ideal tx-line fundamentals -...

Ideal Tx-Line Fundamentals

吳瑞北

Rm. 340, Department of Electrical Engineering

E-mail: [email protected]

google: RBWu

S. H. Hall et al., High-Speed Digital Design, Sec. 3.1-3.41

R. B. Wu

What will you learn

• How signal propagates along interconnects?

• What kinds of structures form Tx-line?

• How to derive Telegrapher eq. from Maxwell eq.?

What’s relation between v(z,t), i(z,t) and E, H?

• What is exact sol. of Telegrapher eq?

How to derive it?

• How to calculate Tx-line capacitance? Can you solve

PDE to obtain capacitance of microstrip line?

• How to estimate major freq. content of digital signal?

2

R. B. Wu

Contents

• Telegrapher’s Equation and Solution

• Tx-Line Parameters Extraction

• Eq. Circuit Modeling

3

Telegrapher Equations & Solutions

4

R. B. Wu

2nd Grand Unification in Physics

0

t

BE

D

H J

B

“The dynamic Theory of

the Electromagnetic

Field,” 1865.

James Clerk Maxwell

(1831-79)

t

D

5

R. B. Wu

Transmission Line Zo

h

w

Coplanar WG

w1

w2

e r

Twisted-pair

Coaxial

b

aStripline

b

w

Common Transmission Lines

(Buried microstrip, if tx-line is

embedded into the dielectric)

Microstrip

6

R. B. Wu

Transmission Lines in PCB

Integrated Circuit

Microstrip

Stripline

Via

Cross section view taken here

PCB substrate

T

W

Cross Section of Above PCB

T

Signal (microstrip)

Ground/Power

Signal (stripline)

Signal (stripline)

Ground/Power

Signal (microstrip)

Copper Trace

Copper Plane

FR4 Dielectric

W

Microstrip

Stripline

7

R. B. Wu

Wave Equations in EM Fields

• Maxwell’s equation in free space:

BE

t

DH J

t

2=

E E

BE

t

2

2

2

22

2i.e., 0

EE J E

t t

EE

t

e e

e

2

2

2Similarly, 0

HH

te

8

R. B. Wu

Voltage & Current in Tx-Line

1

1

( , , , ) ( , ) ( , )

( , , , ) ( , ) ( , )

G

tS

tS

te ds

th d

E x y z t e x y v z t

H x y z t h x y i z t

Electric line

Magnetic line

S

G

9

R. B. Wu

Derivation from Maxwell Equations

2

2

( , )( , ) ( , )

ABCDABCDA S

D

D

dE d B dS

dt

di z tv z z t v z t L z

dt

v iL

z t

2

2

( , )( , ) ( , )

ABCDABCDA S

D

D

dE d B dS

dt


dt

v iL

z t

2

( , ) ( , ) ;

( , )z

d

d DS V

i z t i z z t i

d d di D dS dv C z v z t

dt dt dt

A B

CD

zSz z z

DH

t

dH d H d D dS

dt

2D

i vC

z t

Ampere’s Law

10

R. B. Wu

Derivation from Maxwell Equations

2

2

( , )( , ) ( , )

ABCDABCDA S

D

D

dE d B dS

dt


dt

v iL

z t

2

2

( , )( , ) ( , )

ABCDABCDA S

D

D

dE d B dS

dt


dt

v iL

z t

A B

CD

ABCDABCDA S

BE

t

dE d B dS

dt

2

( , )( , ) ( , ) D


dt

2D

v iL

z t

Faraday’s Law

11

Suitable if simulation

2 2

1

2 D D

fL zC z

R. B. Wu

2 2, versus ,D DL C e

2ˆ

ˆ

G

S

G

S

S

D t

t

t

L h z ds

h z ds

h d

2ˆ

ˆ

S

S

G

S

D t

t

t

C e z d

z e d

e ds

e

e

ˆTEM wave t th z e

2 2D DL C e

2 2

2 22 2( , ) and ( , ) satisfy 0;D D

u uv z t i z t L C

z t

12

R. B. Wu

Solution of Wave Eq. in Phasor Form

• Wave Equation (P.D.E.)

• Wave eq. in freq. domain

2 2

2 2 2

2 2

1( , ) satisfies 0; p

p D D

v vv z t v

z v t L C

2 2

2 2

( )Phasor ( ) satisfies 0;

d V jV z V

dz c

( , ) Re ( ) j tv z t V z e

22

2

Propagation constant

0 ( ) ;

p

j z j z

v

d VV V z V e V e

dz

2 can be found since D

dVI j L I

dz

13

R. B. Wu

Wave Solution in Time Domain

20

0 2

( ; ) ( ) ( ) ; = /

1( ; ) ( ) ( ) ;

j z j z

p

j z j z D

D

V z V e V e v

LI z V e V e Z

Z C

( ) ( ); ( ) ( ) jaf t F f t a F e F F

0

( , ) ( ) ( )

1( , ) ( ) ( )

p p

p p

z zv v

z zv v

V z t V t V t

I z t V t V tZ

Solution in frequency domain

Fourier transform identity

Solution in time domain

14

Tx-Line Parameters Extraction

17

R. B. Wu

• Pure TEM does not exist

• Approximate TEM wave

– Calculation of C2D

– Calculate C0,2D: p.u.l. capacitance

with all dielectrics replaced by

free space

– Determine L2D by L2D C0,2D = 0e0

Tx-Line Parameters

D

r

DDDD

DD

D

D

rD

D

DDDD

p

cCCCcCC

CL

C

LZ

c

C

C

CLCLv

22,022,02

2,02

2

20

2

2,0

2,0222

eff

eff

1

11

e

e

D

Dr

C

C

2,0

2

effe

microstrip line

18

R. B. Wu

Determination of Line Parameters

• Analytical Tech. 1 – Conformal Mapping

– Closed-form solution of Laplace’s equation

– Applicable for several types of lines only

• Analytic Tech. 2 – Eigenfunction Expansion

• Graphical Techniques

– Very rough, but can yield intuitive feeling

• Numerical Techniques

– Calls for numerical techniques

• Empirical Formulas

– Deduce from vast measurement or simulation results

19

R. B. Wu

Analytic Techniques – by conformal mapping

)(2

ABC D

e

'ln'ln2

ABC D

e

Coaxial cable )'/'ln(

22

ABC D

e

AB eAeB ' ;'

Note: pul capacitance keeps invariant in scaling

Theory: pul capacitance keeps invariant in conformal mapping

20

R. B. Wu

Parallel Wire Capacitance

Ref: R. W. Churchill & J. W. Brown, Complex Variable & Applications,

McGraw-Hill, 1984, p.32921

R. B. Wu

Twisted Line Capacitance

1 2

2 21, 1

d dx x

r r

r

2d

1

2d/r

Scaling

22 1

2

1ln coshln 1

D

o

Cd

d dR r

r r

e e e

e e

22

R. B. Wu

Wire above Ground

122 2

lnDC

d

r

e

r

d

DC2 DC2

2

2

2;

2ln

2ln

2

D

D

Cd

r

dL

r

e

DC221

23

R. B. Wu

Analytic Tech.: Governing Eq.

• Field expression:

• Field equation:

• Stored charge:

( , , , ) ( , ) ( , ); where ( , ) satisfies:

( , ) 0

( , ) 1g

s

E x y z t e x y V z t e x y

e x y

e x y ds

ˆ ˆ( , ) ( , ) (F/m)s s

n Ed Q z t n e x y d Ce e

2

( , ) ( , )

1 on 0; ( , )

0 on

s

g

e x y x y

x y

n̂

24

R. B. Wu

Ex: Coaxial Cable

• Laplace eq.

• Sol.:

• Charge:

2

( , ) ( )

1 PDE: 0 for

BC: 1 when ; 0 when

r r

r a r br r r

r a r b

1 1 2 PDE ; ( ) ln

ln BC

ln

r C r C r Cr

b r

b a

2

0

1ˆ ˆ

ln

1 2ˆ

ln lns

e r rr r b a

C n ed ada b a b a

ee e

0

1

ab

0

1

r

r

e e e

25

R. B. Wu

Finte Difference Method

1

4 0 2

3

6

9 5 7

8

26

• Laplace eq.

• FD formula:

• Solve 122 simultaneous equations

10 1 2 3 44

( ) inside material

BC: =0 on ground

=1 on strip

2 22

2 2PDE: 0

x y

1 15 6 8 7 92(1 ) 4

( ) ( ) on material interfacer re

e

re

=0

=1

( )x y

R. B. Wu

Ex. Microstrip Line

• Assume: field can be confined with two side walls

• Laplace eq.

• Appro. Capacitance:

2 22

2 2PDE: 0 unless on metal strip

x y

BC on ground: =0

BC on metal strip:

(i) is continuous;

(ii) ( )y h

y y hxe

/2

/2

(0, )

w

y hwdxQ

CV h

27

R. B. Wu

Ex. Microstrip Line (Cont’)

• PDE + BC(ground) + BC(strip, i):

• BC(strip, ii),

• Solution by Fourier cosine series:

( )

1,3,5,...

cos sinh ; for ; , is odd

cos sinh ; for n

n n n

ny h

n n n n

A x y y h nn

A x h e y h d

Assume ( ) 1 when 2; =0 elsewherex x w

0

1,3,5,...

cos cosh sinh ( )y h

n n n r n ny y hn

A x h h xe e e

1,3,5,...

4sin 2( ) cos n

n n n

n n

wx a x a

d

0

1,3,5,...

... sinh

nn

n n n

n

a wA C

A he

?

28

R. B. Wu

Charge near Conductor Edge

• Laplace eq.

• Solution by separation of variables

• Charge distribution

22

2 2

1 1PDE: 0

BC: (i) on = & =0

rr r r r

V

1

( , ) ( ) ( );

sinn n

n n nn

r R r Y

V a r b r

2

( ) sin ;

( ') ' 0 n n

n n

n n n

Y n

r rR R R a r b r

1 1

0 1

1ˆ n n

n n nnn E E a r b r

r

e e e e

29

R. B. Wu

Charge near Conductor Edge

• Finite charge:

• Near edge (small r):

• A better charge model on microstrip

1 1

0 1on strip: n n

n n nna r b r

e

0finite 0

a

nr

Q dr b

1

13

12

11 11 1

if 3 2 (90 wedge)

if 2 (thin strip)

o

aa r r

r

r

e e

2 2

( )2

Qx

w x

Note: the singularity

is integrable30

R. B. Wu

Graphical Technique

n

j C

n

j

n

j

QQQ

V

n

mD

m

i

ij

n

ijViQmi

j

VVV

QQQ

V

QC

1

1

1

1

1

21

212

11

2

111

0

if curvilinear square

, , ij

m nC C Z

n m

ee

j

iij

V

QC

0

e.g., 26 4

0.154 377

m , n

Z

2D

QC

V

31

R. B. Wu

Parameter Estimation

• Change in orientation of electric flux line

• Effective dielectric constant

• Capacitance

9 27(1) (4) 3.25

9 27 9 27effe

0

143.25 8.85 80.5

5

p

eff

s

nC pF m

ne e

1 2

1 2

tan tan

e e

32

R. B. Wu

Numerical Tech.: Integral Eq.

0

0

Solution :

( , ) ( , ; ', ') ( ', ') '

0; 0

Integral equation:

( , ; ', ') ( ', ') ' ; ( , )

S

y r

S

x y G x y x y x y d

G x y x y x y d V x y S

2

0 r

Green's function: field at ( , ) due to point source at ( ', ')

1DE: ( , ; ', ') ( ') ( ')

BC: G 0 , G 0

t

y

x y x y

G x y x y x x y y e

2 21

In free space: ( , ; ', ') ln ' ' Const.2

G x y x y x x y ye

0

numerical method ( ', ') Q

x y Q ds CV

33

R. B. Wu

Ex: Microstrip Line

+ +

_2 2

2 2

( , ; ', ) ( , ; ', ) ( , ; ', )

( ') ( )1 ln

2 ( ') ( )

newG x y x h G x y x h G x y x h

x x y h

x x y he

( , ; ', )newG x y x h

2 2/2

/2

( ') 4( ')Integral eq.: 1 ln '

2 ' 2

w

w

x x hx wdx x

x x

e

1

Method of moments 1 for 1,2,..., ;

where ( , ; , ) ,

& analytic formula required for

N

mn n

n

mn m n

mm

G m N

G G x h x h x

G

( , ; ', )newG x y x h

34

R. B. Wu

Numerical Techniques

CAP1 CAP2 - microstrip

CAP3 – triplate

Geometry of lines:

width, separation,thickness, height,

substrate, etc.

C2D

Electrical parameters:

capacitance matrix [C]

Inductance matrix [L]

• Based on integral equation formulation

and method of moments

• Employ exact integration formulae

• Four versions are available

W. T. Weeks, “Calculation of coefficients of capacitance of multiconductor

transmission lines in the presence of a dielectric interface,” IEEE Trans.

Microwave Theory Tech., vol. 18, pp. 35-43, Jan. 1970.

CAP4 – e.g. CPW

35

R. B. Wu

CAP2 Example

36

R. B. Wu

Empirical Formula: Commercial Tool

http://web.appwave.com/Products/Microwave_Office/Feature_Guide.php?bullet_id=9 37

R. B. Wu

Simple Microstrip Formula• Parallel Plate Assumptions +

– Large ground plane with zero

thickness

• To accurately predict microstrip

impedance, you must calculate the

effective dielectric constant.

TD

TC

e

WC

From Hall, Hall & McCall:

087 5.98

ln0.81.41r

Zw te

1 1

0.217 12 122 1

r re r

tF

ww

e ee e

2

0.02 1 1 ; for 1

0 ; for 1

r w wF

w

e

Valid when:

0.1 < w < 2.0 and 1 < er < 15

You can’t beat

a field solver

C D

C D

w W T

t T T

38

Various empirical formulae are available, e.g., (3.35) & (3.36b) in textbook.

R. B. Wu

Microstrip Z0 (Rule of Thumb)

• FR4 (Er~4):

50 Ohm line in FR4 has w:h=2:1

• Al2O3氧化鋁(Er~9.9):

50 Ohm line in Al2O3 has w:h=1:1

• GaAs砷化鎵(Er~12.9):

50 Ohm line in GaAs has w:h=0.7:1

W

h

W

h

W

h

http://www.rogerscorporation.com/mwu/mwi_java/Mwij_vp.html

39

R. B. Wu

Simple Stripline Formulas

• Same assumptions as used

for microstrip apply hereTD2

TCe

WCTD1

From Hall, Hall & McCall:

)8.0(67.0

)(4ln

60 110

CC

DD

r

sym

TW

TTZ

e

Symmetric (balanced) Stripline Case TD1 = TD2

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

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

00

000

rCCsymrCCsym

rCCsymrCCsymoffset

TWBZTWAZ

TWBZTWAZZ

ee

ee

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

40

Various empirical formulae are available, e.g., (3.36c) & (3.36d) in textbook.

R. B. Wu

PCB Z0 (Rule of Thumb)

• Microstrip:

50 Ohm line in FR4 has w:h=2:1

• Stripline

50 Ohm line in FR4 has w:b=1:2

• Asymmetric Stripline:

50 Ohm line in FR4 has w:h1=1.25:1, h2=2 h1

W

h

W

W

b

h2

h1

41

R. B. Wu

The “Big Four ” Tx-line Properties

There are four properties affect the performance

of most digital transmission lines:

• Impedance: reflection/distortion

• Time Delay: tx-line concern

• High-Frequency Loss: limit signal bandwidth

and transmission distance

• Crosstalk: coupling

42

Eq. Circuit Modeling

43

R. B. Wu

Transmission Line Concept

PowerPlant

ConsumerHome

Power Frequency (f) is @ 60 HzWavelength (l) is 5 106 m

( Over 3,100 Miles)

44

R. B. Wu

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?

45

In PCB, signal frequency (f) is approaching 10 GHz. Wavelength (l) is 1.5 cm ( 0.6 inches)

R. B. Wu

Short Lines & Tx-Lines

46

R. B. Wu

Rise Time vs. Bandwidth

%10~%90

%90~%10

fromtimefallt

fromtimeriset

f

r

r

H

H

fr

tf

ftt

35.0

35.0

fH : upper cutoff frequency

fallrise

dBt

bandwidth/

3

35.0

Max. slope

= v/T10-90

47

R. B. Wu

Myth of fH = 0.35/tr

3

1( )

1

1

2dB

Hj RC

BWRC

0

0

( ) (1 )

( ) (0.1 0.9)

( ln 0.1 ln 0.9)

2.2

t RC

r

v t V e

v t V

t RC

RC

3

2.2 0.35

2dB

r r

BWt t

Rem.: it is nothing with

the major freq. contents of

the trapezoidal pulse!

t

vi(t)

48

R. B. Wu

Spectrum of Trapezoidal Pulse

0

1

sin sin 2( ) 1 2 cos ;

,

W w r

n w r

Ww r

T

T T

T nx nx ntv t V

T nx nx T

x x

0.44 (3dB)

f0.32

0.32

WT

0.32

49

R. B. Wu

PRBS Power Spectrum

50

2.5 Gb/sec

UI = 400 ps, Tr = 50 ps

Take 28-1 periods

Similar to single pulse response

0 1 2 3 4 5 6 7 8 9 10

UI

-0.5

0

0.5

1

1.5

Vo

ltag

e (

V)

PRBS Pattern

Expect noise level to be lower if resonance occurs at the null

point of power spectrum

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Freq (GHz)

-60

-50

-40

-30

-20

-10

0

No

rma

lize

d P

ow

er

Den

sit

y (

dB

)

PRBS Power Spectrum

PRBS

Pulse

R. B. Wu

Tr and BW

for Ideal Square Wave

f0.32

rt

0.32

WT

spec

trum

• It is suggested to consider significant bandwidth = 1/tr. Then,

as compared with ideal square wave, the neglected part is

even smaller by at least 20log(1/0.32) = 10dB.

• Typical example, clock freq. =1/TW, tr = 0.1TW. Then, the

neglected part is smaller than dc part by at least

20log(1/0.032)+10 = 40dB

Rise Time Bandwidth

10 nsec 100 MHz1 nsec 1 GHz100 psec 10 GHz50 psec 20 GHz

52

R. B. Wu

Did you learn

• How to derive Telegrapher eq. from Maxwell eq.?

What’s relation between v(z,t), i(z,t) and E, H?

• What is the exact solution of Telegrapher eq? How

to derive it?

• How to calculate Tx-line capacitance? Can you

solve PDE to obtain capacitance of microstrip line?

• What is the meaning of 0.35/tr? How to estimate the

major freq. content of a digital signal?

53

R. B. Wu

References & Further Reading

• W. T. Weeks, “Calculation of coefficients of capacitance of multi-conductor transmission lines in the presence of a dielectric interface,” IEEE Trans. Microwave Theory Tech., vol.18, pp. 35-43, Jan. 1970.

54

ideal tx-line fundamentals -...

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