Signal Integrity Analysis of LC lopass Filter
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Transcript of Signal Integrity Analysis of LC lopass Filter
Pg. 1
Signal Integrity Analysis of a 2nd Order
Low Pass Filter
An Intuitive Approach
Andrew Josephson
Overview
• Motivation
• Review simple 2nd Order Low Pass LC Filter
– Develop S-Domain Transfer Function
– Case study: Compare three different 2nd order 2.25GHz LPF
• S-parameters
• Energy rejection mechanism
• Extend analysis to multiple low pass topologies separated by
ideal transmission line
– Eye closure vs. t-line delay
– Impedance (mis)matching in frequency domain and correlation to eye
diagram
• The effect of real lossy transmission lines
Motivation
• 2nd Order system analysis is important in many engineering
disciplines
– They are low order and easy to analyze
– Exhibit complex behavior like overshoot/ringing etc.
• In signal integrity applications, simple LC filters are easily
described using 2nd order system principles
– Since all DC-coupled interconnect is low pass in nature, a thorough
understanding of 2nd order LC circuits leads to significant
understanding and intuition in more complex interconnect problems
Pg. 3
Review of 2nd order LC circuits
Pg. 4
2nd Order Low Pass Filter Analysis Low Pass Filter in High Speed Environments
Transmitter Receiver
L
C
R
R
L
C
LC
10 Intuition tells us
In high speed systems however, there are typically
source and load terminations: Controlled Impedance
2nd Order Low Pass Filter Analysis Developing a Transfer Function
R
R
L
C Vin
Vout
Vin
Vo =
sLRsCR
sCR
)/1(||
)/1(|| =
sLRsRC
R
sRC
R
1
1 =
sLRRLCsCsRR
R
22
Voltage Division in the S-Domain Gives
Eq. 1
2nd Order Low Pass Filter Analysis Developing a Transfer Function
Rewriting Eq.
1 gives
Vin
Vo =
2 2
2
2 n n
n
s s
Eq.2
Eq. 2 has the form of
a typical 2nd order
system with transfer
function
Eq.3
This system has
complex conjugate
poles at
22 1 nns Eq.4
Vin
Vo =
RCRLsRLCs
R
2][ 22 =
2
1
LCL
R
RCss
LC
21
2
2
2nd Order Low Pass Filter Analysis Developing a Transfer Function
Comparing Eqtns 2
& 3 yields the
following system
parameter definitions
LCn
2
L
R
RCn
12
L
CR
C
L
R
1
8
1
or
and Eq.5
Eq.6
Note that for the
special case when C
LR
2
1
8
2
2
1 is the value of the damping coefficient that leads to the
quickest step response without overshoot and ringing. This
has important implications in high speed digital systems as
good interconnect step responses preserve eye opening during
channel propagation.
Eq.7
2nd Order Low Pass Filter Analysis An RF filter designers approach
R
R
L
C Vin
Vout
Question: What physical mechanism prevents the high
frequency energy from getting through this low pass filter
topology? Where does the high frequency energy go?
Answer: The LC filter topology does not contain any resistors.
It can NOT dissipate power. This filter topology reflects power,
through an impedance mismatch, back towards the generator
where it is absorbed by the source termination.
Even a filter where exhibits an impedance
mismatch.
C
LR
2nd Order Low Pass Filter Analysis An RF filter designers approach
• Frequency response of three different 2.25GHz 2nd order low pass filters
– All three have -3dB bandwidths at Fc = 2.25GHz
– Note that for each filter, |S11|=|S21| at Fc
– The impedance matched filter (Zo = 50 Ohms) has the best passband return loss
(steepest slope of S11 up to Fc) Pg. 10
0
0
0
Port1 Port2
Port3 Port4
Port5 Port6
7.33nH
L1
5nH
L2
0.55nH
L3
1pF
C4
2pF
C7
3pF
C8
Zo = √(L/C) = 13.5 Ω
Zo = √(L/C) = 50 Ω
Zo = √(L/C) = 85.6 Ω
0.10 1.00 10.00F [GHz]
-60.00
-50.00
-40.00
-30.00
-20.00
-10.00
0.00
Y1
00_3_low_passXY Plot 1 ANSOFT
Curve Info
dB(S(Port2,Port1))LinearFrequency
dB(S(Port4,Port3))LinearFrequency
dB(S(Port6,Port5))LinearFrequency
dB(S(Port1,Port1))LinearFrequency
dB(S(Port3,Port3))LinearFrequency
dB(S(Port5,Port5))LinearFrequency
Re
turn
& In
se
rtio
n L
os
s (
dB
)
2nd Order Low Pass Filter Analysis Analysis Summary
• In a controlled impedance environment, 2nd order low pass
filters generate impedance mismatches with the source/load
terminations
• The impedance mismatch is frequency dependent and is the
physical mechanism that creates the low pass filter response
• When sqrt(L/C) = Zo, the reflection is minimized but still
present
– Creates the filter topology with the steepest slope in S11 up to Fc
• The return loss of any 2nd order LC filter is -3dB at Fc
Pg. 11
Pg. 12
Extending Analysis to Multiple LC
Circuits Separated by Ideal T-line
Multiple Filter topologies with Prop Delay
• This type of problem is much more interesting in both the frequency and time domains.
• This circuit topology is extremely common in signal integrity analysis where identical reflective discontinuities are often separated by uniform transmission line.
• Examples
– Via ->PWB Route ->Via
– Connector ->Cable -> Connector
– Package -> PWB Route -> Package
• Before investigating the relationship between the periodic impedance mismatch created by the addition of the t-line and the effect on the eye diagram, we will merely observe that the eye can be tuned to local maximum and minimum data dependent jitter as a function of t-line delay
Pg. 13
Transmitter Receiver T-line
Input
Impedance
The Effect of T-line Delay Creating Local Jitter Maximums
• Identical 2Gbps random data pattern (500ps bits)
• The delay of the ideal t-line has been “tuned” in each
case to create a local maximum in DDJ
• This happens approximately when the largest
reflective “blip” occurs near the crosspoint timing
Pg. 14
0
0
0
0
0
0
0 0
0 0
0 0
0
0
00
0
0
V125
V126
V127
50
R128
50
R129
50
R130
V
Name=Vout3
VName=Vout2
VName=Vout1
50
R134
50
R135
50
R136
Z0=50
TD=2ns
Z0=50
TD=1.95ns
Z0=50
TD=1.92ns
7.33nH
5nH
0.55nH
1pF
2pF
3pF3pF
2pF
1pF
0.55nH
5nH
7.33nH
Zo = √(L/C) = 50 Ω
Zo = √(L/C) = 85.6 Ω
Zo = √(L/C) = 13.5 Ω
Zo = 13.5 Ω
Zo = 50 Ω
Zo = 85.6 Ω
“blip maximum”
The Effect of T-line Delay Creating Local Jitter Minimums
Pg. 15
• Identical 2Gbps random data pattern (500ps bits)
• The delay of the ideal t-line has been “tuned” in each
case to create a local minimum in DDJ
• This happens approximately when the reflective blip
minimum is aligned with the crosspoint
“blip minimum”
0
0
0
0
0
0
0 0
0 0
0 0
0
0
00
0
0
V125
V126
V127
50
R128
50
R129
50
R130
V
Name=Vout3
V
Name=Vout2
V
Name=Vout1
50
R134
50
R135
50
R136
Z0=50
TD=2.2ns
Z0=50
TD=2.145ns
Z0=50
TD=2.097ns
7.33nH
5nH
0.55nH
1pF
2pF
3pF3pF
2pF
1pF
0.55nH
5nH
7.33nH
Zo = √(L/C) = 50 Ω
Zo = √(L/C) = 85.6 Ω
Zo = √(L/C) = 13.5 Ω
Zo = 13.5 Ω
Zo = 50 Ω
Zo = 85.6 Ω
The Effect of T-line Delay Periodic Eye Closure
• Focus on filter with largest “reflective blip” (Zo = 13.5 Ohms)
• Sweep T-line delay from 10ps to 2000ps in 10ps steps
• Measure vertical eye opening (mV) and DDJ (ps) for each T-line delay
step
– Periodic response for delays larger t_delay = 500ps • Up until this delay, we have not been able to “fit” a pipelined bit into the t-line
– What can be identified at points of local jitter minimums? Pg. 16
0 0
0 0
00
V126
50
R129
V
Name=Vout2
50
R136
Z0=50
TD=t_delay
0.55nH
3pF3pF
0.55nH
0
200
400
600
800
1000
0 500 1000 1500 2000
Ve
rtic
al E
ye O
pe
nin
g (m
V)
T-line Delay (ps)
Eye Opening vs T-line Delay
0
10
20
30
40
50
60
70
80
0 500 1000 1500 2000
DD
J (p
s)
T-line Delay (ps)
Data Dependent Jitter vs T-line Dealy
Zo = √(L/C) = 13.5 Ω Zo = √(L/C) = 13.5 Ω
0
0 0
000
Port1 Port2
50
R136
Z0=50
TD=t_delay
0.55nH
3pF3pF
0.55nH
50
R197
5.002.001.000.500.200.00
5.00
-5.00
2.00
-2.00
1.00
-1.00
0.50
-0.50
0.20
-0.20
0.000
10
20
30
40
50
60
708090100
110
120
130
140
150
160
170
180
-170
-160
-150
-140
-130
-120
-110-100 -90 -80
-70
-60
-50
-40
-30
-20
-10
05_low_pass_delay_sweep_zSmith Chart 3 ANSOFT
m2
m3
Curve Info
S(Port1,Port1)t_delay='10ps'
S(Port2,Port2)t_delay='200ps'
Name F Ang Mag RX
m2 1.0000 -113.3362 0.4011 0.5675 - 0.4982i
m3 1.0000 102.6638 0.4011 0.6277 + 0.5855i
The Effect of T-line Delay Local Jitter Minimum
• Break circuit to measure input
impedance
– Looking into t-line (blue)
– Looking into LC filter back towards
generator (red)
• At t_delay = 200ps (first DDJ min) the
impedance looking into the delay line
(blue) is near complex conjugately
matched to the terminated filter (red) at
1.0GHz
– 2.0Gbps data rate fundamental
frequency = 1GHz
• Suggests jitter minimums occur near
complex conjugate impedance
matching
– Condition for maximum power transfer
– Expect jitter minimums at t_delay =
200ps + n*1000ps
Pg. 17
0
10
20
30
40
50
60
70
80
0 500 1000 1500 2000
DD
J (p
s)
T-line Delay (ps)
Data Dependent Jitter vs T-line Dealy
0
0 0
000
Port1 Port2
50
R136
Z0=50
TD=t_delay
0.55nH
3pF3pF
0.55nH
50
R197
Input Impedance Looking into T-line
at Local Jitter Minimums
Delay (ps) Mag Z (normalized) Ang Z (deg)
200 0.4011 102.66
450 0.4011 -77.33
700 0.4011 102.66
950 0.4011 -77.33
1200 0.4011 102.66
1450 0.4011 -77.33
1700 0.4011 102.66
1950 0.4011 -77.33
The Effect of T-line Delay Local Jitter Minimum
• Jitter minimums also occur at t_delay =
200ps + n*250ps
• Once the first complex conjugate
matching condition is established
(t_delay = 200ps), local DDJ minima
occur at every additional half bit delay
(+ n*250ps) – Suggest “roundtrip” path delay is important
• To explain the location of the DDJ
maximums, we need to look at what is
happening to the eye in the time
domain first
Pg. 18
The Effect of T-line Delay Local Jitter Maximum
• Beginning with a t_delay = 200ps + n*250ps to establish a local DDJ
minimum at 1200ps, we observe the effect of adding more delay and
sliding the largest reflective “blip” to the right through the eye over a
250ps span Pg. 19
0
10
20
30
40
50
60
70
80
90
100
0
100
200
300
400
500
600
700
800
900
1000
1000 1100 1200 1300 1400 1500
DD
J (p
s)
Ve
rtic
le E
ye O
pe
nin
g (m
V)
T-line Delay (ps)
Eye Closure vs. T-line Delay
Vertical Opening (mv)
Jitter_pk_pk (ps)
Td = 1200 Td = 1320 Td = 1360 Td = 1450Td = 1240
The Effect of T-line Delay Local Jitter Maximum
• The local jitter minimum at
t_delay = 1200ps is explained
through it’s relationship to the
complex conjugate matched
condition which is rooted in
frequency domain impedance
• The local jitter maximum
however is explained in the time
domain from the above reference
time for local DDJ minimum
– It occurs approximately one half
“blip” time later when the reflective
“blip” maximum is aligned with the
crosspoint
– The width of the “blip” is a function of
both the interconnect AND the
ristime of the data pattern
Pg. 20
Jitter Minimum
at Td = 1200
Jitter Maximum
at Td = 1240
Pg. 21
The Effect of Real Lossy T-lines
The Effect of T-line Loss Co-propagating Reflections
• When the generator turns on, the first bit creates a reflection from the first LC filter
– This reflection is immediately absorbed by the source termination
• A filtered version of the data stream then enters the transmission line and propagates in the +Z direction
towards the receiver termination (left to right forwards propagation)
• When the bit gets to the 2nd LC filter, a portion is reflected again and travels right to left in the –Z direction
while most of the un-reflected portion of the bit’s power is delivered to the receiver termination.
• The next bits in the sequence that are being launched into the transmission line at some later time cannot
linearly add with the backwards propagating reflection (exp[- β *z] + exp[+β*z) = (exp[- β *z] + exp[+β*z)
– In order for the reflected “blip” to effect they eye diagram as described in the previous slides, it must reflect again off
the impedance mistmach from the first filter and co-propagate with the next data bits (round trip delay)
– Suggests that controlled impedance attenuator circuits will reduce DDJ since the data sequence is attenuated once
travelling through the attenuator to the load resistor and the blip must be attenuated twice to satisfy the co-
propagating condition
Pg. 22
Transmitter ReceiverT-line
Input
Impedance
The Effect of T-line Loss Co-propagating Reflections
• The following example demonstrates the reduction in DDJ through the addition of controlled
impedance loss (loss with near linear phase response)
– Placing a 2dB attenuator in the middle of the T-line will reduce the magnitude of the reflective “blips”
(reduce DDJ), at the cost of attenuating the vertical eye opening as well
– The same effect is realized with a Zo = 50 Ohm, Td = 2ns lossy stripline designed to have -2dB of
insertion loss at the data rate fundamental frequency (F = 1GHz)
Pg. 23
00
0
0
0
0
0 0 00
0
Z0=50
TD=2ns
3pF
0.55nH
3pF
0.55nH
0.55nH
3pF
0.55nH
3pF
Z0=50
TD=1ns
Z0=50
TD=1ns
215.24
R233
5.73
R234
5.73
R235
0
0
50
R243
50
R244
V245
V246
0
0
50
R253
50
R254
V
Name=Vout1
V
Name=Vout2
P=11.64in
W=4mil
V272
50
R273
0
3pF
0.55nH0.55nH
3pF
00
50
R281
0
V
Name=Vout3
Ideal
Transmission
Line
Ideal
Transmission
Line
Ideal
Transmission
Line
2dB Tee
Attenuator
Real Lossy
Transmission Line
The Effect of T-line Loss Co-propagating Reflections
• Ideal t-line
– Eye Opening = 707mV
– DDJ = 72ps
• Ideal t-line with 2dB attenuator
– Eye Opening = 622mV
– DDJ = 40ps
• Real Lossy t-line
– Eye Opening = 571mV
– DDJ = 51ps
• Why isn’t the -2dB lossy
transmission line as effective
as the tee attenuator in
reducing DDJ?
Pg. 24
Ideal
Transmission
Line
2dB Tee
Attenuator
Real Lossy
Transmission Line
0.00 2.00 4.00 6.00 8.00 10.00F [GHz]
-12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
dB
(S(P
ort
4,P
ort
3))
06_stripline_tuneXY Plot 3 ANSOFT
Curve Info
dB(S(Port4,Port3))
0 0
P=11.64in
W=4mil
V272
50
R273
50
R281
VName=Vout3
Isolated Lossy transmission line
insertion loss (-2dB @ 1GHz)
2Gbps Eye Diagram
The Effect of T-line Loss DDJ of a Single T-line
• The addition of the -2dB tee attenuator
removed 72ps – 40ps = 32ps of DDJ
• The addition of the Fc = -2dB lossy t-line
removed 72ps – 51ps = 21ps of DDJ
• However, the frequency dependent loss of
the t-line by itself generates 9ps of DDJ
• Thus the lossy line reduces DDJ by
attenuating reflections similar to the
attenuator but generates additional DDJ
through it’s transmission response
Pg. 25
Conclusions and Summary
• A signal integrity analysis of 2nd order lowpass LC
filters was given
– The analysis leverages characterization in both time and
frequency domains to develop useful intuition as to how
more generic interconnect discontinuities behave
• As data rates increase, discontinuities from
connectors, PCB vias etc. become electrically larger
requiring higher order lumped element equivalent
circuits
– Their behavior can still be intuited by understanding the
2nd order LC filter.
Pg. 26