1. Interconnect II – Class 17
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
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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 1 1 -  2 3
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) + …)
May do with sum of cosines too.
Interconnect II
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4. 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
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5. 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
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6. 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

7. 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

8. 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
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9. Ramp Square Wave
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10. Power Exponential Square Wave
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11. Sum Cosine Square Wave
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12. Applying Frequency Dependent Effects to Digital FunctionsFT
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

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

17. 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
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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
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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
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
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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.
Ideal 400 MHz waveform
400 MHz switching
200 MHz switching
Interconnect II
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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 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
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23. 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
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24. Topology – the Key to a sound designVs Zo
0-2V
Rs=Zo R1
R 2 J In A A’ B B’ C
In J R1 R2
Interconnect II
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