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

2 RS 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

3 RS 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. Polarizations of UPW obliquely incident on the boundary (1)

4 RS 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. Polarizations of UPW obliquely incident on the boundary (2)

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

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

7 RS 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 Snell’s laws of reflection and refraction (1)

8 RS 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 2 nd medium where they decay exponentially with z. However, the transmitted electric and magnetic fields are 90 o out of phase, so no power is trans- mitted. Snell’s laws of reflection and refraction (2)

9 RS 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 Reflection and transmission coefficients for TE polarization (1)

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

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

12 RS 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

13 RS 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.

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

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

16 RS Total transmission for TM polarization For TM polarization, there exists an incidence angle at which all of the wave is transmitted into the 2 nd 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,

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

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 RS 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 n 2 = a)TM polarization b)TE polarization

20 RS 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.

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

22 RS Distributed-parameter model Types of transmission lines

23 RS 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

24 RS Telegraphist’s equations General transmission lines equations:

25 RS Telegraphist’s equations

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

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

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

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

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

31 RS Characteristic impedance or For lossless line, Characteristic impedance Z 0 is defined as the the ratio of the traveling voltage wave amplitude to the traveling current wave amplitude.

32 RS 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 Power transmission ( lossless: Z 0 = real ) W.

33 RS Power transmission For lossy case: W.

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

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

36 RS 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?

37 RS 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?