Download Effects of Frequency Domain Phenomena on Time Domain Digital

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
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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+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
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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
Tr
0.35
Tr
20dB/decade
Pw
T
40dB/decade
1
T
1
3
5 7 9 …...
Interconnect II
Harmonic Number
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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  et / )
• Setting V=0.1Vinput and V=0.9Vinput allows the calculation of the
10-90% risetime in terms of the time constant
t1090%  t90%  t10%  2.3  0.105  2.195
• The frequency response of a 1 pole network is
F3dB
1
1

 
2
2F3dB
• Substituting into the step response yields
t1090% 
1.09
0.35

F3dB F3dB
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6
Estimating the Frequency Content
t1090%
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
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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…
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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
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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
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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
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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
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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
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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%
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Edge Rate Degradation due to filtering
Remember this equation from a few slides ago?
t1090%  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
t1090%  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
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Additional Effects
Key Topics:
 Serpentine traces
 Bends
 ISI
 Topology
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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
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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
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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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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
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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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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
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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
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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
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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
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