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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 Bifurcation in Deterministic Circuits**

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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**Bob Pease, National Semiconductors**

1. Deterministic Circuit Descriptions A. Meissner, 1913 Bob Pease, National Semiconductors

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**for Electronic Circuits**

Electrical and Electronic Circuits: The Ohm-Kirchhoff-Approach Real Circuit Partitioning b Modelling Circuit-Model Models for Electronic Circuits

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**of Electrical Circuits**

Description of resistive NW elements: Ohm Space Electrical Circuits are defined in Space of all currents and Voltages b Description of connections: Kirchhoff Space State-Space of Electrical Circuits

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**DAE Systems consist of many e.g. (100-) thousand Equations**

Dynamics electronic Circuits (Networks) Capacitors Inductors DAE System: Differential- algebraic System DAE Systems consist of many e.g. (100-) thousand Equations numerical solutions necessary! Special Cases: State-Space Equations (ODEs)

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**Initial value problems suitable the quantitative behavior!**

Deterministic Description: ODEs Initial value problems suitable for studying the quantitative behavior! Are initial value problems suitable for studying the qualitative behavior?

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**p Reformulation of the deterministic Dynamics Qualitative behavior:**

Considering a whole family of systems where generalized Liouville equation Dynamics of a Density Function p ( Frobenius-Perron-Operator ): Set of Initial Values Density Function p

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**2. Stochastic Circuit Descriptions**

Noise Model 2. Stochastic Circuit Descriptions Thermal Noise in linear and nonlinear electrical Circuits with noise sources

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**Deterministic Approach Network Thermodynamics**

Circuits Network Thermodynamics Generalized Liouville Equation Circuit Equations Generalization: (Non)linear Circuits including noise Deterministic Circuits Macroscopic Approach Noisy Circuits Mesoscopic Approaches Microscopic Approach Device Modelling

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**Microscopic Approach: Statistical Physics**

drift movement e.g. Recombination- Generation- Noise Multi-Body System (approx particles) C. Jungemann (see his talk this morning)

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**The Langevin Approach:**

3. Mesoscopic Approaches The Langevin Approach: Noise sources as inputs Stochastic ODE (SODE): Remarks: Fokker-Planck equation as modified generalized Liouville equation

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**Langevin’s 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. Average: =0 Conclusion: First Moment satisfies a determinstic differential equation

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**Compare: In nonlinear systems**

b) Nonlinear Case (van Kampen, 1961) stoch. =0 Average: Coupling of Moments Compare: In nonlinear systems Deterministic nonlinear System: Energetic Coupling of Frequencies Sinusoidal input Coupling of Moments of the probility density Stochastic nonlinear System:

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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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**Contradiction against the second law of thermodynamics**

However: Brillouin‘s 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 Drift Movement internal noise ( cannot switched off) in nonlinear systems ( electrical circuits) phys. No systematic extraction of deterministic equations The entire behavior has to be described as a stochastic process Assumption: description as a Markov process Mesoscopic Approach based on statistical thermodynamics

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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) Types of Markov Processes time domain probability density domain Stochastic differential Equation (SODE) Fokker-Planck equation mathematical equivalent! more general partial differential equations for the General solutions of the Chapman-Kolmogorov equation by the Kramers-Moyal series

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**Perturbation analysis for calculating coefficients**

Derivation of the Kramers-Moyal Coefficients for nonlinear systems by Nonlinear Nonequilibrium Statistical Thermodynamics (Stratonovich): „Thermal noise“: In thermodynamical equilibrium ( ) the density function is known: (stable) Perturbation analysis for calculating coefficients However: Restricted to reciprocal circuits (no transistors!) „Irreversible Statistical Thermodynamics of Circuits“ Weiss; Mathis ( ), Dissertation (Weiss) 1999

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**Statistical Thermodynamics of Thermal Noise **

in Nonlinear Circuit Theory Using Stratonovich‘s Approach: Basic is the Markov Assumption determination

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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:**

not of Fokker-Planck type 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 Current-Voltage Relation Circuit Topology Correct Noise Spectra (if the physical assumptions valid) Note: Assumptions are not satisfied if non-thermal effects are included (hot electron effects)

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**The “Thermodynamic Window”**

of a Circuit Currents and Voltages Microscopic Behavior

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**Dissipation Fluctuation**

Linear RC Networks: Classical Result Stochastic Diff.Equ. ( Noise Source ) - = du C ) u ( K + dt I S dw Signal Noise equivalent SODE Û Fokker-Planck Equation ( distributed Noise ) 2 I C S U - = t ) , U ( p C ) U ( K 2 1 ) t , U ( p + ) t , U ( p our approach Network Equation K(U) = - U / R ) kT / W exp( U ( p C eq - Thermodynamic Equilibrium Nyquist‘s Formular (linear approximation) Dissipation Fluctuation

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our approach (equivalent SODE)

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**Shot Noise! our approach Note: Shot noise has a thermal background**

(see Schottky (1918))

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**known from microscopic analysis (see textbooks):**

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

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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: …) 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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**Idea: „Linearization with respect to a periodic solution“**

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)

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**+ Noise (stochastic processes)**

Third Generation: SDAE Models Noise Analysis by Stochastic Differential Algebraic Equations (Kärtner (1990), Demir, Roychowdhury (2000)) DAE System: Differential- algebraic System + Noise (stochastic processes) Systematic Results in Phase Jitter of Oscillators as well as other nonlinear systems (e.g. PLL), however the onset of oscillations cannot described

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**Barkhausen or Nyquist Criteria**

5. Bifurcation in Deterministic Circuits Given: , Cgs, Cds, RL, y22; Choice: CG, CL (influence of Cgs and Cds „small“) L Cgs//CG Cds//CL RL gm Linearization? Non-reciprocal FET Colpitts Oscillator What is happened if the circuit is non-hyperbolic? x C I Theorem of Hartman-Grobman: The dynamical behavior of state space equations is related to the dynamics of the „linearized“ equations in hyperbolic cases. Barkhausen or Nyquist Criteria

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**Example: Sinusoidal Oscillators**

In certain cases Limit Cycles can be observed damping term Example: Sinusoidal Oscillators Obvious solution: positive State space interpretation: Type of damping Periodic Solution negative

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**Analysis of Systems with Limit Cycles**

Idea: (Poincaré; Mandelstam, Papalexi ) Embedding of an oscillator (equation) into a parametrized family of oscillator (equations) embedding with Example: Van der Pol equation

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**Andronov-Hopf Bifurcation**

Stable equilibrium point Limit Cycle Bifurcation Point State Space Cut plane Cut plane

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**the Jacobi matrix includes a pair of imaginary eigenvalues **

Poincaré-Andronov-Hopf Theorem (1934,1944) Let with for all e in a neighborhood of 0. If the Jacobi matrix includes a pair of imaginary eigenvalues the other eigenvalues have a negative real part the equilibrium point for asymptotic stable Then there exist an asymptotic stable equilibrium point for a stable limit cycle for

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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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**a stable equilibrium point a unstable equilibrium point?**

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: SODE Fokker-Planck stationary

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**For 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 of a family of stochastic dynamics with a ergodic I.M. “Near” we have another I.M with * Arnold, p ** Arnold, p. 473

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**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* There is no D-Bifurcation Remark: There are cases with D-bifurcation but no P-bifurcation* * Arnold, p. 476

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**Case 1: Pitchfork Bifurcation**

Invariant Measures D P

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**Case 2: Andronov Bifurcation**

invariant measures D1 P D2

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**There are stochastic generalizations of geometric theorems**

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 Arnold, Kedai 1993, 1995 Arnold, Imkeller 1997 Remark: Until now a research program (Arnold, ...)

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**7. Examples Meissner oscillator and van der Pol’s equation**

Meissner‘s Tube Oscillator Meissner oscillator and van der Pol’s equation

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**Normalization and Scaling:**

Noisy frequency: (using results from Arnold, Imkeller 1997)

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Linearization: Center Manifold: 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)

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**Meissner oscillator with nonlinear capacitor**

q Duffing-van der Pol equation

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**Normal Form in Resonant Case: (polar coordinates)**

noisy case Normal Form in Resonant Case: (polar coordinates) Solution of stochastic cohomology equation results (results: Arnold, Imkeller, 1997)

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**Ljapunov-Exponents: Stability is lost determinstic Andronov-Hopf**

Pardoux- Wihstutz formula 1988 determinstic Andronov-Hopf bifurcation stochastic P- Andronov-Hopf bifurcation first stochastic D-Andronov-Hopf bifurcation Stability is lost

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**8. Noise Analysis of Phase Locked Loops (PLL)**

F(s) P VCO Phase Detector Lowpass Filter Voltage Controlled Oscillator Equivalent Base-band Modell: F(s) + K A sin(.) F(s)=1 State Space Equation

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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 Formally, one obtains: But:**

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, tn+1] 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), 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), , 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, 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 , Aug 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 , 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 , 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 , 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 , Birkhäuser Verlag, Basel, 2003

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**L. Weiss, M. H. L. Kouwenhoven, A. H. M van Roermund, W**

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 , Crans-Montana, Switzerland, Sept , 1998. L. Weiss, W. Mathis: N-Port Reciprocity and Irreversible Thermodynamics, Proc. ISCAS'98, vol. 3 of 6, pp , Monterey, California, USA, May 31 - June 03, * L. Weiss, W. Mathis, L. Trajkovic: A Generalization of Brayton-Moser's Mixed Potential Function, IEEE CAS I, vol. 45, no. 4, pp , 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 , 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, 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, L. Arnold; P. Imkeller: Normal forms for stochastic differential equations. Probab. Theory Relat. Fields 110, 1998, 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

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