Để tải xuống xem đầy đủ hãy nhấn vào bên trên
The Rectangular Waveguide
635
Similarly, the surface currents are found by the discontinuity
in the tangential components of H to be purely z directed:
Kz(x,y = )=-H(x,y=
K(x, y = b)=
.(x,
0)
kkk2Eo sin kx
2
k,0)2
k k 2Eo
2
= b)-=
y
k
sin kxxcosn
2
)
+k
joCL(kf
kxk 2 Eo
,(x = , y) = jWo(k2
sin ky
+k 2 )
K,(x = 0, y) =
(25)
kk 2 Eo cos mir sin kyy
K/(x = a, y)= -H~,(x = a, y)= -
2
2)k
3owA(kx + ky)
We see that if m or n are even, the surface charges and
surface currents on opposite walls are of opposite sign, while
if m or n are odd, they are of the same sign. This helps us in
plotting the field lines for the various TM,, modes shown in
Figure 8-28. The electric field is always normal and the
magnetic field tangential to the waveguide walls. Where the
surface charge is positive, the electric field points out of the
wall, while it points in where the surface charge is negative.
For higher order modes the field patterns shown in Figure
8-28 repeat within the waveguide.
Slots are often cut in waveguide walls to allow the insertion
of a small sliding probe that measures the electric field. These
slots must be placed at positions of zero surface current so
that the field distributions of a particular mode are only
negligibly disturbed. If a slot is cut along the z direction on
the y = b surface at x = a/2, the surface current given in (25) is
zero for TM modes if sin (ka/2)=0, which is true for the
m = even modes.
8-6-3
Transverse Electric (TE) Modes
When the electric field lies entirely in the xy plane, it is most
convenient to first solve (4) for H,. Then as for TM modes we
assume a solution of the form
H, = Re [I•,(x, y) ei'"'- ~ z ]
which when substituted into (4) yields
2
8 /,
82•H,
x2 +
y2
ax
Oy
/,
- k
w2\
2H
c
= 0
(26)
+
+
+
+
Electric field (-)
-jkk,Eo
E,
k,
cos kx sin ky
-jkykEo
+k2 sin kx cos ky
E,-
E = Eo sin kx sin kyy
-- \A..
TAp
+
--
++
+
x
E, k,tan kx
E, kx tan k,y
(k¢
)
,
2
[cos k ]
TM 11
+
-
dy
dx
cos k,y
Magnetic field (---
+
H=
= const
-)
wk
Eosink,x cosky
2 +k2
H,=-k+k Eocos kxsin ky
I
+tt-
+IIt
I~f
dy
dx
H,
H,
-k,cot kx
k,cot ky
=> sin kx sin kRy = const
5r1
kx= -,
+
++
---
a
a
k
flT
b,
b
WL
=Fw2
-k
2
TM 21
gure 8-28 The transverse electric and magnetic field lines for the TM,I and TM
rely z directed where the field lines converge.
21
modes. The electric fi
637
The Rectangular Waveguide
Again this equation is solved by assuming a product solution
and separating to yield a solution of the same form as (11):
Hz(x, y) = (A, sin k/x + A 2 cos khx)(B, sin ky + B 2 cos ky)
(28)
The boundary conditions of zero normal components of H
at the waveguide walls require that
H,(x = 0, y)= 0,
,(x = a, y)= 0
H,(x, Y= 0)= 0,
H,(x, y = b)= 0
(29)
Using identical operations as in (15)-(20) for the TM modes
the magnetic field solutions are
.=
jkk-Ho
-
k +k,
sin k
kAy,
kcos
mrr
kx -m,
a
cos kx sin ky
=k
,1k2+=kkHoossin
k,
nir
b
(30)
kx + ky
H. = Ho cos k,, cos k,y
The electric field is then most easily obtained from
Ampere's law in (1),
-1
E=- Vx×i
]we
(31)
to yield
j)
az
(ay
2
Sk,k Ho
jwe(k +k ) cos kx sin k,y
, Ho cos kx sin k,y
k-
,=
1-/
H
(32)
kk'Ho
jowiik.2+
= -i-•
Ho sin kxx cos ky
k. +k•
=0
We see in (32) that as required the tangential components
of the electric field at the waveguide walls are zero. The
638
Guided Electromagnetic Waves
surface charge densities on each of the walls are:
-~
l(x= 0,y) =
=(x
(x = a, y)= -e(x
=,
y)=
= a, y)=
Ho
sin ky
(
kYH
cos mvr sin ky
iwj(k, + hk,)
Ck2Hok
'&(x, y = 0) =
,(x, y= 0) =
2
k k
jo(k. + k,)
H).
sin
k.k2Ho
(k +)
'(x,y= b)= -e4,(x, y= b)=
(33)
cos nr sin kAx
For TE modes, the surface currents determined from the
discontinuity of tangential H now flow in closed paths on the
waveguide walls:
K(x = 0, y) = i, x
(x = 0,y)
= iH,t(x = 0, y)- i,H,(x = 0, y)
K(x = a,y)= -i,Xi(x = a,y)
= -iH,(x = a,y)+i,H,(x = a,y)
iK(x, y = 0) i,x I~(x,y = 0)
(34)
= -i'/.(x, y = 0) + i/,(x, y = 0)
K(x, y
= b) = -i,x l(x, y =
b)
= it/,(x, y = b) - i•.,(x,y = b)
Note that for TE modes either n or m (but not both) can be
zero and still yield a nontrivial set of solutions. As shown in
Figure 8-29, when n is zero there is no variation in the fields
in the y direction and the electric field is purely y directed
while the magnetic field has no y component. The TE1l and
TE2 1 field patterns are representative of the higher order
modes.
8-6-4
Cut-Off
The transverse wavenumbers are
k,
m•"
k,=
nlr
(35)
so that the axial variation of the fields is obtained from (10) as
k,,= [!-
-
oe,_)
22
(36)k2
Y
Electric field (-)
E,=
-
____
___
4
-
k• +k
k2
Ho cos kx
Ho sin kx
E2, -=
kA +k,
-E
+
+
k =-
++
+
dy
dx
a
TE,,
~
,
b
a
E,
E,
-ktan kA
k, tan k,y
=>cos k,x cos k,y = c
Magnetic field ( - - -)
jkkHo
-= sin kx cos
/
H,
~
2
k2 + ky
cos kx sin
4, = Ho cos k,,x cos k,y
dy H, k, cot kx
dx H, k, cot k,y
'h '
[sin kxx]( ,I
I e1
I
E21
)
_
c
sin k,y
a) The transverse electric and magnetic field lines for various TE modes. The magnetic field is purely z direc
The TE 0o mode is called the dominant mode since it has the lowest cut-off frequency. (b) Surface current lines
640
Guided Electromagnetic Waves
x
a-
I
I
-
IX_
3-
--
-
.
W
X
3_•
UX
4
2
4
(b)
Figure 8-29
Thus, although Akand k, are real, k can be either pure real or
pure imaginary. A real value of k. represents power flow
down the waveguide in the z direction. An imaginary value of
k, means exponential decay with no time-average power flow.
The transition from propagating waves (kh real) to evanescence (k, imaginary) occurs for k,= 0. The frequency when k,
is zero is called the cut-off frequency w,:
&= [(=C
) 2 + (nI)2]1/2
(37)
This frequency varies for each mode with the mode
parameters m and n. If we assume that a is greater than b, the
lowest cut-off frequency occurs for the TE 1 0 mode, which is
called the dominant or fundamental mode. No modes can
propagate below this lowest critical frequency woo:
TC
o= -
a
~f =
c0
21r
2a
Hz
(38)
If an air-filled waveguide has a = 1cm, then fro=
1.5xl0' 0 Hz, while if a=10m, then f~o=15MHz. This
explains why we usually cannot hear the radio when driving
through a tunnel. As the frequency is raised above oco,
further modes can propagate.
641
The Rectangular Waveguide
The phase and group velocity of the waves are
VP
k
do
Vg
dk,
k'c
= -
2
w
At cut-off, v,=0 and vp = o
constant.
8-6-5
(n
2
(2MW
.)2] 1/2
C22
vp
(39)
v=gy
2
= C
with their product always a
Waveguide Power Flow
The time-averaged power flow per unit area through the
waveguide is found from the Poynting vector:
= 2 Re (E xHI*)
(40)
(a) Power Flow for the TM Modes
Substituting the field solutions found in Section 8-6-2 into
(40) yields
i))e + i
e-ik x (/-*iý+*
= Re [(xi + i,+ !i)
= I Re [(EI,,/
- E4Hi' )i. + E(
i -/4 i,)] ei ' kk
"
]
(41)
where we remember that k. may be imaginary for a particular
mode if the frequency is below cut-off. For propagating modes
where k, is real so that k, = k*, there is no z dependence in (41).
For evanescent modes where k, is pure imaginary, the z
dependence of the Poynting vector is a real decaying
exponential of the form e -21' k". For either case we see from (13)
and (22) that the product of E, with fHxand H, is pure
imaginary so that the real parts of the x- and y-directed time
average power flow are zero in (41). Only the z-directed power
flow can have a time average:
=Eo, 2
=
|2 2) Re [k, e-itk -k*)(k
2(kx +k, )
+kY
sin 2 kx cos 2 kyy)]i.
kýX
COS2
2
2 k'Y
2
cos kX sin k,y
(42)
If k, is imaginary, we have that = 0 while a real k, results
in a nonzero time-average power flow. The total z-directed
642
Guided Electromagnetic Waves
power flow is found by integrating (42) over the crosssectional area of the waveguide:
=
dxdy
oekoabE(
8(k+k)
(43)
where it is assumed that k, is real, and we used the following
identities:
a i 2
mrx
a 1Imrx 1 . 2mrx~
I
m \2 a
4 n a l 0
a
= a/2,
a
m#O
=(44)
0
Cos
a
[-
+- sin
----
dx = -(
mor 2
a
4
a
o
a/2, m#O
a,
m=0
For the TM modes, both m and n must be nonzero.
(b) Power Flow for the TE Modes
The same reasoning is used for the electromagnetic fields
found in Section 8-6-3 substituted into (40):
- 2
Re [(•/-
i,+ fli.) e+ikz
yi,) eik x (• ix +
Re [(ix +
=
-
E,/-H^*)i•
-Hz/
(Ei
-
-
Eyi)] e(k
)z
(45)
Similarly, again we have that the product of H* with E,and
E, is pure imaginary so that there are no x- and y-directed
time average power flows. The z-directed power flow reduces to
=
(•'
,(k cos2 kx sin' k,y
+k' sin 2 k,
cos 2 ky) Re (k, e-i
('
- k*
•)
(46)
Again we have nonzero z-directed time average power flow
only if kRis real. Then the total z-directed power is
sk abH(2
+ k2,
m, n
0
xabHE
, morn=0
(hk+ k )
(47)
The Rectangular Waveguide
643
where we again used the identities of (44). Note the factor of
2 differences in (47) for either the TE1 oor TEo, modes. Both
m and n cannot be zero as the TE0o mode reduces to the
trivial spatially constant uncoupled z-directed magnetic field.
8-6-6
Wall Losses
If the waveguide walls have a high but noninfinite Ohmic
conductivity a-,, we can calculate the spatial attenuation rate
using the approximate perturbation approach described in
Section 8-3-4b. The fields decay as e - ' , where
a= 1I
2
(48)
where is the time-average dissipated power per unit
length and is the electromagnetic power flow in the
lossless waveguide derived in Section 8-6-5 for each of the
modes.
In particular, we calculate a for the TE 0o mode (k. =
ir/a, ky = 0). The waveguide fields are then
(jka
Hao
E=--
s= --a +cos a-ai
i sin
a Ho sin -- i
(49)
Ta
The surface current on each wall is found from (34) as
il(x = 0, y)=
kl(x
= a, y)= -Hoi,
(50)
&TjkIa
i(x, y=0)=-K(x,y= b)= Ho -iL-sin-+i.cos1)With lossy walls the electric field component E, within the
walls is in the same direction as the surface current proportional by a surface conductivity o•8, where 8 is the skin depth
as found in Section 8-3-4b. The time-average dissipated power
density per unit area in the walls is then:
=
-12 Re(Ew.*)I Ho
2 oa8
(51)
=
1 _=- H•
2 o,,,8
k•
)
,ir
2 smin. 2
a
21rX ]
2
+cos21
a
The total time average dissipated power per unit length
required in (48) is obtained by integrating each of the
644
Guided Electromagnetic Waves
terms in (51) along the waveguide walls:
=
[+] dy
+
[+] dx
Hob
,8s
Ho
k_
2[ir)
sin2.
= +
2x
,8
j
2
2 2
C
while the electromagnetic power above cut-off for the TElo
mode is given by (47),
Iphk,abHo
= 4(r/a)
4(7r/a)2
(53)
so that
S
a
2
2 2C2
-
(54)
wjoabk,So8
where
k= -
8-7
/;
->-
a
(55)
DIELECTRIC WAVEGUIDE
We found in Section 7-10-6 for fiber optics that electromagnetic waves can also be guided by dielectric structures
if the wave travels from the dielectric to free space at an angle
of incidence greater than the critical angle. Waves propagating along the dielectric of thickness 2d in Figure 8-30 are
still described by the vector wave equations derived in Section
8-6-1.
8-7-1
TM Solutions
We wish to find solutions where the fields are essentially
confined within the dielectric. We neglect variations with y so
that for TM waves propagating in the z direction the z
component of electric field is given in Section 8-6-2 as
Re [A 2 e- a( x -
d)
e j(It-kz)],
x-d
E,(x,t)= Re [(Al sin k~+B cos k,x) eijt-k-], IxI ld
[Re [As e~(x+d) ej(Wt-kz)],
1
x5 -d
(1)
This site is protected by reCAPTCHA and the Google
Privacy Policy
and
Terms of Service
apply.
10
287 KB
0
0
