notes 6 - waveguides part 3 attenuation
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8/13/2019 Notes 6 - Waveguides Part 3 Attenuation
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Prof. David R. Jackson
Dept. of ECE
Notes 6
ECE 5317-6351
Microwave Engineering
Fall 2011
Waveguides Part 3:
Attenuation
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For most practical waveguides and transmission lines the loss associated with
dielectric loss and conductor loss is relatively small.
To account for these losses we will assume
z k j attenuation constantPhase constant for lossless
wave guide
c d Attenuation
constant due to
conductor loss
Attenuation constant
due to dielectric loss
Attenuation on Waveguiding Structures
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Attenuation due to Dielectric Loss: d
Lossy dielectric complex permittivity
1
(1 tan )
c
c c
cc
c
c
j
j
j
j
2 2 2 (1 tan )c ck j
tan c
c
0
0
rc
rc
complex wavenumber k
ck Note:
k k jk
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Thus
2 2
z d ck j k k
Attenuation due to Dielectric Loss (cont.)
ck
Remember: The value k c is always real, regardless of whether the
waveguide filling material is lossy or not.
Note: The radical sign denotes the principal square root :
Arg z
4
This is an exact formula for
attenuation due to dielectric
loss. It works for both
waveguides and TEM
transmission lines (k c = 0).
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2 (1 tan )
(1 tan )
c
c
k j
j
tan 1
1 1 / 2 1 z z z for
Small dielectric loss in medium:
Approximate Dielectric Attenuation
Use
1 tan / 2ck j
tan / 2
ck
k k
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2 2
2 2
2 2 2
(1 tan )
tan
z d c
c c
c c c
k j k k
j k
k j
2 2 2tan ( )c c ck
1 1
1 / 12 2
z z a z a z a a a z a
a a
for
Small dielectric loss:
Approximate Dielectric Attenuation (cont.)
Use
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2
22
2
2
2
2
2
tan
an
2
t
c
c
cc
z c c c
c
d
k k j
j
k jk
2 tan
2
cd
Attenuation due to Dielectric Loss (cont.)
2 2
c ck 2 2
ck k
2 tan
2d
k
We assume
here that weare above
cutoff.
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d
k
k
z k j k k jk
For TEM mode
Attenuation due to Dielectric Loss (cont.)
tan
2d
k
ck
k jk
8
We can simply put k c = 0 in the previous formulas.
Or, we can start with the following:
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Assuming a small amount of conductor loss:
We can assume fields of the lossy guide are
approximately the same as those for lossless guide,
except with a small amount of attenuation.
We can use a perturbation method to determine c.
Attenuation due to Conductor Loss
Note: Dielectric loss does not change the shape of the fields at all, since the boundary
conditions remain the same (PEC). Conductor loss does disturb the fields slightly.
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Surface Resistance (cont.)
Assume
1/2 1/21
(1 )22
j
k j j j
Note that
1
2k k jk j
2
k k
The we have
Hence
1
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Surface Resistance (cont.)
Denote
Then we have
2k k
1
" p
d k
“skin depth” = “depth of penetration”
2
1' "
pd
k k
At 3 GHz, theskin depth for
copper is about
1.2 microns.
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Surface Resistance (cont.)
2
Example: copper
7
0 4 10 [H/m]
75.8 10 [S/m]
Frequency
1 [Hz] 6.6 [cm]
10 [Hz] 2.1 [cm]
100 [Hz] 6.6 [mm]
1 [kHz] 2.1 [mm]
10 [kHz] 0.66 [mm]
100 [kHz] 21 [mm]
1 [MHz] 66 [m]
10 [MHz] 20.1 [m]
100 [MHz] 6.6 [m]1 [GHz] 2.1 [m]
10 [GHz] 0.66 [m]
100 [GHz] 0.21 [m]
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Surface Resistance (cont.)
where
x y E H
1
2
(1 )2
(1 )
c
s
s
j
j j
j
j
j R
Z
Inside conductor:
2 s R
“Surface resistance ()”
“Surface impedance ()”
1 s s Z j R
15
ˆ
t t E n H
Note: To be more general:
n̂ outward normal
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Surface Resistance (cont.)
(1 )
1
2
s
s s
s
Z
Z j R
R
Summary for a Good Conductor
16
ˆt s t
E Z n H
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ˆt s t
E Z n H
ˆ
t
t
E
H
n
tangential electric field at surface
tangential magnetic field at surface
outward unit normal to conductor surface
n̂
conductor
ˆ
eff
s t J n H
eff t s s
E Z J
“Effective surface current”
The surface impedance gives us the ratio of the
tangential electric field at the surface to the
effective surface current flowing on the object.
Hence we have
Surface Resistance (cont.)
For the “effective”
surface current density
we imagine the actual
volume current density
to be collapsed into a
planar surface current.
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Surface Resistance (cont.)
1
2 s R
Example: copper
7
0 4 10 [H/m]
75.8 10 [S/m]
Frequency Rs
1 [Hz] 2.6110-7 []
10 [Hz] 8.2510-7 []
100 [Hz] 2.6110-6 []
1 [kHz] 8.2510-6 []
10 [kHz] 2.6110-5 []
100 [kHz] 8.2510-5 []
1 [MHz] 2.6110-4 []
10 [MHz] 8.2510-4 []
100 [MHz] 0.00261 []6.6
1 [GHz] 0.00825 []
10 [GHz] 0.0261 []
100 [GHz] 0.0825 []
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Surface Resistance (cont.)
2
0
1
2d s t
R H PIn general,
2 2
*0 0 01 1 1Re ( ) Re
2 2 2d x y z y s y E H H R H P
We then have
For a good conductor, 0
ˆeff
s t J n H
Hence 2
12
eff d s s R J P
This gives us the power dissipated per
square meter of conductor surface, ifwe know the effective surface current
density flowing on the surface.
19
PEC limit:eff
s s J J eff
s s J J Perturbation method : Assume that
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Perturbation Method for c
Power flow along the guide:
2
0( ) z
P z P e
0 (0) P P
Power @ z = 0 is calculated
from the lossless case.
Power loss (dissipated) per unit length:l
dP P
dz
2
0( ) 2 2 ( ) z
l P z P e P z
0
( ) (0)
2 ( ) 2
l l P z P
P z P
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There is a single
conductingboundary.0
(0)
2
l c
P
P
*
0
0
1ˆ
Re 2S z
P E H z dS
2
0
(0)
2
sl s
C z
R P J d
Surface resistance of
metal conductors:2
s R
ˆ s J n H Note :
On PEC conductor
Perturbation Method: Waveguide Mode
C
S
21
For these calculations, we
neglect dielectric loss.
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0
(0)
2
l
c
P
P
*
0
0
2
0
1ˆRe
2
1
2
S z
lossless
P E H z dS
Z I
1 2
2
0
(0) 2 sl s
C C z
R P J d
Surface resistance of
metal conductors:
2
s R
ˆ s J n H Note :
On PEC conductor
2C
S
1C
Perturbation Method: TEM Mode
There are two
conductingboundaries.
22
For these calculations, we
remove dielectric loss.
0 0
lossless Z Z
h l l d l
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0
02
lossless s
c lossless lossless
R dZ
Z d
2C
S
1C
Wheeler Incremental Inductance Rule
23
The Wheeler incremental inductance
rule gives an alternative method for
calculating the conductor attenuation
on a transmission line (TEM mode): It is
useful when Z 0 is already known.
In this formula, d l (for a given conductor) is the distance by which the
conducting boundary is receded away from the field region.
The top plate of a
PPW line is shown
being receded.
The formula is applied for each conductor
and the conductor attenuation from each
of the two conductors is then added.
d
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Example: TEM Mode Parallel-Plate Waveguide
Previously, we showed
2
0
0 * 2
0 0
2
0 *
2
0
1ˆ ˆRe
2
1 1Re
2
1
2
w d
lossless
V P z z dydxd
wV
d
wV
d
0
0
ˆ
ˆ
jkz
jkz
V
E y ed
V H x e
d
0
ˆ
ˆ
top
s
jkz
J y H
V z e
d
0ˆ ˆ
bot jkz
s
V J y H z e
d
y
z w
d , ,
x
On the top plate:
On the bottom plate:
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j
lossless
c
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s
c lossless
R
d
Example: TEM Mode PPW (cont.)
26
Let’s try the same calculation using the Wheeler incremental inductance rule.
0
02
lossless s
c lossless lossless
R Z
Z
0
0
lossless lossless
d Z
w
d Z
w
From previous calculations:top bot
c c c
0 0lossless lossless Z dZ
d
d
w
, ,
0 02 2 22
top lossless s s s sc lossless lossless lossless lossless
lossless
R R R Rd
d Z d w wZ d w
w
0 02 2 2
2
bot lossless s s s sc lossless lossless lossless lossless
lossless
R R R Rd
d Z d w wZ d
w w
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Results for TM/TE Modes (above cutoff): (derivation omitted)
TMn modes of parallel-plate
2, 0 s
cn lossless
k Rn
d
TEn modes of parallel-plate
22
, 0c s
cn lossless
k Rn
k d
c
nk
d
Example: TM z /TE
z Modes of PPW
y
z w
d , ,
x
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