ECE 145A / 218 C, notes set 1: Transmission Line

class notes, M. Rodwell, copyrighted 2009

ECE 145A / 218 C, notes set 1: Transmission Line Properties and Analysis

Mark Rodwell

University of California, Santa Barbara

[email protected] 805-893-3244, 805-893-3262 fax


Transmission Line Analysis

Geometries

Characteristic Impedances

Time Domain Analysis

Lattice Diagrams

Frequency Domain analysis

Reflection coefficients

Movement of Refernece Plane

Impedance vs Position

Smith Chart

Standing Waves

Solving wave equations quickly


types of

transmission lines


Transmission Lines for On-Wafer Wiring

H

W microstrip line

coplanar

waveguide

0V 0V

0V

+V

0V

+V

geometry voltages currents

I

I

H

W

W+2S

I

0

I/2 I/2


Transmission Lines for On-Wafer Wiring

coplanar

strips

slotline

0V

0V

+V

-V

geometry voltages currents

H

G I

0

0V

+V

H

W

W+2S

I

0

I

-V

I


Substrate Microstrip Line

Key advantage: IC interconnects

have very low ground-lead

inductance

Ground-lead inductance:

-leads to ground-bounce

-is Miller-multiplied by IC gain

H

W

Dominant Transmission medium in

III-V microwave & mm-wave ICs

Key problems:

through-wafer grounding holes (vias)

coupling to TM modes in substrate

Via inductance forces progressively

thinner wafers at higher frequencies.


basic theory

L, C, Zo, velocity, Gamma


Transmission Lines

A pair of wires with regular spacing, dielectric loading along the length.

These have inductance per unit length and capacitance per unit length.

Forward and reverse waves propagate.

Reflections will occur if lines are not correctly terminated


Transmission Lines: Basic Theory

Ldz

LZCdz

genV

sZ

LCvCLZ

Z

vztV

Z

vztVtzI

vztVvztVtzV

dtdVCdzdI

dtdILdzdV

o

oo

/1 and /

where

)/()/(),(

)/()/(),(

find which wefrom )/()/(

and )/()/(

:line of analysis nodal basic From


Forward and Reverse Waves

wavereversein current )/(

waveforwardin current )/(

wavereversein voltage)/(

waveforwardin voltage)/(

o

o

Z

vztV

Z

vztV

vztV

vztV


Velocity and Characteristic Impedance

line theofconstant dielectric effective theis

andlight of speed theis where

/

length.unit per quantities here are and

/1 and /

,

,

effr

effr

o

c

cv

CL

LCvCLZ


Reflections

so

os

os

ossgenss

ol

olll

ZZ

ZT

ZZ

ZZVTVV

ZZ

ZZVV

and

1

1 where

:line of beginningAt

1

1 where

:line of endAt LZgenV

sZ

Need good terminations to prevent line reflections and ringing


Total inductance & capacitance in a length of line

olength

o

length

length

ZL

ZC

l

islength in that inductance totaland

islength in that ecapacitanc Then total

islength line totalIf

l

line theon delay"light of speed"/ where vllength


Lumped models of very short transmission lines

L/2 L/2

C

L

C/2 C/2

Pi-model T-model

section or Ta as edapproximat be can line thethen

risetime pulse thanless much is delay line or total

/1 thanless much is /delay line or total

tha waveleng thanless much is length line totalIf

olength

o

length

signallength

length

ZL

ZC

fvl

l


Ladder models of moderately short transmission lines

T-model synthesis

ssimulationcircuit in onsubstitutifrequent a is This

filter. LCanby modelled be can line-siona transmis fashion thisIn

period. signala thanless much is / that such

length of sections into lengthany of linea break can weClearly,

line vll lineline

Pi-model synthesis

L/2L/2

C

L

C/2C/2

L/2L/2

C

L

C/2C/2

L/2L/2

C

L

C/2C/2

L/2L/2

C

L

C/2C/2

L/2L/2

C

L

C/2C/2


Microstrip Lines


Microstrip Line: Approximate Properties (1)

problems) have lines wide:(note

impedance wavespace free theis where, /

light of speed theis /1 where,/

:gives This .dielectricin mostly field line Wide

0

2/1

00

2/1

WHZ

ccv

r

r

H

W


Microstrip Line: Approximate Properties (2)

air.in is field theof proportionupon what depending

,dielectric theof andair ofat between th somewhere lies

/

ly appoximateonly very )2(/

2 widthEffective

eapproximatonly analysis hand narrower, is line theIf

,

2/1

,

2/1

00

effr

effr

r

cv

HWHZ

HW

H

W

~W +2H


Lines in

Time Domain


Lattice Diagrams = Echo Diagrams

so

os

os

ossgenss

oL

oLLL

RZ

ZT

ZR

ZRVTVV

ZR

ZRVV

and 1

1 where

:line of beginningAt

1

1 where

:line of endAt :Recall

reponse. general

find tonconvolutio Use

:Then

response impulsefor Analyze

:First





function.-stepa weregenerator theif

change would waveformshow theconsider pleaseNow


Repeated Reflections→ Ringing or Exponential Decay

ally)(exponentilly geometrica

decay responses pulse

positive, is If sL

-ringing.-sign in alternate

also responses pulse

negative, is If sL

?Why

ringing. RLC toclose

veryappearsBehavior


Time-Domain Analysis

00 / , ZCZL

model eApproximat

charging.circuit RC

inductorneglect

)||()/(

CRRRRL sLsL



00 / , ZCZL

model eApproximat

charging.circuit RL

capacitorneglect

)||()/(

CRRRRL sLsL



00 / , ZCZL

model eApproximat

ringingcircuit RLC

capacitor 2ndneglect

2/ 2/

CRCR SL


L and C are Limiting Cases of High-Z0, low-Z0 lines

00 / , ZCZL

inductor an

ely approximat

. small , large

:line -High 0

CL

Z

capacitor.a

ely approximat

. small , large

:line -Low 0

LC

Z


Lines in

Frequency Domain


Line Analysis in Frequency Domain→ Smith Chart

chart Smith (2)

wavesstanding (1)

impedancesgenerator or load C)(L, reactive easy with

e.intuititiv less

:analysis domain-Frequency

impedancesgenerator or load C)(L, reactive withdifficult very

forth. andback bouncing pulses :clear and eintuititiv

:analysis domain-Time


Line Analysis in Frequency Domain: Phase Constant b

)cos()(

complex. is |||| is where, Re)( :notationPhasor

0

00

os

j

o

tj

s

tVtV

eVVeVtV o

).cos()/cos())/(cos(

,y at velocit traveling wavealcosinusoida For

)./(),/( as travel wavesline, siona transmis On

b

ztvztvzt

v

vztVvztV

constant. npropagatio phase theis /2/ b v


Line Analysis: Exponential Waves

zjtjvzjtjvztj

zjtjvzjtjvztj

tj

tj

o

eeVeVeV

eeVeVeV

eV

eVtV

b

b

/)/(

/)/(

0

00

:direction-z negative thein gpropagatin waveslExponentia

:direction-z positive thein gpropagatin waveslExponentia

. as implicitly written

are wavessinusiodal , Re )cos( Because


Voltages on a Transmission Line

zjzj

tj

tj

eVeV

zVzVzV

ezV

ezVtzV

bb

)0()0(

)()()(

:line theon tagePhasor vol

implicit. dependence time makes )(phasor the withWorking

])(Re[),( :line on Voltage


Voltages and Currents on a Transmission Line

zjzj

zjzj

eVeVzVzVzIZ

eVeVzVzVzV

bb

bb

)0()0()()()(

:line theoncurrent Phasor

)0()0()()()(

:line theon tagePhasor vol

0


Wave Parameters

0

0

0

2*

0

/)()( :amplitude waveReverse

/)()( :amplitude waveFoward

/||)(||)( waveforward inPower

/)( : waveforward inCurrent

)( : waveforward in Voltage

Watt. 1 power wavethen,1|||| if that such amplitude waveDefine

ZzVzb

ZzVza

ZzVIV

ZzV(z)I

zV

aa


Wave Parameters and Power

.quantities R.M.S.use wenotes, theThroughout

)()(/||)(||)( wavereverse inPower

)()(/||)(||)( waveforward inPower

*

0

2*

*

0

2*

zbzbZzVIV

zazaZzVIV


Relfections from the Load

impedance. load *normalized* theis and

t.coefficien reflection load theis 1

1- where

)0()0(

0Z

Z

VV

LL

L

LL

L

Z

Z

Z


Relfections from the Generator

impedance. source *normalized* theis and

t.coefficien reflection source theis 1

1- where

tcoefficien ion transmisssource theis where

)0()0(

0

0

0

Z

Z

ZZ

ZT

VTVV

SS

S

SS

S

S

ssS

Z

Z

Z

moved. been has )0( plane reference that theNote z


Movement of Reference Plane

zj

zj

zj

zjzj

eeV

eV

zV

zVz

eeVzV

zV

zVz

zzVzVzVzV

b

b

b

bb

2

2

)0()0(

)0(

)(

)()( because

)0(1)0()(

tcoefficien reflectiondependent -position theis )(

)( )( where

)(1)()()()(


Position-Dependent Reflection Coefficient

distance. of lengthevery wave

shifts phase degree 360 undergo and because...simply

degrees. 3602 negative

or

radians. 2 negative

or

radians. 22 negative

ofshift phasea throughgone hast coefficien reflection The

)0()(

load. from distancea at t coefficien Reflection

2

VV

l

l

l

el

l

zj

b

b


Impedance vs. Position

)(1

)(1/)()(

pointany at impedance Normalized

)(1

)(1)(

)()()()(

)()()()()(/)()(

pointany at Impedance

0

0

0

z

zZzZz

z

zZzZ

zVzVzVzVZ

zIzIzVzVzIzVzZ

Z


Line Input Impedance

ed.unnormaliz )(1

)(1

)(

)()(

.normalized )(1

)(1

)(

)()(

at impedanceInput

0l

lZ

lI

lVlZ

l

l

lI

lVl

lz

Z


Impedance and Reflection Coefficient vs Postion

tool.graphicala Work withmath. sbut tediou simple,ly Conceptual

impedance normalized )(1

)(1

)(

)(1)(

tscoefficien reflection )0()(

)(/)()(

waves )()0( )()()(

)()0()()()(

0

2

0

z

z

zI

zV

Zz

ez

zVzVz

ezVeVzVzVzIZ

ezVeVzVzVzV

zj

zjzj

zjzj

Z

b

bb

bb


Developing the Smith Chart

graphed. be can iprelationsh This ation. transformconformala

; and s#complex thebetween mapping 1-1a is iprelationsh The

key. is 1

1

1

1 iprelationsh The

Z

Z

ZZ

dot). reda (here,

.pointa by drepresente ist coefficien reflectiona

-plane theof plane ldimensiona-2 theIn


Moving Refernce Planes---on the Smith Chart

line. ion transmiss theon

movement wavelength-half eachfor

plane thein rotation wholeone

2360-

2

angle anby rotates vector the

load, thefromaway distancea move weAs

o

b

l

l

l


Finding Impedances

! plane theon of units Plot the

parts).imaginary and (real impedance normalized and

phase) and (magnitude t coefficien reflection between

iprelationsh 1:1a is This

)(1

)(1

)(

)(1)(

0

Z

Z

Zz

z

zI

zV

Zz


Finding Impedances

01

1

Z

Z

Z

jXRZ

jxr Z

1x

1r3/1r3r

1x

0x

3x

3x

3/1x

3/1x

chart. theon axes

impedance curved the

from read be can

impedance of parts

imaginary and Real


Finding Reflection Coefficient plane-

.*protractora and

rulera using done be

cant measuremen this

angle) and (radiuschart the

from readsimply are of

angle and magnitude The

cursor.a fromt measuremen thedoes softaware CAD today thethough*


Using the Smith Chart

.t coefficien

reflection load thedetermines This

chart. Smith theonpoint thisfind thenWe

./ compute we

, impedance load the withStarting

L

0

ZZ

Z

LL

L

ZLL ,Z


Using the Smith Chart

impedance.input the

off readnow can We

t.coefficien

reflectioninput thelocates This

)./2(360 angle an through

vector therotate thenWe

o l

)(

),(

l

l

in

in

ZZ


Impedance-Admittance Chart

both.or , , for axes have can charts Smith

/ impedance Normalized

/1 Admittance

/ impedance Normalized

Impedance

YZ

Y

Z

jbgYYYZ

jBGZY

jxrZZ

jXRZ

oo

o


Solving Wave

Equations Quickly


Waves and Fourier Transforms (1)

zyxtj

xx

zy

zjkyjkxjktj

xx

zyx

zyx

eeeeEtzyxE

jjkk

EEeeeeEtzyxE

t

E

t

EE

EJ

t

J

t

EE

b

),,,(

writing:complex are s' theSometimes,

) ,for same the(and ),,,(

assume easily, thissolve To

.0 neutrality charge assume , ty,conductivi nonzero Assume

uniform are and if

:equation wavea us give equations sMaxwell'

2

22

1

2

22



statesteady sinusiodal in theequation wave theis This

))(( where

simply becomes This

), ,for same the(and ),,,(

Given

) ,for same the(and

:have weSo

22222

2

2

2

2

2

2

2

2

jjkkkkk

EEeeeeEtzyxE

EEt

E

t

E

z

E

y

E

x

E

zyx

zy

zjkyjkxjktj

xx

zyxxxxx

zyx



VYIIZV

eeItzIeeVtzV

VYz

IIZ

z

V

parallelseries

ztjztj

parallelseries

and

Then

),( and ),(

assume easily, thissolve To

and

line)-ion(transmiss system ldimensiona-1 aconsider Now

////



parallelseriesparallelseries

parallelseriesparallelseries

parallelseries

YZIVZVYIZIV

YZVIYZVI

VYIIZV

/)/( //

: theseDivide

:heseMultiply t

and

0

2

voltage.reverse theof that oppositesign hascurrent reverse the while voltage,forward

theassign same thehascurrent forward that theindicatesroot thebefore The



)/()( ))((

:So

Then

lenght.unit per

econductanc parallel and ecapacitanc parallel has Line

lenght.unit per eresistranc series and inductance series has Line

0 CjGLjRZCjGLjR

CjGYLjRZ

G

C

RL

parallelseries



ncalibratioparameter -Sin sometimesImportant

complex.slightly becomes that Note

2/1

2/1

)/1/()/1(/

)(1)1( Use. and Suppose

)/()( ))((

0

0

0

2

0

Z

CjG

LjR

C

LZ

CjGLjRCjLjZ

ONCjGLR

CjGLjRZCjGLjR

N



b

jLCjGZ

Z

R

LCjCLG

CL

R

CjGLjRLCj

CjGLjRCjLj

ONCjGLR

CjGLjRZCjGLjR

N

22

2

/

/2

)2/1)(2/1(

/1/1))((

)(1)1( Use. and Suppose

)/()( ))((

0

0

2

0

Documents

ECE 145A / 218 C, notes set 1: Transmission Line