Switched

(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 dalibor.biolek@unob.cz 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

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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

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1 Typical problems examples Result: R2*Rz Kv = ------------------------------ R1*Rz +R2*Rz +R2*R1 Simple computations Loaded voltage divider – compute voltage transfer function

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1 Typical problems examples Results: Rx R = R1 R2 Lx = R1 R2 C Simple computations Maxwell-Wien bridge – compute balance condition

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1 Typical problems examples Simple computations Compute all two-port parameters including wave impedances

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1 Typical problems examples Simple computations Transistor amplifier – verify results mentioned below

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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

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1 Typical problems examples Simple computations Resonant circuit – find step response Result: 0.1596*exp(-50000*t)*sin( 626703*t)

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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

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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

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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

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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

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1 Typical problems examples 1MHz bandpass filter – find how CCII nonidealities a  1, b2  1 affect the transfer function Influence of real properties

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Sallen-Key LP filter- influence of OpAmp properties to frequency response 1 Typical problems examples Influence of real properties

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Model of HF transformer with coupled circuits 1 Typical problems examples Special effects

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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,..)

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Example – RC cell 2 (Semi)symbolic versus numerical analysis symbolic and semisymbolic

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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)

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Example – RC cell 2 (Semi)symbolic versus numerical analysis numerical

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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…

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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…

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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 

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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  

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4 Needs Symbolic, Semisymbolic and Numerical links

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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)

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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)

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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

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5 SNAP Symbolic and Numerical Analysis Program Program conception

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5 SNAP Symbolic and Numerical Analysis Program

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5 SNAP Symbolic and Numerical Analysis Program

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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. ?

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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.

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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

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6 Switched linear systems… What is the GTF Mixed S-Z description of circuits with periodically varying parameters

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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

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6 Switched linear systems… What is the GTF Generalized Transfer Function of circuits with periodically varying parameters Sample-Hold

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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

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6 Switched linear systems… Computing the GTF LiSN program (Linear Switched Network) Demonstration of semisymbolic analysis

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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