1 Chapter 8 Maxwell s Equations

Transcription

1 Electromagnetic Waves ECEN 3410 Prof. Wagner Final Review Questions 1 Chapter 8 Maxwell s Equations 1. Describe the integral form of charge conservation within a volume V through a surface S, and give the mathematical form of the equation 2. Give the electromagnetic force experienced by a test charge q moving at a velocity v placed in an electric field E and magnetic field B 3. Explain why a capacitor with an applied AC current violates the DC form of Ampere s law through a judicious choice of the surface over which the current density J is integrated H d l = J(t) ds C and what term Maxwell added to this to generalize to the AC case. 4. Faraday s law in integral form states that E( r, t) d l = t C S S B( r, t) d S Show how the application of STokes theorem leads to the differential form, 5. Explain why there are two current terms J( r, t) = σ( r, t) E( r, t) + ρ( r, t) v( r, t) and what each term means and the units of each 6. What does the lack of Magnetic monopoles mean for the lines of magnetic flux? 7. Show why the divergence of the curl of any field is identically zero implies the conservation of charge from Maxwell s equations. 8. Give the constitutive relations in a homogeneous and isotropic dielectric media. 9. Give the complex form of Maxwell s equations for time harmonic fields. 10. Starting from the integral form of Faraday s law, show how to derive the boundary conditions for the electric field E at an abrupt planar boundary between two disimilar media. 11. Derive the boundary conditions for the electtic flux, D, and magnetic flux, B at the abrupt boundary between two electrically dissimilar media. 12. What are the boundary conditions on E and H if there are no surface currents? 1

2 13. What are the boundary conditions on D and B if there are no surface charges? 14. Is the gradient of a scalar potential field a vector or scalar? Give the form of the gradient in Cartesian coordinates. 15. Is the divergence of a vector field a scalar or vector field? Give the equation of the divergence in cartesian coordinates. 16. Is the curl of a vector field a scalar or vector field? Give the form of the curl in Cartesian coordinates. 17. Give the form of the Laplacian of a scalar field ψ in Cartesian coordinates. 18. For a time-harmonic monochromatic wave write Maxwell s equantions in complex form 19. Derive the Poynting theorem from Maxwell s equations. Explain what the Poynting vector represents. 20. Give the units of the electric field, E, and magnetic field, H, and discuss Poyntings vector and its units. What do Gauss laws tell you about the units of D and B. 21. Poyntings theorem implies a balance between the radiated power applied to a volume and which other power and energy terms integrated throught the volume? 22. Explain the relation between the real Poynting Vector and complex Poynting vector for the time-harmonic case. Give explicit expressions for the time averaged power flow from both. 23. Give the form of an EM plane wave electric and magnetic field propagating along z with linear E-field polarized along x of magnitude E 0 in a medium of impedance η and give the Poynting vector 2 Chapter 9 Plane Waves 1. Derive the vector Wave equation for the electric field from Maxwell s eqns in a homogeneous media.. Express the Laplacian in cartesian coordinates. 2. Derive the wave equation for magnetic field from Maxwell s eqns in a homogeneous, time-invariant, linear, source free medium. 3. Show that traveling wave solutions in the forwards and backwards directions are solutions to the 1-D scalar wave equation 4. Prove that the electric, E, and magnetic fields, H, of a uniform plane wave are perpendicular to each other in a lossless homogeneous media. 5. What are plane waves? What are the key properties of EM plane waves? Write down a general plane wave propagating in an arbitrary direction, with an arbitrary polarization and arbitrary time dependence. How does it get modified for a single frequency ω 2

3 6. For a plane wave propagating in z, show how to find the components of the magnetic field vector H given the components of the electric field E x and E y. Use these to express the form of the Poynting vector assuming the field is monochromatic. 7. For a time dependent plane wave polarized in x and propagating in z, E x (z, t) = E m cos[ωt βz +θ], find the expression for the complex time harmonic phasor notation for the electric and magnetic fields. Find the average Poynting vector. 8. For a wave propagating in the unit direction ˆn at frequency ω in a medium with velocity c/n find an expression for the electric field phasor and time domain representation. What is the constraint on the orientation of the electric field. 9. In a lossy media with conductivity σ what is the form of the solution for a monochromatic plane wave propagating along z? Use γ = jω ɛ j σ = α + jβ ω µ ɛ jσ/ω 10. For a lossy media with impedance η = find the form of the magnetic phasor field for a known electric field polarized along x and propagating in z. Use this to find the time average Poynting vector. 11. For a good dielectric at angular frequency ω find the loss tangent as the ratio of the imaginary part of the displacement current to the real part conductivity current 12. In a good conductor σ ωɛ find the approximate form of the complex propagation constant γ = jω ɛ j σ = α + jβ. What is its phase? ω 13. Ohmic losses in a good conductor with exponentially decaying fields have a disipation in a half space with boundary z = 0 P s = 0 J Edz. Use Poyntings theorem to express what this must equal in terms of the electric and magnetic fields for an x-polarized electric field 14. What is the skin depth and how does it relate to the complex propagation costant γ = α + jβ? How does the skin depth vary with frequency? 15. For a wave propagating along z what is the general state of polarization. Show how you can separate the amplitude and phase information from the polarization. SHow how to express a linear polarization at an arbitrary angle. 16. What is circular polarization and how would you create it? How do you represent circular polarization mathematically? How would you modify this to represent elliptical polarization with its major and minor axis aligned with the x and y axis? 17. Given a linear polarization with the electric field at 45 to the x (and y) axis show how to represent this polarization in the x-y laboratory frame. Now rotate the coordinate axis by 45 degrees to the x y frame and show how to representthis linear polarixation in this frame. 3

4 18. For a nonmagnetic material with relative dielecric constant ɛ r = 4 what is its phase velocity and impedance? If it is also magnetic with relative permeability µ r = 4 give its phase velocity and impedance. 19. Define the phase and group velocities. Sketch an example ω β diagram and graphically illustrate the relation between v p and v g. 20. What is disperion? Is the velocity of an EM pulse envelope always the same as the carrier? Always less than the carrier? Always greater than the carrier? 21. In a plasma with wavenumber β = ω 1 ω2 p find the phase velocity and group velocity c ω 2 for ω > ω p. 22. In a good conductor the wavenumber β = ωµσ/2 Given that the phase velocity is v p = 1, and the group velocity is v β/ω g = 1, find the relation between these two dβ/dω velocities. 3 Chapter 10 Reflection and Refraction 1. State all the boundary conditions that you know. Write compact expressions for them in terms of the unit normal to the boundary ˆn. 2. For the normal incidence refelection of a linear polarization from a PEC at frequency ω, find the time domain representation of the electric and magnetic field standing wave patterns. What are the spatial periods? Where are the nulls? Where and when are the maxima of E and H. 3. For normal incidence on a boundary between disimilar media (ɛ, µ, σ) what are the reflection and transmission coefficients? What is the form of the effective dielectric constant in terms of the dielectric constant, conductivity, and frequency? 4. Show how to decompose an incident wave plus relected wave with reflectivity Γ into a traveling wave plus standing wave. Derive the standing wave ratio as the peak field value to the minimum field value. 5. For a lossless media 1 with a normally-incident linearly-polarized plane wave reflecting off a possibly absorbing media 2, give expressions for the electric and magnetic fields in media 1 and media 2, and find the wave impedance in both media. 6. Describe and mathematically define the plane of incidence of a plane wave incident obliquely with wavevector k i onto a planar boundary with unit normal ˆn. 7. Derive Snell s law for a dielectric boundary between two media with index of refraction n 1 = ɛ r 1 and n 2 = ɛ r 2. Assume non-magnetic media. 8. What happens when a plane wave at an arbitrary angle enters an optically more dense media. Sketch and explain the relevant physics. 4

5 9. For oblique incidence of a plane wave on a planar boundary explain why the boundary condition can result in a sum of the electric field vectors E i + E r E t = g 1ˆn Hi + H r H t = g 2ˆn and under what conditions g 2 = For normal (s) polarized oblique incidence what is the reflection coefficient? Write the time-domain form for the total monochromatic electric field 11. For parallel (p) polarized oblique incidence what is the reflection coefficient? Write the time-domain form for the total monochromatic electric field 12. What is the condition that leads to Brewster s angle and what is the value of Brewster s angle for refraction from a medium of index n 1 = 1.0 into a medium of index n 2? 13. Explain the critical angle for Total Internal Reflection (TIR). 14. Show that the boundary condition for the magnetic flux ( B 1 B 2 ) ˆn = 0 at a planar dielectric boundary can be derived from the electric field boundary condition ( E 1 E 2 ) ˆn = 0 and Snell s law k i ˆn = k r ˆn = k t ˆn 15. Find the standing wave wavelength for the oblique reflection at an angle θ i of a plane wave polarized normal to the plane of incidence (s) from a planar PEC, and identify the planes where a second PEC parallel to the first could be inserted automatically satisfying the boundary condition in order to make a parallel plate waveguide. Find the wavevector along this waveguide and identify the phase velocity down the waveguide. 16. How would you design a coating to maximize the transmission of a monochromatic laser (say red at 670nm) into an abnormally high index glass whose index is n 1 = 1.82 = Chapter 11 Field Analysis of Transmission Lines 1. Derive the telegrapher s equation and voltage wave equation in a lossless coax line for a time-harmonic generator. What are the solutions and what do they mean? 2. Show that a TEM mode in a lossless transmission line where the Electric field is of the form E(x, y, z) = E(x, y, 0)e jβz can be solved for using the transverse component of the operator and the conductor and dielectric geometries, similarly as in the electrostatic case. 3. For a transmission line with lossless conductors (PEC) and transverse field solutions E t (x, y) and H t (x, y), find the Poynting vector. Express in terms of the E t field and identify what impedance is used. 5

6 4. Given transverse fields for a 2-conductor transmission line E t (x, y) and H t (x, y), find expressions for the voltage difference V 12 between the two conductors and current I 1 on the 1st conductor. What is the current on the 2nd conductor, I 2? 5. Characteristic impedance of T.L. is ratio of voltage V 12 to current I, which is charge per unit length Q times speed of light in homogeneous dielectric, v p. Since Capacitance per unit length C is ratio of charge to voltage, find the characteristic impedance, Z 0, in terms of C. 6. What type of wave propagates in a coaxial cable? Derive expressions for the electric field vector, magnetic field vactor, and Poynting vector. 7. Prove that in a coaxial line fed with a matched generator of harmonic voltage V, with current I flowing in the cable, the flux of the vector integrated across the transverse cross-section equals the power V I 8. Find the Capacitance p.u.l C of a coaxial line, and using the expression for a homogeneous line L C = ɛµ determine the Inductance p.u.l. L. 9. The electric and magnetic fields in a coax line of inner radia a and outer radius b are given by E(r, z) = ˆr V m e jβz and H(r, z) = ˆφ V r ln b m 2πrZ 0 e jβz. Since the ratio of the electric a and magnetic fields gives the Wave Impedance Z T EM = µ, find Z ɛ What is the Capacitance p.u.l. of a microstrip line of width w on a dielectric substrate ɛ of height h neglecting fringing fields? Actual microstrip has an inhomogeneous dielectric, but for this model neglecting fringing fields, can we use the homogeneous dielectric T.L. expression L C = ɛµ to find the characteristic impedance, Z 0, of the microstrip fields? Do so. 11. Consider a 2-wire transmission line of radius a and separation d in a dielectric ɛ r. Given a charge p.u.l. Q 1 on wire 1, what is the charge p.u.l. on wire 2? From Gauss law, what is the form of the electrostatic E-field surrounding a wire with charge p.u.l. Q, and using superposition, what does that tell us about the E-field for this 2-wire line? Find the voltage between the two wires, and use that to find the Capacitance p.u.l. C = Q /V for the 2-wire line, and use L C = ɛµ to find the characteristic impedance Z 0 = L /C. 12. Why is it not practically possible to make a coaxial cable with a 500-Ohm characteristic impedance? Can you make a 500-Ohm 2-wire line? 5 Chapter 12 Circuit Analysis of Transmission Lines 1. Consider a lumped element circuit model of a lossless transmission line with a series inductor L = L z and a shunt capacitor L = L z per unit cell. Derive the telegraphers coupled eqns from taking the limit z 0. 6

7 2. Derive the 2nd order wave eqn for the harmonic voltage from the pair of coupled telegrapher s eqns. Find the form of the solutions to the time harmonic form of the voltage wave eqn. 3. Small conductor and dielectric losses R ωl and G ωc in the p.u.l. impedance Z = R + jωl and in the p.u.l. admittance Ỹ = G + jωc allow approximations for the characteristic impedance Z 0 = Z /Y and complex propagation constant γ = Y Z. Find these approximate values. 4. For a tranmsission line of impedance Z 0 and a load Z L at z = 0 with an incident and reflected voltage wave express the boundary conditions for voltage and current at the load. Solve these eqns for the complex reflection coefficient Γ L 5. For a complex load Z L = R L + jx L find the complex reflection coefficient Γ L and express the real and imaginary parts. Find the magnitude and phase. 6. Find the average power transfer down a transmission line of impedance Z 0 in terms of the peak voltage V im and complex reflection coefficient Γ and find the power delivered to a load of complex impedance Z L. Find the load impedance that maximizes the power transfer to the load. 7. For the time harmonic case express the complex reflection coefficient at different positions along a transmission line, Γ (z), and find the impedance along the line Z (z). 8. Show that a quarter wave transformer inverts the line normalized impedance of the load. 9. Show how to impedance match a complex load using an additional transmission line segment and quarter wave transformer to match the real part. 10. Find the input impedance of two transmission lines connected in parallel, where they have characteristic impedances Z 0l, lengths l k, loads Z Lk 11. Derive the input impedance of a shorted transmission line of length l and of characteristic impedance Z 0. Explain how to make an inductor and capacitor out of the shorted stub. 12. Derive the input impedance of an open transmission line of length l and of characteristic impedance Z 0. Explain how to make an inductor and capacitor out of the open stub. 13. Explain how to make a forward only wave in a transmission line. 14. What is the magnitude of the VSWR for which one half of the power of the incident wave is transmitted to the load. 15. For a generator with impedannce R g and a open circuited step function voltage E find the initial voltage launched down the transmission line of characteristic impedance Z 0. 7

8 16. Use the Smith chart to determine the terminating impedance of a 50 Ohm line when the complex reflection coefficient is (a) 0.8 (b).2e jπ/4 (c).5e jπ/3 17. Label the following on the Smith chart (a) Short, Open, Matched loads (b) All purely real impedances and all purely imaginary impedances (c) All impedances that have a real part equal to the characeteristic impedance (d) All impedances that have an imaginary part equal to the characeteristic impedance (e) all points that have a reflection coefficient magnitude of.5 (f) All points that have a VSWR of Find the input impedance at angular radian frequency ω, of a transmission line of characteristic impedance Z 01 = 100Ω and length l 1 =.2λ 1, that feeds a second line of characteristic impedance Z 02 = 50Ω and length l 2 =.1λ 2, with a load of impedance Z L = Z 02 (2 + 1j). 19. A fixed complex impedance Z L is to be connected to a lossline T.L. of characteristic impedance Z 0. Show how to eliminate the reflected wave by connecting an identical shorted line stub of length l at a distance d 1 before the load. How do you choose the distance d 1 and length of the shorted line l. 20. Calculate and plot the magnitude and phase of the reflection coefficient of in open circuited Z 0 = 50Ω transmision line that is λ/4 long at 2GHz as a function of frequency between 1-3 GHz. 21. Show that the sum of an arbitrary shape forward wave propagating at a velocity v given by f(t z/v) is a solution of the wave equation for a transmission line. 2 v(z, t) z 2 1 v 2 2 v(z, t) t 2 = For a T.L. with characteristic impedance Z 0 = 50Ω and a load of 100Ω with a matched generator applying a step function open circuit voltage E at time t =, Eh(t), find the transient response of the transmission line at the load and at the generator. 23. A very small lumped capacitor is inserted into a 2-wire transmission line between the wires. Assume the line is infinite to the right and find the reflection coefficient of a wave incident from the left. 8

9 6 Chapter 13 Waveguides 1. For a infinite planar parallel plate waveguide made by two PECs of separation a what do the boundary conditions enforce upon the transverse wavevector β x? What does this tell us about the wavevector for propagation down the guide β z? 2. At a frequency ω what are the angles of propagation of the T E 0m modes with respect to the propagation axis z of the waveguide? As the frequency is varied below what frequency do these modes experience cutoff? 3. What is the meaning of the modal indices m and n of the field components of a rectangular waveguide for T E mn and T M mn modes 4. What is the assumed shape of the modes of a rectangular waveguide in x, y, z and t at a frequency ω? Plug a cartesian component into the vector wave equation to find a differential equation for the transverse modal profile. Show how to solve this equation for the modal shape. 5. What is a TM mode and what is a TE mode? Is there a TEM mode of a rectangular waveguide? 6. What is the form of the E z component of a TM mode? Explain why the boundary condition E tang = 0 gaurantees this is the correct form for this component. 7. For a TE wave propagating along the z direction provide a derivation of the E z and H z components of the electric and magnetic fields. 8. What is the cutoff frequency of the mnth mode? Is it the same for the TE and TM modes? What is the phase velocity of the mnth mode? Is it the same for the TE and TM modes? 9. Plot the electric and magnetic field distributions in a transverse plane for the T E 10 and T M 11 modes? 10. Derive the relation between the external wavelength, cutoff wavelength, and guided wavelength of the TE 10 mode of a rectangular guide. 11. Sketch the ω β diagram for the first few modes of a 3cm 6cm waveguide and label the salient cutoff frequencies. Show how to find the phase and group velocities, v p and v g, from these diagrams. Why is the T E 10 dominant mode important? 12. X-band covers the GHz band Losses near the dominant T E 10 mode cutoff are avoided by operating the waveguide only down to 1.25fT c E 01. Avoiding the close to propagating regime of the evanescent modes that cut on at a frequency ft c E 20 or ft c E 01 suggests a maximum frequency of.95 times the cuton. Design the waveguide dimensionsi, a b, to meet these conditions. 9

10 13. When calculating the power flow down a waveguide using the integral of the Poynting vector E H across the transverse cross section of the waveguide there will be contributions from each mode. What is the contribution from the modes above cutoff and why? What is the contribution due to the z components of the TE and TM modes and why? When multiple modes are simultaneously present (either propagating or evanescent), what are the contributions due to the cross terms between different modes and why? 14. Which component of what EM field leads to surface charge on the waveguide walls and what is the equation for surface charge density on a PEC ρ s. What is the defining equation for the surface current J s? What is relation between these quantities and why? 15. How would you couple from a coaxial cable to a TM mode of an a b of a rectangular waveguide? How can you avoid coupling to a propagating T E mode? 16. Given the form of the transverse fields of a rectangular waveguide, how can you define and show mode orthogonality? If two modes were not orthogonal what would happen to mode amplitude with propagation and what assumption intrinsic to the modal solutions would that violate? 17. What is the quality factor of a resonator and how is it related to the bandwidth of the resonance? 7 Chapter 14 Antennas 1. For a Hertzian short dipole along z, what components of the magnetic vector potential are produced? Solve for the form of the magnetic vector potential of a dipole driven by current I and length l Given that H = 1 µ A what components of the magnetic field does this yield? What does Maxwells eqns tell us about the Electric field resulting from this magnetic field? 2. What does the rotational symmetry of a Hertzian short dipole, or any straight wire antenna tell us about the nature of the field solutions? If we examine the field solutions of an optical dipole with wavelength of 1 micron, lenth 10 nm at a distance of 1mm, and an RF antenna at a wavelength of 10cm, length of 1mm at a distance of 10m what will we conclude about these field solutions? 3. In the far zone the Electric and Magnetic fields are purely transverse. In the spherical coordinate system surrounding the antenna find the complex Poynting vector in terms of these fields. 4. In the solution for the magnetic vector potential due to an integration across each differential current leads to a spherical wave source e jβr with distance R to the observation position P (r, θ, φ). Explain how we can approximate R in the denominator, R and in the exponential phase factor. 10

11 5. For a straight wire dipole antenna of length l = λ/2, λ, and 1.5λ find the antenna current distrbution. Sketch the polar antenna patterns for each of these cases. 6. The radiation efficiency, η rad expresses the fraction of the power loss due to radiation away from the antenna to total power loss including ohmic losses. For a short dipole of length l and wire radius a, the radiation resistance goes as R rad = 20(βl) 2 [Ω]. Find the power loss due to surface resistance R s = πµf σ and the radiation efficiency. 7. Explain the relation between the far field electric field, E(r, θ, φ), the antenna radiation pattern, F(θ, φ), the radiation pattern, f(θ, φ), the power pattern, p(θ, φ), the power radiated per unit solid angle, U(θ, φ), the antenna directivity, D(θ, φ), the antenna gain, G(θ, φ), and the effective area, A eff (θ, φ). 8. Explain the relation between the antenna radiation pattern, F(θ, φ), and the field pattern of the receiving antenna. 9. For a monopole antenna of length l above a ground plane, describe the radiation pattern at frequencies ω = l [π, 2π, 3π], and relate them to dipole patterns. c 10. Explain the duality between a short electric dipole (Hertzian ) antenna and a small magnetic dipole loop antenna. 11. What is the impedance of a grounded monopole if you know that the impedance of a half-wave dipole in free space is 73 Ohms? 12. What is the directivity of the 300m diameter Arecebio telescope at 3GHz? 11