1. ENE 428 Microwave Engineering
Lecture 8 Rectangular waveguides and cavity resonator

2. TM waves in rectangular waveguides
Finding E and H components in terms of z, WG geometry, and modes.
From
Expanding for z-propagating field for the lossless WG gets
where

3. Method of separation of variables (1)
Assume
where X = f(x) and Y = f(y).
Substituting XY gives
and we can show that
for lossless WG.

4. Method of separation of variables (2)
Let
and
then we can write
We obtain two separate ordinary differential equations:

5. General solutions Appropriate forms must be chosen to satisfy boundary
conditions.

6. Properties of wave in rectangular WGs (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.

7. 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.

8. Properties of wave in rectangular WGs (3)
and
therefore we can write

9. 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 x-
direction
n = number of half-cycle variations of the fields in the y-
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.

10. TM11 field lines
Side view
End view

11. Cutoff frequency and wavelength of TM mode

12. Ex2 A rectangular wg having the interior dimension a = 2
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
Find h, fc, and c for TM11 mode
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
Expanding for z-propagating field for a lossless WG gets
where

14. TE waves in rectangular waveguides (2)
In the x-direction
Since Ey = 0, then from
we have
at x = 0 and x = a

15. TE waves in rectangular waveguides (3)
In the y-direction
Since Ex = 0, then from
we have
at y = 0 and y = b

16. Method of separation of variables (1)
Assume
then we have

17. Properties of TE wave in x-direction of rectangular WGs (1)
in the x-direction
at x = 0,
at x = a,

18. Properties of TE wave in x-direction of rectangular WGs (2)

19. Properties of TE wave in y-direction of rectangular WGs (1)
2. in the y -direction
at y = 0,
at y = b,

20. Properties of TE wave in y-direction of rectangular WGs (2)
For lossless TE rectangular waveguides,

21. Cutoff frequency and wavelength of TE mode

22. TE10 field lines
Side view
End view
Top view

23. 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)

24. 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

25. 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?

26. 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?

27. Rectangular cavity resonators (1)
At microwave frequencies, circuits with the dimension
comparable to the operating wavelength become efficient
radiators
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

28. 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.

29. 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)
similar to Ex(x,y,z).

30. Electric field representation in TMmnp modes (2)
From
Hz vanishes for TM mode, therefore

31. Electric field representation in TMmnp modes (3)
If Ex and Ey depend on sinz then Ez must vary according to cosz, therefore

32. Magnetic field representation in TEmnp modes (1)
Apply similar approaches, namely
transverse components of E vanish at z = 0 and z = d
- require a factor in Ex and Ey as well as Hz.
factor  indicates a negative partial derivative with z.
- require a factor for Hx and Hy
fmnp is similar to TMmnp.

33. Dominant mode
The mode with a lowest resonant frequency is called ‘dominant mode’.
Different modes having the same fmnp are called degenerate modes.

34. Resonator excitation (1)
For a particular mode, we need to
place an inner conductor of the coaxial cable where the
electric field is maximum.
introduce a small loop at a location where the flux of the
desired mode linking the loop is maximum.
source frequency = resonant frequency

35. 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.

36. 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.

37. Ex3 Determine the dominant modes and their frequencies in an air-filled rectangular cavity resonator for
a > b > d
a > d > b
a = b = d