Interconnect II – Class 17 Prerequisite Reading - Chapter 4 12/4/2002

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

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 1 + 2 + 3 1 1 -  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

Frequency Content of Digital Signals The amplitude of the the sinusoid components are used to construct the “frequency envelope” – Output of FT 1 3 5 7 9 …... Harmonic Number 20dB/decade 40dB/decade Pw T Tr Interconnect II 12/4/2002

Estimating the Frequency Content Where does that famous equation come from? It can be derived from the response of a step function into a filter with time constant tau Setting V=0.1Vinput and V=0.9Vinput allows the calculation of the 10-90% risetime in terms of the time constant The frequency response of a 1 pole network is Substituting into the step response yields Interconnect II 12/4/2002

Estimating the Frequency Content 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

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

Ramp Square Wave Interconnect II 12/4/2002

Power Exponential Square Wave Interconnect II 12/4/2002

Sum Cosine Square Wave Interconnect II 12/4/2002

Applying Frequency Dependent Effects to Digital Functions FT Multiply Inverse FFT Frequency Attenuation (V2/V1) Volts Time No AC losses With AC losses AC losses will degrade BOTH the amplitude and the edge rate Input signal into lossy t-line Spectral content of waveform Loss characteristics if t-line Time domain waveform with frequency dependent losses Interconnect II 12/4/2002

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

Edge Rate Degradation due to filtering 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 Remember this equation from a few slides ago? Zo=50 C=5pF tr = 300ps Input edge tout=Output edge Example: Interconnect II 12/4/2002

Additional Effects Key Topics: Serpentine traces Bends ISI Topology Interconnect II 12/4/2002

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 Lp S Interconnect II 12/4/2002

Transmission Line Spice Model 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 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) 1” 2 port Tline model 5 mil 2 port Tline Model 10 port Transmission Line Spice Model Couple length=2 inches Interconnect II 12/4/2002

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

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 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

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

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. Ideal 400 MHz waveform 400 MHz switching 200 MHz switching Interconnect II 12/4/2002

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 Multi-processor FSB) Vs Zo3 Zo2 0-2V Zo1 Rs=Zo Receiver 1 Receiver 2 L2 L3 (L1=L2) 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 This model can be simplified and solved with lattice diagrams Valid when L1=L2 Z=Zo2||Zo3 L1 Interconnect II 12/4/2002

Topology – the Key to a sound design Now, consider the case where L2 and L3 are NOT Equal Zo2 Receiver 1 Rs=Zo L2 Zo1 L3 Vs 0-2V Zo3 Receiver 2 The reflections from the receiver discontinuities will not arrive at the same time; the 2 segment simplification is not applicable This topology will ring with a frequency dependant in L2 and L3 This topology can be solved with a multi-segment lattice diagram Interconnect II 12/4/2002

Topology – the Key to a sound design Vs Zo 0-2V Rs=Zo R1 R 2 J In A A’ B B’ C In J R1 R2 Interconnect II 12/4/2002