Waveguide Types - University of Delawaremirotzni/ELEG413/WaveguidesI.pdf · Rectangular Waveguides....
Transcript of Waveguide Types - University of Delawaremirotzni/ELEG413/WaveguidesI.pdf · Rectangular Waveguides....
Uniform Waveguides
We are interested in finding what electromagnetic field solutions are possible in a uniform infinite waveguide with no sources.
We can always find those solutions by solving:
or
Subject to boundary conditions.
Uniform Waveguides
This is really three PDEs
Subject to boundary conditions.
That’s a lot of work!Are there any shortcuts?
Uniform Waveguides
Because the cross section does not change in the z direction
zjt
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
Uniform Waveguides
Because the cross section does not change in the z direction
zjt
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
[ ] [ ] [ ]
[ ] [ ] [ ] 0),(~),(~),(~),(~
0),(~),(~),(~),(~
22
2
2
2
2
2
22
2
2
2
2
2
=+∂∂
+∂∂
+∂∂
=+∂∂
+∂∂
+∂∂
−−−−
−−−−
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zzzz
zzzz
eyxHkeyxHz
eyxHy
eyxHx
eyxEkeyxEz
eyxEy
eyxEx
ββββ
ββββ
Uniform Waveguides
zjt
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
[ ] [ ] [ ]
[ ] [ ] [ ] 0),(~),(~),(~),(~
0),(~),(~),(~),(~
22
2
2
2
2
2
22
2
2
2
2
2
=+∂∂
+∂∂
+∂∂
=+∂∂
+∂∂
+∂∂
−−−−
−−−−
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zzzz
zzzz
eyxHkeyxHz
eyxHy
eyxHx
eyxEkeyxEz
eyxEy
eyxEx
ββββ
ββββ
[ ] [ ]
[ ] [ ] 0),(~),(~),(~),(~
0),(~),(~),(~),(~
222
2
2
2
222
2
2
2
=+−∂∂
+∂∂
=+−∂∂
+∂∂
−−−−
−−−−
zjt
zjtzt
zjt
zj
zjt
zjtzt
zjt
zj
zzzz
zzzz
eyxHkeyxHyxHy
eyxHx
e
eyxEkeyxEyxEy
eyxEx
e
ββββ
ββββ
β
β
Uniform Waveguideszj
t
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
[ ] [ ] [ ]
[ ] [ ] [ ] 0),(~),(~),(~),(~
0),(~),(~),(~),(~
22
2
2
2
2
2
22
2
2
2
2
2
=+∂∂
+∂∂
+∂∂
=+∂∂
+∂∂
+∂∂
−−−−
−−−−
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zjt
zzzz
zzzz
eyxHkeyxHz
eyxHy
eyxHx
eyxEkeyxEz
eyxEy
eyxEx
ββββ
ββββ
[ ] [ ]
[ ] [ ] 0),(~),(~),(~),(~
0),(~),(~),(~),(~
222
2
2
2
222
2
2
2
=+−∂∂
+∂∂
=+−∂∂
+∂∂
−−−−
−−−−
zjt
zjtzt
zjt
zj
zjt
zjtzt
zjt
zj
zzzz
zzzz
eyxHkeyxHyxHy
eyxHx
e
eyxEkeyxEyxEy
eyxEx
e
ββββ
ββββ
β
β
[ ] [ ] ( )
[ ] [ ] ( ) 0),(~),(~),(~
0),(~),(~),(~
222
2
2
2
222
2
2
2
=−+∂∂
+∂∂
=−+∂∂
+∂∂
yxHkyxHy
yxHx
yxEkyxEy
yxEx
tztt
tztt
β
β
Uniform Waveguideszj
t
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
[ ] [ ] ( )
[ ] [ ] ( ) 0),(~),(~),(~
0),(~),(~),(~
222
2
2
2
222
2
2
2
=−+∂∂
+∂∂
=−+∂∂
+∂∂
yxHkyxHy
yxHx
yxEkyxEy
yxEx
tztt
tztt
β
β
( )( ) 0),(~),(~
0),(~),(~
222
222
=−+∇
=−+∇
yxHkyxH
yxEkyxE
tztt
tztt
β
β
Uniform Waveguides
zjt
zjt
z
z
eyxHzyxH
eyxEzyxEβ
β
−
−
=
=
),(~),,(~),(~),,(~
( )( ) 0),(~),(~
0),(~),(~
222
222
=−+∇
=−+∇
yxHkyxH
yxEkyxE
tztt
tztt
β
β
222zc kk β−=LET
0),(~),(~0),(~),(~
22
22
=+∇
=+∇
yxHkyxH
yxEkyxE
tctt
tctt
Uniform Waveguides
This is really three PDEs
Subject to boundary conditions.
zjt
zeyxEzyxE β−= ),(~),,(~
Uniform Waveguides
This is really three PDEs
Subject to boundary conditions.
This makes it a bunch easier!
Any other short cuts?
zjt
zeyxEzyxE β−= ),(~),,(~
Uniform Waveguides
Three Cases:
(3) Ez=0, Hz = 0 TEM
What do we do with this?Looks like all the fields are zero!
Uniform Waveguides
Three Cases:
(3) Ez=0, Hz = 0 TEM
What do we do with this?Looks like all the fields are zero!
The only way the fields are not all zero is if
Rectangular Waveguides
TM Modes
At the point
the mode changes from evanescent to propagating.
Cutoff frequency
Rectangular Waveguides
TM Modes
The dominate mode is the one that has the lowest cutoff frequency(i.e. first mode to propagate)
What is the dominate mode for this waveguide?
Rectangular Waveguides
TM Modes
The dominate mode is the one that has the lowest cutoff frequency(i.e. first mode to propagate)
What is the dominate TM mode for this waveguide?2222
11 112
12
1
+
=
+
=
babafc µε
ππµεπ
a>b
Rectangular Waveguides
TM Modes
What is wavelength of the guided mode?
222
22
+
−
==
bm
an
nmz
nmb
ππµεω
πβπλ
Rectangular Waveguides
TM Modes
What is wavelength of the guided mode?
µεω
ππ
µεω
π
ππµεω
πβπλ
2
22222
1
222
+
−
=
+
−
==
bm
an
bm
an
nmz
nmb
Rectangular Waveguides
TM Modes
What is wavelength of the guided mode?
( )
2
2
22
2
2
22222
1
211
1
222
−
=
+
−
=
+
−
=
+
−
==
ff
fb
ma
n
f
c
bm
an
bm
an
c
o
nmz
nmb
λ
ππ
µεπ
µεω
ππ
µεω
π
ππµεω
πβπλ
Rectangular Waveguides
TM Modes
What is wavelength of the guided mode?
2222 1
22
−
=
+
−
==
ff
bm
an nm
c
onmz
nmb
λ
ππµεω
πβπλ
Rectangular Waveguides
TM Modes
What is wavelength of the impedance of the guided mode?
nmx
nmy
nmy
nmxnm
w HE
HEZ
−==
Rectangular Waveguides
TM Modes
What is wavelength of the impedance of the guided mode?
ωε
ππµεω
ωεβ
222
−
−
===b
ma
n
HEZ
nmz
nmy
nmxnm
w
Rectangular Waveguides
TM Modes
What is wavelength of the impedance of the guided mode?
2
222
1
−=
−
−
===f
fbm
an
HEZ
nmc
nmz
nmy
nmxnm
w εµ
ωε
ππµεω
ωεβ
Rectangular Waveguides
TM Modes
What is wavelength of the impedance of the guided mode?
2
1
−==
ff
HEZ
nmc
onmy
nmxnm
w η
Rectangular Waveguides
TM Modes
What is wavelength of the phase velocity of the guided mode?
nmz
nmpv
βω
=
Rectangular Waveguides
TM Modes
What is wavelength of the phase velocity of the guided mode?
222
−
−
==
bm
an
v nmz
nmp
ππµεω
ωβω
Rectangular Waveguides
TM Modes
What is wavelength of the phase velocity of the guided mode?
2222 1
−
=
−
−
==
ff
c
bm
an
vnm
c
nmz
nmp
ππµεω
ωβω