transmission line basics ii - class 6 prerequisite reading assignment: ch2 acknowledgements: intel...

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

Agenda

The Transmission Line ConceptTransmission line equivalent circuits

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

network theory come later)


4Two Transmission Line Viewpoints

Steady state ( most historical view)Frequency domain

TransientTime 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 () is 5 106 m

( Over 3,100 Miles)


6

PC Transmission Lines

Integrated Circuit

Microstrip

Stripline

Via

Cross section view taken herePCB substrate

T

W

Cross 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 () 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)

+

-

+

+ +-

- -

-


9Segment 2: Transmission line equivalent circuits and relevant equations

Physics of transmission line structures

Basic transmission line equivalent circuit

?Equations for transmission line propagation

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


11

Transmission 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.

I

VZ 0

r

cv


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 lineImplies equivalent circuit model must contain inductor.

V

I

I

E

+

-

+

-

+

-

+

-

V + DV

I + DI

I + DI

V

IH

IH

V + DV

I + DI

I + DI


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) [/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 line

Perfect (lossless) conductor (R00)

Perfect (lossless) dielectric (G00)

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 1/10th the signal rise time.

lL0

lC0


15 Signal Frequency and Edge Rate vs. Lumped or Tline Models

In 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


18

When 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 When do we need to use to use

transmission line transmission line analysis analysis

techniques vs. 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 equation

jCjGLjR ))((

)(

)(0

CjG

LjRZ

Characteristic Impedance equation

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

is the attenuation (loss) factor

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.

is electric permittivity 0= 8.85 X 10-12 F/m (free space)

ri s relative dielectric constant

is magnetic permeability 0= 4p X 10-7 H/m (free space)

r is relative permeability

.;

;;1

;;

;;

00

000

0000

00

00d0

00

rr

LCv

TZLZ

TC

CLTC

LZ

Don’t forget these relationships and what they mean!Don’t forget these relationships and what they mean!


22

Parallel Plate Approximation Assumptions

TEM conditions

Uniform dielectric ( ) between conductorsTC<< TD; WC>> TD

T-line characteristics are function of:

Material electric and magnetic propertiesDielectric Thickness (TD)

Width of conductor (WC) Trade-off

TD ; C0 , L0 , Z0 WC ; C0 , L0 , Z0

TD

TC

WC

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

d

PlateAreaC

* Base

equation

C0 WC

TD

F

m

8.85 r

WC

TD

pF

m

L0 TD

WC

F

m

0.4 r

TD

WC

H

m

Z0 377TD

WC

r

r


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

WC

From Hall, Hall & McCall:

CC

D

r TW

TZ

8.0

98.5ln

41.1

870

DC

Cr

C

D

rre

TW

TF

W

T1217.0

1212

1

2

1

2

1102.0

D

Cr

T

WF

1D

C

T

Wfor

0 1D

C

T

Wfor

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

TC

WCTD1

From Hall, Hall & McCall:

)8.0(67.0

)(4ln

60 110

CC

DD

r

symTW

TTZ

Symmetric (balanced) Stripline Case TD1 = TD2

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

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

00

000

rCCsymrCCsym

rCCsymrCCsymoffset

TWBZTWAZ

TWBZTWAZZ

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 =sign(S11)*|S11|If at load or source the reflection may be called gamma (L or s)Time domain 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


26

Reflection and Transmission

Incident

Reflected

Transmitted

Reflection CoeficientTransmission Coeffiecent

1 "" "" 1Zt Z0Zt Z0

Zt Z0Zt Z0

2 Zt

Zt Z0


27

Special Cases to Remember

1ZoZo

0ZoZo

ZoZo

100

ZoZo

Vs

ZsZo Zo

A: Terminated in Zo

Vs

ZsZo

B: Short Circuit

Vs

ZsZo

C: Open Circuit


28Assignment – Building the SI Tool Box

Compare 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, r = 4

Symmetric Stripline:

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

Basic pulse launching onto transmission lines


classic load conditions


30

Review: 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….

Why do we care for? Next page….


31Solving Transmission Line Problems

The 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:

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

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

3.Calculation of source reflection coefficient and resultant source voltage

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

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!

VsZo

RsVs

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 + B(Vi )t=2TD, V=Vi + B(Vi) + AB)(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

Vreflected = (Vincident)

VB = Vincident + Vreflected


34Voltage Reflected Back to the Source

VsZo

RsVs

0

TD

Rt

A B

t=0, V=Vi

t=TD, V=Vi + B (Vi )t=2TD,

V=Vi + B

(Vi) + A

B)(Vi )

(signal is reflected)

(initial voltage)

A B


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

Vreflected = (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 ,T0

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

S = 0, V(x,) = 0.5VA(1+ L)

Uncharged lineV2(0) = 0, I2(0) = 0

Open circuit means RL= L = / = 1

V1() = V2() = 0.5VA(1+1) = VA

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


38Step-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 reflected

V1(2T) = VA, I1(2T) = 0

V2(2T) = VA, I2(2T) = 0

Solution


39

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

Z0 ,T0

V1 V2

l

I2I1

VSOpen

RS

Cur

rent

(A)

2T Time (ns)3TT 4T0

0.5IA

0.25IA

IA

0.75IA

I1

I2

Vol

tage

(V)

2T Time (ns)3TT 4T0

0.5VA

0.25VA

VA

0.75VA

V1

V2

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

S = 0, V(x,) = 0.5VA(1+ L)

Uncharged lineV2(0) = 0, I2(0) = 0

RL= 3Z0

L = (3Z0 -Z0) / (3Z0 +Z0) = 0.5

V1() = 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 source

V1(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


42

Waveshapes for Problem 1b

Z0 ,T0

V1 V2

l

I2I1

VS RL

RS

Cur

rent

(A)

2T Time (ns)3TT 4T0

0.5IA

0.25IA

IA

0.75IA

I1

I2

Vol

tage

(V)

2T Time (ns)3TT 4T0

0.5VA

0.25VA

VA

0.75VA

I1

I2Note that a Note that a properly properly terminated wave terminated wave settle out at 0.5 settle out at 0.5 VVSolution

Solution


43

Transmission line step response

Introduction to lattice diagram analysis


classic load impedances Solving multiple reflection problems

Introduction to lattice diagram analysis


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.

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

Vs

Rs

ZoV(source) V(load)

TD = N ps0

Vs

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

Time V(source) V(load)

a

source load

bA

c

A’

B’

C’

dB

e

0

N ps

2N ps

3N ps

4N ps

5N ps


45Lattice Diagram Analysis – Detail

V(source) V(load)

Vlaunch

source

load

Vlaunch load

Vlaunch

0

Vlaunch(1+load)

Vlaunch(1+load +load source)

Time

0

2N ps

4N ps

Vlaunch loadsource

Vlaunch loadsource

Vlaunch load

source

Vlaunch(1+load+loadsource+

loadsource)

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

Zs

ZoV(source) V(load)


150

50

2.05075

5075

8.05075

50)2(

ZoZl

ZoZl

ZoZs

ZoZs

ZoZs

ZoVsV

load

source

initial

0.8v

2.0source 1load

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

0

2 v

Response fr om lattice diagram

0

0.5

1

1.5

2

2.5

0 2 50 500 750 1000 1250

Tim e , ps

Vo

lts

Source

Load


47Transient Analysis – Under Damped

150

50

33333.05025

5025

3333.15025

50)2(

ZoZl

ZoZl

ZoZs

ZoZs

ZoZs

ZoVsV

load

source

initial

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

Zs

ZoV(source) V(load)

TD = 250 ps0

2 v


1.33v

3333.0source

1load

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

Vo

lts

Source

Load


48Two Segment Transmission Line Structures

Vs

RsZo1 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

T

T

ZRt

ZRt

ZZ

ZZ

ZZ

ZZ

ZRs

ZRs

ZRs

ZVv

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

hfdcAC

dcaB

aA

lkigebC

igebB

ebA

'

'

'

A

B

C

A’

B’

C’

1 2 3 43T 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 ,T01

V1 V2

l1

I2I1

VS

RSZ02 ,T02

V3

l2

I3

Short

Previous examples are the preparation

transmission line basics ii - class 6 prerequisite reading assignment: ch2 acknowledgements: intel...

Documents

transmission lines class

transmission line operation

transmission line basics

transmission line behavior

tem mode transmission

data slide

history slide

michael leddige slide