Notes 1 - Transmission Line Theory

8/20/2019 Notes 1 - Transmission Line Theory

1/82

Prof. David R. JacksonDept. of ECE

Notes 1

ECE 5317-6351Microwave Enineerin

Fall

2011

!rans"ission #ine !$eor%

1


2/82

& wave'idin str'ct're is one t$at carriesa sina( )or power* fro" one point toanot$er.

!$ere are t$ree co""on t%pes+

!rans"ission (ines ,ier-optic 'ides ave'ides

ave'idin /tr'ct'res

Note+ &n a(ternative to wave'idin str'ct'res is wire(esstrans"ission 'sin antennas. )antenna are disc'ssed inECE 5310.*

2


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!rans"ission #ine

as two cond'ctors r'nnin para((e(an propaate a sina( at an% fre2'enc% )in t$eor%*eco"es (oss% at $i$ fre2'enc%an $and(e (ow or "oderate a"o'nts of poweroes not $ave sina( distortion4 'n(ess t$ere is (ossa% or "a% not e i""'ne to interference

oes not $ave E z or H z co"ponents of t$e e(ds )!EM z *

Properties

Coaia( ca(e )coa*

!win (ead )s$own connected to a +1

i"pedance-transfor"ina('n*3


4/82

!rans"ission #ine )cont.*

C&! 5 ca(e)twisted pair*

!$e two wires of t$e trans"ission (ine are twisted to red'ceinterference and radiation fro" discontin'ities.

4


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Microstrip

h

w

ε r

ε r

w

/trip(ine

h

rans"ission (ines co""on(% "et on printed-circ'it oards

Cop(anar strips

hε r

w w

Cop(anar wave'ide )CP*

hε r

w

5


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!rans"ission (ines are co""on(% "et on printed-circ'it oards.

& "icrowave interatedcirc'it

Microstrip line

6


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,ier-8ptic 9'ide

Properties

:ses a die(ectric rod Can propaate a sina( at an% fre2'enc% )in t$eor%* Can e "ade ver% (ow (oss as "ini"a( sina( distortion ;er% i""'ne to interference

Not s'ita(e for $i$ power as ot$ E z and H z co"ponents of t$e e(ds

7


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,ier-8ptic 9'ide )cont.*

!wo t%pes of er-optic 'ides+

1* /in(e-"ode er


9/82

,ier-8ptic 9'ide )cont.*

$ttp+@@en.wikipedia.or@wiki@8ptica(Aer

i$er inde core reion

9


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ave'ides

as a sin(e $o((ow "eta( pipe

Can propaate a sina( on(% at $i$ fre2'enc%+ ω > ω c$e widt$ "'st e at (east one-$a(f of a wave(ent$

as sina( distortion4 even in t$e (oss(ess case=""'ne to interferenceCan $and(e (are a"o'nts of power

as (ow (oss )co"pared wit$ a trans"ission (ine*as eit$er E z or H z co"ponent of t$e e(ds )!M z or !E z *

Properties

$ttp+@@en.wikipedia.or@wiki@ave'ideA)e(ectro"anetis"*10


11/82

Lumped circuits: resistors, capacitors, inductors

ne(ect ti"e de(a%s)p$ase*

acco'nt for

propaation and ti"ede(a%s )p$ase c$ane*

!rans"ission-#ine !$eor%

Distributed circuit elements: transmission lines

e need trans"ission-(ine t$eor% w$enevert$e (ent$ of a (ine is sinicant co"pared wit$a wave(ent$.

11


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ransmission Line

2 conductors

4 per!unit!len"t# parameters:

C $ capacitance%len"t# &F/m'

L $ inductance%len"t# &H/m'

R $ resistance%len"t# &Ω/m'

G $ conductance%len"t# & / m or S/m' Ω

∆ z

12


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ransmission Line (cont)*

z ∆

( ),i z t

+ + + + + + +! ! ! ! ! ! ! ! ! ! ( ),v z t

13

R∆ z L∆ z

G∆ z C ∆ z

z

v( z +∆ z ,t )

+

!

v( z ,t )

+

!

i( z ,t ) i( z +∆ z ,t )


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( , )( , ) ( , ) ( , )

( , )( , ) ( , ) ( , )

i z t v z t v z z t i z t R z L z t

v z z t i z t i z z t v z z t G z C z

t

∂= + ∆ + ∆ + ∆∂

∂ + ∆= + ∆ + + ∆ ∆ + ∆

∂

ransmission Line (cont)*

14

R∆ z L∆ z

G∆ z C ∆ z

z

v( z +∆ z ,t )

+

!

v( z ,t )

+

!

i( z ,t ) i( z +∆ z ,t )


15/82

-ence

( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )( , )

v z z t v z t i z t Ri z t L

z t

i z z t i z t v z z t Gv z z t C

z t

+ ∆ − ∂= − −∆ ∂

+ ∆ − ∂ + ∆= − + ∆ −

∆ ∂

.o/ let ∆ z → 0:

v i

Ri L z t

i vGv C

z t

∂ ∂= − −∂ ∂

∂ ∂= − −

∂ ∂

ele"rap#ers

uations

M ransmission Line (cont)*

15


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o combine t#ese, tae t#e deriatie o t#e irst one /it#

respect to z :2

2

2

2

v i i R L

z z z t

i i R L z t z

v R Gv C

t

v v L G C

t t

∂ ∂ ∂ ∂ = − − ÷∂ ∂ ∂ ∂

∂ ∂ ∂ = − − ÷∂ ∂ ∂ ∂ = − − −

∂ ∂ ∂ − − − ∂ ∂

/itc# t#e

order o t#e

deriaties)


16


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( )

2 2

2 2( ) 0

v v v

RG v RC LG LC z t t

∂ ∂ ∂

− − + − = ÷∂ ∂ ∂

#e same euation also #olds or i.

-ence, /e #ae:

2 2

2 2

v v v v R Gv C L G C z t t t ∂ ∂ ∂ ∂ = − − − − − − ∂ ∂ ∂ ∂


17


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( )2

2

2( ) ( ) 0

d V RG V RC LG j V LC V

dz ω ω − − + − − =

( )2 2

2 2( ) 0

v v v RG v RC LG LC

z t t

∂ ∂ ∂ − − + − = ÷

∂ ∂ ∂


ime!-armonic aes:

18


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.ote t#at

$ series impedance%len"t#

( ) ( )2

2

2

( )d V

RG V j RC LG V LC V dz

ω ω = + + −

2( ) ( ) ( ) RG j RC LG LC R j L G j C ω ω ω ω + + − = + +

Z R j L

Y G j C

ω

ω

= +

= + $ parallel admittance%len"t#

#en /e can /rite:

2

2( )

d V ZY V

dz =


19


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Let

onention:

olution:

2γ = ZY

( ) z z V z Ae Beγ γ − += +

[ ]1/2

( )( ) R j L G j C γ ω ω = + +

= principal suare root

2

2

2

( )d V

V dz

γ =

#en


γ is called t#e ;propa"ation constant);

/2 j z z e θ =

π θ π − < <

jγ α β = +

0, 0α β ≥ ≥

α

β

==attenuationcontant

p#aseconstant

20


21/82


0 0( ) z z j z

V z V e V e eγ α β + + − + − −

= =

,orward trave((in wave )a wave trave(in in t$e positive z direction*+

( ){ }

( ){ }( )

0

0

0

( , ) Re

Re

cos

z j z j t

j z j z j t

z

v z t V e e e

V e e e e

V e t z

α β ω

φ α β ω

α ω β φ

+ + − −

+ − −

+ −

=

=

= − +

g λ

0t =

z

0

z V e α + −

2

g

π β

λ =

2 g

βλ π =

!$e wave Brepeats w$en+

ence+

21


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P$ase ;e(ocit%

!rack t$e ve(ocit% of a ed point on t$e wave )a point of constant

p$ase*4 e..4 t$e crest.

0( , ) cos( ) z v z t V e t z α ω β φ + + −= − +

z

v p )p$ase ve(ocit%*

22


23/82

P$ase ;e(ocit% )cont.*

0

constantω β

ω β

ω

β

− =

− =

=

t z

dz

dt

dz

dt

/et

ence pv ω

β =

[ ]{ }1/2

Im ( )( ) p

v R j L G j C

ω

ω ω =

+ +

=n epanded for"+

23


24/82

C$aracteristic ="pedance Z

0

( )

( )

V z Z

I z

+

+≡

0

0

( )

( )

z

z

V z V e

I z I e

γ

γ

+ + −

+ + −

=

=

so 00

0

V Z

I

+

+=

+V +( z )

-

I + ( z )

z

& wave is trave(in in t$e positive z direction.

) Z is a n'"er4 not a f'nction of z .*

24


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26/82

>rom t#is /e #ae:


27/82

( )

00

0 0

j z j j z

z z

z j z V e e

V z V e V

V e e e

e

eφ α

γ γ

β β φ α −+

+ + −

−+ + − +

+

+

= +

= +

( ) ( ){ }

( )

( )0

0 cos

c

, R

os

e j t

z

z

V e t

v z t V z

z

V z

e

e t

ω

α

α

ω β

ω β φ

φ − +

+ +

−

−

+

=

+

+

+

−=

.ote:/ae in + z direction /ae in - z

direction

9enera( Case )aves in ot$Directions*

27


28/82

?ac/ard!raelin" ae

0

( )

( )

V z Z

I z

−

− =− 0( )

( )

V z Z

I z

−

− = −so

+V -( z )

-

I - ( z )

z

@ /ae is traelin" in t#e ne"atie z direction)

Note+ !$e reference directions for vo(tae and c'rrentare t$e sa"e as for t$e forward wave.

28


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Aeneral ase

0 0

0 0

0

( )1

( )

z z

z z

V z V e V e

I z V e V e Z

γ γ

γ γ

+ − − +

+ − − += +

= −

@ "eneral superposition o or/ard and

bac/ard traelin" /aes:

Most "eneral case:

Note+ !$ereferencedirections forvo(tae and c'rrentare t$e sa"e forforward and

ackward waves.29

+V ( z )

-

I ( z )

z


30/82

( )

( )

( ) ( )1

2

12

0

0 0

0 0

0 0

z z

z z

V z V e V e

V V

I z e e Z

j R j L G j C

R j L

Z G j

Z

C

γ γ

γ γ

γ α β ω ω

ω

ω

+ − − +

+ −− +

=

= +

+ = + +

+=

=

÷

−

+

[ ]2

m g π

λ β

=

[m/s] pv ω

β =

"uided /aelen"t# ≡ λ g

p#ase elocitB ≡ v p

ummarB o ?asic L ormulas

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#oss(ess Case

0, 0 R G= =

[ ]1/ 2

( )( ) j R j L G j C

j LC

γ α β ω ω

ω

= + = + +

=

so 0

LC

α β ω

==

1/2

0

R j L Z

G j C

ω

ω

+= ÷+

0

L Z

C =

1 pv

LC =

pv ω β

=

)indep. of fre2.*)rea( and indep) o re)*31

( C


32/82

#oss(ess Case)cont.*1

pv

LC

=

=n t$e "edi'" etween t$e two cond'ctors is$o"oeneo's )'nifor"* and is c$aracteriFed % )ε 4 µ *4t$en we $ave t$at

LC µε =

!$e speed of (i$t in a die(ectric"edi'" is

1d c

µε =

ence4 we $avet$at

p d v c=

!$e p$ase ve(ocit% does not depend on t$e fre2'enc%4 and it isa(wa%s t$e speed of (i$t )in t$e "ateria(*.

)proof iven

(ater*

32


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( ) 0 0 z z V z V e V eγ γ + − − += +

#ere do /e assi"n z = 0C

#e usual c#oice is at t#e load)

erminatin" impedance (load*

@mpl) o olta"e /ae

propa"atin" in ne"atie z

direction at z = 0.

@mpl) o olta"e /ae

propa"atin" in positie z

direction at z = 0.

erminated ransmission Line

Note+ !$e (ent$ l "eas'res distance fro" t$e (oad+ z = −(33

i d i i Li ( *


34/82

#at i /e no/

@V V z + − = −(and

( ) ( )0 0V V V e γ + + + −= = − ((

( ) ( ) ( ) ( ) ( ) z z

V z V e V eγ γ − + ++ −= − + −( (( (

( ) ( )0V V e γ − − −− = ((

( ) ( )0 0V V V eγ − − −⇒ = = − ((

erminated ransmission Line (cont)*

( ) 0 0 z z

V z V e V eγ γ + − − +

= +

-ence

an /e use z = - l asa reerence planeC


34

( *


35/82

( ) ( ) ( ) ( ) ( )( ) ( ) z z V z V e V eγ γ − − − − −+ −= − + −( (( (


( ) ( ) ( )0 0 z z V z V e V eγ γ + − − += +

ompare:

Note+ !$is is si"p(% a c$ane of reference p(ane4 fro" z = 0

to z = -l.


35

i t d i i Li ( t *


36/82

( ) 0 0 z z

V z V e V e

γ γ + − − +

= +

#at is V (-l )?

( ) 0 0V V e V eγ γ + − −− = +( ((

( ) 0 0

0 0

V V I e e

Z Z

γ γ + −

−− = −( ((

propa"atin"

or/ards

propa"atin"

bac/ards


l ≡ distance a/aB rom load

#e current at z = - l is t#en


36

i t d i i Li ( t *


37/82

( ) ( )200

1 LV

I e e

Z

γ γ +

−− = − Γ ( ((

( ) 2000

0 0 1 V

V eV V e eeV

V γ γ γ γ

−

+− −+ − +− =

= + + ÷( (( ((

otal olt) at distance l

rom t#e load

@mpl) o olt) /ae prop)

to/ards load, at t#e load

position ( z = 0*)

imilarlB,

@mpl) o olt) /ae prop)

a/aB rom load, at t#e

load position ( z = 0*)

( )0 21 LV e eγ γ + −= + Γ ( (

Γ L ≡ Load relection coeicient


Γ l ≡ election coeicient at z = - l

37

( *


38/82

( ) ( )

( ) ( )

( ) ( )( )

2

0

2

2

0

0

2

0

11

1

1

L

L

L

L

V V e e

V I e e

Z

V e Z Z I e

γ γ

γ γ

γ

γ

+ −

+

−

−

−

− = + Γ

− + Γ − = = ÷− −

− = − Γ

Γ

( (

(

(

(

(

((

(

(

(

=nput impedance seen looin" to/ards load

at z = -l .


( ) Z −(

38

i d i i Li ( *


39/82

@t t#e load (l = 0*:

( ) 01

01

L L

L

Z Z Z + Γ

= ÷− Γ ≡

#us,

( )

20

0

0

20

0

1

1

L

L

L

L

Z Z e Z Z

Z Z Z Z

e Z Z

γ

γ

−

−

−+ ÷ ÷+ ÷− = ÷ −

− ÷ ÷ ÷+

(

(

(


0

0

L L

L

Z Z

Z Z

−⇒ Γ =

+

( )2

0 2

1

1

L

L

e Z Z

e

γ

γ

−

−

+ Γ − = ÷− Γ

(

((ecall

39

i d i i Li ( *


40/82

impliBin", /e #ae

( ) ( )

( )0

0

0

ta!

ta!

L

L

Z Z Z Z

Z Z

γ

γ

+− = ÷ ÷+

((

(


( ) ( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )

20

2

0 0 0

0 0 2

2 0 00

0

0 00

0 0

0

0

0

1

1

cos! s"!

cos! s"!

L

L L L

L L L

L

L L

L L

L

L

Z Z e

Z Z Z Z Z Z e Z Z Z

Z Z Z Z e Z Z e

Z Z

Z Z e Z Z e Z Z Z e Z Z e

Z Z Z

Z Z

γ

γ

γ

γ

γ γ

γ γ

γ γ

γ γ

−−

−−

+ −

+ −

−+ ÷ ÷ + + + − ÷− = = ÷ ÷ ÷ + − − − − ÷ ÷ ÷+

+ + −= ÷ ÷+ − −

+= +

(

(

(

(

( (

( (

(

( (

( ( ÷÷

-ence, /e #ae

40

i t d L l i i Li


41/82

( ) ( )

( ) ( )

( )

2

0

20

0

2

0 2

1

1

1

1

j j

L

j j

L

j

L

j

L

V V e e

V I e e

Z

e Z Z

e

β β

β β

β

β

+ −

+−

−

−

− = + Γ

− = − Γ

+ Γ − = ÷− Γ

( (

( (

(

(

(

(

(

=mpedance is periodic

/it# period λ g /2

2

/ 2

g

g

β π

π

π λ

λ

=

=

⇒ =

(

(

(

erminated Lossless ransmission Line

j jγ α β β = + =

Note+ ( ) ( ) ( )ta! ta! ta j jγ β β = =( ( (

tan repeats w$en

( ) ( )

( )0

0

0

ta

ta

L

L

Z jZ Z Z

Z jZ

β

β

+− = ÷ ÷+

((

(

41

i t d L l i i Li


42/82

,or t$e re"ainder of o'r trans"ission (ine disc'ssion we wi((ass'"e t$at t$e trans"ission (ine is (oss(ess.

( ) ( )

( ) ( )

( ) ( )

( )

( )

( )

2

0

20

0

2

0 2

0

0

0

1

1

1

1

ta

ta

j j

L

j j

L

j

L

j

L

L

L

V V e e

V I e e

Z

V e Z Z

I e

Z jZ Z

Z jZ

β β

β β

β

β

β

β

+ −

+−

−

−

− = + Γ

− = − Γ

− + Γ − = = ÷− − Γ

+= ÷ ÷

+

( (

( (

(

(

(

(

((

(

(

(

0

0

2

L L

L

g

p

Z Z

Z Z

v

π λ

β

ω

β

−Γ =

+

=

=

erminated Lossless ransmission Line

( ) Z −(

42

Matc#ed Load


43/82

Matc$ed (oad+ ( Z L= Z 0)

0

0

0 L L L

Z Z

Z Z

−Γ = =

+

>or anB l

.o relection rom t#e load

&

Matc#ed Load

( ) Z −(

( ) 0 Z Z ⇒ − =(

( )

( )

0

0

0

j

j

V V e

V I e

Z

β

β

+ +

++

⇒ − =

− =

(

(

(

(

43

#ort ircuit Load


44/82

/$ort circ'it (oad+ ) Z L = 0*

( ) ( )

0

0

0

0 10

ta

L Z Z

Z jZ β

−Γ = = −+

⇒ − =( (

@l/aBs ima"inarBE.ote:

2

g

β π

λ

=( (

( ) sc Z j ⇒ − =(

)) can become an F))

/it# a λ g /# trans) line

#ort!ircuit Load

( )0 ta sc Z β = (

44


45/82


46/82

( )

( )

( )

0

00

ta

ta

L

i! L

Z jZ d

Z Z d Z Z jZ d

β

β

+

= − = ÷ ÷+

( ) i!"H i! "H

Z V d V

Z Z

⇒ − = ÷+

ample

>ind t#e olta"e at anB point on t#e line)

46

l ( t *


47/82

Note+ ( ) ( )02

1 j

L

jV V e e

β β + −+ Γ =− ( ((

0

0

L L

L

Z Z

Z Z −Γ =+

( ) ( )20 1 j d j d i!

"H

i! "H

LV d Z

Z

e V

Z

V e β β + −− = + Γ

= ÷

+

( ) ( )2

2

1

1

j j d i! L

"H j d

# "H L

Z eV V e

Z Z e

β β

β

−− −

−

+ Γ − = ÷ ÷+ + Γ

((

(

@t l = d :

-ence

ample (cont)*

0 2

1

1

j d i!"H j d

i! "H L

Z V V e

Z Z e

β

β

+ −−

⇒ = ÷ ÷+ + Γ

47

ample (cont *


48/82

ome al"ebra: ( )2

0 2

1

1

j d

Li! j d

L

e Z Z d Z

e

β

β

−

−

+ Γ = − = ÷

− Γ

( )

( ) ( )

( )( ) ( )

( )

2

20 20

2 22

0

0 2

2

0

2

0 0

2

0

20 0

0

1

11

1 11

1

1

1

1

j d

L j d j d

L L

j d j d j d

L "H L L

"H j d L

j d

L

j d

"H L "H

j d

L

j d "H "H L

"H

i!

i! "H

e Z

Z ee

Z e Z ee

Z Z e

Z e

Z Z e Z Z

e Z

Z

Z

Z Z

Z Z Z e Z Z

Z

β

β β

β β β

β

β

β

β

β

−

−−

− −−

−

−

−

−

−

+ Γ ÷ + Γ − Γ ⇒ = =

+ Γ + − Γ + Γ

+ ÷− Γ

+ Γ =

+ + Γ

+

−

+ Γ = ÷

+ − + Γ ÷+

= ( )2

0

20 0

0

1

1

j d

L

j d "H "H L

"H

e

Z Z Z Z e

Z Z

β

β

−

−

+ Γ ÷+ − − Γ ÷+

ample (cont)*

48

ample (cont *


49/82

( ) ( )2

0

2

0

1

1

j j d L

"H j d

"H $ L

Z eV V e

Z Z e

β β

β

−− −

−

+ Γ − = ÷ ÷+ − Γ Γ

((

(

20

2

0

1

1

j d i! L

j d

i! "H "H $ L

Z Z e

Z Z Z Z e

β

β

−

− + Γ = ÷ ÷+ + − Γ Γ

/#ere 0

0

"H $

"H

Z Z

Z Z

−Γ =

+

ample (cont)*

#ereore, /e #ae t#e ollo/in" alternatie orm or t#e result:

-ence, /e #ae

49

ample (cont *


50/82

( ) ( )2

0

2

0

1

1

j j d L

"H j d

"H $ L

Z eV V e

Z Z e

β β

β

−− −

−

+ Γ − = ÷ ÷+ − Γ Γ

((

(

ample (cont)*

Holta"e /ae t#at /ould eist i t#ere /ere no relections rom

t#e load (a semi!ininite transmission line or a matc#ed load*)

50

ample (cont *


51/82

( )

( )

( ) ( ) ( ) ( )

2 2

2 2 2 20

0

1 j d j d

L L $

j d j d j d j d

"H L $ L L $ L $

"H

e e

Z V d V e e e e Z Z

β β

β β β β

− −

− − − −

+ Γ + Γ Γ

− = + Γ Γ Γ + Γ Γ Γ Γ ÷ + +

G

ample (cont)*

ae!bounce met#od (illustrated or l % d *:

51

ample (cont *


52/82

ample (cont)*

( )( ) ( )

( ) ( )

22 2

22 2 20

0

1

1

j d j d

L $ L $

j d j d j d

"H L L $ L $

"H

e e

Z V d V e e e

Z Z

β β

β β β

− −

− − −

+ Γ Γ + Γ Γ + − = + Γ + Γ Γ + Γ Γ + ÷ +

+

G

G

G

Aeometric series:

2

0

11 , 1

1

!

!

z z z z z

∞

=

= + + + = <−∑ G

( )

( )

( ) ( ) ( ) ( )

2 2

2 2 2 20

0

1 j d j d

L L $

j d j d j d j d "H L $ L L $ L $

"H

e e

Z V d V e e e e Z Z

β β

β β β β

− −

− − − −

+ Γ + Γ Γ

− = + Γ Γ Γ + Γ Γ Γ Γ ÷ + +

G

2 j d

L $ z e β −= Γ Γ

52

ample (cont *


53/82

ample (cont)*

or

( )2

0

20

2

1

1

1

1

j d L s

"H

j d "H

L j d

L s

e Z V d V

Z Z e

e

β

β

β

−

−−

− Γ Γ − = ÷ + + Γ ÷− Γ Γ

( )2

0

2

0

1

1

j d

L"H j d

"H L s

Z eV d V

Z Z e

β

β

−

−

+ Γ − = ÷ + − Γ Γ

#is a"rees /it# t#e preious result (settin" l % d *)

Note+ !$is is a ver% tedio's "et$od H not reco""ended.

-ence

53

ime! @era"e Io/er >lo/


54/82

@t a distance l rom t#e load:

( ) ( ) ( ){ }

( ) ( )$

$

2

0 2 2 $ 2

$

0

1Re 1 1

1 R

2

e2

L L

V e e

Z

V I

e

&

α γ γ

+− −

− =

= + Γ − Γ

− −

( ( (

( ( (

( ) ( )2

20 2 #

0

11

2 L

V & e e

Z

α α

+−− ≈ − Γ ( ((

=f Z 0 ≈ rea( )(ow-(osstrans"ission (ine*


( ) ( )( ) ( )

2

0

20

0

1

1

L

L

V V e e

V I e e

Z

j

γ γ

γ γ

γ α β

+ −

+−

− = + Γ − = − Γ

= +

( (

( (

(

(

( )

$2 $ 2

$2 2

L L

L L

e e

e e

γ γ

γ γ

− −

− −

Γ − Γ

= Γ − Γ

=

( (

( (

pure imaginary

Note+

54



55/82

Lo/!loss line

( ) ( )2

20 2 #

0

2 2

20 02 2

$ $

0 0

11

2

1 1

2 2

L

L

V & d e e

Z

V V e e

Z Z

α α

α α

+−

+ +−

− ≈ − Γ

= − Γ

( (

( (

1 7< 73 1 7 7 < 7 7 3power inforwardwave power in -ackwardwave

( ) ( )2

20

0

11

2 L

V & d

Z

+

− = − Γ

Lossless line (α = 0*

ime @era"e Io/er >lo/

55

Juarter!ae ransormer


56/82

00

0

tata

L " i! "

" L

Z jZ Z Z Z jZ

β β

+= ÷+ ((

2

# # 2

g g

g

λ λ π π β β

λ = = =(

00

" i! "

L

jZ Z Z

jZ

⇒ = ÷

0

2

00

0i! i!

"

L

Z Z Z

Z Z

Γ = ⇒ =⇒ =

Juarter!ae ransormer

2

0" i!

L

Z Z

Z =

so

[ ]1/2

0 0" L Z Z Z =

ence

#is reuires Z L to be real)

Z L Z 0 Z 0%

Z i!

56

Holta"e tandin" ae atio


57/82

( ) 20 1 L j j

LV V e e

φ β + −− = + Γ ((

( ) ( )( )

2

0

2

0

1

1 L

j j

L

j j j

L

V V e e

V e e e

β β

φ β β

+ −

+ −− = + Γ = + Γ

( (

( (

(

( )

( )

ma& 0

m" 0

1

1

L

L

V V

V V

+

+

= + Γ

= − Γ

( ) ma&

m"

V V

=;o(tae /tandin .ave Ratio ;/.R

Holta"e tandin" ae atio

1

1

L

L

+ Γ =

− Γ

;/.R

z

1+ LΓ

1

1- L

Γ

0

( )V z

V +

/ 2 z λ ∆ =0 z =

57

oaial able


58/82

oaial able

e we present a Bcase st'd% of one partic'(ar trans"ission (ine4 t$e coaia( ca(

'

( ,r ε σ

,ind C) L) G) R

e wi(( ass'"e no variation in t$e z direction4 and take a (ent$ ofone "eter in t$e z direction in order top ca(c'(ate t$e per-'nit-(ent$para"eters.

58

,or a !EM z "ode4 t$e s$ape of t$e e(ds is independent of fre2'enc%4and $ence we can perfor" t$e ca(c'(ation 'sin e(ectrostatics and"anetostatics.

oaial able (cont)*


59/82

oaial able (cont)*

- ρ * 0

ρ * 0

'

(

r ε 0 0

0

' '2 2 r

E ρ ρ

ρ ρ π ε ρ π ε ε ρ

= = ÷ ÷

( (

,ind C )capacitance @(ent$*

Coaia(ca(e

h = 1 [m]

r ε

>rom Aausss la/:

0

0

2

B

AB

A

(

r '

V V E dr

( E d

' ρ

ρ ρ

π ε ε

= = ×

= = ÷

∫

∫ (

59

oaial able (cont)*


60/82

- ρ * 0

ρ * 0

'

(

r ε

Coaia(ca(e

h = 1 [m]

r ε

( )00

0

1

2 r

C V (

'

ρ

ρ

π ε ε

= = ÷ ÷

(

(

ence

e t$en$ave

0 F/m2

[ ]

r C (

'

π ε ε =

÷

oaial able (cont)*

60

oaial able (cont)*


61/82

'2

I H φ

π ρ

= ÷

,ind L )ind'ctance @

(ent$*>rom @mperes la/:

Coaia(

ca(e

h = 1 [m]

r µ

I

0'

2r

I B φ µ µ

π ρ

= ÷

(1)

(

'

B d φ ψ ρ = ∫ $

h

I

I z center cond'ctorMa"netic lu:

oaial able (cont)*

61

.ote: e i"nore internal inductance

#ere, and onlB loo at t#e ma"netic ield

between t#e t/o conductors (accurate

or #i"# reuencB)

oaial able (cont)*


62/82

Coaia(

ca(e

h = 1 [m]

r µ

I

( ) 0

0

0

1

2

2

(

r

'

(

r

'

r

H d

I d

I (

'

φ ψ µ µ ρ

µ µ ρ πρ

µ µ

π

=

=

= ÷

∫ ∫

0

1

2r

( L

I '

ψ µ µ

π

= = ÷

0 H/m [ ]2

r ( L'

µ µ

π

= ÷

ence

oaial able (cont)*

62

oaial able (cont)*


63/82

0 H/m [ ]2

r ( L'

µ µ

π

= ÷

8servation+

0 F/m2

[ ]

r C (

'

π ε ε =

÷

( )0 0 r r LC µε µ ε µ ε = =

!$is res'(t act'a((% $o(ds for an% trans"ission (ine.

oaial able (cont)*

63

oaial able (cont)*


64/82

0

H/m [ ]2

r (

L '

µ µ

π

= ÷

,or a (oss(ess ca(e+

0

F/m

2

[ ]

r

C (

'

πε ε

= ÷

0

L Z C =

0 0

1

[ ]2

r

r

(

Z '

µ

η ε π

= Ω ÷

00

0

*.*0 [ ] µ

η ε

= = Ω

oaial able (cont)*

64

oaial able (cont)*


65/82

- ρ * 0

ρ * 0

'

(

σ 0 0

0

' '2 2 r

E ρ ρ

ρ ρ π ε ρ π ε ε ρ

= = ÷ ÷

( (

,ind G )cond'ctance @(ent$*

Coaia(ca(e

h = 1 [m]

σ

>rom Aausss la/:

0

0

2

B

AB

A

(

r '

V V E dr

( E d

' ρ

ρ ρ

π ε ε

= = ×

= = ÷

∫

∫ (

oa a ab e (co *

65

oaial able (cont)*


66/82

- ρ * 0

ρ * 0

'

(

σ , E σ =

e t$en$ave

*e' I GV

=

[ ]

0

0

(1) 2

2

2 2

*e'- '

'

r

I , '

' E

' '

ρ ρ

ρ ρ

π

π σ

ρ π σ π ε ε

=

=

=

=

= ÷

(

0

0

0

0

22

2

r

r

''

G(

'

ρ π σ π ε ε

ρ

π ε ε

÷ =

÷

(

(

2[S/m]

G(

'

πσ =

÷

or

( *

66

oaial able (cont)*


67/82

8servation+

F/m2

[ ]

C (

'

πε =

÷

G C σ ε

= ÷

!$is res'(t act'a((% $o(ds for an% trans"ission (ine.

2[S/m]

G(

'

πσ =

÷

0 r ε ε ε =

( *

67

oaial able (cont)*


68/82

G C σ ε

= ÷

!o e "ore

enera(+

taG

C

σ δ

ω ωε

= = ÷

taGC

δ ω =

Note+ =t is t$e (oss tanent t$at is 's'a((%)approi"ate(%* constant for a "ateria(4 over awide rane of fre2'encies.

( *

&s I'st

derived4

!$e (oss tanent act'a((%arises fro" ot$ cond'ctivit%(oss and po(ariFation (oss)"o(ec'(ar friction (oss*4inenera(.

68

#is is t#e loss tan"ent t#at /ould

arise rom conductiitB eects)

oaial able (cont)*


69/82

9enera( epression for (oss

tanent+

( )

c

c c

j

j j

j

σ ε ε

ω

σ ε ε

ω ε ε

= − ÷

′ ′′= − − ÷

′ ′′= −

ta c

c

σ ε

ε ω δ

ε ε

′′+ ÷′′ ≡ =′′

E?ective per"ittivit% t$at acco'nts for cond'ctiv

#oss d'e to "o(ec'(ar friction #oss d'e to cond'ctivit%

( *

69

oaial able (cont)*


70/82

,ind R )resistance @ (ent$*

Coaia(ca(e

h = 1 [m]

( *

,( r(σ µ

'

(

σ

,' r'σ µ

' ( R R R= +

1

2' s' R R

'π

= ÷

1

2( s( R R

(π

= ÷

1

s'' ' R σ δ =

1

s(( ( R σ δ =

0

2'

r' '

δ ωµ µ σ

=0

2(

r( (

δ ωµ µ σ

=

R s = s'rface resistance of "eta(

70

Aeneral ransmission Line >ormulas


71/82

taG

C δ

ω =

( )0 0 r r LC µε µ ε µ ε ′ ′= =

0

*.ss*ess L Z

C

= = c$aracteristic i"pedance of (ine )ne(ectin (oss*)1*

)


72/82

Aeneral ransmission Line >ormulas (cont)*

( ) taG C ω δ =

0

*ss*ess L Z µε ′=

0/*ss*essC Z µε ′=

R R=

&( fo'r per-'nit-(ent$ para"eters can e

fo'nd fro"

0 ,*.ss*ess Z R

72

ommon ransmission Lines


73/82

0 0

1

[ ]2

*.ss*ess r

r

(

Z '

µ

η ε π

= Ω ÷

Coa

!win-(ead

100 cos! [ ]

2

*.ss*ess r

r

h Z

'

η µ

π ε

− = Ω ÷

2

1 2

1

2

s

h

' R R

' h

'

π

÷ = − ÷

1 1

2 2 s' s( R R R

' (π π

= + ÷ ÷

'

(

,r r ε µ

h

,r r ε µ

' '

73

ommon ransmission Lines (cont)*


74/82

( *

Microstrip

( ) ( ) ( )

( )

( )

( )0 0

1 00

0 1

e// e//

r r

e// e//

r r

/ Z / Z

/

ε ε

ε ε

−= ÷ ÷−

( )( ) ( ) ( )( )

0

1200

0 / 1. 0.++* / 1.###e// r

Z

w h w h

π

ε

= ′ ′+ + +

( / 1)w h ≥

21

t hw w

t π

′ = + + ÷ ÷

h

w

ε r

t

74

ommon ransmission Lines (cont)*


75/82

Microstrip

( / 1)w h ≥

h

w

ε r

t

( )

2

1.

(0)(0)

1 #

e//

r r e// e//

r r / 0

ε ε ε ε −

− ÷= + ÷+

( )( )

1 1 11 /0

2 2 #.+ /1 12 /

e// r r r r

t h

w hh w

ε ε ε ε

+ − − ÷= + − ÷ ÷ ÷ ÷ +

2

0

# 1 0. 1 0. 1r h w

0 h

ε λ

= − + + + ÷ ÷ ÷ ÷ ÷

75

Limitations o ransmission!Line #eorB


76/82

&t $i$ fre2'enc%4 discontin'it% e?ects can eco"e

i"portant.

end

incident

re>ected

trans"itted

!$e si"p(e !# "ode( does not acco'nt for t$e end. Z "H

Z L Z 0K-

76

Limitations o ransmission!Line #eorB (cont)*


77/82

&t $i$ fre2'enc%4 radiation e?ects can eco"e

i"portant.

$en wi(( radiationocc'rL

e want ener% to trave( fro" t$e enerator to t$e (oad4 wit$o'tradiatin.

Z "H

Z L Z 0K-

77



78/82

r ε '

( z

!$e coaia( ca(e is aperfect(% s$ie(ded s%ste" Ht$ere is never an% radiationat an% fre2'enc%4 or 'nderan% circ'"stances.

!$e e(ds are conned to t$ereion etween t$e twocond'ctors.

78



79/82

!$e twin (ead is an open t%pe of

trans"ission (ine H t$e e(ds etend o'tto innit%.

!$e etended e(ds "a%ca'se interference wit$

near% oIects. )!$is "a%e i"proved % 'sinBtwisted pair.*

K -

n e(ds t$at etend to innit% is not t$e sa"e t$in as $avin radiation4 $owe

79



80/82

!$e innite twin (ead wi(( not radiate % itse(f4 reard(ess of

$ow far apart t$e (ines are.

h

incident

re>ected

!$e incident and re>ected waves represent an eactso('tion to Mawe((s e2'ations on t$e innite (ine4 at an%fre2'enc%.

( )$1

'Re H 02

t

$

& d$ ρ = × × = ÷ ∫

$

No atten'ation on an innite (oss(ess (ine

80



81/82

& discontin'it% on t$e twin (ead wi(( ca'se radiation to occ'r.

Note+ Radiatione?ects increase as

t$e fre2'enc%increases.

h

=ncident wavepipe

8stac(e

Re>ected wave

end h

=ncident wave

end

Re>ected wave 81



82/82

!o red'ce radiation e?ects of t$e twin (ead at

discontin'ities+

h

1* Red'ce t$e separation distance h (keep h 11 λ ).

Documents

Notes 1 - Transmission Line Theory