© H. Heck 2008 Section 2.3 1

Module 2: Transmission LinesTopic 3: Reflections

OGI ECE564

Howard Heck

© H. Heck 2008 Section 2.3 2

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64 Where Are We?

1. Introduction

2. Transmission Line Basics1. Transmission Line Theory

2. Basic I/O Circuits

3. Reflections

4. Parasitic Discontinuities

5. Modeling, Simulation, & Spice

6. Measurement: Basic Equipment

7. Measurement: Time Domain Reflectometry

3. Analysis Tools

4. Metrics & Methodology

5. Advanced Transmission Lines

6. Multi-Gb/s Signaling

7. Special Topics

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64 Contents

Reflections at Discontinuities Reflection and Transmission Coefficients Examples

Open Circuit Short Circuit Matched Load

Summary References

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64 Reflections At Discontinuities

Define the reflection coefficients:

[2.4.1a]

i

rI I

I [2.4.1b]

Vi

Vr

Vt

It

Ii

-Ir

Z01 Z02

I

z

Important: Be sure to define a direction for positive current flow & then use it.

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64 Reflection Coefficient Derivation

Apply KCL at z: I I Ii r t [2.4.3]

Use Ohm’s law: V IZ 0 [2.4.4]

Combine [2.4.3] & [2.4.4]:V

Z

V

Z

V

Zi r t

01 01 02

[2.4.5]

Combine [2.4.2] & [2.4.5]:V

Z

V

Z

V V

Zi r i r

01 01 02

[2.4.6]

Z V V V V Z

V Z Z V Z Z

i r i r

i r

02 01

02 01 02 01

Vi

r

ZZ

ZZ

V

V

0102

0102 [2.4.7]

Apply Ohm’s law to [2.4.7] Vi

r

i

r

i

rI V

V

V

Z

Z

V

I

I

= 01

01

[2.4.8]

V V Vi r t [2.4.2] Apply KVL at z:

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64 Transmission Coefficient

Define transmission coefficients:

[2.4.9a] i

tV V

V

[2.4.9b] i

tI I

I

Divide [2.4.2] by Vi: Vi

r

i

i

i

t

V

V

V

V

V

V 1

[2.4.10] VVi

t

V

V 1

[2.4.11] IIi

ri

i

t

I

II

I

I

1

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64

IIII iir 1

VVVV ivt 2111

Example: Open Circuit

001 ZZ

02Z

RS = Z0

Z0

2V

VVZZ

ZVi

2

00

0

IZ

VI i

0

1

i

Vi = V Vt = 2V

Vr = V

Ii = I It = 0

-Ir = I

1

v

21 vv

01 ii

VVVV ivr 1

0111 III iit

I

0z

lz

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64

I

I

2I

zl

Wave Propagation: Open Circuit

l z

V

V

l

I

I

2I

z

l

I

I

z

z

V

V

2V

l

t0

t0 < t1 < dl

dl < t2

z

V

l

V

RS = Z0Z0

2V0z

lz

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64 Waveforms: Open Circuit

0 d

d

I

2I

0

z = 0z = l

z = 0z = l

V

2V

0 d

d

0

RS = Z0Z0

2V0z

lz

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64 Example: Short Circuit

VVZZ

ZVi

2

00

0

IZ

VI i

0

10

0

Z

Zi

10

0

Z

Zv

01 vv

21 ii

VVVV ivr 1

IIII iir 1

0111 VVV ivt

IIII ivt 2111

RS = Z0

Z0

2V

Vi = V

Vr = -V

Vt = 0

Ii = I

-Ir = -I

It = 2I

I

001 ZZ

002 Z

0z

lz

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64

V

z

V

2V

l

Wave Propagation: Short Circuit

t0

t0 < t1 < dl

dl < t2

z

V

l

V

l

I

I

2I

z

l

I

I

2I

z

l

I

I

zz

V

l

V

RS = Z0Z0

2V 0z

lz

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64 Waveforms: Short Circuit

0 d

d

I

2I

0

z = 0z = l

V

2V

0 d

d

0

z = 0z = l

RS = Z0Z0

2V 0z

lz

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64 Special Case: Matched Circuit

VVZZ

ZVi

2

00

0

IZ

VI i

0

02

0

0

Zi

02

0

0

Zv

11 vv 11 ii

00 VVV ivr

00 III iir

VVVV ivt 011

IIII iit 011

RS = Z0

2V

RTT = Z0

Z0

Vi = V

Vr = 0

Vt = V

Ii = I

-Ir = 0

It = I

I

001 ZZ

002 ZZ

0z

lz

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64

V

z

V

2V

l

Wave Propagation: Matched Circuit

t0

t0 < t1 < dl

d < t2l

z

V

l

V

l

I

I

2I

z

l

I

I

2I

z

l

I

I

zz

V

l

V

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64

dd0

V

2V

0

Waveforms: Matched Circuit

0 d

d

I

2I

0

z = 0z = l

z = 0z = l

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64 Summary

Reflected voltage and current waves are generated when the incident waves encounter a discontinuity in the transmission line.

The magnitude of the reflection is determined by the impedances of the lines and by the amplitude of the incident signal.

Special cases:Open circuits fully reflect the voltage signal. Short circuits reflect the incident signal with equal

magnitude but opposite sign.Matched circuits do not generate reflections.

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64 References

S. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.

W. Dally and J. Poulton, Digital Systems Engineering, Cambridge University Press, 1998.

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

Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, 1985.

H.B.Bakoglu, Circuits, Interconnections, and Packaging for VLSI, Addison Wesley, 1990.

K.M. True, “Reflections: Computations and Waveforms,” National Semiconductor Application Note AN-807, March 1993.

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

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