© H. Heck 2008 Section 3.1 1

Module 3: Analysis TechniquesTopic 1: Lattice Diagrams

OGI EE564

Howard Heck

© H. Heck 2008 Section 3.1 2

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Where Are We?

1. Introduction

2. Transmission Line Basics

3. Analysis Tools1. Lattice Diagrams

2. Bergeron Diagrams

4. Metrics & Methodology

5. Advanced Transmission Lines

6. Multi-Gb/s Signaling

7. Special Topics

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Contents

Lattice Diagram Construction Axes Wave vectors Voltages & Currents

Example Circuit Lattice Diagram Waveforms

Summary References

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Construction #1 The lattice diagram is a tool for keeping track of the

voltage & current waves as they travel back & forth along the transmission line:

0

td

2td

3td

4td

t

0 lz ==

The time axis runs down the page & is usually expressed in terms of the propagation delay (td) of the transmission line.

The z- (distance) axis runs across the page and extends from the source to the farthest load.

For convenience, the reflection coefficients at each discontinuity are often placed at the top of the diagram.

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To show the propagation of the signal:

Construction #2

0

td

td

td

td

td

6td

7td

8td

td

10td

11td

12td

t

V(z=0) I(z=0) V(z=l) I(z=l)

= =

0 lz

V0

V1

V2

V3

V5

V1

V1

V11

V4

V6

V8

V10

I0

I1

I2

I3

I5

I7

I9

I11

I4

I6

I8

I10

Do the same for current.

Use the initial wave and the reflection coefficients to calculate the voltage amplitude for each successive reflected wave

Use a “vector” to show the travel of the signal along the line.

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Track the signals at the discontinuities:

Construction #3

0

td

2td

3td

4td

5td

6td

td

8td

9td

10td

11td

12td

t

V(z=0) I(z=0) V(z=l) I(z=l)

= =

0 lz

V0

V1

V2

V3

V5

V1

V1

V11

V4

V6

V8

V10

I0

I1

I2

I3

I5

I7

I9

I11

I4

I6

I8

I10

Recall from superposition that the total signal (V or I) at any point on the network at a given time is the sum of all waves that have reached that point since the last signal transition (t=0).

V0- I0-

V0- I0-V0 I0

V0+V1 I0-I1

V0+V1

+V2

I0-I1

+I2 V0+V1

+V2+V3

I0-I1

+I2-I3

etc.etc.

We can now construct the voltage and current waveforms.

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Example Circuit

Reflection coefficients: lZR

ZRl I

L

LV

333.050100

50100

0

0

0333.05025

50250

0

0I

S

SV ZR

ZR

Initial wave:

mA

V

Z

VI

VVVRZ

ZV S

S

7.6650

333.30,00,0

333.352550

500,0

0

0

0

Analyze the low-high transition

Final value:

mA

V

RR

VI

VVVRR

RV

SL

SSL

L

40125

000.5,0,0

000.4525100

100,0

Z0 = 50, D = 3 ns

z =

0

z =

lIRS = 25

VS = 5.0VRL= 100

Z0 = 50

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Example Lattice

0

t d

2t d

td

td

5td

6t d

7t d

8t d

t d

10t d

V(z=0) I(z=0) V(z=l) I(z=l)

(z=l) = 1/3(z=0) = -1/3

0 l

z

t

0.000V 0.00mA

0.000V 0.00mA3.333V 66.7mA

4.444V 44.5mA

3.333V

66.7mA

1.111V

22.2mA-0.370V

-7.41mA-0.123V

-2.47mA0.041V

0.82mA

0.27mA

0.013V

-0.005V

-0.09mA

-0.03mA

-0.002V

<0.001V

<0.01mA

4.074V 37.1mA

3.992V 40.4mA

4.000V 40.0mA

4.000V 40.0mA

3.951V 39.6mA

4.005V 40.1mA

4.002V 40.0mA

RS = 25

RT = 100

Z0 = 50 ,

D = 3 ns

VS = 5.0V z

= 0

z =

l

I

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Example Waveforms

0 1 2 3 4 5 6time [td]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0v

olt

ag

e [

V]

V(z=0)V(z=l)

0 1 2 3 4 5 6time [td]

0

10

20

30

40

50

60

70

cu

rre

nt

[mA

]

I(z=0)

I(z=l)

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Summary

Lattice diagrams provide a useful tool for analyzing the voltage and current at points along the interconnect circuit as a function of time.They track voltage and current wave components,

and reflections at discontinuities.

The voltage and current waveforms can be easily constructed from the lattice diagrams.

Lattice diagrams are of limited use for complex topologies.

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References S. Hall, G. Hall, and J. McCall, High Speed Digital System

Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.

R. Poon, Computer Circuits Electrical Design, Prentice Hall, 1st edition, 1995.

H. Johnson and M. Graham, High Speed Digital Design: A Handbook of Black Magic, PTR Prentice Hall, 1993.

“Line Driving and System Design,” National Semiconductor Application Note AN-991, April 1995.

K.M. True, “Data Transmission Lines and Their Characteristics,” National Semiconductor Application Note AN-806, February 1996.

“Transmission Line Effects in PCB Applications,” Motorola Application Note AN1051, 1990.

W.R. Blood, MECL System Design Handbook, Motorola, Inc., 4th edition, 1988.