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(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.czdalibor.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 Finding DC voltages and currents Harmonic steady state responses Voltage gain of loaded divider Balance condition of DC or AC bridge Computing two-port parameters Finding gain formula of transistor amplifier Finding oscillation condition in the circuit Compute step response of resonant circuit … Verifying formulas of transfer functions of filters and amplifiers, containing OpAmps, current conveyors, etc. Verifying impedance and admittance formulas of synthetic elements … Verification of circuit principle Simple computations Studying how real properties of active and passive elements affect circuit behavior and finding the ways how to compensate them … Influence of real properties Necessity to work with behavioral models of new circuit elements like CDBA, various types of current and voltage conveyors, CDTAs etc. which still are not commercially available … Working with new circuit elements Necessity to model special dependencies among circuit parameters by means of manipulating data in computer memory … Special effects

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1 Typical problems examples Result: R2*Rz R2*Rz Kv = ------------------------------ R1*Rz +R2*Rz +R2*R1 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 Results:

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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 R 1 = 1344 , R 2 = 123 , R 3 = 672 , R 4 = 116 , R 5 = 685 , R 6 = 94 , R 7 = R 8 = 1k , C 1 = 110pF, C 2 = 25pF, C 3 = 113pF, C 4 = 24pF, C 5 = 156pF, C 6 = 16.5pF, C 7 = 15pF, C 8 = 12pF, C 9 = 8pF frequency response Results:_______________zeros__________________ 5 x 0 _______________poles__________________ -3.04559956366840E6 ± j 6.27631020418348E7 -1.76830262858284E6 ± j 6.70757952287423E7 -1.33796432873573E6 ± j 5.93101288578915E7

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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 Z2 Zin= --------------- gm1 b1 Z1 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 Result: s*( C2*R1*a*b2 ) s*( C2*R1*a*b2 ) Kv = ------------------------------------------------------------- a*b2^(2) + s*( R1*C4 ) + s^(2)*( R3*C2*R1*C4 ) a*b2^(2) + s*( R1*C4 ) + s^(2)*( R3*C2*R1*C4 ) b2=0.95..1.05

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Sallen-Key LP filter- influence of OpAmp properties to frequency response 1 Typical problems examples Influence of real properties from symbolic analysis: frequency responses ideal 1-pole model 2-pole model

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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 symbolic analysis semisymbolic analysis fraction line 1k 10n

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Example – RC cell 2 (Semi)symbolic versus numerical analysis symbolic and semisymbolic 1k 10n _______________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) response to Heaviside step no zeros pole -100000 response to Dirac impulse

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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 Simplification of symbolic results SAG, SBG, SDG

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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 outputinput equivalent signal period of parameter alternation 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. Depending on , various GTFs can represent network behaviour

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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 output input Modified nodal analysis: Solving for initial condition response impulse response

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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 GTF modeling „discrete-time“ behaviour modeling „continuous-time“ behaviour Laplace transform and arrangement

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

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6 Switched linear systems… Computing the GTF Mixed S-Z description of circuits with periodically varying parameters GTF modeling „discrete-time“ behaviour modeling „continuous-time“ behaviour Algorithmic GTF computation:..by numerical integration..solving eigenvalue problem..by a special procedure 1 Finding g*, g x 2 Finding z-domain zeros and poles 3 Finding s-domain zeros and poles

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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 … how to improve SNAP http://snap.webpark.cz …and other programs

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