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- 1. May 2nd 2013 Copyright 2013 Piero Belforte1Prediction of rise time errors of a cascade ofequal behavioral cells.IntroductionIn this paper the effects of finite rise time of time-domain stepresponse on a chain of equal behavioral cells are analyzed. Thechain output delay and rise time are obtained by time-domainsimulation using the SWAN/DWS (1) wave circuit simulator andthe Spicy SWAN application available on the WEB and on mobiledevices (2,https://www.ischematics.com/webspicy/portal.py#).Two situations are considered:Ramp shape and erfc (complementary error function) shape ofbehavioral time-domain description of the single cell withequivalent rise times.RAMP SHAPED TRANSFER FUNCTIONThe situation of Fig.1 is simulated to get the response at even celloutputs of a 10-cell chain. A single block is modeled as aSWAN/DWS VCVS (Voltage Controlled Voltage Source) whosestatic transfer function is linear and of unit value. The dynamicportion s(t) of the VCVSs control link is described with a two-breakpoints PWL behavior corresponding to a ramp with a totalrise time of 20ps. The last two parameters of the VCVS controllink are the delay set to 0ns (will be approximated as a unit time
- 2. May 2nd 2013 Copyright 2013 Piero Belforte2step, 20fs by DWS) and the output resistance of the VS (set to 0ohms).Figure 1: circuital configuration to calculate the step response of a 10 equalblocks with ramp step response.Figure 2: Spicy SWAN voltage waveforms at even tap outputs of circuit of Fig.1The simulation results with a time step of 100 femtoseconds on awindow of 200ps are shown in Figure 2 (1000 samples per
- 3. May 2nd 2013 Copyright 2013 Piero Belforte3waveform). The increasing 50% point delay and rise time valuesare easily measurable as function of tap number.2-port S-parameter blocksA situation like that of figure 1 can be also modeled by DWSusing a chain of equal 2-port S-parameters blocks (Fig.3). Eachblock is both symmetrical and reciprocal and can be characterizedby its time-domain S11 and S21 behaviors ( BTM: BehavioralTime Model). For sake of semplicity S11 is assumed to be zero.S21 has a ramp shape with a 20 ps rise time described by a 2-point PWL behavior. Due to DWS stability no particularrequirement is needed for S11-S21 relationship. To get voltagevalues similar to those of circuit of fig.1 a 2V step input isrequired because the S-parameters are related to a 50 ohmimpedance and the chain is terminated by a 50 ohm resistance(R0).Figure 3: circuit configuration to get the step response of a 10 equal S-parametersblocks with ramp step response of S21
- 4. May 2nd 2013 Copyright 2013 Piero Belforte4Figure 4: Voltage waveforms at even tap outputs of circuit of Fig.3Figure 5 : Calculation of the ratio between the output rise time (10-90%) and unit-cell rise time
- 5. May 2nd 2013 Copyright 2013 Piero Belforte5As can be easily noticed from Fig.4 the tap waveforms are exactlythe same of that of Fig.2. A perfect equivalence is observedbetween the transfer function blocks and impedance matched S-parameter block implementations. The results got from transferfunction implementation are still applicable to a cascade of S-parameters blocks. This is particularly interesting because chainof BTM cells can be utilized to model interconnections like cablesand p.c.b traces (3).The total delay of the chain (101ps) is pointed out by a cursor onthe simulated waveform. This delay approaches the half of ramptotal rise time (20ps) multiplied by the number of cells (10). Theextra 1ps delay is the error due to simulation time step (100fs *10).Fig. 5 reports the calculation of the ratio between the output risetime (47ps,10%-90%) and the unit-cell rise time (16ps,10%-90%).This ratio (2.95) approaches the square root of the number ofcells (10).
- 6. May 2nd 2013 Copyright 2013 Piero Belforte6CASCADE of 1000 CELLSThanks to DWS speed it easy to extend this investigation to asituation where 1000 unit cells are connected in a chain (Fig.6).Figure 6: 1000-cell of transfer function blocks using the CHAIN utility of DWSFiFIgure 7: Output rise time calculation after 1000 cells
- 7. May 2nd 2013 Copyright 2013 Piero Belforte7As can easily verified in the plot of figure 7 the rule of the squareroot of the number of cells still apply to the chain out rise timeeven in this case.ERFC SHAPED TRANSFER FUNCTIONTo point out the unit-cell transfer function shape effects, acouple of equivalent rise time generators has been built up.Two set of generators related to 10%-90% and 20%-80%equivalent rise times respectively are built up to compare theirwaveforms (Fig.8). An extra delay has been added to rampgenerators to compensate erfc higher delay at 50% of its swing.Figure 8: Equivalent 50% point delay and rise time ramp and erfc shapesFigure 9 shows the waveforms of the 4 generators superimposed.
- 8. May 2nd 2013 Copyright 2013 Piero Belforte8The equivalence of 50% point delay and rise times is pointed outin Fig.9.Figure 9: Wave shape comparison of the generators of Fig.8Fig. 10 reports the total delay at the output of 1000 erfc shapedunit cells. The previous rule of calculation (half rise timemultiplied by the number of cells) is still verified.Figure 10: Total delay (50% point) of 1000 cells with erfc shaped transfer function
- 9. May 2nd 2013 Copyright 2013 Piero Belforte9Fig. 11 reports the rise time at the ouput of 1000 erfc-shapedunit cells. The previous rule of calculation (Unit cell rise timemultiplied by the square root of number of cells) is still verifiedwith an error of 10% (556ps instead of 506ps).Figure 11: 10%-90% rise time at the output of 1000 cells having erfc shapedtransfer function.Previous plots (Fig.10 and Fig. 11) are obtained with a cell erfcshape modeled with its PWL (Piece Wise Linear) approximationbehavior using 10 breakpoints (Fig.12).
- 10. May 2nd 2013 Copyright 2013 Piero Belforte10Figure12: 10-breakpoint PWL approximation of erfc behavior used in the unit celltransfer function (BTM)Figure 12 shows the Spicy SWAN schematic related to thecomparison between the pulse response of two cascades of 1000cells having equivalent delay an 10%-90% rise times but with erfcand linear (ramp) shapes respectively.Figure 13: 1000-cell outputs, comparison between equivalent ramp and erfcshapes of unit-cell transfer function
- 11. May 2nd 2013 Copyright 2013 Piero Belforte11As can be easily noticed in Figure 13 the erfc shaped outputs aresimilar with a 50% point delay difference of 70ps (erfc moredelayed) for a total delay of about 16ns corresponding to a +.4%.The rise time difference is 75ps (erfc slower) over a risetime ofabout 500ps (+15% for the erfc shape).Figure 14: 1000 cell outputs waveforms , comparison between equivalent rampand erfc shapes of unit-cell transfer function
- 12. May 2nd 2013 Copyright 2013 Piero Belforte12Concluding remarksPrevious simulations demonstrate that cascading N equal blockeach showing a step response rise time tr, the chain ouput showsa total rise time TRT that is about :Eq. 1 TRT= tr * SQRT(N)The total 50% point delay of the chain, TDT, is about:Eq. 2 TDT= td * Nwhere td is the 50% point delay of the single cell.Equations 1 and 2 applies to both ramp and erfc TransferFunction of the unit cell with a small difference in overall outputwave shapes shown in Fig.13 in the case of a cascade of 1000cells.Equations 1 and 2 applies to cascades of both time-domain(BTM) transfer function blocks and BTM 2-port S-parameterblocks.
- 13. May 2nd 2013 Copyright 2013 Piero Belforte13The above considerations are to be taken into account whencascading several equal blocks starting from the Time-domaintransfer function of each block obtained experimentally (eg.from TDR measures) or by simulation (eg. from 2D-3D fieldsolvers).The previous situation is very common for fast and accuratemodeling of physical interconnects (cables, p.c.b. traces etc.).In this case the response of a total length L is obtained from acascade of N equal cells related to a sub-multiple length l of thesame interconnect (L= N*l where N is an integer). If the responseof the unit cell of length l is obtained from a band-limitedinstrument (like a TDR having a 20ps rise time pulse) or a band-limited numerical method (like a 3D full wave field solver) therewill be a delay error and a rise time error (or bandwidth error) onthe overall response as previously shown .The rise time (bandwidth) absolute error increases with thesquare root of the number of cascaded cells.The good thing is that the ratio between rise time error andphysical delay of the interconnect (relative error) decreases withthe length of the interconnect by a factor proportional to thesquare root of the number of cells utilized.
- 14. May 2nd 2013 Copyright 2013 Piero Belforte14WEB REFERENCES1) http://www.slideshare.net/PieroBelforte1/dws-84-manualfinal270120132) https://www.ischematics.com/webspicy/portal.py#3) http://www.slideshare.net/PieroBelforte1/2009-pb-dwsmultigigabitmodelsoflossycoupledlinesNOTE : some of Spicy SWAN circuits shown in this paper areavailable in the public libraries available on line at Ischematicswebsite (https://www.ischematics.com/). All simulationsrelated to previous circuits run in few seconds (SWAN mode).

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