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 z i
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.
Fields do extend into the 2nd medium where they decay
exponentially with z. However, the transmitted electric and
magnetic fields are 90o out of phase, so no power is trans-
mitted. 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. Average power conservation for TE polarization
It should be noted that in terms of power conservation, we only consider power directed normal to the boundary. For TE polarization, we have RS

13. 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

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

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

16. Total transmission for TM polarization
For TM polarization, there exists an incidence angle at which all of the wave is transmitted into the 2nd medium.
This known as the Brewster’s angle, i = BA and it can be found by first setting the numerator of the reflection coeff. equal to zero; that is,
Using Snell’s law of refraction and do some algebraic manipulation, RS

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

18. When a randomly polarized wave (such as light) is incident on a material at the Brewster’s angle, the TM polarized portion is totally transmitted but at TE component is partially reflected.
This principle is employed in gas lasers, where quartz windows at each end of the laser tube are set at the Brewster’s angle to produce linearly polarized laser output.
p = parallel
s = senkrecht (german)
= perpendicular

19. 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

20. 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

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

22. Distributed-parameter model
Types of transmission lines RS

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

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

25. Telegraphist’s equations
Applying Kirchoff’s voltage law and
We’ll get
Divide both sides by z and take the limit as z goes to zero,
A similar expression can be found by applying Kirchoff’s current law at node a
and using for a capacitor and take the limit as z goes to zero, RS

26. Telegraphist’s time-harmonic wave equations
Time-harmonic waves on transmission lines
Take of eqn (1),
Substitute from eqn (2), we’ll get
After arranging we have
(1)
(2)
where

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

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

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

30. Low loss transmission line (2)
After the binomial series expansion, we’ll get
Therefore, we get RS

31. 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

32. Power transmission (lossless: Z0 = real)
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

33. Power transmission
For lossy case: W. RS

34. 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

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

36. 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

37. 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