Transmission Line (TL): Electrical conductors carrying an electrical signal from one place to another Practical types Coaxial cable Planar lines: Microstrip, Stripline, Coplanar waveguide Balanced lines: Twisted pair, Star quad, Twin-lead, Lecher lines Single-wire line Waveguide, Dielectric waveguide, Optical fibre _ Ch.12 - .hwp - 1 - Chapter 12 Plane Wave Reflection and Dispersion In Ch. 11, How to mathematically represent uniform plane waves as functions of orientation frequency , medium properties, and How to calculate the wave velocity, attenuation, and power In Ch. 12, Wave reflection and transmission at planar boundaries between different media ☞ For any orientation between the wave and boundary ☞ For multiple boundaries Study on the practical case of waves that carry power over a finite band of ’s in a modulated carrier _ Ch.12 - .hwp - 2 - In dispersive media, in which some parameter that affects propagation (e.g. ) varies with Effect of a dispersive medium on a signal : ☑ The signal envelope will change its shape as it propagates. As a result, detection and faithful representation of the original signal at the receiving end become problematic. Consequently, dispersion and attenuation must both be evaluated when establishing maximum allowable transmission distances. _ Ch.12 - .hwp - 3 - 12.1 Reflection of Uniform Plane Waves at Normal Incidence Incident wave 𝓔 cos In phasor form, , = real (1) Fig. 12.1 (2) & = complex for ′′ (or ) ≠ 0 Normal incidence (11-37) ; ′ ′′ ′ ′′ (11-42) ; ′ ; Transmitted wave : _ Ch.12 - .hwp (3) (4) - 4 - ′ ′′ ′ BC’s at = 0 : ; ′ ′′ 𝓔 cos Re Re 1) ) 2) ( (∴) ~ unable to satisfy the BC’s with only an incident and a transmitted wave. [Emphasis!] We require the reflected wave. Reflected wave : = complex ; (5) (6) for the Poynting vector : × must be in the direction. _ Ch.12 - .hwp - 5 - At = 0, ; = Total or or ; ( ) (7) ( ) (8) (8) → (7) : .hwp or _ Ch.12 - = Total continuous - 6 - Reflection coefficient, : Ratio of the amplitudes of the reflected and incident electric fields or = complex ⇒ (9) = complex, and so a reflective phase shift, Transmission coefficient, : Relative amplitude of the transmitted (9) → (7) : _ Ch.12 - .hwp (7) (10) - 7 - Region 1 : Perfect dielectric Region 2 : Perfect conductor ( → ∞ ) Intrinsic impedance ⇒ (11-48) ⇒ ′ ′′ ′ With ′′ , (10) ′ → ′ → No time-varying fields can exist in the perfect conductor. The skin depth is zero. (9) ; _ Ch.12 - .hwp & ⇒ & - 8 - The incident and reflected fields are of equal amplitude, and so all the incident energy is reflected by the perfect conductor. At the boundary (or at the moment of reflection), the reflected field is shifted in phase by 180o relative to the incident field. in region 1, Total In the perfect dielectric, sin Real instantaneous form : sin sin (11) (12) ≡ Re Re cos sin Re sin cos sin ⇒ Standing wave, obtained by combining two waves of equal amplitude traveling in opposite directions _ Ch.12 - .hwp - 9 - cos Incident wave, (13) ☑ : traveling in the direction at sin sin Standing wave : (12) Whenever , at all positions. Spatial nulls in the standing wave pattern occur for all times wherever , ± ± … . & : cos cos [ or ] (14) cos sin sin Standing wave : Max amplitude at the positions where 90o out of time phase with everywhere Avg. power = 0 in the forward and backward directions _ Ch.12 - .hwp - 10 - ⟨⟩ Re × [ W/m2 ] ∗ (11-77) sin cos cos sin sin sin Region 1 & 2 : Perfect dielectric Fig. 12.2 & : both real positive quantities & Calculate and find in terms of the incident field . and In region 2, _ Ch.12 - ⇒ .hwp ⇒ Find and . ⇒ (9) (10) - 11 - General rule on the transfer of power through reflection and transmission Assumption : complex impedances Incident power density, ∗ ∗ Re Re Re ⟨ ⟩ ∗ ∗ Reflected power density, ∗ ∗ ∗ Re Re Re ⟨ ⟩ ∗ ∗ Relation between the reflected and incident power: ⟨⟩ ⟨⟩ (15) Transmitted power density: ∗ ∗ ∗ Re Re Re ⟨ ⟩ ∗ ∗ Relation between the incident and transmitted power densities ∗ Re ∗ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ∗ ∗ Re (16) ⟨ ⟩ ⟨ ⟩ _ Ch.12 - .hwp (17) - 12 - 12.2 Standing Wave Ratio Total electric field phasor in region 1, (18) ⇐ = real > 0 for a lossless medium ; = complex [cf.] If region 2 is a perfect conductor, = 0, and so . If = real < , . ; (18) _ Ch.12 - ⇒ .hwp If = real > , . ′ (19) - 13 - Max and min field amplitudes in (19) ~ -dependent and subject to measurement (19) ☑ Their ratio ~ Standing wave ratio ( ) [cf.] Voltage amplitudes in TL’s [Sec. 10.10], max ☑ Max when each term [ ] in (19) has the same phase angle; so, for positive (20) and real, (19) max ← ( ± ± … ) (22) (21) max An electric field max is located at the boundary plane ( ) if ( when = real > 0). : This occurs for real and when . With , maxima at max . _ Ch.12 - .hwp - 14 - For the perfect conductor , and maxima at max , … ⇒ (19) min ← or max , … . Minima where the phase angles of in (19) differ by 180o : min (23) ( ± ± … ) max (24) (25) Minima ~ separated by multiples of (as maxima), and for the perfect conductor the first min. occurs when , or at the conducting surface In general, min. at whenever ; _ Ch.12 - .hwp if and both are real. - 15 - Find the total field in region 1: Max amplitude cos traveling wave cos cos standing wave observed in region 1 : (26) Min amplitude : The standing wave achieves a null, leaving only the traveling wave amplitude of . max min ⇒ (27) ⇒ ≥ ∞ Reflected amplitude = Incident amplitude, so all the incident energy is reflected. Planes separated by multiples of : at all times. Midway between these planes, has a max amplitude twice that of the incident wave. _ Ch.12 - .hwp - 16 - : no energy is reflected. : the max and min amplitudes are equal. If one-half the incident power is reflected, , (27) , and . _ Ch.12 - .hwp - 17 - 12.3 Wave Reflection From Multiple Interfaces Steady-state situation: (1) An overall fraction of the incident wave is reflected; (2) An overall fraction of the incident wave is transmitted; (3) A net backward wave exists in #2; (4) A net forward wave exists in #2. Five waves to consider Fig. 12.4 _ Ch.12 - .hwp - 18 - Consider the two waves in region 2. -polarized electric fields: ; (28a) and = complex -polarized magnetic field (29) ; _ Ch.12 - .hwp (28b) (31a) ; (30) (31b) - 19 - Wave impedance, : -dependent ratio of the total electric field to the total magnetic field cos sin cos sin cos sin cos sin cos sin cos sin BC’s : and ~ continuous across the boundary tangential @ : _ Ch.12 - (32) .hwp (33a) (34a) @ (33b) (34b) input impedance, - 20 - (34) ⇒ (35) Input impedance at cos sin cos sin (32) cos sin cos sin (35) & (36) (36) ⇒ Net reflected wave amplitude and phase from two parallel interfaces between lossless media Total transmission when , or when . The input impedance is matched to that of the incident medium. _ Ch.12 - .hwp - 21 - Half-wave matching : Suppose that . or (37) ( is an integer) [ ] → ⇒ Rendering the 2nd region immaterial Equivalently, a single-interface problem involving and cos sin cos sin A matched input impedance, and No net reflected wave [Applications] Radomes, ANT housings on airplanes which form a part of the fuselage _ Ch.12 - .hwp - 22 - The half-wave matching condition no longer applies as we deviate from the wavelength that satisfies it. When this is done, the device reflectivity increases (with increased wavelength deviation), so it ultimately acts as a BPF. Refractive index (or just index), : (38) At optical frequencies (on the order of 1014 Hz) Since ~ complex in lossy media, ~ complex Under lossless conditions, ′′ Phase velocity (39) ; (41) ; and Wavelength (40) (42) = wavelength in free space _ Ch.12 - .hwp - 23 - In optics, Fabry-Perot interferometer, A single block of glass or other transparent material of index , whose thickness, , is set to transmit wavelengths which satisfy l As a narrow-band filter (transmitting a desired wavelength and a narrow spectrum around this wavelength) if the spectrum to be filtered is narrower than the free spectral range. Quart-wave matching : Suppose , or an odd multiple of . _ Ch.12 - .hwp ( …) cos sin cos sin - 24 - (44) (45) For total transmission, , (46) Antireflective coatings for optical devices _ Ch.12 - .hwp - 25 - Impedance transformation Effective impedance at the first interface, , cos sin ; cos sin cos sin cos sin Power transmitted into #4 = Motivation for using multiple layers to reduce reflection : Antireflection coat a camera lens With a large number of layers fabricated in this way, the situation begins to approach (but never reaches) the ideal case. _ Ch.12 - .hwp Fig. 12.5 - 26 - 12.4 Plane Wave Propagation in General Directions How to mathematically describe Fig. 12.6 uniform plane waves that propagate in any direction Oblique incidence : In a lossless medium, = Direction of Propagation direction = Direction of the Poynting vector = phase shift per unit distance along that direction Phase, _ Ch.12 - .hwp ; ⋅ ⋅ (49) - 27 - In our 2-D case, position vector, , ⇒ ⋅ Angle of propagation from tan the axis, Wavelength, (50) → Phase velocity, → and . _ Ch.12 - .hwp - 28 - 12.5 Plane Wave Reflection at Oblique Incidence Angles The incident wave propagates at some angle to the surface. 1) How to determine the relation between incident, reflected, and transmitted angles 2) How to derive reflection and transmission coefficients that are functions of the incident angle and wave polarization Fig. 12.7 _ Ch.12 - .hwp - 29 - Wavevector Incident wave Angle : Reflected Transmitted wave wave ′ and a line that is angle between normal to the surface (the axis) ′ Lossless dielectrics ( ) & Nonmagnetic materials ( ) : Intrinsic impedances : ) [Emphasis!] Dielectric constant ( ) or Refractive indices ( _ Ch.12 - .hwp - 30 - Plane of incidence : and the plane spanned by normal to the surface // Parallel polarization or -polarized ( incidence plane): : polarized in the plane of the page : perpendicular to the page and pointing outward ☑ parallel (or transverse) to the interface Transverse magnetic (TM) polarization _ Ch.12 - .hwp - 31 - Perpendicular polarization, or -polarized ~ in the plane of incidence ~ perpendicular to the plane ~ parallel to the interface ☑ Transverse electric (TE) polarization The reflection and transmission coefficients will differ for the two polarization types, but reflection and transmission angles will not depend on polarization. Any other field direction can be constructed as some combination of and waves. -polarized : senkrecht, German for perpendicular -polarized : parallel, German for parallel _ Ch.12 - .hwp - 32 - ⋅ (51) ⋅ (52) ⋅ (53) cos sin (54) cos′ sin′ (55) cos sin (56) (57) : _ Ch.12 - .hwp - 33 - BC_[1] : continuous tangential electric field ( ) (at ) cos sin ⋅ cos (58) cos′ sin′ ⋅ cos ′ (59) ⋅ sin cos cos sin cos sin′ cos ′ (60) sin cos (61) , , and ~ all constants (independent of ) (61) must hold for all values of (everywhere on the interface). All the phase terms in (61) are equal. sin sin′ sin _ Ch.12 - .hwp - 34 - ′ ⇒ sin sin′ sin sin sin sin sin (62) (63) Snell’s law of refraction ! for nonmagnetic dielectrics ( ) BC_[2] : at tangential continuity of Magnetic field vectors for the -polarized wave : all negative -directed cos sin cos ′ (64) cos cos cos (65) sin′ sin cos ′ ; sin sin′ sin & cos & _ Ch.12 - .hwp ⇒ cos cos cos (66) - 35 - Effective impedances, valid for -polarization : ≡ cos → → (64) → (67) ≡ cos (69) cos cos → cos cos cos cos cos cos _ Ch.12 - (65) (70) cos cos cos cos → cos cos cos cos → cos cos → → cos cos cos (68) .hwp - 36 - Electric field vectors for the -polarized wave : all positive -directed Continuous tangential electric field : sin cos sin′ cos ′ cos sin ′ ; sin sin′ sin ⇒ cos cos ′ cos & ; cos & ⇒ cos cos cos cos cos → → sec sec → → → _ Ch.12 - .hwp - 37 - → (71) (72) effective impedances for -polarization ≡ sec (73) ; ≡ sec (74) When the second medium is a perfect conductor, . , and : Total reflection occurs, regardless of the incident angle or polarization. _ Ch.12 - .hwp - 38 - 12.6 Total Reflection and Total Transmission of Obliquely Incident Waves Necessary condition for total power reflection : ⇒ possibly, For the incident medium, & ~ real and positive ′ For the second medium, & ~ involving cos and cos cos sin sin (75) sin sin (63) Snell’s law For sin , cos , and hence and , become imaginary. For parallel polarization, conditions of imaginary , (69) ⇒ ; : _ Ch.12 - .hwp Total Power Reflection, - 39 - whenever = imaginary or (⇒ sin ) Condition for total reflection sin ≥ (76) Critical angle of total reflection, sin (77) ≥ (78) Total reflection condition : The wave must be incident from a medium of higher refractive index than that of the medium beyond the boundary. ~ Total Internal Reflection Beam-steering prisms; optical Fig. 12.9 waveguides _ Ch.12 - .hwp - 40 - Total transmission ( ) For -polarization: sin For -polarization: If , sin sin sin (79) ⇐ sin sin sin _ Ch.12 - .hwp & (69) → sin sin sin Brewster angle or Polarization angle sin ⇒ (∴) no value of (71) sin If , , or sec sec → → sin sin sin → → sin - 41 - Polarization angle (Brewster angle) : If light having both - and -polarization components is incident at , the component will be totally transmitted, leaving the partially reflected light entirely -polarized. At angles that are slightly off , the reflected light is still predominantly -polarized. Polaroid sunglasses to reduce glare Blocking the transmission of horizontally polarized light Passing vertically polarized light ※ Scottish physicist Sir David Brewster _ Ch.12 - .hwp - 42 - Summary: Fig. 12.10 For , & → ±1 at At , → imaginary (and are not from (69) shown) but nevertheless for because all curves cross the axis. No in the functions because for when . from (71) (69) & (71) ≡ cos (67) & (68) ( ) ; ≡ sec _ Ch.12 - .hwp (73) & (74) - 43 - 12.7 Wave Propagation in Dispersive Media Complex permittivity of the medium ~ Upon frequency Oscillating bound charges in a material ~ Harmonic oscillators that have resonant frequencies associated with them. When the frequency of an incoming EM wave is at or near a bound charge resonance, the wave will induce strong oscillations. ⇒ Effect of depleting energy from the wave in its original form. ⇒ The wave thus experiences absorption, and it does so to a greater extent than it would at a frequency that is detuned from resonance. ⇒ Real part of the dielectric constant ~ different at frequencies near resonance than at frequencies far from resonance In short, resonance effects → ′ and ′′ varying continuously with frequency ⇒ Complicated frequency dependence in and _ Ch.12 - .hwp ′′ ′ Re ′ (11-44) ′′ ′ Im ′ (11-45) - 44 - Consider the effect of a frequency-varying dielectric constant (or refractive index) on a wave as it propagates in an otherwise lossless medium. Significant refractive index variation can occur at frequencies far away from resonance, where absorptive losses are negligible. [e.g.] Separation of white light into its component colors by a glass prism Frequency-dependent refractive index → Different angles of refraction for the different colors—hence the separation Color separation effect produced by the prism ~ Angular Dispersion or Chromatic angular dispersion. Dispersion ⇒ Separation of distinguishable components of a wave In the prism, the components are the various colors that have been spatially separated.: Spectral power, dispersed by the prism Power detector with a very narrow aperture It measure the power in a “spectral packet,” or a very narrow slice of the total power spectrum. _ Ch.12 - .hwp - 45 - Lossless nonmagnetic medium where varies with frequency. Phase constant of a uniform plane wave in this medium: Fig. 12.11 (80) = a monotonically increasing function of frequency _ Ch.12 - .hwp - 46 - Suppose two waves at two frequencies, and , which are co-propagating in the material and whose amplitudes are equal. The electric fields of the two waves are linearly polarized in the same direction (along , for example), while both waves propagate in the forward direction. The waves will thus interfere with each other, producing a resultant wave whose field function can be found simply by fields of the two waves. adding the _ Ch.12 - .hwp Fig. 12.12 - 47 - We must use the full complex forms (with frequency dependence retained) as opposed to the phasor forms, since the waves are at different frequencies.: cos and (81) Real instantaneous form of (81) : 𝓔 Re cos cos If ≪ , (82) (82) as a carrier wave at that is sinusoidally modulated at frequency Fig. 12.13 _ Ch.12 - .hwp - 48 - For the phase velocities of the carrier wave and the modulation envelope, (carrier velocity) (envelope velocity) (83) (84) Carrier phase velocity = Slope of the straight line that joins the origin to the point on the curve whose coordinates are and Envelope velocity = Quantity that approximates the slope of the curve at the location of an operation point specified by ( , ) Group velocity function for the material, : lim → (85) At a specified frequency , it represents the velocity of a group of frequencies within a spectral packet of vanishingly small width, centered at frequency . _ Ch.12 - .hwp - 49 - Each frequency component (or packet) is associated with a group velocity at which the energy in that packet propagates. Since the slope of the curve changes with frequency, group velocity will obviously be a function of frequency. The group velocity dispersion of the medium is, to the first order, the rate at which the slope of the curve changes with frequency. _ Ch.12 - .hwp - 50 - 12.8 Pulse Broadening in Dispersive Media To see how a dispersive medium affects a modulated wave, let us consider the propagation of an EM pulse. Pulses are used in digital signals.: The effect of the dispersive medium on a pulse is to broaden it in time. To see how this happens, we consider the pulse spectrum, found through the Fourier transform of the pulse in time domain. Suppose the pulse shape in time is Gaussian, and electric field given at exp exp (86) position 0 : ~ constant; ~ carrier frequency; the pulse envelope ~ characteristic half-width of exp exp exp At , the pulse intensity, or magnitude of the Poynting vector, falls to of its max value (note that intensity ∝ ). _ Ch.12 - .hwp - 51 - Frequency spectrum of the pulse ⇐ Fourier exp (87) transform of (86) ☑ Frequency displacement from at which the spectral intensity (proportional to ) falls to of its max ~ . exp ; exp ∞ cos exp ∞ ∞ ≡ (Real( ) > 0) (7.4.6) ∞ ∞ ∞ [Abramowitz, M. and cos sin Stegun, I. A. (Eds.). ∞ ∞ ∞ Handbook of cos × cos Mathematical Functions ∞ with Formulas, Graphs, × exp exp and Mathematical Tables, 9th printing. exp New York: Dover, p. 302, eq. 7.4.6, 1972.] _ Ch.12 - .hwp - 52 - ∙ Gaussian intensity spectrum of the pulse, centered at , where and are the spectral intensity positions ∙ curve for the medium ∙ 3 lines tangent to the curve at , , and : slopes of the lines = group velocities as , , and Figure 12.14 Pulse spreading in time ~ resulting from the differences in propagation times of the spectral energy packets that make up the pulse spectrum Pulse spectral energy ~ highest at the center frequency, _ Ch.12 - .hwp - 53 - Difference in arrival times (group delays) between and after propagating through a distance (88) of the medium: Medium ~ Temporal prism Instead of spreading out the spectral energy packets spatially, it is spreading them out in time. ⇨ A new temporal pulse envelope: Its width is based fundamentally on the spread of propagation delays of the different spectral components. (88): ☑ If we assume that the curve is smooth and has fairly uniform curvature, using a Taylor series expansion about the carrier frequency, : _ Ch.12 - .hwp - 54 - ≐ (89) (90) (91) If the curve were a straight line, then the first two terms in (89) would precisely describe . Third term in (89), involving ~ describing the curvature and ultimately the dispersion. , , and ~ constants (92) → (88) : (91) : (92) (88) (93) dispersion parameter ☑ [ time2/distance ] = [ time/bandwidth/distance = time×time/distance ] _ Ch.12 - .hwp - 55 - In optical fibers the units most commonly used are psec2/km. can be determined when we know how varies with frequency, or it can be measured. If initial pulse width << , then the broadened pulse width at location will be simply . If initial pulse width ~ , then the pulse width at can be found through the convolution of the initial Gaussian pulse envelope of width with a Gaussian envelope whose width is . Thus, the pulse width at location : _ Ch.12 - .hwp ′ (94) - 56 - By-product of pulse broadening through chromatic dispersion: The broadened pulse is chirped. The instantaneous frequency of the pulse varies monotonically (either increases or decreases) with time over the pulse envelope. The spectral components at different frequencies are spread out in time as they propagate at different group velocities. Group delay, ~ a function of frequency: ~ a linear function of frequency and higher (92) (95) frequencies will arrive at later times if > 0. Positive chirp if the lower frequencies lead the higher frequencies in time [requiring a positive in (95)] Negative chirp if the higher frequencies lead in time (negative ) _ Ch.12 - .hwp - 57 - Pulse bandwidth, = From the Fourier transform of the pulse envelope (86); (87) = constant, and so the only time variation Broadening effect; Chirping phenomenon Figure 12.15 arose from the carrier wave and the Gaussian envelope. Such a pulse, whose frequency spectrum is obtained only from the pulse envelope ~ Transform-limited _ Ch.12 - .hwp - 58 - In general, additional frequency bandwidth may be present since may vary with time for one reason or another (such as phase noise that could be present on the carrier). In these cases, pulse broadening : (96) = net spectral bandwidth arising from all sources Transform-limited pulses are preferred in order to minimize broadening because these will have the smallest spectral width for a given pulse width. _ Ch.12 - .hwp - 59 -
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