1. ENE 429 Antenna and Transmission Lines Theory
Lecture 10 Waveguides

2. A pair of conductors is used to guide TEM wave
Microstrip
Parallel plate
Two-wire TL
Coaxial cable

3. The use of waveguide
Waveguide refers to the structure that does not support TEM mode, bring up “the cutoff frequency”

4. General wave behaviors along uniform guiding structures (1)
The wave characteristics are examined along straight guiding structures with a uniform cross section such as rectangular waveguides
We can write in the instantaneous form as
We begin with Helmholz’s equations:
assume WG is filled in with a charge-free
lossless dielectric or
where

5. General wave behaviors along uniform guiding structures (2)
We can write and in phasor forms as
and

6. Use Maxwell’s equations to show and in terms of z components (1)
From
we have

7. Use Maxwell’s equations to show and in terms of z components (2)
We can express Ex, Ey, Hx, and Hy in terms of z-component
by substitution so we get for lossless media  = j,

8. Propagating waves in a uniform waveguide
Transverse ElectroMagnetic (TEM) waves, no Ez or Hz
Transverse Magnetic (TM), non-zero Ez but Hz = 0
Transverse Electric (TE), non-zero Hz but Ez = 0

9. Transverse ElectroMagnetic wave (TEM)
Since Ez and Hz are 0, TEM wave exists only when
A single conductor cannot support TEM

10. Transverse Magnetic wave (TM)
From
We can solve for Ez and then solve for other components from (1)-(4) by setting Hz = 0, then we have
Notice that  or j for TM is not equal to that for TEM .

11. Eigen values
We define
Solutions for several WG problems will exist only for real
numbers of h or “eigen values” of the boundary value problems, each eigen value determines the characteristic of
the particular TM mode.

12. Cutoff frequency From The cutoff frequency exists when  = 0 or or
We can write

13. a) Propagating mode (1) or and  is imaginary Then
This is a propagating mode with a phase constant :

14. a) Propagating mode (2) Wavelength in the guide,
where u is the wavelength of a plane wave with a frequency
f in an unbounded dielectric medium (, )

15. a) Propagating mode (3)
The phase velocity of the propagating wave in the guide is
The wave impedance is then

16. b) Evanescent mode or Then Wave diminishes rapidly with distance z.
ZTM is imaginary, purely reactive so there is no power flow

17. Transverse Electric wave (TE)
From
We can solve for Hz and then solve for other components from (1)-(4) by setting Ez = 0, then we have
Notice that  or j for TE is not equal to that for TEM .

18. TE characteristics
Cutoff frequency fc,, g, and up are similar to those in TM mode.
But
Propagating mode f > fc
Evanescent mode f < fc

19. Ex1 Determine wave impedance and guide wavelength (in terms of their values for the TEM mode) at a frequency equal to twice the cutoff frequency in a WG for TM and TE modes.

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

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

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

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

24. 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.

25. 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.

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

27. 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.

28. Cutoff frequency and wavelength of TM mode

29. Ex2 A rectangular wg having interior dimension a = 2
Ex2 A rectangular wg having 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.