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Interconnect II – Class 22 Prerequisite Reading - Chapter 4 12/4/2002 2 Effects of Frequency Domain Phenomena on Time Domain Digital Signals Key Topics: Frequency Content of Digital Waveforms Frequency Envelope Incorporating frequency domain effects into time domain signals Interconnect II 12/4/2002 3 Decomposing a Digital Signal into Frequency Components • Digital signals are composed of an infinite number of sinusoidal functions – the Fourier series The Fourier series is shown in its progression to approximate a square wave: 1+2+3 1+2 1 1 - 0 2 3 1+2+3+4+5 1+2+3+4 Square wave: Y = 0 for - < x < 0 and Y=1 for 0 < x < Y = 1/2 + 2/pi( sinx + sin3x/3 + sin5x/5 + sin7x/7 … + sin(2m+1)x/(2m+1) + …) 1 2 3 4 5 May do with sum of cosines too. Interconnect II 12/4/2002 4 Frequency Content of Digital Signals • The amplitude of the the sinusoid components are used to construct the “frequency envelope” – Output of FT Tr 0.35 Tr 20dB/decade Pw T 40dB/decade 1 T 1 3 5 7 9 …... Interconnect II Harmonic Number 12/4/2002 5 Estimating the Frequency Content 0.35 come from? Tr • It can be derived from the response of a step function into a filter with time constant tau • Where does that famous equation F V Vinput (1 et / ) • Setting V=0.1Vinput and V=0.9Vinput allows the calculation of the 10-90% risetime in terms of the time constant t1090% t90% t10% 2.3 0.105 2.195 • The frequency response of a 1 pole network is F3dB 1 1 2 2F3dB • Substituting into the step response yields t1090% 1.09 0.35 F3dB F3dB Interconnect II 12/4/2002 6 Estimating the Frequency Content t1090% 0.35 F3dB Edge time factor This equation says: The frequency response of the network with time constant tau will degrade a step function to a risetime of t10-90% The frequency response of the network determines the resulting rise time ( or transition time) The majority of the spectral energy will be contained below F3dB • This is a good “back of the envelope” way to estimate the frequency response of a digital signal. • Simple time constant estimate can take the form L/R, L/Z0, R*C or Z0*C. Interconnect II 12/4/2002 Examining Frequency Content of Digital Signals • The frequency dependent effects described earlier in this class can be applied to each sinusoidal function in the series Digital signal decomposed into its sinusoidal components Frequency domain transfer functions applied to each sinusoidal component Modified sinusoidal functions are then re-combined to construct the altered time digital signal • There are several ways to determine this response Fourier series (just described) Fast Fourier transform (FFT) Widely available in tools such as excel, Mathematica, MathCad… Interconnect II 12/4/2002 7 3 Method of Generating a Square Wave Ramp pulses Use Heavy Side function Used for first pass simulations Power Exponential Pulses Realistic edge that can match silicon performance Used for behavioral simulation that match silicon performance. Sum of Cosines Text book identity. Used to get a quick feel for impact of frequency dependant phenomena on a wave. Interconnect II 12/4/2002 8 9 Ramp Square Wave Ramp Pulse Train RUe( tt) tt ( tt ) tt 1 tt 1 onepulse( tt k) ( RUe( tt period k) RUe( tt pw k period ) ) number_of_pulses Rpt( t ) onepulse( t k) k 0 1 Rpt t i 0 0 50 100 150 ti ns Interconnect II 12/4/2002 10 Power Exponential Square Wave Power Exponetial Pulse Train 2.5 ( tt ) edge( tt) Va 1 e Power Exponential Edge Ppt( tt) edge( tt ) ( tt ) onepulse( tt k) ( Ppt( tt period k) Ppt ( tt pw k period ) ) number_of_pulses Ppt ( t) onepulse( t k) k 0 1 Ppt t i 0 0 50 100 150 ti ns Interconnect II 12/4/2002 11 Sum Cosine Square Wave Sum Cosine Pulse Wave C period 2 Tedge a Establish Fourier Spectrum C period freq ( n ) n 2 period n 1 3 10 Define pulse train of fourier coef i.e. pulse = sum of cosines Fpt ( tt) 1 2 4 2 1 ( 1 2 a) n n 2 cos n a cos ( freq ( n ) tt ) Va 1.5 1 Fpt t i 0.5 0 0.5 0 50 100 150 ti ns Interconnect II 12/4/2002 Applying Frequency Dependent Effects to Digital Functions Input signal into lossy t-line 12 Spectral content of waveform Volts FT 0.35 Trise FFT F Frequency Loss characteristics if t-line Time domain waveform with frequency dependent losses With AC losses Inverse FFT No AC losses Volts Attenuation (V2/V1) Time Time Frequency AC losses will degrade BOTH the amplitude and the edge rate Interconnect II 12/4/2002 13 Assignment Use MathCad to create a pulse wave with Sum of sine waves Sum of ramps Sum of realistic edge waveforms Exponential powers Use MathCad to determine edge time factor for exponential and Gaussian wave, 10% - 90% 20% - 80% Interconnect II 12/4/2002 14 Edge Rate Degradation due to filtering Remember this equation from a few slides ago? t1090% t90% t10% 2.3 0.105 2.195 This equation says: • If a step response is driven into a filter with tine constant tau, the output edge rate is t10-90% • However, realistic edge rates are not step functions • RSS the input edge rate with the filter response Input edge Example: tr = 300ps Zo=50 C=5pF tout=Output edge t1090% 2.195 2.195( Zo C ) 2.195(50 5 pF ) 548 ps tout Tr Tout (300 ps) 2 (548 ps) 2 625 ps 2 2 Interconnect II 12/4/2002 15 Additional Effects Key Topics: Serpentine traces Bends ISI Topology Interconnect II 12/4/2002 16 Effects of a Serpentine Trace • Serpentine traces will exhibit 2 modes of propagation • Typical “straight line” mode • Coupled mode via the parallel sections • Causes the signal to “speed up” because a portion of the signal will propagate perpendicular to the serpentine • ”Speed up” is dependent on the spacing and the length 0.65 0.55 0.45 Volts 0.35 Lp 0.25 S=5 S=15 0.15 S 0.05 4.0E-09 3.5E-09 3.0E-09 Time, s 2.5E-09 2.0E-09 1.5E-09 Interconnect II 1.0E-09 5.0E-10 -0.15 0.0E+00 -0.05 12/4/2002 17 Modeling Serpentines Assignment – Find a the uncoupled trace length that matches the delay of the serpentine route below Use Maxwell Spice/2D modeling of serpentine vs. equal length wave. Trace route on PWB 1” 2 port Tline model 1 oz copper 5 mil space 5mil width 5 mil distance to ground plane Symmetric stripline Use 50 ohm V source w/ 1ns rise time (do for ramp and Gaussian) 5 mil 2 port Tline Model 10 port Transmission Line Spice Model Couple length=2 inches Interconnect II 12/4/2002 18 Rules of Thumb for Serpentine Trace • The following suggestions will help minimize the effect of serpentine traces • Make the minimum spacing between parallel section (s) at least 3-4H, this will minimize the coupling between parallel sections • Minimize the length of the parallel sections (Lp) as much as possible • Embedded microstrips and striplines exhibits less serpentine effects than normal m9ictrostirpsd Interconnect II 12/4/2002 19 Effects of bends • Virtually every PCB design will exhibit bends • The excess area caused by a 90o bend will increase the self capacitance seen at the bend • Empirically inspired model of a 90o bend is simply 1 square of excess capacitance C90o _ bend C11W Capacitance of 1 extra square • Measurements have shown increased delays due to the current components “hugging” the corner increasing the mean length • 2 rights do not necessarily equal a left and a right, especially for wide traces • 45o bends, round and chamfered bends exhibit reduced effects Interconnect II 12/4/2002 20 Inter Symbol Interference • Inter symbol interference (ISI) is reflection noise that effects both amplitude and timing The nature of this interference is cause by a signal not settling to a steady stated value before the next transition occurs. Can have an effect similar to crosstalk but has completely different physics Volts Ideal waveform beginning transition from low to high with no reflections or losses Timing difference Waveform beginning transition from low to high with unsettled noise cased by reflections. Receiver switching threshold Time Different starting point due to ISI Interconnect II 12/4/2002 21 Inter Symbol Interference • ISI can dramatically affect the signal quality Depending on the switching rate/pattern, significant differences in waveform shape can be realized – one or two patterns won’t produce worst case If the designer does not account for this effect, switching patterns that are unaccounted for result in latent product defects. 400 MHz switching 200 MHz switching 4 3 Volts 2 1 0 -1 Ideal 400 MHz waveform -2 1.E-08 9.E-09 8.E-09 7.E-09 6.E-09 5.E-09 4.E-09 3.E-09 2.E-09 1.E-09 0.E+00 Time, s Interconnect II 12/4/2002 22 Topology – the Key to a sound design • What about the case where there is more than one receiver, or more than one driver (e.g., a Multiprocessor FSB) Rs=Zo Vs 0-2V L1 Zo1 Zo2 Receiver 1 (L1=L2) Zo3 Receiver 2 • There will be an impedance discontinuity at the junction The equivalent input impedance looking into the junction will be the parallel combination of Zo2 and Zo1 Z || Z Z o1 o 2 o3 Z o 2 || Z o 3 Z o1 • This model can be simplified and solved with lattice diagrams Valid when L1=L2 Rs=Zo Vs 0-2V L1 Zo1 L2 Z=Zo2||Zo3 Interconnect II 12/4/2002 23 Topology – the Key to a sound design • Now, consider the case where L2 and L3 are NOT Equal Zo2 Rs=Zo Vs 0-2V Zo1 Receiver 1 Zo3 Receiver 2 2.5 Voltage at receiver 2.0 The reflections from the receiver discontinuities will not arrive at the same time; the 2 segment simplification is not applicable 1.5 1.0 This topology will ring with a frequency dependant in L2 and L3 0.5 This topology can be solved with a multi-segment lattice diagram 0.0 0.0 2.0 4.0 6.0 8.0 10.0 Time, ns Interconnect II 12/4/2002 24 Topology – the Key to a sound design Rs=Zo In Vs 0-2V 1 0 J R1 Zo R2 A A’ B B’ C In J Zo 1 Zo Rs Zo Zo 1 2 2 3 Zo 3 Zo 2 2 T2 T3 1 3 2 2 4 A 3 3 3 2 2 2 2 8 B 3 3 9 9 9 2 2 2 2 2 2 2 2 16 C 3 3 9 9 3 3 9 9 9 2 2 4 A' 3 3 3 2 2 2 2 2 2 20 B' 3 3 3 3 9 9 9 Vinitial 2 Zo T3 T2 Zo 2 3 R1 R2 Interconnect II 12/4/2002