interconnect ii – class 22

12/4/2002

Interconnect II – Class 22

Prerequisite Reading - Chapter 4

12/4/2002Interconnect II

2

Key Topics:

Frequency Content of Digital WaveformsFrequency Envelope Incorporating frequency domain effects into

time domain signals

Effects of Frequency Domain Phenomena on Time Domain Digital Signals


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) + …) 1 2 3 4 5May do with sum of cosines too.

1

0- 2 3

1

1 + 2 1 + 2 + 3

1 + 2 + 3 + 4 1 + 2 + 3 + 4 + 5

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:


4

• The amplitude of the the sinusoid components are used to construct the “frequency envelope” – Output of FT

Frequency Content of Digital Signals

1 3 5 7 9 …...Harmonic Number

20dB/decade

40dB/decade

rT

35.0

Pw

T

T

1

Tr


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

TrF

35.0

)1( /tinput eVV

• Setting V=0.1Vinput and V=0.9Vinput allows the calculation of the 10-90% risetime in terms of the time constant

195.2105.03.2%10%90%9010 ttt

• The frequency response of a 1 pole network is

dBdB F

F3

3 2

1

2

1

• Substituting into the step response yields

dBdB FFt

33%9010

35.009.1


6

Estimating the Frequency Content

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.

dBFt

3%9010

35.0

Edge time factor


7

• The frequency dependent effects described earlier in this class can be applied to each sinusoidal function in the seriesDigital signal decomposed into its sinusoidal

componentsFrequency domain transfer functions applied to

each sinusoidal componentModified sinusoidal functions are then re-

combined to construct the altered time digital signal

• There are several ways to determine this responseFourier series (just described)Fast Fourier transform (FFT)

Widely available in tools such as excel, Mathematica, MathCad…

Examining Frequency Content of Digital Signals


83 Method of Generating a Square Wave

Ramp pulsesUse Heavy Side functionUsed for first pass simulations

Power Exponential PulsesRealistic edge that can match silicon performanceUsed for behavioral simulation that match silicon performance.

Sum of CosinesText book identity.Used to get a quick feel for impact of frequency dependant phenomena on a wave.


9

Ramp Square WaveRamp Pulse Train

RUe tt( ) tt tt( ) tt1

tt

1

onepulse tt k( ) RUe tt period k( ) RUe tt pw k period( )( )

Rpt t( )

0

number_of_pulses

k

onepulse t k( )

0 50 100 150

0

1

Rpt ti

ti

ns


10

Power Exponential Square WavePower Exponetial Pulse Train

edge tt( ) Va 1 e tt( )

2.5

Power Exponential Edge

Ppt tt( ) edge tt( ) tt( )

onepulse tt k( ) Ppt tt period k( ) Ppt tt pw k period( )( )

Ppt t( )

0

number_of_pulses

k

onepulse t k( )

0 50 100 150

0

1

Ppt ti

ti

ns


11

Sum Cosine Square WaveSum Cosine Pulse Wave

Establish Fourier SpectrumC

period

2Tedge a

C

period

freq n( )n 2period

n 1 3 10

Define pulse train of fourier coef i.e. pulse = sum of cosines

Fpt tt( )1

2

4

2

1 2 a( ) n

1

n2

cos n a cos freq n( ) tt( )

Va

0 50 100 1500.5

0

0.5

1

1.5

Fpt ti

ti

ns


12Applying Frequency Dependent Effects to Digital Functions

FT

Multiply

Inverse FFT

FFT

Frequency

Att

enua

tion

(V

2/V

1)

Frequency

Vol

ts

Time

Vol

tsTime

No AC lossesWith 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

riseTF

35.0


13

Assignment

Use MathCad to create a pulse wave with

Sum of sine wavesSum of rampsSum of realistic edge waveforms

Exponential powers

Use MathCad to determine edge time factor for exponential and Gaussian wave,

10% - 90%20% - 80%


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

195.2105.03.2%10%90%9010 tttRemember this equation from a few slides ago?

Zo=50 C=5pF

tr = 300ps

pspspsTTt

pspFCZot

outrout 625)548()300(

548)550(195.2)(195.2195.2

2222

%9010

Input edge

tout=Output edgeExample:


15

Key Topics:

Serpentine tracesBends ISITopology

Additional Effects


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

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.0E+

00

5.0E-10

1.0E-09

1.5E-09

2.0E-09

2.5E-09

3.0E-09

3.5E-09

4.0E-09

Time, s

Vo

lts

S=5S=15


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.

10 portTransmission Line Spice

Model Couple length=2 inches

Trace route on PWB

5 mil 2 port Tline Model

1” 2 port Tline model

1 oz copper5 mil space5mil width5 mil distance to ground

planeSymmetric striplineUse 50 ohm V source w/ 1ns

rise time (do for ramp and Gaussian)


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


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 capacitanceWCC

bendo 11_90

• 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

Capacitance of 1 extra square


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

Waveform beginning transition from low to high with unsettled noise cased by reflections.

Different starting point due to ISI

Volts

Time

Ideal waveform beginning transition from low to high with no reflections or losses

Timing difference

Receiver switching threshold


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.

-2

-1

0

1

2

3

4

0.E

+00

1.E

-09

2.E

-09

3.E

-09

4.E

-09

5.E

-09

6.E

-09

7.E

-09

8.E

-09

9.E

-09

1.E

-08

Time, s

Vo

lts

Ideal 400 MHz waveform

400 MHz switching200 MHz switching


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=ZoReceiver 1

Receiver 2

L2L3

(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

132

132

||

||

ooo

ooo

ZZZ

ZZZ

• This model can be simplified and solved with lattice diagrams Valid when L1=L2

Vs Z=Zo2||Zo30-2V Zo1

Rs=Zo L2

L1

L1


23


• Now, consider the case where L2 and L3 are NOT Equal

VsZo3

Zo2

0-2VZo1

Rs=ZoReceiver 1

Receiver 2

L2L3

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 diagram0.0

0.5

1.0

1.5

2.0

2.5

0.0

Vo

ltag

e at

rec

eive

r

2.0 4.0 6.0 8.0 10.0

Time, ns


24


A

B

01 2 33T 2T

C

9

20

9

2

9

2

3

2

3

2

3

2

3

2'

3

4

3

2

3

2'

9

16

9

2

9

2

3

2

3

2

9

2

9

2

3

2

3

29

8

9

2

9

2

3

2

3

23

4

3

2

3

23

21

3

1

2

2

12

32

32

B

A

C

B

A

TT

ZoZo

ZoZoRsZo

ZoVinitial

B’

A’

In J R1 R2

Vs Zo

Zo

0-2VZo

Rs=ZoR1

R 2

JIn

interconnect ii – class 22

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