Transmission Line

Line

Microwave Engineering

CHO, Yong Heui

  Communication service: telegraph

Ocean

Telegraph

  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.

} Re{

} Re{

) cos(

) , (

) (

t j z j j

o

z t j o

o

e e

e V

e V

z t

V t

z V

































 

  Lumped element: R, L, C

  Distributed element: tx line

  ^{, [ ], [ 1 / ], [ ]}

: Length

direction aˆ

in the n

Propagatio

F z C z

G H

z L z

R z

z













 

: infinitesimal approach

lim

) ( )

( ))

( )

( 2 (

1 2

1

) 2 (

1 2

) 1 (

) ( : Phasor

z V z

V z

I z

I z L j

z R

z I z L j

z R z

V

e e V z

V

s s

s s

s s

z j j

o s









 



 



   



 

 



   













0

, 0

As : ) 2 )(

) ) (

( ) (

(      



 

 

V

z G j C z V

z z V

z

I 

0

, 0

As : 2 (z)

1 2

) 1 ( ) ) (

(      



 



 







 



s s

s

s

R j L I z R j L I z I

z z

V  





) ( )

) (

( R j L I z

dz z dV

s

s

   

) ( )

) (

( G j C V z

dz z dI

s

s

   





Tx line modeling

  Traveling wave solution - Voltage:

- Current:

z z

s

z V e V e

V ( ) 



z z

s

z I e I e

I ( ) 



) ( )

)(

) ( (

2 2

z V

C j

G L

j dz R

z V

d

s

s

    

) )(

( R j L G j C

j   



     

  Important parameter in tx line:

-

-

C j

G

L j

Z R







 

0













0 0 0

0

0

I

V I

Z V

Z

LC j

j  

  

C Z

 L

z j z

j

s

z V e V e

V ( ) 



z j z

j

s

e

Z e V

Z z V

I

0 0 0

)

(

 













 

 



 



 



 















C G L

R LC j

j

C j

G L

j R

j



 











1 2

) )(

(

C Z

 L





 

 



 



2

1 GZ

Z

 R



LC Z j

j  R 



    

0

C Z

 L





0 0 0

11

| |

Z Z

Z Z

V e V

L L j o



 











  Voltage wave continuity conditions

 Current wave continuity conditions

  



0 2

* 0

2 1 2 Re

1   





Z VI V

 P

  SWR: field theory

 VSWR (Voltage SWR): tx line theory

|

| 1

|

| 1

min max







 

 V s V

Experiment

s s

xs ys

s s

s s

ys xs

ys xs

y ys s

x xs s

s s

V C j

dz G E dI

dz j dH

j

I L j

dz R H dV

dz j dE

z H

E

H E

j

) (

) (

) (

) (

only.

of functions are

and where

and

Set

, comparison For































































E H

a H

a E

H E





) )(

(

,

) (

,

. and

for those

to same the

are

and on

conditions boundary

The

and ,

and ,

and ,

and :

analogy direct

A

0 0

C j

G L

j R

j e

V V

j j

jk e

E E

H I

E V

E V

C G

H I

z s

jkz x

xs

ys s

xs s

xs s

ys s























    



















v LC

j j

G R

1 ,

) 0 (

line lossless

a For

























  2  v



C j

G

L j

Z R j

j Z

Z e I V

E e

H









 







 

 









0 0

0 0 0

impedance) stic

characteri :

(

01 02

01 02

0 11

1 2

1 2

0 0

11

| |

Z Z

Z Z

V e V

E

E

x x



 









 

 

















l jZ

l Z

l jZ

l Z Z

Z

l j

l

l j

l

z Z

Z z

Z Z

l z

I V

in in

s s

1 2

1 1

1 1

1 2

1

1 2

1 1

1 1

1 2

1

1 2

sin cos

sin cos

sin cos

sin cos

: 0 for

and 0

for when

at to

of ratio The























 





 





 











 



  Equivalence: field theory and tx line theory

 Simulation tool: Micro-stripes based on TLM method

Bluetooth antennas

Current density

  Antenna impedance (not infinity) matching

 No reflection, power efficiency

b d C

Z L

R b b

L d

d C b

d G C b

d b

ext

c



































0

ext

2

, , Assume





a d b

V

L LD

Q d

a L b a L

b

L o

L S

L L

' ln ln 2

' 2 '

2 ' 2

2

2



 



 















 































a E

a D

S D

  ^F/m

a C b

) / ln(

2 



Cross section





 ^m 

a G b

C G G

RC  C     1 / 

) / ln(

, 2

, 













 ^{H m} 

a L b

a b Id

ext

ln /

, 2

2 ln 





 _





 

a b C

Z L

b m R a

R b R a

ext

c c

c

2 ln 1

coax a

of impedance stic

characteri The

1 / 1

2 1 2

, 1 2

1

0

outer inner





















 



 

  















