Formula sheet: Waveguides | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

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Electromagnetics: W a v eguides F orm ula Sheet for
GA TE
W a v eguide Basics
• Definition : Hollo w metallic or dielectri c structures used to guide electromagnetic
w a v es.
• T yp es : Rectangular, cylindrical, and dielectric w a v eguides.
• W a v e Num b er:
k =
2p
?
=
?
c
, where c=3×10
8
ms
-1
where ? is angular frequency , ? is w a v elength in free space.
Mo des of Propagation
• TE Mo des (T ransv erse Electric) : E
z
=0 , H
z
?=0 .
• TM Mo des (T ransv erse Magnetic) : H
z
=0 , E
z
?=0 .
• TEM Mo des : E
z
=0 , H
z
=0 (not supp orted in hollo w w a v eguides).
• Mo de Indices : F or rectangular w a v eguides, mo des are denoted as TE
mn
or TM
mn
,
where m and n are m o de n um b ers.
Rectangular W a v eguides
• Cutoff F requency : F requency b elo w whic h a mo de cannot propagate.
f
c
=
c
2
v
(
m
a
)
2
+
(
n
b
)
2
where a and b are the w a v eguide dimensions (a>b ), c is the sp eed of ligh t.
• Cutoff W a v elength :
?
c
=
c
f
c
=
2
v
(
m
a
)
2
+
(
n
b
)
2
• Propagation Constan t :
ß =
v
k
2
-k
2
c
, k
c
=
2p
?
c
=
v
(
mp
a
)
2
+
(
np
b
)
2
where k =
2p
?
, ß is the phase constan t.
• W a v eguide W a v elength :
?
g
=
2p
ß
=
?
v
1-
(
fc
f
)
2
1
Page 2


Electromagnetics: W a v eguides F orm ula Sheet for
GA TE
W a v eguide Basics
• Definition : Hollo w metallic or dielectri c structures used to guide electromagnetic
w a v es.
• T yp es : Rectangular, cylindrical, and dielectric w a v eguides.
• W a v e Num b er:
k =
2p
?
=
?
c
, where c=3×10
8
ms
-1
where ? is angular frequency , ? is w a v elength in free space.
Mo des of Propagation
• TE Mo des (T ransv erse Electric) : E
z
=0 , H
z
?=0 .
• TM Mo des (T ransv erse Magnetic) : H
z
=0 , E
z
?=0 .
• TEM Mo des : E
z
=0 , H
z
=0 (not supp orted in hollo w w a v eguides).
• Mo de Indices : F or rectangular w a v eguides, mo des are denoted as TE
mn
or TM
mn
,
where m and n are m o de n um b ers.
Rectangular W a v eguides
• Cutoff F requency : F requency b elo w whic h a mo de cannot propagate.
f
c
=
c
2
v
(
m
a
)
2
+
(
n
b
)
2
where a and b are the w a v eguide dimensions (a>b ), c is the sp eed of ligh t.
• Cutoff W a v elength :
?
c
=
c
f
c
=
2
v
(
m
a
)
2
+
(
n
b
)
2
• Propagation Constan t :
ß =
v
k
2
-k
2
c
, k
c
=
2p
?
c
=
v
(
mp
a
)
2
+
(
np
b
)
2
where k =
2p
?
, ß is the phase constan t.
• W a v eguide W a v elength :
?
g
=
2p
ß
=
?
v
1-
(
fc
f
)
2
1
Phase and Group V elo cities
• Phase V elo cit y : Sp eed of the phase of the w a v e.
v
p
=
?
ß
=
c
v
1-
(
fc
f
)
2
• Group V elo cit y : Sp eed of the signal or energy propagation.
v
g
=c
v
1-
(
f
c
f
)
2
Note: v
p
v
g
=c
2
.
Characteristic Imp edance
• TE Mo de Imp edance :
Z
TE
=
?
v
1-
(
fc
f
)
2
, ? =
v
µ
0
?
0
˜377?
• TM Mo de Imp edance :
Z
TM
=?
v
1-
(
f
c
f
)
2
where ? is the in trinsic imp edance of free space.
A tten uation in W a v eguides
• A tten uation Constan t : Due to conductor losses or dielectric losses.
a=a
c
+a
d
where a
c
is conductor loss, a
d
is dielectric loss.
• P o w er Loss : Expressed in dBm
-1
.
Key Notes
• Dominan t mo de in rectangular w a v eguides: TE
10
(lo w est c utoff frequency , f
c
=
c
2a
).
• Ensure f >f
c
for propagation; if f <f
c
, the w a v e is ev anescen t.
• T ypical v alues: c=3×10
8
ms
-1
, ?˜377? .
• F or GA TE, fo cus on cutoff frequency , w a v eguide w a v elength, and v elo cit y calculations.
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