Signal Integrity Analysis of LC lopass Filter

Pg. 1

Signal Integrity Analysis of a 2nd Order

Low Pass Filter

An Intuitive Approach

Andrew Josephson

[email protected]

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


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


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






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 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


• Ideal t-line

– Eye Opening = 707mV

– DDJ = 72ps

• Ideal t-line with 2dB attenuator


– DDJ = 40ps

• Real Lossy t-line


– 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

Technology

Signal Integrity Analysis of LC lopass Filter