1. ENE 428 Microwave Engineering
Lecture 7 Waveguides RS

2. Review Impedance matching to minimize power reflection from load
Lumped-element tuners
Single-stub tuners
Microstrip lines
The most popular transmission line
Knowing the characteristic impedance and the relative dielectric constant of the material helps determine the strip line configuration and vice versa.
Attenuation
conduction loss
dielectric loss
radiation loss

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

4. The use of waveguide
Waveguide refers to the structure that does not support TEM mode.
They are unable to support wave propagation below a certain frequency, termed the cutoff frequency.
Figures (a) and (b) are metallic waveguide. The rectangular one is for high-power microwave applications but is limited in frequency range and tends to suffer from dispersion.
The circular waveguide has higher power handling capability than the rectangular one. The dielectric waveguide has a smaller loss than metallic waveguide at high freq. Optical fiber has wider bandwidth and provides good signal isolation between adjacent fibers.

5. Rectangular waveguide fundamentals
A typical cross section is shown. Propagation is in the +z direction. The conducting walls are brass, copper or aluminum. The inside is electroplated with silver or gold and smoothly polished to reduce loss.
The interior dimensions are “a x b”, the longer side is “a”.
“a” determines the frequency range of the dominant, or lowest order mode.
Higher modes have higher attenuation and be difficult to extract from the guide.
“b” affects attenuation; smaller “b” has higher attenuation.
If “b” is increased beyond “a/2”, the next mode will be excited at a lower frequency, thus decreasing the useful frequency range.
In practice, “b” is chosen to be about “a/2”.

6. Rectangular waveguide fundamentals
Waveguide can support TE and TM modes.
The order of the mode refers to the field configuration in the guide and is given by “m” and “n” integer subscripts, as TEmn and TMmn.
The “m” subscript corresponds to the number of half-wave variations of the field in the x direction.
The “n” subscript is the number of half-wave variations in the y direction.
“m” and “n” determines the cutoff frequency for a particular mode.
The relative cutoff freq for the first 12 modes of waveguide with “a” = “2b” are shown.

9. Wave Propagation
(a) A y-polarized TEM plane wave propagating in the +z direction.
(b) Wavefront view of the propagating wave
Note: the waves propagate at a velocity Uu, where u subscript indicates media unbounded by guide wall.
In air, Uu = c

10. Now a pair of identical TEM waves, labeled as U+ and U- are superimposed.
The horizontal lines can be drawn on the superimposed waves that correspond to zero total field. Along these lines, the U+ wave is always 180o out of phase with the U- wave.
Next, replace the horizontal lines with perfect conducting walls.

11. “a” is the wall separation and is determined by the angle  and wavelength 
For a given wave velocity Uu, the frequency is f = Uu/ 
If we fix the wall separation at “a” and change the frequency, we must then also change the angle  if we are to maintain a propagating wave.
The edge of a +Eo wavefront (point A) will line up with edge of a –Eo wavefront (point B), and two fronts must be /2 apart for the m = 1 mode. For any value of m, we can write or

12. The waveguide can support propagation as long as the wavelength is smaller than a critical value that occurs at  = 90o, or
where fc is the cutoff frequency for the propagating mode. We can relate the angle  to the operating frequency (f) and the cutoff frequency (fc) by
The time tAC it takes for the wavefront to move from A to C is
Meanwhile, a constant phase point moves along the wall from A to D. Calling the phase
velocity, Up: so

13. Since the times tAD and tAC must be equal,
we have
The phase velocity can be considerably faster than the velocity of the wave in unbounded media, tending toward infinity as f approaches fc.
The phase constant associated with the phase velocity is
Where u is the phase constant in unbounded media. The wavelength in the guide is related to this phase velocity by  = 2

14. The propagation velocity of the superposed wave is given by the group velocity UG.
The group velocity is slower than that of an unguided wave, which is to be expected since the guided wave propagates in a zig-zag path, bouncing off the waveguide walls.
Waveguide Impedance
The ratio of the transverse electric field to the transverse magnetic field for a propagating mode at a particular frequency.
where u is the intrinsic impedance of the propagating media. In air u = o = 120 

15. Waveguide impedance of the TE11 and TM11 modes vs frequency for WR90

16. Solution: at 10 GHz, only the TE10 mode is supported, so
Example: determine the TE mode impedance looking into a 20-cm long section of
shorted WR90 waveguide operating at 10 GHz.
Solution: at 10 GHz, only the TE10 mode is supported, so
We can find Zin from
for a shorted line.
Now  is found from
So, l in the Zin equation is
And Zin is then calculated as

17. 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 Helmholtz’s equations:
assume WG is filled in with a charge-free
lossless dielectric

18. General wave behaviors along uniform guiding structures (2)
We can write and in the phasor forms as
and
Note that are functions of x and y only. The z-
dependency comes from the term e-z. In other words, for example
and so on ….
so that

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

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

21. 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

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

23. 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 .

24. 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.

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

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

27. 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 (, )

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

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

30. Transverse Electric wave (TE)
From
Expanding for z-propagating field gets
where
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 .

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

32. Ex2 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.