analysis (MATHEMATICA, MAPLE)
44
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.
?
initial P.D.F.
Behavior of P.D.F. p near a stable equilibrium
point
Behavior of P.D.F. p near a unstable equilibrium
point?
45
Dynamical Equation
46
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
47
Question Is there any relationship between these
types of bifurcation
In general, there is not!
Arnold, p. 476
48
Case 1 Pitchfork Bifurcation
49
Case 2 Andronov Bifurcation
50
Main Questions There are stochastic
generalizations of geometric theorems
• Hartman-Grobman
• Poincaré-Andronov-Hopf
• Center Manifold
• Poincarè Normal Form

Remark Until now a research program (Arnold, ...)
51
7. Examples
Meissner oscillator and van der Pols equation
52
Normalization and Scaling
53
Linearization
Center Manifold
54
Meissner oscillator with nonlinear capacitor
q
Duffing-van der Pol equation
55
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56
Ljapunov-Exponents
57
8. Noise Analysis of Phase Locked Loops (PLL)
Equivalent Base-band Modell
F(s)1
58
2nd order PLL with ideal Integrator
Equivalent
Base-band Modell

System State Space Equation
59
Noisy state equation
noise
Formally, one obtains
But How do we interpret this equation, if n(t)
is not exactly known? - need
generic results
60
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
61
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

62
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
63
The Euler-Maruyama scheme
Consistency with Ito-calculus
Noise term in the EM scheme approximates the Ito
stochastic integral over interval tn, tn1 by
evaluating its integrand at the lower end point
of this interval, that is
64
Phase-acquisition time
• Time to reach locked state from an initial state

65
Transient PDF Lock-in
66
Meantime between cycle slips
67
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
68
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

69
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), 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
Australia, July 11-15, 1999.
• L. Weiss, D. Abbott, B. R. Davis 2-stage RC
• 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

70
• 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,