Switched

Information about Switched

Published on January 16, 2008

Author: Viola

Source: authorstream.com

Content

(Semi) symbolic computer analysis of continuous-time and switched linear systems:  (Semi) symbolic computer analysis of continuous-time and switched linear systems Dalibor Biolek, Dept. of Microelectronics, FEEC Brno University of Technology, Czech Republic [email protected] http://user.unob.cz/biolek 1 Typical problems 2 (Semi)symbolic versus numerical analysis 3 Needs versus reality 4 Needs 5 SNAP 6 Switched linear systems – generalized s-z transfer functions 7 Instead of Conclusion Slide2:  1 Typical problems to solve in the area of linear analogue systems Verification of circuit principle Simple computations Influence of real properties Working with new circuit elements Special effects Slide3:  1 Typical problems examples Result: R2*Rz Kv = ------------------------------ R1*Rz +R2*Rz +R2*R1 Simple computations Loaded voltage divider – compute voltage transfer function Slide4:  1 Typical problems examples Results: Rx R = R1 R2 Lx = R1 R2 C Simple computations Maxwell-Wien bridge – compute balance condition Slide5:  1 Typical problems examples Simple computations Compute all two-port parameters including wave impedances Slide6:  1 Typical problems examples Simple computations Transistor amplifier – verify results mentioned below Slide7:  1 Typical problems examples Results: h21e=C2/C1=100, then wosc=sqrt[(1+h21)/(L*C2)], fosc=wosc/(2*pi)=715 kHz. Simple computations Colpitts oscillator – derive oscillation condition Slide8:  1 Typical problems examples Simple computations Resonant circuit – find step response Result: 0.1596*exp(-50000*t)*sin( 626703*t) Slide9:  1 Typical problems examples FDNR in series with resistance Result: Zin=R1/2+1/(D*s^2) D=2*R3*C1^2 Verification of circuit principle Slide10:  1 Typical problems examples DC precise LP filter. Frequency response looks good, but... Result: filter poles: -971695 + j484850 -971695 - j484850 -321953 195172 + j461620 195172 - j461620 FILTER IS UNSTABLE! Verification of circuit principle Slide11:  1 Typical problems examples 10MHz bandpass filter containing CDBA elements- find zeros and poles of current transfer function and frequency response Working with new circuit elements R1 = 1344W, R2 = 123W, R3 = 672W, R4 = 116W, R5 = 685W, R6 = 94W, R7 = R8 = 1kW, C1 = 110pF, C2 = 25pF, C3 = 113pF, C4 = 24pF, C5 = 156pF, C6 = 16.5pF, C7 = 15pF, C8 = 12pF, C9 = 8pF Slide12:  1 Typical problems examples Impedance converter/inverter with two CTTA elements with parameters b1,gm1, b2, gm2. Derive input impedance. Working with new circuit elements Result: Z2 Zin= --------------- gm1 b1 Z1 Slide13:  1 Typical problems examples 1MHz bandpass filter – find how CCII nonidealities a  1, b2  1 affect the transfer function Influence of real properties Slide14:  Sallen-Key LP filter- influence of OpAmp properties to frequency response 1 Typical problems examples Influence of real properties Slide15:  Model of HF transformer with coupled circuits 1 Typical problems examples Special effects Slide16:  Forms of the analysis outputs 2 (Semi)symbolic versus numerical analysis SYMBOLIC: math. formula which includes symbols of circuit parameters SEMISYMBOLIC: numerical values are instead of some symbols, the complex frequency s or z (freq. domain) or the time variable t or k (time domain) is also present in the formula NUMERICAL: numerical results (poles and zeros, waveform points,..) Slide17:  Example – RC cell 2 (Semi)symbolic versus numerical analysis symbolic and semisymbolic Slide18:  Example – RC cell 2 (Semi)symbolic versus numerical analysis symbolic and semisymbolic _______________zeros__________________ none _______________poles__________________ -1.00000000000000E+0005 ___________step response______________ 1.00000000000000E+0000 -1.00000000000000E+0000*exp(-1.00000000000000E+0005*t) ___________pulse response_____________ 1.00000000000000E+0005*exp(-1.00000000000000E+0005*t) Slide19:  Example – RC cell 2 (Semi)symbolic versus numerical analysis numerical Slide20:  Limitations of typical commercial circuit simulators 3 Needs versus reality Only numerical analysis, not symbolic and semisymbolic  no formulas Zeros and poles are not available Too complicated models, it is hard to study influence of partial component parameters Too primitive sensitivity analysis when it is available Too expensive… Slide21:  Wanted: new software tool for analysis of large linear systems 3 Needs versus reality Symbolic and semisymbolic analysis, numerical analyses in frequency/time domains Zeros and poles, waveform equations, symbolic-based sensitivity analysis Special effects (Dependences editor), export of equations into Matlab, MathCad etc. User-modified behavioral models based on MNA Free of charge… Slide22:  Why (semi)symbolic computation? 3 Needs versus reality Equations = more information than those from numerical results (they include them) Equations = important connections between the system and its behavior Equations = important data for verification of system principle Equations = important data for system optimization pro – and – con  Slide23:  Why NOT (semi)symbolic computation? 3 Needs versus reality CPU time- and memory-expensive algorithms Serious numerical problems must be overcome in some cases Complexity and non-transparency of symbolic results while analyzing large systems pro – and – con   Slide24:  4 Needs Symbolic, Semisymbolic and Numerical links Slide25:  4 Needs Computing system eigenvalues Numerical way (5): large circuits, problematic precision; QR, QZ,.., “optional precision” Semisymbolic way (4,3): moderate-size to large-size systems, problematic precision; FFT, Faddeyev algorithm (4), Laguer, method of accompanying matrix, “optional precision” (3) Symbolic way (1,2,3): small-size to moderate-size systems, excellent precision; ? (1), utilization of “optional precision” (2,3) Slide26:  4 Needs Computing time responses Numerical way (5): large circuits, good precision; classical integration formulas Semisymbolic way (4,3): moderate-size to large-size systems, precision depends on computing eigenvalues; partial fraction expansion, “optional precision” (3) Slide27:  5 SNAP Symbolic and Numerical Analysis Program Symbolic and semisymbolic analysis, numerical analyses in frequency/time domains Zeros and poles, waveform equations, symbolic-based sensitivity analysis Special effects (Dependences editor), export of equations into Matlab, MathCad etc. User-modified behavioral models based on MNA Free on http://snap.webpark.cz Slide28:  5 SNAP Symbolic and Numerical Analysis Program Program conception Slide29:  5 SNAP Symbolic and Numerical Analysis Program Slide30:  5 SNAP Symbolic and Numerical Analysis Program Slide31:  6 Switched linear systems… How to analyze in the frequency domain… Linear systems with periodically varying parameters Switched Capacitor and Switched Current circuits Sample-Hold circuits Switched DC-CD converters… ………. Classical harmonic steady-state does not exist in these circuits. AC analysis, frequency responses, … are based on harmonic steady state. ? Slide32:  6 Switched linear systems… What is the GTF Generalized Transfer Function of circuits with periodically varying parameters Equivalent signal: - interpolates samples v(kT+T) - its spectral components fall to the spectral area of w(t). There is infinite number of equivalent signals for <0,1) GTF is the ratio of Fourier/Laplace transformations of equivalent output signal and input signal. Slide33:  6 Switched linear systems… What is the GTF Generalized Transfer Function of circuits with periodically varying parameters Sample-Hold Evaluation of the dynamic error of sampling process by GTF: frequency responses Slide34:  6 Switched linear systems… What is the GTF Mixed S-Z description of circuits with periodically varying parameters Slide35:  6 Switched linear systems… What is the GTF Mixed S-Z description of circuits with periodically varying parameters …recurrent formula of linear periodically varying system …formula for equivalent signal Slide36:  6 Switched linear systems… What is the GTF Generalized Transfer Function of circuits with periodically varying parameters Sample-Hold Slide37:  6 Switched linear systems… Computing the GTF Mixed S-Z description of circuits with periodically varying parameters Algorithmic GTF computation: ..by numerical integration ..solving eigenvalue problem ..by a special procedure Slide38:  6 Switched linear systems… Computing the GTF LiSN program (Linear Switched Network) Demonstration of semisymbolic analysis Slide39:  6 Instead of Conclusion ? The rational arithmetic (RA) Contemporary problems …. ? The “optional precision” and “infinite precision” arithmetic (OPA, IPA) ? Solving the eigenvalue problem by means of RA, OPA, and IPA ? Topological methods of matrix deflation ? Solving the polynomial roots from symbolic results by means of OPA ? Special methods (SBE) of approximate symbolic analysis …and other programs

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