Electromagnetic waves Unifrom Plane Slides, Lecture notes of Classical Mechanics

Download Electromagnetic waves Unifrom Plane Slides and more Lecture notes Classical Mechanics in PDF only on Docsity! Department of Electrical and Computer Engineering ECSE 352 Electromagnetic Waves Module 2: The uniform plane wave 2.1 EM Wave propagation in free space Reference: H&B, Section 11.1 1-2 Overview In this class we will investigate the way that electromagnetic waves arise from Maxwell's equations. All waves require a source and we will see that when electric and magnetic dipoles oscillate at high frequencies, waves are radiated. We will then follow the derivation of the source-free wave equations from Maxwell's equations. In many cases of practical interest, the frequency of a wave-source remains constant and this leads to the concept of harmonic waves that have a single well-defined frequency. This assumption greatly simplifies the mathematical treatment of EM waves and results in a time-independent formulation of the wave equations called the vector Helmholtz equation. We will investigate the concept of phasor representation of EM waves and will see that simple cosine functions are solutions of this equation. Finally we will examine the relationship between the electric and the magnetic fields. 1-5 Rem: Transmission line propagation Lossless transmission line: Equivalent lumped element model: Charging speed goes as dIV L dt  dVI C dt  H-field E-field 1v LC  V0 RL + - L C L C L L C V+=V0- + V0 RL + - z I+ 1-6 James Clerk Maxwell (1831-1879) • Professor of Physics at Aberdeen and subsequently Cambridge University. • Unified theories of electricity and magnetism • Predicted the existence of EM waves 20 years before they were observed (Hertz) Maxwell’s Equations Constitutive relations D = E B = H 0 t t                  DH J BE D H 1-10 Free-space WE: Interpretation Change of E in time involves a change of H in space E.g. Dipole with E=Exax t y and/or z Ex Hz Hy 0 0 0 0 t t                 EH HE E H ˆ ˆ ˆx y z x y z x y z H H H         a a a H 1-11 Plane wave 1-12 0 0 0 0 t t                 EH HE E H Plane wave Assume that a plane wave is a solution to Maxwell’s equations Example: A constant H-field H=Hyay that fills the x-y plane 0 t     EH In a plane wave format, E and H must be orthogonalTEM ˆ ˆ ˆx y z x y z x y z H H H         a a a H 0 y x x x H E z t        a a 1-15 Sinusoidal solutions An example of solution for the e-field is the cos function Where the wavenumber and free-space wavelength       ' 0 1 0 2 ' 0 0 1 0 0 2 , cos cos cos cos x x x p p x x z zE z t E t E t v v E t k z E t k z                                              0 0k c rad m 02 k  1-16 Complex solutions Keeping in mind that there are 2 solutions, backward and forward, we generally work with only the forward solution to simplify the analysis. As well, the complex notation is used              01 0 0 0 0 1 1 02 1 0 02 1 2 , cos Re Re x x j t k zj x j t k z j t k z x x j t j t xs xs E z t E t k z E e e cc E e cc E e E e cc E e                           '0 0 1 0 0 2, cos cosx x xE z t E t k z E t k z         0 1 0 0 0 jk z xs x j x x E E e E E e          E-field amplitude Time-independent form 1-17 Maxwell’s equations in cpx form The time-independent form is most often used. For example with previous equation becomes and more generally valid for any E, H fields (even beyond planar waves) 0 y x x x H E z t        a a 0 1 0 0 0 j t x xs jk z xs x j x x E E e E E e E E e             0 0 0 0 t t                      EH HE E H 0 ys x xs x H j E z      a a 0 0 0 0 s s s s s s j j             H E E H E H Time-independent Maxwell’s equations 1-20 Transverse ElectroMagnetic (TEM) wave E and H are always orthogonal following the RH rule         0 0 0 0 0 , cos , cos x y z t E t k z Ez t t k z            E a H a E y z H x Right hand rule ExH 1st  E 2nd  H Thumb  Direction of propagation 1-21 Before next class Problems D11.1, D11.2 Examples 11.1, 11.2 Problems 11.1, 11.3, 11.5 Read section 11.2 Assignment #4 has been released, due Monday next week before class (11:35) Department of Electrical and Computer Engineering ECSE 352 Electromagnetic Waves Module 2: The uniform plane wave 2.2 EM wave propagation in dielectrics Reference: H&B, Section 11.2 1-25 EM plane waves in free-space 2 2 0 0k  E E Helmholtz equation (Time independent wave equation) Free- space harmonic wave k0=/c0 Plane wave with e.g. propagation along z-axis E-field in x-axis What about the magnetic field? Maxwell’s Equations 0 0 0 J t t                    EH HE E HRE VI EW la st le ct ur e z/2  23/2 E0 -E0 0  Vector wave equation    0 0, cosxz t E t k z  E a 1-26 Transverse ElectroMagnetic (TEM) wave E and H are always orthogonal following the RH rule         0 0 0 0 0 , cos , cos x y z t E t k z Ez t t k z            E a H a E y z H x Right hand rule ExH 1st  E 2nd  H Thumb  Direction of propagation R EV IE W la st le ct ur e 1-27 Magnetic field solution The H-field is provided from the E-field solution into and provides Where the ratio of electric field to magnetic field is the intrinsic impedance 0 0 0 0 s s s s s s j j             H E E H E H 0 0' 0 0 jk z jk z xs x xE E e E e   0 0 s s xs y ys y j dE j H dz        E H a a 0 0 0 0 '0 0 0 0 0 0 ' 0 0 0 0 1 1 jk z jk z ys x x jk z jk z x x H E e E e E e E e             0 0 0 377     zyx zyx AAA zyx        aaa A ˆˆˆ R EV IE W la st le ct ur e 1-30 Helmholtz equation in dielectrics Keeping the forward-propagating wave solution in a real instantaneous form 0 jkz xs xE E e   jk j    0 coszx xE E e t z     1 2 3 4 5 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 Lossless medium z Ex Lossy medium 1 2 3 4 5 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 z Ex ze   cos z t=0 0 0 jkz jkz xs x xE E e E e     1-31 Wave incident on lossy medium (there might also be reflection, ignored for now)  0 coszx xE E e t z     z Ex 1-32 Attenuation constant Attenuation constant is quantified in Nepers/m (Np/m) 1 Np/m means the wave decreases by a factor of Amplitude 1/e (=0.368) every meter Power 1/e2 (=0.135) every meter The attenuation is also expressed in dB/m. On one hand we have, On the other hand thus      1010 0A dB log z    P P      20 z Np mz e P P      8 69 A dB dB m . Np m z    1-35 Origins of losses Dielectrics: E-field causes bound charges to oscillate (with damping loss) – Model with complex permittivity Conductors: Oscillating E- field induces surface currents – Model with a finite conductivity Ex Js conductor Ex dielectric + - - - - -- - -- - -- - 1-36 Complex dielectric constant The effect of losses is modeled in a dielectric by modifying the dielectric constant  From Ampere’s law for time harmonic waves   ' " d c c j j j j j j                                  H J J E E E  ’’ always positive Dielectrics have ’’>0 = Conduction curr. + Displacement curr. 1-37 Wavenumber ' "c j j          jk j    ' " ' 1 "/ 'ck j j                  1/2 2 1/2 2 ' "Re 1 1 2 ' ' "Im 1 1 2 ' jk jk                               Attenuation coefficient Phase constant Loss tangent So we have Since and we get 1-40 Impedance in a dielectric The intrinsic impedance of a dielectric is also modified with respect to vacuum So the electric field to magnetic field ratio of a TEM wave changes in a dielectric with respect to free-space. Also in a lossless medium What would a complex impedance imply?   1 ' " ' 1 " 'j j              0 1 ' 1 " ' r rj          Intrinsic impedance is reduced in a nonmagnetic dielectric Waves from free-space to some dielectric Ei, Er and Etrans n=, N2=No/3 1-41 1-42 Example With a phase constant of 391.5m-1, snow has a permittivity of =0(3.30-1.65j). Express the electric and magnetic fields propagating in this medium in the real instantaneous form. The electric field amplitude is 1V/m polarized along the z axis and propagating along the -y axis.     1/2 2 1/2 2 0 1/2 2 1 0 ' " 3.31 1 1 0.5 1 62.8 2 ' 2 3.3 1 0.5 1 92.4 2 Grad s c m c                            0 1 1 196 0.23 3.3 1 0.5 rad j          68.7 68.7 ˆcos 47 391.5 ˆ0.0051 cos 47 391.5 0.23 y z z y x x E e Gt y a V m H e Gt y a A m       Check if right hand rule works 1-45 Conceptual Questions 2. Which of the following statements are true for dielectrics? a) EM waves in a dielectric have shorter wavelength compared to the same EM wave in free space. b) In lossy dielectrics, E and H fields are generally out of phase. c) The loss of a dielectric is generally frequency dependent. d) Damping loss is mainly responsible for E-field loss in dielectrics. 1-46 Before next class Problems Examples 11.3, 11.4 D11.3 Read section 11.2 Assignment #4 has been released, due Monday next week before 11:35 There is no class on Monday Department of Electrical and Computer Engineering ECSE 352 Electromagnetic Waves Module 2: The uniform plane wave 2.3 The loss tangent Reference: H&B, Section 11.2 1-50 Module 2: The Uniform Plane Wave Module 1: Transmission Lines Module 2: The Uniform Plane Wave 1. EM wave propagation in free space 2. EM wave propagation in dielectrics 3. The loss tangent 4. EM wave power and the Poynting vector 5. The skin effect 6. EM wave polarization Module 3: Waves at boundaries Module 4: Waveguides Module 5: Antennas 1-51 Revision: Model for loss EJ c 1. Model a conduction current   cj j     H E E2. Modify Ampere’s law ' "c j j         3. Introduce complex permittivity c 4. Modify wavenumber "' 1 'c jk j j j             1/2 1/2 2 2' " ' "Re 1 1 Im 1 1 2 ' 2 ' jk jk                                      5. Extract  and  jk j   z Ex Lossy medium EJc R EV IE W la st le ct ur e 1-52 Helmholtz in dielectrics Keeping the forward-propagating wave solution it becomes a real instantaneous form 0 jkz xs xE E e   jk j    0 coszx xE E e t z     1 2 3 4 5 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 Lossless medium z Ex Lossy medium 1 2 3 4 5 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 z Ex ze   cos z t=0 0 0 jkz jkz xs x xE E e E e     R EV IE W la st le ct ur e 1-55 Loss tangent Return to Ampere’s law The complex permittivity is and thus the lossy current (associated with ’’) is taken as a conduction current ' " 'c j j        S d I d t        DH L S H Integral form   d c j j j j                     H J J E E E = Conduction curr. + Displacement curr. 1-56 Loss tangent Use ratio of conduction to displacement currents to determine the degree of loss They are 90° out of phase in time. Displacement current offsets total current by an angle . The tangent of this angle is the loss tangent. A nonzero loss tangent implies losses " ' 'd J J j j        "tan ' '        J E 'd jJ E d J J J  1/2 2' "1 1 2 '               1-57 Loss tangent vs frequency (theoretical) This graph assumes that  is constant over frequency, which is a good but approximate assumption. log (f) lo g 1 0 (     " ' '      1-60 Example: Power absorbed in an oven A microwave oven irradiates water with an E-field amplitude of 100 V/m. What is the average power loss per unit volume taking into account that water has r=78-7j at fmw=2.50 GHz?  9 0 " 2 2.5 10 7 0.97 /S m             2 2 3 1 2 1 0.97 100 2 4.87 P E kW m    1-61 Conductors and insulators Materials are classified as conductors or insulators according to the magnitude of the loss tangent. When the loss tangent is small (’) – Low propagation losses – Insulator/dielectric When the loss tangent is large – High propagation losses – Conductor Waves incident on conductors is a topic of section 11.4 " ' ' eff    1-62 Typical dielectric material For many dielectrics, the loss tangent model is only approximate as is strongly frequency-dependent. See Appendix D for more details.   ’ ’’ µ-wave IR UV Absorption resonance peaks 1-65 Waves in materials: special cases (Derived in sec. 11.5) Lossless medium (=0) Dielectrics (/=/<<1) Conductors (/=/>0.1) Units = 0 Np/m = rad/m =  vp= m/s = m 2 '    f  21' 1 8 '              f   1' 2 ' j         1  j   1/  21 11 8 ''            4f /  /c f /c f/c f 1-66 Conceptual Questions 1. Which of the following statements are true? a) Depending on the frequency of operation, some materials sometimes act as dielectrics and sometimes as conductors. b) The frequency of the microwave oven has been designed at the frequency of strongest absorption coefficient by water. c) The loss tangent is always inversely proportional to frequency d) The frequency of a EM wave is always constant, independently of the permittivity of the dielectric or conductor. 1-67 Before next class Problems Examples 11.4, 11.5 (find the 2 typos) D11.4, D11.5 11.7, 11.11, 11.14 Read section 11.3 Assignment #5 has been released, due Monday next week before class (11:35) 1-70 Module 2: The Uniform Plane Wave Module 1: Transmission Lines Module 2: The Uniform Plane Wave 1. EM wave propagation in free space 2. EM wave propagation in dielectrics 3. The loss tangent 4. EM wave power and the Poynting vector 5. The skin effect 6. EM wave polarization Module 3: Waves at boundaries Module 4: Waveguides Module 5: Antennas 1-71 Contents • The relationship between electric field amplitude and power. • The Poynting vector • Instantaneous and time averaged electromagnetic power • Power loss in a lossy medium • Maximum permitted levels of exposure to EM radiation 1-72 EM waves transport energy Electromagnetic waves transport energy and thus deliver power The sun radiates a power density of ~1.1kW/m2 at the earth’s surface 1-75 Power and field values What is the potential difference across a laser beam ?! *1 Re 2 P VI    1-76 Energy transport from a volume E and H fields carry energy in and out of a reference volume through S Energy is also dissipated inside the volume, in the form of Joule heating V: volume S: surface S: Power flow S V S 1-78 Derivation of the Poynting vector Start with Maxwell equation Take scalar product with E Use identity Faraday’s law Rearrange       t t t t t t t                                                  DH J DE H J E E DH E E H J E E B DH E H J E E D BE H J E E H   1 1 2 2t t                     E H J E D E B H         E H E H H E t      BE 1-81 Time averaged power density In general, the most convenient power quantity is the the time averaged power density flowing across a surface       0 2 20 0 2 20 1 , 1 cos cos 2 2 2 cos 2 T T z z z z z z t dt T E e t z dt T E e                         S S a a   2 20 cos 2 z z E z e   S a Average power density [W/m2] 1-82 Time-averaged power: phasors           0 0 00 2 2 2 * 2 0 0 0 0 * 2 * * 2 2 20 0 0 0 0 * 2 * 2 0 0 0 0 1 , 1 1 4 1 4 4 T T z j z j t z j z j t x y z j z j t z T z z z j z j t z zz z z t dt T E e e e cc H e e e cc dt T E H e e e E H e dt T E H e E H e e e E H e E H e                                               S S a a a a =     0 * 0 z j z s E jz j z s H z E e e Ez e e e            E a H a cc   *1 2 s s z   S E H= Re 1-83 Time average and instantaneous power density S(z,t) (W/m2) <S(z)> (W/m2) E(z,t) (V/m) Distance z (m) S(z,t): Instantaneous power density <S(z)>: Time averaged power density E(z,t): Electric Field 1-88 Solution Calculate the power density just entering the tree and just about to pass through the tree, and take the difference S1 S2 Ein Eout   1 2 2 21 1 0 2 21 2 2 21 1 cos 2 1 cos 2 1 cos 1 2 z z z z L L S S S ES e ES e ES e                          1-89 Waves in materials: special cases (Derived in sec. 11.5) Lossless medium (=0) Dielectrics (/=/<<1) Conductors (/=/>>1) Units = 0 Np/m = rad/m =  vp= m/s = m 2 '    f  21' 1 8 '              f   1' 2 ' j         1  j   1/  21 11 8 ''            4f /  /pv f /pv f/pv f 1-90 Solution   4 0 2 tan 1.11 10 1 266.6 0.029 ' 2 ' 0.0148 Np/m 2 ' 1 1 1 0.971 54.7 W/m 2 266.6 r S m j S                              Thus the incoming power over a 1 m2 surface is 1.875 mW. After propagation through 1m deep of tree, there is 54.7 uW of power that has been absorbed. f=1GHz E0=1V/m r=2.0 tan=’’/’=/’=0.001 Low loss dielectric   2 211 cos 1 2 LES e     S1 1-93 Before next class Problems D11.6 11.18 Read section 11.4 Assignment #5 has been released, due Monday next week before class (11:35) Next week’s lab: Lab 3 theory document is available on MyCourses Midterm on sections 1 and 2 (chapters 10-11) on Monday Feb 26 Department of Electrical and Computer Engineering ECSE 352 Electromagnetic Waves Module 2: The uniform plane wave 2.5 Waves in good conductors: the skin effect References: Hayt and Buck 11.4 Increasing frequency 1-95 Overview In this class, we investigate propagation of waves in conductors. This leads to the concept of the skin depth; i.e. the distance which a wave penetrates into a conductive medium. We will see that this results in significant implications for conductor design. 1-98 Wavenumber So we have Since and Therefore ' "c j j          jk j    ' " ' 1 "/ 'k j j                  1/2 2 1/2 2 ' "Re 1 1 2 ' ' "Im 1 1 2 ' jk jk                               Attenuation coefficient Phase constant Loss tangent R EV IE W 1-99 Waves in good conductors In a good conductor, ‘ and thus ' 1 "/ 'jk j j j          ' 1 / ' 1 2 jk j j j j            2    Good conductors approximation 1-100 Attenuation is abrupt at the interface of a conductor The skin depth or penetration depth  [m] quantifies the distance over which a wave amplitude is attenuated by a factor of e=2.72, left with 1/e2 = 13.5% in power At and above microwave frequencies, EM waves penetrate over a shallow depth within the surface of metals. Skin depth 1 1 f     