1. Applied Electromagnetic Waves Waveguiding Structures
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes 19
Waveguiding Structures

2. Waveguiding Structures
A waveguiding structure is one that carries a signal (or power) from one point to another.
There are three common types:
Transmission lines
Fiber-optic guides
Waveguides
An alternative to a waveguiding system is a wireless system using antennas.

3. (nonmagnetic filling)
Transmission lines
Properties
Has two conductors running parallel
Can propagate a signal at any frequency (in theory)
Becomes lossy at high frequency
Can handle low or moderate amounts of power
Does not have signal distortion, unless there is loss
May or may not be immune to interference
Does not have Ez or Hz components of the fields (TEMz)
Lossless:
(nonmagnetic filling)

4. Fiber-Optic Guide Properties Has a single dielectric rod
Can propagate a signal at any frequency (in theory)
Can be made very low loss
Has minimal signal distortion
Very immune to interference
Not suitable for high power
Has both Ez and Hz components of the fields (“hybrid mode”)

5. Fiber-Optic Guide (cont.)
Two types of fiber-optic guides:
1) Single-mode fiber
Carries a single mode, as with the mode on a waveguide. Requires the fiber diameter to be small relative to a wavelength.
2) Multi-mode fiber
Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

6. Multi-Mode Fiber

7. Multi-Mode Fiber (cont.)
There is a maximum angle max at which the beam can be incident from outside the rod.
Assume cladding is air
At left end of rod:

8. Multi-Mode Fiber (cont.)
At top boundary with air:

9. Rectangular waveguide
Properties
Has a single hollow metal pipe
Can propagate a signal only at high frequency:  > c
The width must be at least one-half of a wavelength
Has signal distortion, even in the lossless case
Immune to interference
Can handle large amounts of power
Has low loss (compared with a transmission line)
Has either Ez or Hz component of the fields (TMz or TEz)
Inside microwave oven
Rectangular waveguide

10. Waveguides (cont.)
Cutoff frequency property for lossless guide (derived later)
In a waveguide:
where
(wavenumber (real) of material inside waveguide)
We can write:
(definition of cutoff frequency c)
(propagation)
(evanescent decay)

11. Field Expressions of a Guided Wave
Statement:
All six field components of a guided wave can be expressed in terms of the two fundamental field components Ez and Hz.
"Guided-wave theorem"
Assumption:
(This is the definition of a guided wave.)
A proof of this statement is given in the Appendix.

12. Field Expressions of a Guided Wave (cont.)
Summary of Fields

13. TEMz Wave TEMz Assume a TEMz wave:
To avoid having a completely zero field,
Hence,
TEMz

14. TEMz Wave (cont.) Examples of TEMz waves:
A wave in a transmission line (no conductor loss)
A plane wave
In each case the fields do not have a z component !
Plane wave 
Coax

15. Wave Impedance Property of TEMz Mode
TEMz Wave (cont.)
Wave Impedance Property of TEMz Mode
Faraday's Law:
Take the x component of both sides:
Assume that the field varies as
Hence,
Therefore, we have

16. TEMz Wave (cont.) Now take the y component of both sides: Hence,
Therefore, we have
Hence,

17. TEMz Wave (cont.) Summary:
These two equations may be written as a single vector equation:
The electric and magnetic fields of a TEMz wave are perpendicular to each other, and the amplitudes of them are related by .

18. “Quasi-TEM” (TEM-like at low frequency)
TEMz Wave (cont.)
Examples
Plane wave 
Coax
Microstrip
The fields look like a plane wave in the central region.
“Quasi-TEM” (TEM-like at low frequency)

19. In a waveguide, the fields cannot be TEMz.
Proof:
Assume a TEMz field
waveguide
PEC boundary C
(property of flux line)
contradiction!
(Faraday's law in integral form)

20. Waveguide (cont.) TMz: Hz = 0, Ez  0 TEz: Ez = 0, Hz  0
In a waveguide (hollow pipe of metal), there are two types of modes:
TMz: Hz = 0, Ez  0
TEz: Ez = 0, Hz  0
Each type of mode can exist independently.

21. Proof of Guided Wave Theorem
Appendix
Proof of Guided Wave Theorem
(illustrated for Ey) or
Now solve for Hx :

22. Appendix (cont.)
Substituting this into the equation for Ey yields the result
Next, multiply by

23. The other three components Ex, Hx, Hy may be found similarly.
Appendix (cont.)
This gives us or
Solving for Ey, we have:
The other three components Ex, Hx, Hy may be found similarly.