1. ENE 428 Microwave Engineering
Lecture 4 Reflection and Transmission at Oblique Incidence, Transmission Lines RS RS

2. Plane wave propagation in general dielectrics
Assume lossless medium
The propagation directions are and
The plane of incidence is defined as the plane containing both normal to the boundary and the incident wave’s propagation direction.
The angle of incidence i is the angle the incident field makes with a normal to the boundary RS

3. Polarizations of UPW obliquely incident on the boundary (1)
Perpendicular polarization or transverse electric (TE)
polarization
is normal to the plane
of incidence and tangential
to the boundary.
Only the x component
of the magnetic field is
tangential. RS

4. Polarizations of UPW obliquely incident on the boundary (2)
Parallel polarization or transverse magnetic (TM)
polarization
is normal to the plane
of incidence and tangential
to the boundary.
Only the x component
of the electric field is
tangential. RS

5. TE polarization x i z
We can write
and RS

6. Reflected and transmitted fields for TE polarization
Reflected fields
Transmitted fields RS

7. Snell’s laws of reflection and refraction (1)
Tangential boundary condition for the electric field
at z = 0
for this equality to hold,
Snell’s law of reflection
Snell’s law of refraction or RS

8. Snell’s laws of reflection and refraction (2)
the critical angle for total reflection
If i  cri, then it is total reflection and no power can be
transmitted, these fields are referred as evanescent waves. RS

9. Reflection and transmission coefficients for TE polarization (1)
From the electric field’s B.C. with phases matched, we have
Tangential B.C. for the magnetic field considering matched phase and equal incident and reflected angles is RS

10. Reflection coefficient for TE polarization
Solving Eqs. (1) and (2) gets or RS

11. Transmission coefficient for TE polarization
Solving Eqs. (1) and (2) gets or
Notice that RS

12. Ex2 A 2 GHz TE wave is incident at 30 angle of incidence from air on to a thick slab of nonmagnetic, lossless dielectric with r = 16. Find TE and TE. RS

13. Fields for TM polarization
Incident fields
Reflected fields
Transmitted fields RS

14. Reflection and transmission coefficients for TM polarization
Solving B.C.s gets
and
Notice that RS

15. Total transmission for TM polarization
Brewster’s angle for total transmission
For lossless, non-magnetic media, we have RS

16. Ex3 A uniform plane wave is incident from air onto glass at an angle from the normal of 30. Determine the fraction of the incident power that is reflected and transmitted for a) and b). Glass has refractive index n2 = 1.45.
TM polarization
TE polarization RS

17. Transmission lines (1)
Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies.
Examples:
Transmitter and antenna
Connections between computers in a network
Interconnects between components of a stereo system
Connection between a cable service provider and aTV set.
Connection between devices on circuit board
Distances between devices are separated by much larger order of wavelength than those in the normal electrical circuits causing time delay. RS

18. Transmission lines (2) Properties to address: time delay reflections
attenuation
distortion RS

19. Distributed-parameter model
Types of transmission lines RS

20. Distributed-parameter model
The differential segment of the transmission line
R’ = resistance per unit length
L’= inductance per unit length
C’= capacitor per unit length
G’= conductance per unit length RS

21. Telegraphist’s equations
General transmission lines equations: RS

22. Telegraphist’s time-harmonic wave equations
Time-harmonic waves on transmission lines
After arranging we have
where RS

23. Traveling wave equations for the transmission line
Instantaneous form
Phasor form RS

24. Lossless transmission line
lossless when R’ = 0 and G’ = 0
and RS

25. Low loss transmission line (1)
low loss when R’ << L’ and G’ << C’
Expanding in binomial series gives
for x << 1 RS

26. Low loss transmission line (2)
Therefore, we get RS

27. Characteristic impedance
Characteristic impedance Z0 is defined as the
the ratio of the traveling voltage wave amplitude to the traveling current wave amplitude. or
For lossless line, RS

28. Power transmission
Power transmitted over a specific distance is calculated.
The instantaneous power in the +z traveling wave at any point along the transmission line can be shown as
The time-averaged power can be shown as W. RS

29. Power ratios on the decibel scale (1)
A convenient way to measure power ratios
Power gain (dB)
Power loss (dB)
1 Np = dB dB dB RS

30. Power ratios on the decibel scale (2)
Representation of absolute power levels is the dBm scale
dBm RS

31. Ex1 A 12-dB amplifier is in series with a 4-dB attenuator
Ex1 A 12-dB amplifier is in series with a 4-dB attenuator. What is the overall gain of the circuit? Ex2 If 1 W of power is inserted into a coaxial cable, and 1 W of power is measured 100m down the line, what is the line’s attenuation in dB/m? RS

32. Ex3 A 20 m length of the transmission line is known to produce a 2 dB drop in the power from end to end,
what fraction of the input power does it reach the output?
What fraction of the input power does it reach the midpoint of the line?
What is the attenuation constant? RS