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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