Semi symbolic computer analysis of continuoustime and switched

(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 2 3 4 5 6 7 Typical problems (Semi)symbolic versus numerical analysis Needs versus reality Needs SNAP Switched linear systems – generalized s-z transfer functions Instead of Conclusion

1 Typical problems to solve in the area of linear analogue systems Simple computations Verification of circuit principle Influence of real properties Working with new circuit elements Special effects Finding DC voltages and currents Harmonic steady state responses Voltage gain of loaded divider Balance condition or ACfunctions bridge Verifying formulasofof. DC transfer Computing parameters of filters two-port and amplifiers, containing Finding gain formula of transistoretc. amplifier Op. Amps, current conveyors, Finding oscillation condition in the circuit Verifying impedance and admittance Studying how real properties of active and Compute stepofresponse resonant circuit formulas syntheticofelements passive elements affect circuit behavior and … finding the ways how to compensate them … 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 Necessity to model special dependencies available among circuit parameters by means of … manipulating data in computer memory …

1 Typical problems examples Simple computations Loaded voltage divider – compute voltage transfer function Result: R 2*Rz Kv = ---------------R 1*Rz +R 2*R 1

1 Typical problems examples Simple computations Maxwell-Wien bridge – compute balance condition Results: Rx R = R 1 R 2 Lx = R 1 R 2 C

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

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

1 Typical problems examples Simple computations Colpitts oscillator – derive oscillation condition Results: h 21 e=C 2/C 1=100, then wosc=sqrt[(1+h 21)/(L*C 2)], fosc=wosc/(2*pi)=715 k. Hz.

1 Typical problems examples Simple computations Resonant circuit – find step response Result: 0. 1596*exp(-50000*t)*sin( 626703*t)

1 Typical problems examples Verification of circuit principle FDNR in series with resistance Result: Zin=R 1/2+1/(D*s^2) D=2*R 3*C 1^2

1 Typical problems examples Verification of circuit principle DC precise LP filter. Frequency response looks good, but. . . Result: filter poles: -971695 + j 484850 -971695 - j 484850 -321953 195172 + j 461620 195172 - j 461620 FILTER IS UNSTABLE!

1 Typical problems frequency response examples Working with new circuit elements 10 MHz bandpass filter containing CDBA elementsfind zeros and poles of current transfer function and frequency response Results: ________zeros_________ 5 x 0 R 1 = 1344 W, R 2 = 123 W, R 3 = 672 W, R 4 = 116 W, ________poles_________ R 5 = 685 W, R 6 = 94 W, R 7 = R 8 = 1 k. W, C 1 = 110 p. F, C 2 = 25 p. F, C 3 = 113 p. F, C 4 = 24 p. F, C 5 = 156 p. F, C 6 = 16. 5 p. F, C 7 = 15 p. F, -3. 04559956366840 E 6 ± j 6. 27631020418348 E 7 C 8 = 12 p. F, C 9 = 8 p. F -1. 76830262858284 E 6 ± j 6. 70757952287423 E 7 -1. 33796432873573 E 6 ± j 5. 93101288578915 E 7

1 Typical problems examples Working with new circuit elements Impedance converter/inverter with two CTTA elements with parameters b 1, gm 1, b 2, gm 2. Derive input impedance. Result: Z 2 Zin= -------gm 1 b 1 Z 1

1 Typical problems examples Influence of real properties b 2=0. 95. . 1. 05 1 MHz bandpass filter – find how CCII nonidealities a 1, b 2 1 affect the transfer function Result: s*( C 2*R 1*a*b 2 ) Kv = ------------------------------a*b 2^(2) + s*( R 1*C 4 ) + s^(2)*( R 3*C 2*R 1*C 4 )

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

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

2 (Semi)symbolic versus numerical analysis Forms of the analysis outputs 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, . . )

2 (Semi)symbolic versus numerical analysis Example – RC cell symbolic and semisymbolic 1 k 10 n fraction line symbolic analysis semisymbolic analysis

2 (Semi)symbolic versus numerical analysis Example – RC cell symbolic and semisymbolic 1 k 10 n ________zeros_________ none no zeros ________poles_________ -1. 0000000 E+0005 pole -100000 ______step response_______ response to Heaviside step 1. 0000000 E+0000 -1. 0000000 E+0000*exp(-1. 0000000 E+0005*t) ______pulse response_______ 1. 0000000 E+0005*exp(-1. 0000000 E+0005*t) response to Dirac impulse

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

3 Needs versus reality Limitations of typical commercial circuit simulators 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…

3 Needs versus reality Wanted: new software tool for analysis of large linear systems 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, Math. Cad etc. User-modified behavioral models based on MNA Free of charge…

3 Needs versus reality pro – and – con Why (semi)symbolic computation? 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

3 Needs versus reality pro – and – con Why NOT (semi)symbolic computation? 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 SAG, SBG, SDG Simplification of symbolic results

4 Needs Symbolic, Semisymbolic and Numerical links

4 Needs Symbolic way (1, 2, 3): small-size to moderatesize systems, excellent precision; ? (1), utilization of “optional precision” (2, 3) Semisymbolic way (4, 3): moderate-size to largesize systems, problematic Computing system eigenvalues precision; FFT, Faddeyev algorithm (4), Laguer, method of accompanying matrix, “optional precision” (3) Numerical way (5): large circuits, problematic precision; QR, QZ, . . , “optional precision”

4 Needs Semisymbolic way (4, 3): moderate-size to large. Computing time responses size systems, precision depends on computing eigenvalues; partial fraction expansion, “optional precision” (3) Numerical way (5): large circuits, good precision; classical integration formulas

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, Math. Cad etc. User-modified behavioral models based on MNA Free on http: //snap. webpark. cz

5 SNAP Symbolic and Numerical Analysis Program conception

5 SNAP Symbolic and Numerical Analysis Program

5 SNAP Symbolic and Numerical Analysis Program

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. ?

6 Switched linear systems… What is the GTF Generalized Transfer Function of circuits with periodically varying parameters period of parameter alternation output input Equivalent signal: - interpolates samples v(k. T+ T) - its spectral components fall to the spectral area of w(t). There is infinite number of equivalent signals for <0, 1) equivalent signal Depending on , various GTFs can represent network behaviour GTF is the ratio of Fourier/Laplace transformations of equivalent output signal and input signal.

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

6 Switched linear systems… What is the GTF Mixed S-Z description of circuits with periodically varying parameters Modified nodal analysis: input output Solving for impulse response initial condition response

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

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

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

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

6 Instead of Conclusion Contemporary problems …. … how to improve SNAP http: //snap. webpark. cz …and other programs ? Special methods (SBE) of approximate symbolic analysis ? The rational arithmetic (RA) ? The “optional precision” and “infinite precision” arithmetic (OPA, IPA) ? Topological methods of matrix deflation ? Solving the eigenvalue problem by means of RA, OPA, and IPA ? Solving the polynomial roots from symbolic results by means of OPA