© H. Heck 2008 Section 2.1 1

Module 2: Transmission LinesTopic 1: Theory

OGI EE564

Howard Heck

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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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Contents

Propagation Velocity Characteristic Impedance Visualizing Transmission Line Behavior General Circuit Model Frequency Dependence Lossless Transmission Lines Homogeneous and Non-homogeneous Lines Impedance Formulae for Transmission Line Structures Summary References

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Propagation Velocity

Physical example:

Wave propagates in z direction

Circuit: L = [nH/cm]C = [pF/cm]

t

ILdzdz

z

V

Total voltage change across Ldz (use ): V L dI

dt

Total current change across Cdz (use ): dt

dVCI

t

VCdzdz

z

I

[2.1.1]

[2.1.2]

Simplify [2.1.1] & [2.1.2] to get the Telegraphist’s Equations [2.1.3a]

t

IL

z

V

t

VC

z

I

[2.1.3b]

I

V

Ldz

Cdz

dz

V+ dzdVdz

I+ dzdIdz

z

xy

V, I

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Propagation Velocity (2)

Phase velocity definition: vLC

1

[2.1.7]

Equation in terms of current:2

2

22

2

2

2 1

t

I

t

ILC

z

I

[2.1.8]

Equate [2.1.4] & [2.1.5]: [2.1.6]2

2

22

2

2

2 1

t

V

t

VLC

z

V

Differentiate [2.1.3b] by z: [2.1.5]zt

IL

z

V

2

2

2

Differentiate [2.1.3a] by t: [2.1.4]2

22

t

VC

tz

I

Equation [2.1.6] is a form of the wave equation. The solution to [2.1.6] contains forward and backward traveling wave components, which travel with a phase velocity.

An alternate treatment of propagation velocity is contained in the appendix.

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Characteristic Impedance (Lossless)

The input impedance (Z1) is the impedance of the first

inductor (Ldz) in series with the parallel combination of the impedance of the capacitor (Cdz) and Z2.

Ldz

Cdx

Z1 Z2 Z3

Ldz

Cdz

Ldz

Cdz

dz dz

V1 V3V2 to

a

fed

cb

dz

dz = segment length

C = capacitance per segment

L = inductance per segment

[2.1.9]

CdzjZ

CdzjZLdzjZ

/1

/1

2

21

0/1/1/1 2221 lCjZlCjZlLjlCjZZ

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Characteristic Impedance (Lossless) Assuming a uniform line, the input impedance should

be the same when looking into node pairs a-d, b-e, c-f, and so forth. So, Z2 = Z1= Z0.

0/1/1/1 0000 CdzjZCdzjZlLdzjCdzjZZ [2.1.10]

Cdzj

LdzjdzLZjZ

Cdzj

Z

Cdz

LdzdzLZj

Cdzj

ZZ

0

20

00

020 0

Allow dz to become very small, causing the frequency dependent term to drop out:

0020

C

LdzLZjZ [2.1.11]

020

C

LZ [2.1.12]

Solve for Z0:

C

LZ 0

[2.1.13]

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Visualizing Transmission Line Behavior

Water flow Potential = Wave height [m] Flow = Flow rate [liter/sec]

I

I

V

+++++++

- - - - - - -

Transmission Line Potential = Voltage [V] Flow = Current [A] =

[C/sec]

Just as the wave front of the water flows in the pipe, the voltage propagates in the transmission line. The same holds true for current. Voltage and current propagate as waves in the transmission line.

h

f

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Visualizing Transmission Line Behavior #2 Extending the analogy

The diameter of the pipe relates the flow rate and height of the water. This is analogous to electrical impedance.

Ohm’s law and the characteristic impedance define the relationship between current and potential in the transmission line.

Effects of impedance discontinuities What happens when the water encounters a ledge or a

barrier? What happens to the current and voltage waves when the

impedance of the transmission line changes? The answer to this question is a key to understanding

transmission line behavior. It is useful to try visualize current/voltage wave propagation

on a transmission line system in the same way that we can for water flow in a pipe.

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General Transmission Line Model (No Coupling) Transmission line parameters are distributed (e.g.

capacitance per unit length). A transmission line can be modeled using a network

of resistances, inductances, and capacitances, where the distributed parameters are broken into small discrete elements.

R L

G C

R L

G C

R L

G C

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General Transmission Line Model #2

Parameter Symbol Units

Conductor Resistance R •cm-1

Self Inductance L nH•cm-1

Total Capacitance C pF•cm-1

Dielectric Conductance G -1•cm -1

Parameters

Characteristic Impedance

ZR j L

G j C0

[2.1.14]

Propagation Constant jCjGLjR [2.1.15]

= attenuation constant = rate of exponential attenuation = phase constant = amount of phase shift per unit length

pPhase Velocity [2.1.16]

In general, and are frequency dependent.

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Frequency Dependence

From [2.1.14] and [2.1.15] note that: Z0 and depend on the frequency content of the

signal. Frequency dependence causes attenuation and edge

rate degradation.

Attenuation

Edge rate degradation

Output signal from lossytransmission line

Signal at driven end oftransmission line

Output signal fromlossless transmission line

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Frequency Dependence #2

R and G are sometimes negligible, particularly at low frequencies Simplifies to the lossless case: no attenuation & no

dispersion

In modules 2 and 3, we will concentrate on lossless transmission lines.

Modules 5 and 6 will deal with lossy lines.

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Lossless Transmission Lines

Quasi-TEM AssumptionQuasi-TEM Assumption The electric and magnetic fields are perpendicular to

the propagation velocity in the transverse planes.

x

zy

HE

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Lossless Line Parameters

Lossless line characteristics are frequency independent. As noted before, Z0 defines the relationship between

voltage and current for the traveling waves. The units are ohms [].

defines the propagation velocity of the waves. The units are cm/ns. Sometimes, we use the propagation delay, d (units are ns/cm).

C

LZ 0

vLC

1

Characteristic ImpedanceCharacteristic Impedance

Propagation VelocityPropagation Velocity

[2.1.17]

[2.1.18]

Lossless transmission lines are characterized by the following two parameters:

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Lossless Line Equivalent Circuit

The transmission line equivalent circuit shown on the left is often represented by the coaxial cable symbol.

L

C

L

C

L

C

Z0, v, lengthZ0, , length

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Homogeneous Media

A homogeneous dielectric medium is uniform in all directions. All field lines are contained within the dielectric.

For a transmission line in a homogeneous medium, the propagation velocity depends only on material properties:

vLC

c cm ns

r r r

1 1 300

/

[2.1.19]

0 r Dielectric Permittivity

cmFx 14

0 10854.8 Permittivity of free space

cmHx 8

0 10257.1 Magnetic Permeability

0 Permeability of free space

r is the relative permittivity or dielectric constant.

Note: only Note: only rr

is required to is required to calculate calculate ..

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Non-Homogeneous Media

A non-homogenous medium contains multiple materials with different dielectric constants.

For a non-homogeneous medium, field lines cut across the boundaries between dielectric materials.

In this case the propagation velocity depends on the dielectric constants and the proportions of the materials. Equation [2.1.19] does not hold:

11

LC

v

In practice, an effective dielectric constant, r,eff is often used, which represents an average dielectric constant.

© H. Heck 2008 Section 2.1 19

Some Typical Transmission Line Structures

And useful formulas for Z0

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rR

r

Coax Cable Impedance

2 3 4 5 6 7 8 9 10

R/r

20

40

60

80

100

120

140

Z0 [

]

r = 1

r = 4r = 3.5r = 3r = 2.5r = 2

Z0, v, lengthZ0, , length

r

RZ ln

2

10

[2.1.20]

r

RC

ln

2

[2.1.21]

r

RL ln

2

[2.1.22]

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Centered Stripline Impedance

wtw

hZ

r 8.067.0

4ln

60 20

w

t

h1

h2

r

Source: Motorola application note AN1051.

35.02

th

wValid for

25.02h

t

0.003 0.005 0.007 0.009 0.011 0.013 0.015

w [in]

10

15

20

25

30

35

40

45

50

55

60

Z0

[]

0.0700.0600.0500.0400.0300.0250.020

h2

t = 0.0007”r = 4.0

[2.1.23]

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Dual Stripline Impedance

w

t

h2

h1

r

w

t

h1

ZY

YZZ

20

wtw

hY

r 8.067.0

8ln

60 1

wtw

hhZ

r 8.067.0

8ln

60 21

tw

ththh

h

Zr

8.0

29.1ln

4180

121

1

0

.115.0 hwh

Source: Motorola application note AN1051.

OR

0.003 0.005 0.007 0.009 0.011 0.013 0.015

w [in]

10

20

30

40

50

60

70

80

90

100

110Z

0 [

]

0.020”0.018”0.015”0.012”0.010”0.008”

0.005”

2h1 + h2 + 2t = 0.062” t = 0.0007”r = 4.0

h1

[2.1.24]

[2.1.27]

[2.1.25]

[2.1.26]

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Surface Microstrip Impedancew

t

h

r

0

d

hZ

eff

4ln

2

10

twd 67.0536.0

067.0475.0 reff

Source: National AN-991.

Source: Motorola MECL Design Handbook.

tw

hZ

r8.0

98.5ln

41.1

870

0.003 0.005 0.007 0.009 0.011 0.013 0.015

w [in]

20

40

60

80

100

120

140

160Z

0 [

]

0.025”0.020”0.015”0.012”0.009”0.006”0.004”

h

t = 0.0007”

r = 4.0

[2.1.28]

[2.1.29]

[2.1.30]

[2.1.31]

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Embedded Microstrip

t

h1

r

0

w

h2

tw

hKZ

r8.0

98.5ln

2805.01

0

6560 where K

tw

hZ

r8.0

98.5ln

41.1

87 10

1255.11 hhrr e

67.0475.0017.1 r

Or

0.003 0.005 0.007 0.009 0.011 0.013 0.015

w [in]

0

20

40

60

80

100

120

140

Z0

[] 0.015”

0.012”0.010”0.008”0.006”0.005”

0.003”

h2 - h1 = 0.002“ t= 0.0007”r = 4.0

h1

[2.1.32]

[2.1.33]

[2.1.34]

[2.1.35]

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Summary

System level interconnects can often be treated as lossless transmission lines.

Transmission lines circuit elements are distributed. Voltage and current propagate as waves in

transmission lines. Propagation velocity and characteristic impedance

characterize the behavior of lossless transmission lines.

Coaxial cables, stripline and microstrip printed circuits are the typical transmission line structures in PCs systems.

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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.

H. Johnson and M. Graham, High-Speed Signal Propagation: Advanced Black Magic, Prentice Hall, 2003, 1st edition, ISBN 0-13-084408-X.

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

R.E. Matick, Transmission Lines for Digital and Communication Networks, IEEE Press, 1995.

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

H.B.Bakoglu, Circuits, Interconnections, and Packaging for VLSI, Addison Wesley, 1990, ISBN 0-201-060080-6.

B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1st edition, ISBN 0-13-028904-3.

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Phase Constant (Lossless Case)

Recall the basic voltage divider circuit:R1

R2V1

+

V2

-

I

We want to find the ratio of the input voltage, V1, to the output voltage, V2.

Now, we apply it to our transmission line equivalent circuit...

0211 IRIRV21

1

RR

VI

21

2122 RR

RVIRV

2

1

2

21

2

1 1R

R

R

RR

V

V

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Phase Constant (Lossless Case) #2

The analogous transmission line circuit looks like this:

The phase shift is the ratio of V1 to V2:

Substitute the expressions for ZC, ZL, and Z0:

00

0

00

0

2

1 11111

ZZZ

ZZ

ZZZ

ZZ

Z

ZZ

ZZZ

V

V

CL

C

CL

C

L

C

CL

LZR 1

0

002 ZZ

ZZZZR

C

CC

CdzjZC

1

LdzjZL

CLZ 0

LCdzjLCdzjLdzj

CdzjCdzjLdzj

ZZZ

V

V

CL

222222

02

1 1111

1

LCdzjLCdzV

V 22

2

1 1

Ldz

CdzV1

+

V2

-

Z0

I

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Phase Constant (Lossless Case) #3

The amplitude of the phase constant is:

The phase angle, denoted as tanl, is:

Now, we make the assumption that dz is small enough that the applied frequency, , is much smaller than the resonant frequency, , of each subsection, so that:

LCdzLCdzV

V 22222

2

1 1

LCdz

LCdzl 221

tan

LCdz1

122 LCdz

The phase angle becomes:LCdzl tan

Since , tanl is, very small. Therefore: LCdzll tan

122 LCdz

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Phase Constant (Lossless Case) #4 The phase shift per unit length is:

p l represents the amount by which the input voltage, V1,

leads the output voltage, V2. We can simplify the amplitude ratio by using the

condition of small l:

So, there is no decrease in the amplitude of the voltage along the line, for the lossless case. Only a shift in phase.

From our definition of phase velocity in equation [2.1.16] we get

LCdz

l

11 22222

2

1 LCdzLCdzV

V

CL

p