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

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

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

TE polarization x z i We can write and RS

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

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

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

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

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

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

Fields for TM polarization Incident fields Reflected fields Transmitted fields RS

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

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

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

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

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

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

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

Distributed-parameter model Types of transmission lines 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 RS

Telegraphist’s equations General transmission lines equations: RS

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

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

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

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

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

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

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

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

Power transmission For lossy case: W. RS

Power ratios on the decibel scale (1) A convenient way to measure power ratios Power gain (dB) Power loss (dB) 1 Np = 8.686 dB dB dB RS

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

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

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