```ENE 428
Microwave
Engineering
Lecture 8 Rectangular waveguides
and cavity resonator
1
TM waves in rectangular waveguides
• Finding E and H components in terms of z, WG geometry, and
modes.
From
2xy Ez  ( 2  u2 )Ez  0,
Expanding for z-propagating field for the lossless WG gets
 2 Ez  2 Ez
2
2


(



) Ez  0
u
2
2
x
y
where
Ez  Ez ( x, y)e j z
2
Method of separation of variables (1)
Assume
Ez ( x, y)  XY
where X = f(x) and Y = f(y).
Substituting XY gives
d2X
d 2Y
2
2
Y

X

(



) XY  0
u
2
2
dx
dy
and we can show that
for lossless WG.
2
2
1
d
Y
1
d
X
2
2
u   

.
Y dy 2
X dx 2
3
Method of separation of variables (2)
Let
and
2
1
d
X
2
x  
X dx 2
2
1
d
Y
 y2  
Y dy 2
then we can write
  u2   x2   y2 .
We obtain two separate ordinary differential equations:
d2X
2


x X 0
2
dx
d 2Y
2


yY  0
2
dy
4
General solutions
X ( x)  c1 cos   x x   c2 sin   x x 
Y ( y )  c3 cos   y y   c4 sin   y y 
Appropriate forms must be chosen to satisfy boundary
conditions.
5
Properties of wave in rectangular WGs
(1)
1. in the x-direction
Et at the wall = 0 Ez(0,y) and Ez(a,y) = 0
and X(x) must equal zero at x = 0, and x = a.
Apply x = 0, we found that C1 = 0 and X(x) = c2sin(xx).
Therefore, at x = a, c2sin(xa) = 0.
  x a  m
(m  0,1, 2,3,...)
m
x 
.
a
6
Properties of wave in rectangular WGs
(2)
2. in the y-direction
Et at the wall = 0 Ez(x,0) and Ez(x,b) = 0
and Y(y) must equal zero at y = 0, and y = b.
Apply y = 0, we found that C3 = 0 and Y(y) = c4sin(yy).
Therefore, at y = a, c4sin(yb) = 0.
  yb  n
(n  0,1, 2,3,...)
n
y  .
b
7
Properties of wave in rectangular WGs
(3)
 m   n 
   u2  



a
b

 

2
 m   n 
h



a
b

 

2
and
2
2
therefore we can write
Ez  E0 sin
m x
n y  j z
sin
e
V /m
a
b
8
TM mode of propagation
 Every combination of integers m and n defines possible
mode for TMmn mode.
 m = number of half-cycle variations of the fields in the xdirection
 n = number of half-cycle variations of the fields in the ydirection
 For TM mode, neither m and n can be zero otherwise Ez
and all other components will vanish therefore TM11 is the
lowest cutoff mode.
9
TE10 field lines
End view
Side view
Top view
10
TM11 field lines
End view
Side view
11
Cutoff frequency and wavelength of TM
mode
f c ,mn 
h
2 
c ,mn 

2
1
2 
m n
 a   b 
   
2
2
m n
 a   b 
   
2
2
Hz
m
12
Ex2 A rectangular wg having the interior dimension
a = 2.3cm and b = 1cm filled with a medium
characterized by r = 2.25, r = 1
a)
Find h, fc, and c for TM11 mode
b)
If the operating frequency is 15% higher than the cutoff
frequency, find (Z)TM11, ()TM11, and (g)TM11. Assume
the wg to be lossless for propagating modes.
13
TE waves in rectangular waveguides (1)
• Ez = 0
From
2xy H z  (u2   2 )H z  0
Expanding for z-propagating field for a lossless WG gets
2 H z 2 H z
2
2


(



) H z ( x, y)  0
u
2
2
x
y
where
H z  H z ( x, y)e j z
14
TE waves in rectangular waveguides (2)
• In the x-direction
Since Ey = 0, then from
j H z
j  Ez
Ey  2
 2
2
u   x u   2 y
we have
H z
0
x
at x = 0 and x = a
15
TE waves in rectangular waveguides (3)
• In the y-direction
Since Ex = 0, then from
 j H z
j  Ez
Ex  2
 2
2
u   y u   2 x
we have
H z
0
y
at y = 0 and y = b
16
Method of separation of variables (1)
H z ( x, y)  XY
Assume
then we have
X ( x)  c1 cos   x x   c2 sin   x x 
Y ( y )  c3 cos   y y   c4 sin   y y 
17
Properties of TE wave in x-direction of
rectangular WGs (1)
1. in the x-direction
at x = 0,
H z
0
x
dX ( x)
  x c1 sin   x x    x c2 cos   x x   0
dx
 c2  0.
at x = a,
H z
0
x
dX ( x)
  x c1 sin   x x   0
dx
18
Properties of TE wave in x-direction of
rectangular WGs (2)
  x a  m
(m  0,1, 2,3,...)
m
x 
.
a
19
Properties of TE wave in y-direction
of rectangular WGs (1)
2. in the y -direction
H z
0
y
dY ( y )
   y c3 sin   y y    y c4 cos   y y   0
dy
at y = 0,
 c4  0
at y = b,
H z
0
y
dY ( y )
   y c3 sin   y y   0
dy
20
Properties of TE wave in y-direction
of rectangular WGs (2)
  yb  n
(n  0,1, 2,3,...)
n
y  .
b
For lossless TE rectangular waveguides,
m x
n y  j z
H z  H0 cos
cos
e
a
b
A/ m
21
Cutoff frequency and wavelength of TE
mode
f c ,mn 
h
2 
c ,mn 

2
1
2 
m n
 a   b 
   
2
2
m n
 a   b 
   
2
2
Hz
m
22
A dominant mode for TE waves
•
For TE mode, either m or n can be zero, if a > b,
is a smallest eigne value and fc is lowest when m = 1
and n = 0 (dominant mode for a > b)
h
( fc )TE10

a
up


2a  2a
1
Hz
(c )TE10  2a m
23
A dominant mode for TM waves
•
For TM mode, neither m nor n can be zero, if a > b, fc
is lowest when m = 1 and n = 1
( f c )TM11 
(c )TM11 
2
1 1
 a  b 
   
1
2 
2
2
1 1
 a  b 
   
2
2
Hz
m
24
Ex1 a) What is the dominant mode of an axb
rectangular WG if a < b and what is its cutoff
frequency?
b) What are the cutoff frequencies in a square WG
(a = b) for TM11, TE20, and TE01 modes?
25
Ex2 Which TM and TE modes can propagate in
the polyethylene-filled rectangular WG (r = 2.25,
r = 1) if the operating frequency is 19 GHz given
a = 1.5 cm and b = 0.6 cm?
26
Rectangular cavity resonators (1)
 At microwave frequencies, circuits with the dimension
comparable to the operating wavelength become efficient
 An enclose cavity is preferred to confine EM field, provide
large areas for current flow.
 These enclosures are called ‘cavity resonators’.
There are both TE and TM modes
but not unique.
b
d
a
27
Rectangular cavity resonators (2)
 z-axis is chosen as the reference.
 “mnp” subscript is needed to designate a TM or TE standing
wave pattern in a cavity resonator.
28
Electric field representation in
TMmnp modes (1)
 The presence of the reflection at z = d results in a standing
wave with sinz or cozz terms.
Consider transverse components Ey(x,y,z),
from B.C. Ey = 0 at z = 0 and z = d
 1) its z dependence must be the sinz type
2)  
p
d
( p  0,1, 2,...)
similar to Ex(x,y,z).
29
Electric field representation in
TMmnp modes (2)
From
 j H z
j  Ez
Ex  2
 2
2
u   y u   2 x
j H z
j  Ez
Ey  2
 2
2
u   x u   2 y
Hz vanishes for TM mode, therefore
j
Ex   2
h
j
Ey   2
h
E z
x
E z
y
30
Electric field representation in
TMmnp modes (3)
If Ex and Ey depend on sinz then Ez must vary according
to cosz, therefore
Ez ( x, y, z)  Ez ( x, y)cos
p z
V /m
d
m x
n y
p z
 E0 sin
sin
cos
V /m
a
b
d
f mnp
u p  m 2  n 2  p 2
 resonant frequency 


2  a   b   d 
Hz
31
Magnetic field representation in
TEmnp modes (1)
 Apply similar approaches, namely
1) transverse components of E vanish at z = 0 and z = d
- require a sin
p z
factor in Ex and Ey as well as Hz.
d
2) factor  indicates a negative partial derivative with z.
p z
- require a cos
factor for Hx and Hy
d
p z
 H z ( x, y, z)  H z ( x, y)sin
A/ m
d
m x
n y
p z
 H0 cos
cos
sin
a
b
d
fmnp is similar to TMmnp.
A/ m
32
Dominant mode
• The mode with a lowest resonant frequency is called
‘dominant mode’.
• Different modes having the same fmnp are called
degenerate modes.
33
Resonator excitation (1)
For a particular mode, we need to
1) place an inner conductor of the coaxial cable where the
electric field is maximum.
2) introduce a small loop at a location where the flux of the
desired mode linking the loop is maximum.
source frequency = resonant frequency
34
Resonator excitation (2)
For example, TE101 mode, only 3 non-zero components are
Ey, Hx, and Hz.

insert a probe in the center region of the top or bottom
face where Ey is maximum or place a loop to couple
Hx maximum inside a front or back face.

Best location is affected by impedance matching
requirements of the microwave circuit of which the
resonator is a part.
35
Coupling energy method

place a hole or iris at the appropriate location

field in the waveguide at the hole must have a component
that is favorable in exciting the desired mode in the
resonator.
36
Ex3 Determine the dominant modes and their
frequencies in an air-filled rectangular cavity
resonator for
a)
a>b>d
b)
a>d>b
c)
a=b=d
37
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