Kein Folientitel

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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
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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

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• 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

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1. Deterministic Circuit Descriptions
A. Meissner, 1913
Bob Pease, National Semiconductors
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Electrical and Electronic Circuits The
Ohm-Kirchhoff-Approach
Models for Electronic Circuits
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Dynamics electronic Circuits (Networks)
DAE System Differential- algebraic System
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Deterministic Description ODEs
Initial value problems suitable for studying the
quantitative behavior!
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Reformulation of the deterministic Dynamics
Qualitative behavior
Considering a whole family of systems
Density Function p
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2. Stochastic Circuit Descriptions
Thermal Noise in linear and nonlinear electrical
Circuits with noise sources
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Deterministic Approach
Network Thermodynamics
Generalized Liouville Equation
Circuit Equations
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Microscopic Approach Statistical Physics
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3. Mesoscopic Approaches
The Langevin Approach
Noise sources as inputs
Remarks Fokker-Planck equation as modified
generalized Liouville equation
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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)
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Physical Interpretation of SODE (Langevin, 1908)
a) Linear Case
stoch.
Conclusion First Moment satisfies a determinstic
differential equation
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b) Nonlinear Case
(van Kampen, 1961)
stoch.
0
Average
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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
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However Brillouins Paradoxon of nonlinear
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))
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Nonlinear Electronic Circuits


Electrical current is related to noisy electron
transport



internal
noise
(
cannot
switched
off)



in
nonlinear
systems
(
electrical
circuits)
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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
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Derivation of the Kramers-Moyal Coefficients for
nonlinear systems by Nonlinear
Nonequilibrium Statistical Thermodynamics
(Stratonovich)
(stable)
Perturbation analysis for calculating coefficients
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Statistical Thermodynamics of Thermal Noise in
Nonlinear Circuit Theory
Using Stratonovichs Approach Basic is the
Markov Assumption
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Nonlinear Circuits (Weiss und Mathis (1995-1999))
Starting Point
Complete Reciprocal Circuits Brayton-Moser
Description
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Linear Approximation
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Quadratic Approximation
2
2
3
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Cubic Approximation
Noise cannot be determined thermodynamical!
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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)
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The Thermodynamic Window of a Circuit
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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
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our approach (equivalent SODE)
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our approach
Shot Noise!
Note Shot noise has a thermal background
(see Schottky (1918))
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(simple model)
our approach
known from microscopic analysis (see textbooks)
known from
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known from microscopic analysis (e.g. van der
Ziel (1962)
our approach
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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
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• 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)
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• Third Generation SDAE Models
• Noise Analysis by Stochastic Differential
  Algebraic Equations
• (Kärtner (1990), Demir, Roychowdhury (2000))

DAE System Differential- algebraic System
Noise (stochastic processes)
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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.
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Analysis of Systems with Limit Cycles
Idea (Poincaré Mandelstam, Papalexi - 1931)
Embedding of an oscillator (equation) into a
parametrized family of oscillator (equations)
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Andronov-Hopf Bifurcation
State Space
Cut plane
Cut plane
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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

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Transient Behavior of a Sinusoidal
Oscillator (Center Manifold Mc)
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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)
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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?
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Dynamical Equation
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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
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Question Is there any relationship between these
types of bifurcation
In general, there is not!
Arnold, p. 476
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Case 1 Pitchfork Bifurcation
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Case 2 Andronov Bifurcation
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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, ...)
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7. Examples
Meissner oscillator and van der Pols equation
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Normalization and Scaling
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Linearization
Center Manifold
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Meissner oscillator with nonlinear capacitor
q
Duffing-van der Pol equation
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Ljapunov-Exponents
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8. Noise Analysis of Phase Locked Loops (PLL)
Equivalent Base-band Modell
F(s)1
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2nd order PLL with ideal Integrator
Equivalent
Base-band Modell

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

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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
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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
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Phase-acquisition time

• Time to reach locked state from an initial state

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Transient PDF Lock-in
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Meantime between cycle slips
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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
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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

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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
  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

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• 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