Time and frequency domain analysis of a 2nd order low pass filter for signal integrity applications

1. Signal Integrity Analysis of a 2nd Order
Low Pass Filter
An Intuitive Approach
Andrew Josephson
ajosephson@comcast.net
Pg. 1

2. 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

3. 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

4. Review of 2nd order LC circuits
Pg. 4

5. 2nd Order Low Pass Filter Analysis
Low Pass Filter in High Speed Environments
L
1
Intuition tells us 0 
C LC
In high speed systems however, there are typically
source and load terminations: Controlled Impedance
R L
C
Transmitter R Receiver

6. 2nd Order Low Pass Filter Analysis
Developing a Transfer Function
R L
Vout
C
Vin R
Voltage Division in the S-Domain Gives
 R 
 
Vo
=
R || (1 / sC )
=  sRC  1  =
R
Vin R || (1 / sC )  R  sL  R  R  sR 2 C  s 2 RLC  R  sL Eq. 1
   R  sL
 sRC  1 

7. 2nd Order Low Pass Filter Analysis
Developing a Transfer Function
 2 
 
Rewriting Eq. Vo R 1  LC 
= =
1 gives Vin s RLC  s[ L  R C ]  2 R 2  2
2 2
 1 R 2  Eq.2
 s  s   
  RC L  LC 

Eq. 2 has the form of Vo n 2
a typical 2nd order = 2 Eq.3
Vin s  2 n s   n
2
system with transfer
function
This system has
complex conjugate
s   n   2

1 n
2
Eq.4
poles at

8. 2nd Order Low Pass Filter Analysis
Developing a Transfer Function
Comparing Eqtns 2
2  1 R
& 3 yields the n  and 2 n     Eq.5
following system LC  RC L 
parameter definitions
or
1 1 L C
   R  Eq.6
R C
8 L

L 2 1
Note that for the R    Eq.7
C 8 2
special case when
1
  is the value of the damping coefficient that leads to the
2
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.

9. 2nd Order Low Pass Filter Analysis
An RF filter designers approach
R L
Vout
C
Vin R
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.
L
Even a filter where R exhibits an impedance
mismatch. C

10. 2nd Order Low Pass Filter Analysis
An RF filter designers approach
L1 XY Plot 1 00_3_low_pass ANSOFT
Port1 Port2 0.00
7.33nH
C4
1pF
Zo = √(L/C) = 85.6 Ω -10.00
0
Return & Insertion Loss (dB)
-20.00 Curve Info
dB(S(Port2,Port1))
L2 LinearFrequency
Port3 Port4 dB(S(Port4,Port3))
5nH LinearFrequency
C7 dB(S(Port6,Port5))
LinearFrequency
Y1
2pF -30.00 dB(S(Port1,Port1))
Zo = √(L/C) = 50 Ω
LinearFrequency
dB(S(Port3,Port3))
LinearFrequency
0 dB(S(Port5,Port5))
LinearFrequency
-40.00
L3
Port5 Port6
0.55nH -50.00
C8
3pF
Zo = √(L/C) = 13.5 Ω
-60.00
0 0.10 1.00 10.00
F [GHz]
• 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

11. 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

12. Extending Analysis to Multiple LC
Circuits Separated by Ideal T-line
Pg. 12

13. Multiple Filter topologies with Prop Delay
Input
Impedance
T-line
Transmitter Receiver
• 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

14. The Effect of T-line Delay
Creating Local Jitter Maximums
Z0=50
Zo = √(L/C) = 85.6 Ω V
TD=1.92ns
R130 Name=Vout1
7.33nH 7.33nH
50 R134
V127 0 0
1pF 1pF 50
Zo = 85.6 Ω
0 0 0 0
Z0=50
Zo = √(L/C) = 50 Ω V
TD=1.95ns
R129 Name=Vout2
5nH 5nH
50 R136
V126 0 0
2pF 2pF 50
0 0 0 0
Z0=50
Zo = 50 Ω
Zo = √(L/C) = 13.5 Ω TD=2ns V
R128 Name=Vout3
0.55nH 0.55nH
50 R135
V125 0 0
3pF 3pF 50
0 0 0 0
• 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 Zo = 13.5 Ω
• This happens approximately when the largest
reflective “blip” occurs near the crosspoint timing
“blip maximum” Pg. 14

15. The Effect of T-line Delay
Creating Local Jitter Minimums
Z0=50
V
Zo = √(L/C) = 85.6 Ω
TD=2.097ns
R130 Name=Vout1
7.33nH 7.33nH
50 R134
V127 0 0
1pF 1pF 50
Zo = 85.6 Ω
0 0 0 0
Z0=50
V
Zo = √(L/C) = 50 Ω
TD=2.145ns
R129 Name=Vout2
5nH 5nH
50 R136
V126 0 0
2pF 2pF 50
0 0 0 0
Z0=50
Zo = 50 Ω
Zo = √(L/C) = 13.5 Ω V
TD=2.2ns
R128 Name=Vout3
0.55nH 0.55nH
50 R135
V125 0 0
3pF 3pF 50
0 0 0 0
• 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 Zo = 13.5 Ω
• This happens approximately when the reflective blip
minimum is aligned with the crosspoint
“blip minimum” Pg. 15

16. The Effect of T-line Delay
Periodic Eye Closure
Z0=50
TD=t_delay V
Zo = √(L/C) = 13.5 Ω Zo = √(L/C) = 13.5 Ω
R129 Name=Vout2
0.55nH 0.55nH
50 R136
V126 0 0
3pF 3pF 50
0 0 0 0
Eye Opening vs T-line Delay Data Dependent Jitter vs T-line Dealy
1000 80
Vertical Eye Opening (mV)
70
800
60
50
DDJ (ps)
600
40
400 30
20
200
10
0 0
0 500 1000 1500 2000 0 500 1000 1500 2000
T-line Delay (ps) T-line Delay (ps)
• 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

17. The Effect of T-line Delay
Local Jitter Minimum
Z0=50
TD=t_delay • Break circuit to measure input
Port1 Port2 impedance
0.55nH 0.55nH
R197
50 3pF
0 0
3pF
R136
50
– Looking into t-line (blue)
– Looking into LC filter back towards
0 0 0 0
generator (red)
Smith Chart 3 05_low_pass_delay_sweep_z ANSOFT
• At t_delay = 200ps (first DDJ min) the
110
100 90
1.00
80
70
Curve Info
S(Port1,Port1)
impedance looking into the delay line
120 60 t_delay='10ps'
S(Port2,Port2)
t_delay='200ps'
(blue) is near complex conjugately
130 0.50 2.00 50
matched to the terminated filter (red) at
140 40
Name
m2
F Ang Mag
1.0000 -113.3362 0.4011 0.5675 - 0.4982i
RX
1.0GHz
150 m3 1.0000 102.6638 0.4011 0.6277 + 0.5855i 30
m3 – 2.0Gbps data rate fundamental
160 0.20 5.00 20 frequency = 1GHz
170 10 • Suggests jitter minimums occur near
0.00
180 0.00
0.20 0.50 1.00 2.00 5.00
0
complex conjugate impedance
matching
-170 -10 – Condition for maximum power transfer
-160 -0.20
m2
-5.00 -20 – Expect jitter minimums at t_delay =
200ps + n*1000ps
-150 -30
-140 -40
-130 -0.50 -2.00 -50
-120 -60
-110 -1.00 -70
-100 -90 -80
Pg. 17

18. The Effect of T-line Delay
Local Jitter Minimum
Input Impedance Looking into T-line
at Local Jitter Minimums
Data Dependent Jitter vs T-line Dealy Delay (ps) Mag Z (normalized) Ang Z (deg)
80
200 0.4011 102.66
70 450 0.4011 -77.33
60 700 0.4011 102.66
50 950 0.4011 -77.33
DDJ (ps)
1200 0.4011 102.66
40
1450 0.4011 -77.33
30 1700 0.4011 102.66
20 1950 0.4011 -77.33
10
0
0 500 1000 1500 2000
• Jitter minimums also occur at t_delay =
T-line Delay (ps) 200ps + n*250ps
• Once the first complex conjugate
Z0=50
TD=t_delay
matching condition is established
Port1 Port2
(t_delay = 200ps), local DDJ minima
0.55nH 0.55nH
R197
50
0 0
R136
50
occur at every additional half bit delay
3pF 3pF
(+ n*250ps)
0 0 0 0 – 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

19. The Effect of T-line Delay
Local Jitter Maximum
Eye Closure vs. T-line Delay
1000 100
Vertical Opening (mv)
900 90
Verticle Eye Opening (mV)
Jitter_pk_pk (ps)
800 80
700 70
600 60
DDJ (ps)
500 50
400 40
300 30
200 20
100 10
0 0
1000 1100 1200 1300 1400 1500
T-line Delay (ps)
Td = 1200 Td = 1240 Td = 1320 Td = 1360 Td = 1450
• 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

20. 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
Jitter Minimum
complex conjugate matched
at Td = 1200 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
Jitter Maximum crosspoint
at Td = 1240
– The width of the “blip” is a function of
both the interconnect AND the
ristime of the data pattern
Pg. 20

21. The Effect of Real Lossy T-lines
Pg. 21

22. The Effect of T-line Loss
Co-propagating Reflections
Input
Impedance
T-line
Transmitter Receiver
• 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

23. The Effect of T-line Loss
Co-propagating Reflections
Z0=50
TD=2ns V
R243 Name=Vout1
0.55nH 0.55nH
50 R254
V245 0 0
3pF 3pF 50
Ideal
Transmission
0 0 Line 0 0
Z0=50 Z0=50
TD=1ns 2dB Tee TD=1ns V
Attenuator
R244 R235 R234 Name=Vout2
0.55nH 0.55nH
50 5.73 5.73 R253
V246 0 0 0 0
3pF R233 3pF 50
Ideal Ideal
215.24
Transmission Transmission
0 0
Line Line 0 0
0
V
R273 Name=Vout3
0.55nH 0.55nH
50 W=4mil R281
V272 P=11.64in 50
3pF 3pF
Real Lossy
Transmission Line
0 0 0 0
• 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

24. The Effect of T-line Loss
Co-propagating Reflections
• Ideal t-line
Ideal – Eye Opening = 707mV
Transmission
Line – DDJ = 72ps
• Ideal t-line with 2dB attenuator
– Eye Opening = 622mV
– DDJ = 40ps
• Real Lossy t-line
2dB Tee
Attenuator
– Eye Opening = 571mV
– DDJ = 51ps
• Why isn’t the -2dB lossy
transmission line as effective
as the tee attenuator in
reducing DDJ?
Real Lossy
Transmission Line
Pg. 24

25. The Effect of T-line Loss
DDJ of a Single T-line
V
R273 Name=Vout3
50 W=4mil R281
V272 P=11.64in 50
2Gbps Eye Diagram
0 0
XY Plot 3 06_stripline_tune ANSOFT
0.00
Curve Info
-2.00 dB(S(Port4,Port3))
-4.00 • The addition of the -2dB tee attenuator
dB(S(Port4,Port3))
removed 72ps – 40ps = 32ps of DDJ
-6.00
• The addition of the Fc = -2dB lossy t-line
-8.00
removed 72ps – 51ps = 21ps of DDJ
-10.00
Isolated Lossy transmission line • However, the frequency dependent loss of
insertion loss (-2dB @ 1GHz)
the t-line by itself generates 9ps of DDJ
•
-12.00
0.00 2.00 4.00
F [GHz]
6.00 8.00 10.00
Thus the lossy line reduces DDJ by
attenuating reflections similar to the
attenuator but generates additional DDJ
through it’s transmission response
Pg. 25

26. 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