transmission line

Transmission Line

Microwave EngineeringMicrowave EngineeringCHO, Yong HeuiCHO, Yong Heui

Microwave Microwave EngineeringEngineering

EM Wave EM Wave LabLab

2

Telegrapher’s equation

Communication service: telegraph

Ocean

Telegraph

1. Tx line theory



World wide transmission lines

1. Tx line theory



4

Introduction

The differential equations which the voltage or current must satisfy on a uniform transmission line.

Circuit model: tie field theory and circuit theory together.

Our circuit model contain the inductance L, capacitance C, shunt conductance G, and series resistance R associated with an incremental length of line.

Coaxial transmission line containing a dielectric.

1. Tx line theory



5

Wave propagation

}Re{

}Re{

)cos(),()(

tjzjjo

ztjo

o

eeeV

eV

ztVtzV

1. Tx line theory



6

Distributed element

Lumped element: R, L, C

Distributed element: tx line

1. Tx line theory



7

][ ],/1[ ],[ , :Length

direction a in then Propagatio

FzCzGHzLzRzz

Circuit model

: infinitesimal approachlim0z

1. Tx line theory



8

Transmission line equation

)()())()((2

1

2

1

)(2

1

2

1)(

)( :Phasor

zVzVzIzIzLjzR

zIzLjzRzV

eeVzV

ssss

ss

zjjos

0 ,0 As :))(2

)()((

)(

ss

ss VzCjG

zVzV

z

zI

0 ,0 As :(z)2

1

2

1)()(

)(

ssss IzILjRzILjRz

zV

1. Tx line theory



9

Transmission line equation

)()()(

zILjRdz

zdVs

s

)()()(

zVCjGdz

zdIs

s

Tx line

Tx line modeling

1. Tx line theory



10

Wave solution

Traveling wave solution- Voltage:

- Current:

zzs eVeVzV 00)(

zzs eIeIzI 00)(

)())(()(

2

2

zVCjGLjRdz

zVds

s

))(( CjGLjRj

1. Tx line theory



11

Characteristic impedance

Important parameter in tx line:

-

-

CjG

LjRZ

0

0

0

0

00 I

V

I

VZ

1. Tx line theory

0Z



12

Lossless line

LCjj

C

LZ 0

zjzjs eVeVzV 00)(

zjzjs e

Z

Ve

Z

VzI

0

0

0

0)(

1. Tx line theory



13

Low-loss line

C

G

L

RjLCj

CjGLjRj

21

))((

C

LZ 0

0

02

1GZ

Z

R

1. Tx line theory



14

Distortionless line

LCjZ

Rj

0

C

LZ 0

1. Tx line theory



15

Reflection coefficient

0

0

011 ||

ZZ

ZZ

V

Ve

L

Loj

Voltage wave continuity conditions Current wave continuity conditions

1. Tx line theory



16

Wave power

2

0

2

0* 12

Re2

1

Z

VVIP

1. Tx line theory



17

SWR (Standing Wave Ratio)

SWR: field theory VSWR (Voltage SWR): tx line theory

||1

||1min

max

V

Vs

Experiment

1. Tx line theory



1818

Comparison with circuit theory

1. Tx line theory



1919


1. Tx line theory



2020


1. Tx line theory



21

Comparison with field theory

ss

xsys

ss

ss

ysxs

ysxs

yyssxxss

ss

VCjGdz

dIEj

dz

dH

j

ILjRdz

dVHj

dz

dE

zHE

HE

j

)()(

)(

)(

only. of functions are and where

and Set

,comparisonFor

EH

aHaE

HE

1. Tx line theory



22

Comparison with field theory

))(( ,

)( ,

. and for those tosame theare

and on conditionsboundary The

and , and , and , and :analogydirect A

0

0

CjGLjRjeVV

jjjkeEE

HI

EV

EVCGHI

zs

jkzxxs

yss

xss

xssyss

LCvLCjj

GR

p

1 ,

)0( line lossless aFor

pv

2

1. Tx line theory



23

CjG

LjRZ

j

j

Z

eZ

VIe

EH zj

sjkzx

ys

0

0

0

00

impedance) sticcharacteri :(

0102

0102

011

12

12

0

011 ||

ZZ

ZZ

V

Ve

E

E oj

x

x

Reflection coefficient

1. Tx line theory



24

Input impedance

ljZlZ

ljZlZZZ

ljl

ljl

zZZzZZ

lzIV

in

in

ss

1211

11121

1211

11121

12

sincos

sincos

sincos

sincos

:0for and 0for when

at to of ratio The

1. Tx line theory



25

TLM (Transmission Line Matrix)

Equivalence: field theory and tx line theory Simulation tool: Micro-stripes based on TLM method

Bluetooth antennas

Current density

1. Tx line theory



26

Antenna impedance

Antenna impedance (not infinity) matching No reflection, power efficiency

Handy phone antenna

1. Tx line theory



27

2. Tx line parameters

Planar line

b

d

C

LZ

bR

b

dL

d

bCG

d

bC

db

ext

c

0

ext 2

,

,

Assume

Cross section



28

Coaxial line

a

bdV

LLDQd

La

bL

a

b

Lo

L

S

LL

ln'2

ln'2'2

'2

2

2

aE

aDSD

F/mab

C)/ln(

2

Cross section

Coaxial line + connector




29

mab

GCGG

CRC /1

)/ln(

2,,

Coaxial line

mHa

bL

a

bIdext / ln

2,ln

2

a

b

C

LZ

mba

Rb

Ra

R

ext

ccc

ln2

1

coax a of impedance sticcharacteri The

/11

2

1

2

1,

2

1

0

outerinner


transmission line

Technology

s z vs z

tx line theory vs

tx line theory lossless

e z0 z

tx line theory wave

tx line theory comparison

v0 e z

tx line theory swr