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Module 3: Analysis Techniques Topic 1: Lattice Diagrams

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Where Are We?

- Introduction
- Transmission Line Basics
- Analysis Tools
- Lattice Diagrams
- Bergeron Diagrams
- Metrics & Methodology
- Advanced Transmission Lines
- Multi-Gb/s Signaling
- Special Topics

Section 3.1

Contents

- Lattice Diagram Construction
- Axes
- Wave vectors
- Voltages & Currents
- Example
- Circuit
- Lattice Diagram
- Waveforms
- Summary
- References

Section 3.1

Construction #1

0

l

z

r

r

=

=

0

t

d

2t

d

3t

d

4t

d

t

- The lattice diagram is a tool for keeping track of the voltage & current waves as they travel back & forth along the transmission line:

- 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.

Section 3.1

Construction #2

0

l

z

r

r

=

=

V(z=0)

I(z=0)

V(z=l)

I(z=l)

V0

0

I0

V1

t

d

I1

I5

I7

I9

I11

I3

V2

2t

d

I2

I4

I10

I6

I8

V3

3t

d

V4

4t

d

V5

5t

d

V6

6t

d

V1

7t

d

V8

8t

d

V1

9t

d

V10

10t

d

V11

11t

d

12t

d

t

- To show the propagation of the signal:

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

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

- Do the same for current.

Section 3.1

Construction #3

0

l

z

r

r

=

=

V(z=0)

I(z=0)

V(z=l)

I(z=l)

V0-

V0

V0+V1

V0+V1 +V2

I0-I1 +I2

I0

I0-

I0-I1

V0

0

V0-

I0-

I0

V1

t

d

I11

I7

I9

I3

I1

I5

V2

2t

d

I10

I2

I8

I6

I4

V3

3t

d

V0+V1 +V2+V3

I0-I1 +I2-I3

V4

4t

d

V5

5t

d

V6

6t

d

V1

7t

d

V8

8t

d

V1

9t

d

V10

10t

d

V11

11t

d

12t

d

t

- Track the signals at the discontinuities:

- 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).

- We can now construct the voltage and current waveforms.

etc.

etc.

Section 3.1

Example Circuit

I

RS = 25W

z = l

Z0 = 50W

W

t

Z

= 50

,

= 3 ns

0

D

VS = 5.0V

z = 0

RL= 100W

Analyze the low-high transition

- Reflection coefficients:

- Initial wave:

- Final value:

Section 3.1

Example Lattice

I

z = l

W

t

Z

= 50

,

= 3 ns

0

D

z = 0

0

l

z

W

R

= 25

r

r

(z=0) = -1/3

(z=l) = 1/3

S

V(z=0)

I(z=0)

V(z=l)

I(z=l)

V

= 5.0V

0.000V

0.00mA

S

3.333V

0

66.7mA

R

= 100

W

0.000V

0.00mA

T

t

3.333V

66.7mA

1.111V

d

22.2mA

2t

4.444V

44.5mA

-0.370V

d

-7.41mA

3t

4.074V

37.1mA

-0.123V

d

-2.47mA

0.041V

4t

3.951V

39.6mA

d

0.82mA

5t

3.992V

40.4mA

0.013V

d

0.27mA

-0.005V

6t

4.005V

40.1mA

d

-0.09mA

7t

4.000V

40.0mA

-0.002V

d

-0.03mA

8t

<0.001V

4.002V

40.0mA

d

<0.01mA

9t

4.000V

40.0mA

d

10t

d

t

Section 3.1

Example Waveforms

5.0

V(z=0)

4.5

V(z=l)

4.0

3.5

3.0

voltage [V]

2.5

2.0

1.5

1.0

0.5

0.0

0

1

2

3

4

5

6

time [td]

70

I(z=0)

60

I(z=l)

50

40

current [mA]

30

20

10

0

0

1

2

3

4

5

6

time [td]

Section 3.1

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.

Section 3.1

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.

Section 3.1

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