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

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

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

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

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

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

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

12. drift movement e.g. Recombination- Generation- Noise Multi-Body System (approx. 1023 particles) C. Jungemann (see his talk this morning) Microscopic Approach: Statistical Physics

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

14. Langevin’s Approach: Deterministic Circuit (without inputs) output Noisy input Applications: e.g in Communication Systems Transmission of noisy signals through a deterministic channel (Mathematics: Transformation of stochastic processes)

15. =0 Average: Physical Interpretation of SODE (Langevin, 1908) a) Linear Case stoch. Conclusion: First Moment satisfies a determinstic differential equation

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

17. Alternative: Analyzing nonlinear circuits including noise Extraction ofNoise 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. 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))

19. 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 Nonlinear Electronic Circuits Electrical current is related to noisy electron transport · internal noise ( cannot switched off) · in nonlinear systems ( electrical circuits)

20. 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 probability density domain “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. „Thermal noise“: In thermodynamical equilibrium ( ) equilibrium the density function is known: • „Irreversible Statistical Thermodynamics of Circuits“ • Weiss; Mathis (1995-2001), Dissertation (Weiss) 1999 However: Restricted to reciprocal circuits (no transistors!) Derivation of the Kramers-Moyal Coefficients for nonlinear systems by Nonlinear Nonequilibrium Statistical Thermodynamics(Stratonovich): (stable) Perturbation analysis for calculating coefficients

22. determination Statistical Thermodynamics of Thermal Noise in Nonlinear Circuit Theory Using Stratonovich‘s Approach: Basic is theMarkov Assumption

23. Nonlinear Circuits (Weiss und Mathis (1995-1999)) Starting Point: Complete Reciprocal Circuits: Brayton-Moser Description

24. Linear Approximation:

25. not of Fokker-Planck type Quadratic Approximation: 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 assumptions valid) Current-Voltage Relation Circuit Topology Note: Assumptions are not satisfied if non-thermal effects are included (hot electron effects)

28. Microscopic Behavior Currents and Voltages The “Thermodynamic Window” of a Circuit

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

34. 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)

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

36. Linearization? L What is happened if the circuit is non-hyperbolic? gm RL C I x x Cgs//CG Cds//CL Barkhausen or Nyquist Criteria x 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.

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

38. Example: Van der Pol equation embedding with Analysis of Systems with Limit Cycles Idea: (Poincaré; Mandelstam, Papalexi - 1931) Embedding of an oscillator (equation) into a parametrized family of oscillator (equations)

39. Stable equilibrium point Limit Cycle Bifurcation Point Andronov-Hopf Bifurcation State Space Cut plane Cut plane

40. Then there exist • an asymptotic stable equilibrium point for • a stable limit cycle for 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 forasymptotic stable

41. Transient Behavior of a Sinusoidal Oscillator (Center Manifold Mc)

42. 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)

43. 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?

44. SODE Fokker-Planck stationary Dynamical Equation:

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

46. 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* Question:Is there any relationship between these types of bifurcation In general, there is not! * Arnold, p. 476

47. Invariant Measures D P Case 1: Pitchfork Bifurcation

48. invariant measures D2 D1 P Case 2: Andronov Bifurcation

49. 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, 1995 Arnold, Imkeller 1997 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, ...)