eele 3332 electromagnetic ii chapter 12...
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EELE 3332 – Electromagnetic II
Chapter 12
Waveguides
Islamic University of Gaza
Electrical Engineering Department
Dr. Talal Skaik
2012 1
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Waveguides
Waveguides are used at high frequencies since they have larger
bandwidth and lower signal attenuation than transmission lines.
Waveguides are used at high power applications.
Waveguides can operate above certain frequencies. (act as high
pass filter).
Normally circular or rectangular.
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Waveguides
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Waveguides
Dr. Talal Skaik 2012 IUG
Rectangular waveguide Waveguide to coax adapter
E-tee Waveguide bends
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Waveguide Filter
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Transmission Lines, Waveguides - Comparison
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Transmission Lines, Waveguides - Comparison
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12.2 Rectangular Waveguides
Assume a rectangular waveguide filled with lossless dielectric
material and walls of perfect conductor, Maxwell equations in
phasor form become,
2 2
2 2
E E 0 where
H H 0
s s
s s
kk
k
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Rectangular Waveguides
2 2
2 2 22
2 2 2
2
Applying on z-component:
0
0
Solving by method of Separation of Variables:
( , , ) ( ) ( ) ( )
from where we obtain:
zs zs
zs zs zszs
z
'' '' ''
E k E
E E Ek E
x y z
E x y z X x Y y Z z
X Y Zk
X Y Z
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1 2
3 4
cos sin
cos sin
( , , ) ( ) ( ) ( ) ( )
x x
y y
zs
X(x) c k x c k x
Y(y) c k y c k y
E x y z X x Y y Z z Z z c
5 6
1 2 3 4 5 6
1 2 3 4
1
cos sin cos sin
Assume wave propagates along waveguide in direction:
cos sin cos sin
Similarly for the magnetic field,
cos
z z
z z
zs x x y y
z
zs x x y y
zs
e c e
E c k x c k x c k y c k y c e c e
z
E A k x A k x A k y A k y e
H B
2 3 4sin cos sin zx x y yk x B k x B k y B k y e 10
Rectangular Waveguides
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Other Components
From Maxwell’s equations, we can determine the other components Ex , Ey , Hx , Hy .
2 2
2 2
2 2
2 2
2 2 2 2 2
zs zsxs
zs zsys
zs zsxs
zs zsys
x y
E HjE
h x h y
E HjE
h y h x
E HjH
h y h x
E HjH
h x h y
where
h k k k
*So once we know
Ez and Hz, we can
find all the other
fields.
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From these equations we notice that there are different field patterns,
each of these field patterns is called a mode.
• Ezs=Hzs=0 (TEM mode): transverse electromagnetic mode. Both E
and H are transverse to the direction of propagation. From previous
equations we notice that all field components vanish for Ezs=Hzs=0.
→Rectangular waveguide can’t support TEM mode.
• Ezs=0, Hzs≠0 (TE modes) transverse electric
The electric field is transverse to the direction of propagation.
• Ezs ≠ 0, Hzs= 0 (TM modes) transverse magnetic
The magnetic field is transverse to the direction of propagation.
• Ezs ≠ 0, Hzs ≠ 0 (HE modes) hybrid modes
All components exist. 12
Modes of Propagation
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Transverse Magnetic (TM) mode
1 2 3 40, cos sin cos sin
0 at 0 (bottom and top walls)
0 at 0 (left and right walls)
Applying boundary conditio
Boundar
ns at ( 0 and
y
Conditio s
0
n
z
z zs x x y y
zs
zs
H E A k x A k x A k y A k y e
E y ,b
E x ,a
y x
zs 1 3
0 0 2 4
zs
) to E 0
sin sin ( )
Applying boundary conditions at ( and ) to E
sin 0, sin 0 , This implies that :
, 1, 2,3,...
,
z
zs x y
x y
x
y
A A
E E k x k y e E A A
y b x a
k a k b
k a m m
k b n n
0
1,2,3,...
,
sin sin
x y
z
zs
m nor k k
a b
m x n yE E e
a b
Tangential components are continuous
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Transverse Magnetic (TM) mode
• Other components are
2
2
2
2
zsx
zsy
zsx
zsy
EE
h x
EE
h y
EjH
h y
EjH
h x
sin sin , 0zzs o zsm n
E E x y e Ha b
2
2
2
2
cos sin
sin cos
sin cos
cos sin
z
xs o
z
ys o
z
xs o
z
ys o
m m x n yE E e
h a a b
n m x n yE E e
h b a b
j n m x n yH E e
h b a b
j m m x n yH E e
h a a b
2 2
2 2 2 x ym n
where h k ka b
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Transverse Magnetic (TM) mode
2 2
2 2
2
2 2
2
Propagation constant: ,
,
h k
m nh k
a b
m n
a b
•Each set of integers m and n gives a different field pattern or mode.
•Integer m equals the number of half cycle variations in the x-direction.
•Integer n is the number of half cycle variations in the y-direction.
•Note that for the TM mode, if n or m is zero, all fields are zero. Hence,
TM11 is the lowest order mode of all the TMmn modes.
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Example: Field configuration for TM21 mode
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Transverse Magnetic (TM) mode 2 2
2m n
a b
2 2
2
2 2
,
then 0
1 1or
2
No propagation takes pla
The cuttoff occurs when
ce at this frequenc
:
y
c
cm n
m nj
a b
m nf
a b
2 2
2When and 0
No wave propagation at all. (everything is attenuated)
So when
Evanescent m
, all field components will decay exponantially
odes :
.c
m n
a b
f f
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Transverse Magnetic (TM) mode
2 2
2 and 0
This is the case we are interested since is when the wave
is allowed to travel t
Propaga
hrough the guide.
, at a given operating fre
tion occurs
que
whe
nc
n
m nj
a b
So
y f, only those modes with
will propagate.f fc
fc,mn
attenuation Propagation
of mode mn
The cutoff frequency is the
frequency below which attenuation
occurs and above which propagation
takes place. (High Pass)
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2 2
,
2 2
2 2
2
:
1 1
2
' 1 , where '
2
T can be written in term
The cutoff Frequency is
he phase co s of asnst t :an
cm n
cmn
c
m nf
a b
u m nor f u
a b
f
m n
a b
2 2
2
2
1
' 1 , where ' / 'c
m n
a b
fk u
f
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Transverse Magnetic (TM) mode
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Transverse Magnetic (TM) mode
2 2
2 2
2 2 2 ' ', but ' , '
'' 1 1
:- (varies with freq
The gu
uency)
1 ' 1
ide wavelength is:
Intrinsic Impedance
, w
g g
c c
x c cTM TM
y
u
ff f
f f
E f f
H f f
' /
', ', ', and ' are parameters for unguided wave propagating
in the same dielectric medium ( , ) unbounded by the waveguide.
(i.e. waveguide removed and entire space is filled with diele
here
u
ctric.)
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Transverse Electric (TE) modes
1 2 3 4
2
0, cos sin cos sin
0 at 0 (bottom and top walls)
0 at 0 (le
Bounda
ft and right walls)
0 at
ry
Conditions
0
z
z zs x x y y
xs
ys
zs zsxs
E H B k x B k x B k y B k y e
E y ,b
E x ,a
j H HE y ,b
h y y
2
0 1 3
0 at 0
From this we conclude
cos cos ( = )
zs zsys
z
zs o
j H HE x ,a
h x x
m x nH H y e H B B
a b
Tangential components are continuous
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Other components are
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Transverse Electric (TE) modes
2
2
2
2
cos sin
sin cos
sin cos
cos sin
z
xs o
z
ys o
z
xs o
z
ys o
j n m x n yE H e
h b a b
j m m x n yE H e
h a a b
j m m x n yH H e
h a a b
j n m x n yH H e
h b a b
0, cos cos zz zs om n
E H H x y ea b
2
2
2
2
zxs
zys
zxs
zys
HjE
h y
HjE
h x
HH
h x
HH
h y
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Example: Field configuration for TE32 mode
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• The cutoff frequency is the same expression as for the TM
mode
• For TE modes, (m,n) may be (0,1) or (1,0) but not (0,0). Both
m and n cannot be zero at the same time because this will force
the field components to vanish.
• Hence, the lowest mode can be TE10 or TE01 depending on the
values of a and b.
• It is standard practice to have a>b, thus TE10 is the lowest
mode.
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22
2
'
b
n
a
muf
mnc
TE modes - Cuttoff
10 '/ 2cf u a
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TE modes
• The dominant mode is the mode with lowest cutoff frequency.
The cutoff frequency of the TE10 mode is lower than that of TM11
mode. Hence, TE10 is the dominant mode.
If more than one mode is propagating, the waveguide is overmoded.
Single mode propagation is highly desirable to reduce dispersion.
This occurs between cutoff frequency for TE10 mode and cuttoff
frequency of next higher mode.
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2
2 2
he phase constant is the same as TM mode:
The intrinsic impedance of the TE mod
T
' 1 , where ' / '
1 ' , w ' /
1 1
e is:
c
xTE TE
yc c
fu
f
Ehere
H f f
f f
2
2Note that ' ' 1 cTE TM TMf
f
TE modes
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For TE mode, cos cos
For TE mode, cos
In the time domain: =Re
cos cos
,
j z
zs o
j z
zs o
j t
z zs
z o
y
m nH H x y e
a b
xH H e
a
H H e
or
xH H t z
a
Similarly
E
0
0
sin sin
sin sin
0
x
z x y
a xH t z
a
a xH H t z
a
E E H
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TE10 mode
Variation of the field components with x for TE10 mode.
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TE10 mode
Field lines for TE10 mode
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TE/TM modes Wave in the dielectric medium Inside the waveguide
/'
'/' u
2
1
'
f
fc
TE
2
'
1 cf
f
/
1'
2
f
f
u
c
p
2
1'
f
fc
fu /''
/1'/' fu
2
, ' 1 cTMf
f
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2 2
2 2 2 2
2 2
The cutoff frequency is given by:
' 1 , u'=
2 2
Hence
4 4 2.5 10 1 10
cmn
r r
cmn
u m n c cf
a b
c m n c m nf
a b
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Example 12.1
• Example: A rectangular waveguide with dimensions a=2.5 cm,
b=1 cm is to operate below 15.1 GHz. How many TE and TM
modes can the waveguide transmit if the guide is filled with a
medium characterized by σ=0, ε=4 ε0, µr=1? Calculate the
cutoff frequencies of the modes.
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2 2
2 2
01 01
02 02
03 03
10 10
20 10
4 2.5 10 1 10
For TE mode ( =0, =1), 7.5 GHz
For TE 15 GHz
For TE 22.5 GHz
For TE 3 GHz
For TE 6 GH
cmn
c m nf
m n fc
fc
fc
fc
fc
30 10
40 10
50 10
60 10
z
For TE 9 GHz
For TE 12 GHz
For TE 15 GHz
For TE 18 GHz
fc
fc
fc
fc
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Example 12.1 - solution
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2 2
2 2
11 11 11
21 21 21
31 31 31
4 2.5 10 1 10
For TE , TM modes , 8.078 GHz
For TE , TM modes , 9.6 GHz
For TE , TM modes , 11.72 GHz
cmn
c m nf
fc
fc
fc
41 41 41
12 12 12
Modes with cutoff frequencies les
For TE , TM modes , 14.14 GHz
s than or equal 15.1 GHz
will be tran
For TE
smitt
, TM modes , 15
ed. (11 TE modes
.3
an
GHz
d 4 TM
modes)
fc
fc
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Example 12.1 - solution
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Example 12.1 - solution
Cutoff frequencies of rectangular waveguide with a 2.5b; for Example 12.1.
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Example 12.3
• Example: in a rectangular waveguide for which a=1.5 cm,
b=0.8 cm, σ=0, µ=µ0, ε=4ε0.
• Determine:
• (a) the mode of operation.
• (b) the cutoff frequency
• (c) the phase constant β.
• (d) the propagation constant γ.
• (e) the intrinsic wave impedance η.
113
2sin cos sin 10 A/mxx y
H t za b
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13 13 13
2 2
2 2
13 2 2
( ) the guide is operating at TM or TE . Suppose we choose TM .
' 1( ) , '
2 2
1 3 28.57 GHz
4 1.5 10 0.8 10
(c) ' 1
cmn
r r
c
c
a
u m n c cb f u
a b
cHence f
f
f
13
2 2 2
11
211
8
2 2
TM
1 1
2 10 50 GHz
10 4 28.571 1718.81 rad/m
3 10 50
( ) = 1718.81/ m
377 28.57( ) ' 1 1 154.7
50
rc c
c
r
f f
f c f
f f
d j j
fe
f
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Example 12.3 - Solution
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