Transmission Lines Fundamentals

VII 1

Transmission Lines

(a) Parallel-plate transmission line (b) Two-wiretransmission line

(c) Coaxialtransmission line

Metal strip

Groundedconducting plane

Dielectric subtrate

Groundedconducting plane

Metal stripGroundedconducting plane

Dielectric subtrate

Two types of microstrip lines

VII 2

TEM-Waves along a Parallel-Plate Transmission Line

dx

y

z

w

Lossless case:

( )term e always omittedj tω

r r r

r r r

E E e E e e

H H eE

e e

j

y yz

y

x xz

x

= ⋅ = ⋅ ⋅

= ⋅ = − ⋅ ⋅

= =

−

−

0

0

γ

γ

γ ω µε µε

Γ

Γ

in order to find the charge density and the current density we use:

D D D e E E en n y yz

2 1 0− = → ⋅ = → = ⋅ = ⋅ −σ σ σ ε ε γr r

σ: free surface charge

H H J e H J J e H e

Eet t s y s s z x z

z2 1

0− = → × = → = − ⋅ = ⋅ −r r r r r r

Γγ

Js: free surface currentd dsΙ

VII 3

Fields, Charge and Current Distribution along a Coaxial Transmission Line

B

E

xx xxxxx

x x

xx xxxxx

x x

xx xxxxx

x x

xx xxxxx

x x

λ

Current

Displacement Current

BEB E

x

+ +++ + - --- -

+ +++ + + +++ +

+ +++ +

- --- - + +++ +

- --- -

- --- -

- --- -

+ +++ +- --- -

VII 4

Parallel-Plate Transmission Line in Terms of L and C

Lossless case term e always omittedj tω( )

∇ × = −

=

=∫ ∫

r rE j H

dE

dzj H

ddz

E dy j H dy

yx

y

d

x

d

ωµ

ωµ

ωµ0 0

+( )

= ( ) ⋅

=

( ) ⋅( )

= ⋅ ( )= ⋅ [ ]

dV z

dzj J z d

jdw

J z w

j L z

Ldw

H m

sz

sz

ωµ

ω µ

ω

µ

Ι

∇ × =

=

=∫ ∫

r rH j E

dHdz

j E

ddz

H dx j E dx

xy

x

w

y

w

ωε

ωε

ωε0 0

− ( ) = − ( ) ⋅

=

− ( ) ⋅( )

= ( )

= [ ]

d zdz

j E z w

jwd

E z d

j CV z

Cwd

F m

y

y

Ι ωε

ω ε

ω

ε

VII 5

d V zdz

LCV z2

22( ) = − ( )ω

d z

dzLC z

2

22Ι

Ι( )

= − ( )ω

V z V e V ej LC z j z( ) = ⋅ = ⋅− −0 0

ω ω µε

Ι Ι Ιz e ej LC z j z( ) = ⋅ = ⋅− −0 0

ω ω µε

Phase velocity: u

LCp = = =ωω µε µε

1 1

Characteristicimpedance:

ZV z

zLC0 = ( )

( )=

Ι

VII 6

Lossy Parallel-Plate Transmission Line

Conductance between the two conductors:

Compare with the analogy of resistance and capacitance

ε κc

R= ⋅

case a case b

⇒ = = = ⋅ = ⋅G

RC

wd

wd

1 κε

κε

ε κ

G

wd

S m= ⋅ [ ]κ

VII 7

Ohmic power dissipated in the plates

r r rS e E e HLoss z z x x= × ⋅ * Power flux density flowing into the plates ( )

rey

Def. Surface impedance Z

EJs

t

s

= J free surface current

ddxs

z= Ι

Z R j Xs s s= + ⋅

R

lengthcross tion w ds

c

= ⋅ =⋅

1 1κ κsec

l

= ⋅ ⋅ =1

2 2κκ µ ω µ ω

κc

c c c

cw wl l

effective series resistance per unit length

R

w wf

mc

c

c

c

= = ⋅ [ ]22

2µ ωκ

µ πκ

Ω /

d penetration depth

c c

= = 2

κ µ ω

VII 8

Equivalent Circuit of a Differential Length ∆∆∆∆ z

of a Two-Conductor Transmission Line

G ∆z•

R ∆z• L ∆z•

C ∆z•

VII 9

Distributed Parameters of Transmission Lines

Parameter Parallel Plate Two-Wire Line Coaxial Line Unit

R

L

G

C

2w

f c

c

π µκ

µ d

w

κ w

d

ε w

d

w=widthd=separation

R

as

π

µπ

cosh−

1

2

D

a

πκcosh /− ( )1 2D a

πεcosh /− ( )1 2D a

R

fs

c

c

= π µκ

a=radiusD=distance

cosh /

ln /

/

− ( )≈ ( )

( ) >>

1

2

2

2 1

D a

D a

if D a

R

a bs

2

1 1

π+

µπ2

lnba

2πκln /b a( )

2πεln /b a( )

R

fs

c

c

= π µκ

a=radiuscenter cond.

b=radiusouter cond.

Ω / m

H m/

S m/

F m/

VII 10

Wave Equation for Lossy Transmission Lines

− ( ) = +( ) ( )

− ( ) = +( ) ( )

( ) = ( )

( ) = ( )

dV zdz

R j L z

d zdz

G j C V z

d V zdz

V z

d zdz

z

ω

ω

γ

γ

Ι

Ι

Ι Ι

2

22

2

22

γ α β ω ω= + = + +j R j L G j C( )( )

VII 11

Waveguides

x

y

z A uniform waveguide with an arbitrary cross section

Time-harmonic waves in lossless media:

∆r rE E+ =ω µ ε2 0

r rE x y z t E x y e j t k zz, , , ,( ) = ( ) ⋅ − ⋅( )0 ω

∇ + ∇( ) = ∇ −xy z xy zE E k E2 2 2 2r r r

∇ + −( ) =xy zE k E2 2 2 0

r rω µε

∇ + −( ) =xy zH k H2 2 2 0

r rω µε

VII 12

From x E j H we get∇ = −r r

ωµ : From xH j E we get∇ =r r

ωε :

∂∂

+ = −

− − ∂∂

= −

∂∂

− ∂∂

= −

Ey

jk E j H

jk EEx

j H

E

xEy

j H

zz y x

z xz

y

y xz

00 0

00

0

0 00

ωµ

ωµ

ωµ

∂∂

+ =

− − ∂∂

=

∂∂

− ∂∂

=

Hy

jk H j E

jk HHx

j E

H

xHy

j E

zz y x

z xz

y

y xz

00 0

00

0

0 00

ωε

ωε

ωε

Hh

jkHx

jEy

Hh

jkHy

jEx

Eh

jkEx

jHy

Eh

jkEy

jH

x zz z

y zz z

x zz z

y zz z

02

0 0

02

0 0

02

0 0

02

0 0

1

1

1

1

= − ∂∂

− ∂∂

= − ∂∂

+ ∂∂

= − ∂∂

+ ∂∂

= − ∂∂

− ∂∂

ωε

ωε

ωµ

ωµxx

h kz2 2 2= −ω µε

VII 13

Three Types of Propagating Waves

Transverse electromagnetic waves TEM : EZ = 0 & HZ = 0

Transverse magnetic waves TM : EZ ­ 0 & HZ = 0

Transverse electric waves TE : EZ = 0 & HZ ­ 0

VII 14

TEM - Waves

H E k kz z z TEM z TEM= = → − + = → =0 0 02 2& ω µε ω µε

Phase velocity uk

Wave impedance ZEH k

pTEMz

TEMx

y zTEM

= =

= = =

ωµε

ωµ µε

1

0

0

for hollow single-conductor

waveguides:

H there is only H and H

div H H fields must form closed loops

EDt

rot H J TEM waves cannot exist in

gle conductor hollow waveguides

z x y

zz

= →

= → −

= → ∂∂

=

= →−

0

0

0 0

r

r r

sin

VII 15

TM-Waves

Ejk

kEx

Ejk

kEy

Hj

kEy

Hj

kEx

xz

z

z

yz

z

z

xz

z

yz

z

= −−

∂∂

= −−

∂∂

=−

∂∂

= −−

∂∂

ω µε

ω µε

ωεω µε

ωεω µε

2 2

2 2

2 2

2 2

Wave equation

∂∂

+ ∂∂

+ −( ) =2

2

2

22 2 0

Ex

Ey

k Ez zz zω µε

VII 16

TM-Modes in Rectangular Waveguides

a

x

y

b

z

boundary conditions

E y and E a y in the x direction

E x and E x b in the y direction

z z

z z

0 0 0

0 0 0

, ,

, ,

( ) = ( ) =

( ) = ( ) =

separation of variables

E x y E k x k yz x y, sin sin( ) = ( ) ( )0

k

ma

and knb

m n are egersx y= =π π( , int )

VII 17

Solution

E x yjk

kE

ma

ma

xnb

y

E x yjk

kE

nb

ma

xnb

y

H x yj

kE

nb

ma

xn

xz

z

yz

z

xz

, cos sin

, sin cos

, sin cos

( ) = −−

( ) = −−

( ) =−

ω µεπ π π

ω µεπ π π

ωεω µε

π π π

2 2 0

2 2 0

2 2 0 bby

H x yj

kE

ma

ma

xnb

yzz

( ) = −−

, cos sin

ωεω µε

π π π2 2 0

TM13 mode means m=1, n=3

(if m=0 or n=0 then E=H=0)

k

ma

nbz

2 22 2

= −

−

ω µε π π

ω µε π π

π µεπ π

c

c

ma

nb

fma

nb

cut off frequency

22 2

2 2

0

12

−

+

=

=

+

if f < fc then jkz is real no wave propagation

VII 18

Field Lines for TM11 Mode in Rectangular Waveguide

1,0

0,5

00

π/2 βzπ 2π3π/2

y/b

x

xxx

x

xx

x

xx

x

x

x

xx

Magnetic field lines

x/a

y/b

O

Electric field lines

VII 19

TE-Waves

Ej

kHy

Ej

kHx

Hjk

kHx

Hjk

kHy

xz

z

yz

z

xz

z

z

yz

z

z

= −−

∂∂

=−

∂∂

= −−

∂∂

= −−

∂∂

ωµω µε

ωµω µε

ω µε

ω µε

2 2

2 2

2 2

2 2

wave equation

∂∂

+ ∂∂

+ −( ) =2

2

2

22 2 0

Hx

Hy

k Hz zz zω µε

VII 20

TE-Modes in Rectangular Waveguides

boundary condition

∂∂ ( ) = ∂

∂ ( ) = ( ) =

∂∂ ( ) = ∂

∂ ( ) = ( ) =

Hx

y andHx

a y in the x direction E

Hy

x andHy

x b in the y direction E

z zy

z zx

0 0 0 0

0 0 0 0

, ,

, ,

separation of variables

H x y H

ma

xnb

yz , cos cos( ) =

0

π π

VII 21

Solution:

E x yj

kH

nb

ma

xnb

y

E x yj

kH

ma

ma

xnb

y

xz

yz

, cos sin

, sin cos

( ) =−

( ) = −−

ωµω µε

π π π

ωµω µε

π π π

2 2 0

2 2 0

H x yjk

kH

ma

ma

xnb

y

H x yjk

kH

nb

ma

xnb

y

xz

z

yz

z

, sin cos

, cos sin

( ) =−

( ) =−

ω µεπ π π

ω µεπ π π

2 2 0

2 2 0

TE01 mode means m = 0, n = 1

k

ma

nbz

2 22 2

= −

−

ω µε π π

f

ma

nbc =

+

12

2 2

π µεπ π

cut off frequency

if f < fc then jkz is real no wave propagation

VII 22

Field Lines for TE10 Mode in Rectangular Waveguide

π/2 βzπ 2π3π/2

xx

x

x

x

x

x

x

x

x

x

x

xx

x

x

x

x

x

1,0

0,5

00

y/b

1,0

00

x/a

x/a

y/b

O

x

Electric field lines

Magnetic field lines

π/2 βzπ 2π3π/2

x

x

x

x

x

xx

xx

x

x

x x

x

x