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## Thermal Noise in Nonlinear Devices and Circuits

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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**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**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**1. Deterministic Circuit Descriptions**A. Meissner, 1913 Bob Pease, National Semiconductors**Real Circuit**Circuit-Model Modelling b b Partitioning Electrical and Electronic Circuits: The Ohm-Kirchhoff-Approach Models for Electronic Circuits**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**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**Are initial value problems suitable for studying the**qualitative behavior? Deterministic Description: ODEs Initial value problems suitable for studying the quantitativebehavior!**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**Noise Model**2. Stochastic Circuit Descriptions Thermal Noise in linear and nonlinear electrical Circuits with noise sources**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**drift movement**e.g. Recombination- Generation- Noise Multi-Body System (approx. 1023 particles) C. Jungemann (see his talk this morning) Microscopic Approach: Statistical Physics**Stochastic ODE**(SODE): 3. Mesoscopic Approaches The Langevin Approach: Noise sources as inputs Remarks: Fokker-Planck equation as modified generalized Liouville equation**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)**=0**Average: Physical Interpretation of SODE (Langevin, 1908) a) Linear Case stoch. Conclusion: First Moment satisfies a determinstic differential equation**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:**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**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))**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)**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**„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**determination**Statistical Thermodynamics of Thermal Noise in Nonlinear Circuit Theory Using Stratonovich‘s Approach: Basic is theMarkov Assumption**Nonlinear Circuits (Weiss und Mathis (1995-1999))**Starting Point: Complete Reciprocal Circuits: Brayton-Moser Description**not of**Fokker-Planck type Quadratic Approximation: 2 2 3**Cubic Approximation:**Noise cannot be determined thermodynamical!**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)**Microscopic**Behavior Currents and Voltages The “Thermodynamic Window” of a Circuit**=**- + 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**our approach**(equivalent SODE)**our**approach Shot Noise! Note: Shot noise has a thermal background (see Schottky (1918))**(simple model)**our approach known from microscopic analysis (see textbooks): known from**known from microscopic analysis (e.g. van der Ziel (1962):**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 - …) 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**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)**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)**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.**In certain cases Limit Cyclescan be observed**Example: Sinusoidal Oscillators damping term Obvious solution: State space interpretation: Type of damping positive Periodic Solution negative**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)**Stable equilibrium**point Limit Cycle Bifurcation Point Andronov-Hopf Bifurcation State Space Cut plane Cut plane**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**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 Symbolic Analysis (MATHEMATICA, MAPLE)**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?**SODE**Fokker-Planck stationary Dynamical Equation:**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**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**Invariant Measures**D P Case 1: Pitchfork Bifurcation**invariant**measures D2 D1 P Case 2: Andronov Bifurcation**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, ...)

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