Transmission Line Basics

Transmission Line Basics II - Class 6

Prerequisite Reading assignment: CH2

Acknowledgements: Intel Bus Boot Camp: Michael Leddige

Transmission Lines Class 6

2

Real Computer Issues

Dev a Dev b

Clk Switch Threshold

Signal Measured here

An engineer tells you the measured clock is non-monotonic and because of this the flip flop internally may double clock the data. The goal for this class is to by inspection determine the cause and suggest whether this is a problem or not.

data


3

AgendaThe Transmission Line ConceptTransmission line equivalent circuits

and relevant equationsReflection diagram & equationLoadingTermination methods and comparison Propagation delaySimple return path ( circuit theory,

network theory come later)


4

Two Transmission Line ViewpointsSteady state ( most historical view)

Frequency domainTransient

Time domainNot circuit element Why?

We mix metaphors all the timeWhy convenience and history


5

Transmission Line Concept

PowerPlant

ConsumerHome

Power Frequency (f) is @ 60 Hz Wavelength (λ) is 5× 106 m

( Over 3,100 Miles)


6

PC Transmission Lines

Integrated Circuit

Microstrip

Stripline

Via

Cross section view taken herePCB substrate

T

WCross Section of Above PCB

T

Signal (microstrip)

Ground/PowerSignal (stripline)Signal (stripline) Ground/Power

Signal (microstrip)

Copper Trace

Copper Plane

FR4 Dielectric

W

Signal Frequency (f) is approaching 10 GHz

Wavelength (λ) is 1.5 cm ( 0.6 inches)

Micro-Strip

Stripline


7

Key point about transmission line operation

The major deviation from circuit theory with transmission line, distributed networks is this positional dependence of voltage and current!

Must think in terms of position and time to understand transmission line behaviorThis positional dependence is added when the assumption of the size of the circuit being small compared to the signaling wavelength

( )( )tzfI

tzfV,,

==

V1 V2

dz

I2I1

Voltage and current on a transmission line is a function of both time and position.


8

Examples of Transmission Line Structures- I Cables and wires

(a) Coax cable(b) Wire over ground(c) Tri-lead wire (d) Twisted pair (two-wire line)

Long distance interconnects

(a) (b)

(c) (d)

+-

+

+ +-

- -

-


9

Segment 2: Transmission line equivalent circuits and relevant equations

Physics of transmission line structures

Basic transmission line equivalent circuit

?Equations for transmission line propagation


10

Remember fields are setup given an applied forcing function.

(Source)

How does the signal move from source to load?

E & H Fields – Microstrip Case

The signal is really the wave propagating between the

conductors

Electric field

Magnetic field

Ground return path

X

Y

Z (into the page)

Signal path

Electric field

Magnetic field

Ground return path

X

Y

Z (into the page)

Signal path


11Transmission Line “Definition” General transmission line: a closed system in which

power is transmitted from a source to a destination

Our class: only TEM mode transmission linesA two conductor wire system with the wires in close proximity, providing relative impedance, velocity and closed current return path to the source.Characteristic impedance is the ratio of the voltage and current waves at any one position on the transmission line

Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium.

IVZ =0

r

cvε

=


12

Presence of Electric and Magnetic Fields

Both Electric and Magnetic fields are present in the transmission lines

These fields are perpendicular to each other and to the direction of wave propagation for TEM mode waves, which is the simplest mode, and assumed for most simulators(except for microstrip lines which assume “quasi-TEM”, which is an approximated equivalent for transient response calculations).

Electric field is established by a potential difference between two conductors.

Implies equivalent circuit model must contain capacitor. Magnetic field induced by current flowing on the line

Implies equivalent circuit model must contain inductor.

V

I

I

E

+

-

+

-

+

-

+

-

V + ∆ V

I + ∆ I

I + ∆ IV

IH

IH

V + ∆ V

I + ∆ I

I + ∆ I


13

General Characteristics of Transmission Line

Propagation delay per unit length (T0) time/distance [ps/in]Or Velocity (v0) distance/ time [in/ps]

Characteristic Impedance (Z0) Per-unit-length Capacitance (C0) [pf/in]Per-unit-length Inductance (L0) [nf/in]Per-unit-length (Series) Resistance (R0) [Ω/in]Per-unit-length (Parallel) Conductance (G0) [S/in]

T-Line Equivalent Circuit

lL0lR0

lC0lG0


14

Ideal T Line Ideal (lossless) Characteristics of

Transmission LineIdeal TL assumes:

Uniform linePerfect (lossless) conductor (R0→0)Perfect (lossless) dielectric (G0→0)

We only consider T0, Z0 , C0, and L0.

A transmission line can be represented by a cascaded network (subsections) of these equivalent models.

The smaller the subsection the more accurate the model

The delay for each subsection should be no larger than 1/10 th the signal rise time.

lL0

lC0


15 Signal Frequency and Edge Rate vs. Lumped or Tline ModelsIn theory, all circuits that deliver transient power from one point to another are transmission lines, but if the signal frequency(s) is low compared to the size of the circuit (small), a reasonable approximation can be used to simplify the circuit for calculation of the circuit transient (time vs. voltage or time vs. current) response.


16

T Line Rules of Thumb

Td < .1 Tx

Td < .4 Tx

May treat as lumped Capacitance Use this 10:1 ratio for accurate modeling of transmission lines

May treat as RC on-chip, and treat as LC for PC board interconnect

So, what are the rules of thumb to use?


17

Other “Rules of Thumb”

Frequency knee (Fknee) = 0.35/Tr (so if Tr is 1nS, Fknee is 350MHz)

This is the frequency at which most energy is below

Tr is the 10-90% edge rate of the signal Assignment: At what frequency can your thumb be

used to determine which elements are lumped?Assume 150 ps/in


18When does a T-line become a T-Line? Whether it is a

bump or a mountain depends on the ratio of its size (tline) to the size of the vehicle (signal wavelength)

When do we need to When do we need to use transmission line use transmission line

analysis techniques vs. analysis techniques vs. lumped circuit lumped circuit

analysis? analysis?

TlineWavelength/edge rate

Similarly, whether or not a line is to be considered as a transmission line depends on the ratio of length of the line (delay) to the wavelength of the applied frequency or the rise/fall edge of the signal

Equations & Formulas

How to model & explain transmission line behavior


20

Relevant Transmission Line Equations

Propagation equationβαωωγ jCjGLjR +=++= ))((

)()(0

CjGLjRZ

ωω

++=

Characteristic Impedance equation

In class problem: Derive the high frequency, lossless approximation for Z0

α is the attenuation (loss) factorβ is the phase (velocity) factor


21

Ideal Transmission Line Parameters Knowing any two out of Z0,

Td, C0, and L0, the other two can be calculated.

C0 and L0 are reciprocal functions of the line cross-sectional dimensions and are related by constant me.

ε is electric permittivity ε0= 8.85 X 10-12 F/m (free space) εri s relative dielectric constant

µ is magnetic permeability µ0= 4p X 10-7 H/m (free space) µr is relative permeability

.;

;;1

;;

;;

00

000

0000

00

00d0

00

εεεµµµ

µεµε

rr

LCv

TZLZTC

CLTCLZ

==

==

==

==

Don’t forget these relationships and what they mean!


22Parallel Plate Approximation Assumptions

TEM conditionsUniform dielectric (ε ) between conductorsTC<< TD; WC>> TD

T-line characteristics are function of:

Material electric and magnetic propertiesDielectric Thickness (TD)Width of conductor (WC)

Trade-offTD ; C0 , L0 , Z0 WC ; C0 , L0 , Z0

TD

TC

WC

ε

To a first order, t-line capacitance and inductance can be approximated using the parallel plate approximation.

dPlateAreaC *ε= Base

equationC0 ε

WCTD

⋅Fm

⋅ 8.85 ε r⋅WCTD

⋅pFm

⋅

L0 µTDWC

⋅Fm

⋅ 0.4 π⋅ µ r⋅TDWC

⋅µ Hm

⋅

Z0 377TDWC

⋅µ rε r

⋅ Ω⋅


23

Improved Microstrip Formula Parallel Plate Assumptions +

Large ground plane with zero thickness

To accurately predict microstrip impedance, you must calculate the effectiveeffective dielectric constant.

TD

TC

ε

WC

From Hall, Hall & McCall:

++≈

CC

D

r TWTZ

8.098.5ln

41.187

0ε

( )DC

Cr

C

D

rre

TWTF

WT

1217.01212

12

1 −−++

−++= εεεε

( )2

1102.0

−−

D

Cr

TWε

=F1<

D

C

TW

for

0 1>D

C

TW

for

Valid when: 0.1 < WC/TD < 2.0 and 1 < r < 15

You can’t beat a field solver


24

Improved Stripline Formulas Same assumptions as

used for microstrip apply here TD2

TCε

WC TD1

From Hall, Hall & McCall:

+

+≈)8.0(67.0

)(4ln60 110

CC

DD

rsym

TWTTZ

πε

Symmetric (balanced) Stripline Case TD1 = TD2

),,,2(),,,2(),,,2(),,,2(2

00

000

rCCsymrCCsym

rCCsymrCCsymoffset

TWBZTWAZTWBZTWAZZ

εεεε

+⋅≈

Offset (unbalanced) Stripline Case TD1 > TD2

Valid when WC/(TD1+TD2) < 0.35 and TC/(TD1+TD2) < 0.25

You can’t beat a field solver


25

Refection coefficient Signal on a transmission line can be analyzed by

keeping track of and adding reflections and transmissions from the “bumps” (discontinuities)

Refection coefficientAmount of signal reflected from the “bump”Frequency domain ρ=sign(S11)*|S11|If at load or source the reflection may be called gamma (ΓL or Γs)Time domain ρ is only defined a location

The “bump”Time domain analysis is causal.Frequency domain is for all time.We use similar terms – be careful

Reflection diagrams – more later


26Reflection and Transmission

ρ

1+ρIncident

Reflected

Transmitted

Reflection Coeficient Transmission Coeffiecentτ 1 ρ+( ) "" ""→ τ 1

Zt Z0−Zt Z0+

+ρZt Z0−Zt Z0+

τ2 Zt⋅

Zt Z0+


27

Special Cases to Remember

1=+∞−∞=

ZoZoρ

0=+−=

ZoZoZoZoρ

100 −=

+−=

ZoZoρ

VsZs

Zo Zo

A: Terminated in Zo

VsZs

Zo

B: Short Circuit

VsZs

Zo

C: Open Circuit


28

Assignment – Building the SI Tool BoxCompare the parallel plate approximation to the improved microstrip and stripline formulas for the following cases:Microstrip:

WC = 6 mils, TD = 4 mils, TC = 1 mil, εr = 4Symmetric Stripline:

WC = 6 mils, TD1 = TD2 = 4 mils, TC = 1 mil, εr = 4

Write Math Cad Program to calculate Z0, Td, L & C for each case.

What factors cause the errors with the parallel plate approximation?


29

Transmission line equivalent circuits and relevant equations

Basic pulse launching onto transmission lines

Calculation of near and far end waveforms for classic load conditions


30Review: Voltage Divider Circuit Consider the

simple circuit that contains source voltage VS, source resistance RS, and resistive load RL.

The output voltage, VL is easily calculated from the source amplitude and the values of the two series resistors.

RS

RLVS VL

RSRL

RLVSVL +

=

Why do we care for? Next page….


31

Solving Transmission Line ProblemsThe next slides will establish a procedure that

will allow you to solve transmission line problems without the aid of a simulator. Here are the steps that will be presented:

3.Determination of launch voltage & final “DC” or “t =0” voltage

4.Calculation of load reflection coefficient and voltage delivered to the load

5.Calculation of source reflection coefficient and resultant source voltage

These are the steps for solving all t-line problems.


32

Determining Launch Voltage

Step 1 in calculating transmission line waveforms is to determine the launch voltage in the circuit.

The behavior of transmission lines makes it easy to calculate the launch & final voltages – it is simply a voltage divider!

Vs ZoRsVs

0

TD

Rt

A B

t=0, V=Vi(initial voltage)

RSZ0

Z0VSVi +

=RSRt

RtVSVf +=


33

Voltage Delivered to the Load

Vs ZoRsVs

0

TD

Rt

A B

t=0, V=Vi

t=TD, V=Vi + ρB(Vi )t=2TD, V=Vi + ρB(Vi) + ρA(ρB)(Vi )

(signal is reflected)

(initial voltage)

Step 2: Determine VB in the circuit at time t = TD The transient behavior of transmission line delays the

arrival of launched voltage until time t = TD. VB at time 0 < t < TD is at quiescent voltage (0 in this case)

Voltage wavefront will be reflected at the end of the t-line VB = Vincident + Vreflected at time t = TD

ZoRtZoRt

+−=ρΒ

Vreflected = ρΒ (Vincident)

VB = Vincident + Vreflected


34

Voltage Reflected Back to the Source

VsZo

RsVs

0

TD

Rt

A B

t=0, V=Vi

t=TD, V=Vi + ρB (Vi )t=2TD, V=Vi + ρB (Vi) + ρA

(ρB)(Vi )

(signal is reflected)

(initial voltage)

ρA ρB


35Voltage Reflected Back to the Source

Step 3: Determine VA in the circuit at time t = 2TD The transient behavior of transmission line delays the

arrival of voltage reflected from the load until time t = 2TD. VA at time 0 < t < 2TD is at launch voltage

Voltage wavefront will be reflected at the source VA = Vlaunch + Vincident + Vreflected at time t = 2TD

In the steady state, the solution converges to VB = VS[Rt / (Rt + Rs)]

ZoRsZoRs

+−=ρΑ

Vreflected = ρΑ (Vincident)

VA = Vlaunch + Vincident + Vreflected


36

Problems Consider the circuit

shown to the right with a resistive load, assume propagation delay = T, RS= Z0 . Calculate and show the wave forms of V1(t),I1(t),V2(t), and I2(t) for (a) RL= ∞ and (b) RL= 3Z0

Z0 ,Τ 0

V1 V2

l

I2I1

VS RL

RS

Solved Homework


37

Step-Function into T-Line: Relationships

Source matched case: RS= Z0

V1(0) = 0.5VA, I1(0) = 0.5IA

ΓS = 0, V(x,∞) = 0.5VA(1+ ΓL) Uncharged line

V2(0) = 0, I2(0) = 0 Open circuit means RL= ∞

ΓL = ∞ /∞ = 1V1(∞) = V2(∞) = 0.5VA(1+1) = VA

I1(∞) = I2 (∞) = 0.5IA(1-1) = 0 Solution


38

Step-Function into T-Line with Open Ckt

At t = T, the voltage wave reaches load end and doubled wave travels back to source end

V1(T) = 0.5VA, I1(T) = 0.5VA/Z0

V2(T) = VA, I2 (T) = 0 At t = 2T, the doubled wave reaches the

source end and is not reflectedV1(2T) = VA, I1(2T) = 0V2(2T) = VA, I2(2T) = 0

Solution


39

Waveshape:Step-Function into T-Line with Open Ckt

Z0 ,Τ 0

V1 V2

l

I2I1

VSOpen

RS

Cur

rent

(A)

2Τ Time (ns)3ΤΤ 4Τ0

0.5IA

0.25IA

IA

0.75IA

I1I2

Vol

tage

(V)


0.5VA

0.25VA

VA

0.75VA

V1V2

This is called This is called “reflected wave “reflected wave switching”switching”

Solution


40

Problem 1b: Relationships Source matched case: RS= Z0

V1(0) = 0.5VA, I1(0) = 0.5IA

ΓS = 0, V(x,∞) = 0.5VA(1+ ΓL) Uncharged line

V2(0) = 0, I2(0) = 0 RL= 3Z0

ΓL = (3Z0 -Z0) / (3Z0 +Z0) = 0.5V1(∞) = V2(∞) = 0.5VA(1+0.5) = 0.75VA

I1(∞) = I2(∞) = 0.5IA(1-0.5) = 0.25IA Solution


41

Problem 1b: Solution At t = T, the voltage wave reaches load end

and positive wave travels back to the sourceV1(T) = 0.5VA, I1(T) = 0.5IA

V2(T) = 0.75VA , I2(T) = 0.25IA

At t = 2T, the reflected wave reaches the source end and absorbed

V1(2T) = 0.75VA , I1(2T) = 0.25IA

V2(2T) = 0.75VA , I2(2T) = 0.25IA

Solution


42Waveshapes for Problem 1b

Z0 ,Τ 0

V1 V2

l

I2I1

VS RL

RS

Curr

ent (

A)


0.5IA

0.25IA

IA

0.75IA

I1I2

Volta

ge (V

)


0.5VA

0.25VA

VA

0.75VA

I1I2

Note that a Note that a properly terminated properly terminated wave settle out at wave settle out at 0.5 V0.5 V

SolutionSolution


43Transmission line step response

Introduction to lattice diagram analysis

Calculation of near and far end waveforms for classic load impedances

Solving multiple reflection problems

Complex signal reflections at different types of transmission line “discontinuities” will be analyzed in this chapter. Lattice diagrams will be introduced as a solution tool.


44

Lattice Diagram Analysis – Key Concepts

Diagram shows the boundaries (x =0 and x=l) and the reflection coefficients (GL and GL )

Time (in T) axis shown vertically

Slope of the line should indicate flight time of signal

Particularly important for multiple reflection problems using both microstrip and stripline mediums.

Calculate voltage amplitude for each successive reflected wave

Total voltage at any point is the sum of all the waves that have reached that point

VsRs

ZoV(source) V(load)

TD = N ps0Vs

RtThe lattice diagram is a tool/technique to simplify the accounting of reflections and waveforms

Time V(source) V(load)

a

sourceρ loadρ

bA

c

A’

B’

C’

dB

e

0

N ps

2N ps

3N ps

4N ps

5N ps


45Lattice Diagram Analysis – DetailV(source) V(load)

Vlaunch

sourceρ load

ρ

Vlaunch ρload

Vlaunch

0

Vlaunch(1+ρload)

Vlaunch(1+ρload +ρload ρsource)

Time

0

2N ps

4N ps

Vlaunch ρloadρsource

Vlaunch ρ2loadρsource

Vlaunch ρ2loadρ2

source

Vlaunch(1+ρload+ρ2loadρsource+ ρ2

loadρ2source)

Time

N ps

3N ps

5N psVs

RsZoV(source) V(load)

TD = N ps0Vs

Rt


46Transient Analysis – Over DampedAssume Zs=75 ohms Zo=50ohmsVs=0-2 voltsVs

ZsZoV(source) V(load)


15050

2.050755075

8.05075

50)2(

=+∞−∞=

+−=

=+−=

+−=

=

+=

+=

ZoZlZoZl

ZoZsZoZs

ZoZsZoVsV

load

source

initial

ρ

ρ0.8v

2.0=sourceρ 1=loadρ

0.8v0.8v

0.16v

0v

1.6v

1.92v

0.16v1.76v

0.032v

TD = 250 ps

0

500 ps

1000 ps

1500 ps

2000 ps

2500 ps

02 v

Response from lattice diagram

0

0.5

1

1.5

2

2.5

0 250 500 750 1000 1250

Tim e , ps

Vol

tsSource

Load


47Transient Analysis – Under Damped

15050

33333.050255025

3333.15025

50)2(

=+∞−∞=

+−=

−=+−=

+−=

=

+=

+=

ZoZlZoZl

ZoZsZoZs

ZoZsZoVsV

load

source

initial

ρ

ρ

Assume Zs=25 ohms Zo =50ohmsVs=0-2 voltsVs

ZsZoV(source) V(load)

TD = 250 ps02 v


1.33v

3333.0−=sourceρ 1=loadρ

1.33v1.33v

-0.443v

0v

2.66v

1.77v

-0.443v2.22v

0.148v

0

500 ps

1000 ps

1500 ps

2000 ps

2500 ps 1.920.148v

2.07

Response from lattice diagram

0

0.5

1

1.5

2

2.5

3

0 250 500 750 1000 1250 1500 1750 2000 2250

Time, ps

Volts

SourceLoad


48Two Segment Transmission Line Structures

VsRs

Zo1 RtZo2

X X

a

bc

d e

f g

h i

j k

l

33

22

2

24

21

213

12

122

1

11

1

1

1

1

ρ

ρ

ρ

ρ

ρ

ρ

+=

+=

+−=

+−=

+−=

+−=

+=

T

T

ZRtZRt

ZZZZ

ZZZZ

ZRsZRs

ZRsZVv

o

o

oo

oo

oo

oo

o

o

o

osi

23

32

4

1

23

32

4

1

2

2

hTik

iThj

gi

fh

dTeg

eTdf

be

cd

ac

aTb

va i

+=

+=

=

=

+=

+=

=

=

=

=

=

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

hfdcACdcaB

aA

++++=++=

=

lkigebCigebB

ebA

+++++=+++=

+=

'''

A

B

C

A’

B’

C’

1ρ 2ρ 3ρ 4ρ3T 2T

TD TD

TD

3TD

2TD

4TD

5TD


49

Assignment Consider the two segment

transmission line shown to the right. Assume RS= 3Z01

and Z02= 3Z01 . Use Lattice diagram and calculate reflection coefficients at the interfaces and show the wave forms of V1(t), V2(t), and V3(t).

Check results with PSPICE

Z01 ,Τ 01

V1 V2

l1

I2I1

VS

RSZ02 ,Τ 02

V3l2

I3

Short

Previous examples are the preparation

Documents

Transmission Line Basics