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EEE 498/598 Overview of Electrical Engineering Lecture 10: Uniform Plane Wave Solutions to Maxwell’s Equations 1 Lecture 10 Objectives To study uniform plane wave solutions to Maxwell’s equations: In the time domain for a lossless medium. In the frequency domain for a lossy medium. 2 Lecture 10 Overview of Waves A wave is a pattern of values in space that appear to move as time evolves. A wave is a solution to a wave equation. Examples of waves include water waves, sound waves, seismic waves, and voltage and current waves on transmission lines. 3 Lecture 10 Overview of Waves (Cont’d) Wave phenomena result from an exchange between two different forms of energy such that the time rate of change in one form leads to a spatial change in the other. Waves possess no mass energy momentum velocity 4 Lecture 10 Time-Domain Maxwell’s Equations in Differential Form Kc Ki B E K t D H J t D qev B qmv Jc Ji 5 Lecture 10 Time-Domain Maxwell’s Equations in Differential Form for a Simple Medium D E B H J c E Kc m H H E m H K i t E H E J i t 6 E qev H qmv Lecture 10 Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, and Lossless Medium J i K i 0 qev qmv 0 m 0 H E t E H t E 0 H 0 7 Lecture 10 Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, and Lossless Medium Obviously, there must be a source for the field somewhere. However, we are looking at the properties of waves in a region far from the source. 8 Lecture 10 Derivation of Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium 0 E E E 2 H 2 E 2 t t 0 H H H 2 E H 2 t t 2 9 Lecture 10 Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium E E 2 0 t 2 2 H H 2 0 t 2 2 10 The wave equations are not independent. Usually we solve the electric field wave equation and determine H from E using Faraday’s law. Lecture 10 Uniform Plane Wave Solutions in the Time Domain A uniform plane wave is an electromagnetic wave in which the electric and magnetic fields and the direction of propagation are mutually orthogonal, and their amplitudes and phases are constant over planes perpendicular to the direction of propagation. Let us examine a possible plane wave solution given by E aˆ x Ex z, t 11 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The wave equation for this field simplifies to 2 Ex 2 Ex 0 2 2 z t The general solution to this wave equation is Ex z, t p1 z v pt p2 z v pt 12 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The functions p1(z-vpt) and p2 (z+vpt) represent uniform waves propagating in the +z and -z directions respectively. Once the electric field has been determined from the wave equation, the magnetic field must follow from Maxwell’s equations. 13 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The velocity of propagation is determined solely by the medium: vp 1 The functions p1 and p2 are determined by the source and the other boundary conditions. 14 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) Here we must have H aˆ y H y z, t where H y z, t 1 p z v t p z v t 1 p 15 2 p Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) is the intrinsic impedance of the medium given by Like the velocity of propagation, the intrinsic impedance is independent of the source and is determined only by the properties of the medium. 16 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) In free space (vacuum): v p c 3 10 m/s 8 120 377 17 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) Strictly speaking, uniform plane waves can be produced only by sources of infinite extent. However, point sources create spherical waves. Locally, a spherical wave looks like a plane wave. Thus, an understanding of plane waves is very important in the study of electromagnetics. 18 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) Assuming that the source is sinusoidal. We have p1 z v p t C1 cos z v p t C1 cost z v p p2 z v p t C2 cos z v p t C2 cost z v p vp 19 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The electric and magnetic fields are given by E x z, t C1 cost z C2 cost z H y z, t 1 C1 cost z C2 cost z 20 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The argument of the cosine function is the called the instantaneous phase of the field: z, t t z 21 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The speed with which a constant value of instantaneous phase travels is called the phase velocity. For a lossless medium, it is equal to and denoted by the same symbol as the velocity of propagation. t 0 t z 0 z dz 1 vp dt Lecture 10 22 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The distance along the direction of propagation over which the instantaneous phase changes by 2 radians for a fixed value of time is the wavelength. 2 23 2 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) The wavelength is also the distance between every other zero crossing of the sinusoid. Function vs. position at a fixed time 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 2 4 24 6 8 10 12 14 16 18 Lecture 10 20 Uniform Plane Wave Solutions in the Time Domain (Cont’d) Relationship between wavelength and frequency in free space: c f Relationship between wavelength and frequency in a material medium: vp f 25 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) is the phase constant and is given by vp rad/m 26 Lecture 10 Uniform Plane Wave Solutions in the Time Domain (Cont’d) In free space (vacuum): 0 0 c k0 2 0 free space wavenumber (rad/m) 27 Lecture 10 Time-Harmonic Analysis Sinusoidal steady-state (or time-harmonic) analysis is very useful in electrical engineering because an arbitrary waveform can be represented by a superposition of sinusoids of different frequencies using Fourier analysis. If the waveform is periodic, it can be represented using a Fourier series. If the waveform is not periodic, it can be represented using a Fourier transform. 28 Lecture 10 Time-Harmonic Maxwell’s Equations in Differential Form for a Simple, Source-Free, Possibly Lossy Medium E j H H j E E 0 H 0 j j m j j 29 Lecture 10 Derivation of Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium 0 E E E 2 j H 2 E 0 H H H 2 2 j E 2 H 30 2 Lecture 10 Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium E E 0 2 2 H H 0 2 2 31 The Helmholtz equations are not independent. Usually we solve the electric field equation and determine H from E using Faraday’s law. Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain Assuming a plane wave solution of the form E aˆ x Ex z The Helmholtz equation simplifies to 2 d Ex 2 Ex 0 2 dz 32 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) The propagation constant is a complex number that can be written as j j 2 (m-1) attenuation constant (Np/m) 33 phase constant (rad/m) Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) is the attenuation constant and has units of nepers per meter (Np/m). is the phase constant and has units of radians per meter (rad/m). Note that in general for a lossy medium 34 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) The general solution to this wave equation is Ex z C1e C1e z C2 e z j z e z z C2 e e j z Ex z Ex z • wave traveling in the -z-direction • wave traveling in the +z-direction 35 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Converting the phasor representation of E back into the time domain, we have Ex z, t Re Ex z e C1e z jt cost z C2e cost z z • We have assumed that C1 and C2 are real. 36 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) The corresponding magnetic field for the uniform plane wave is obtained using Faraday’s law: E E j H H j 37 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Evaluating H we have H y z Ce 1 1 E 1 x 38 z C2 e z z Ex z Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) We note that the intrinsic impedance is a complex number for lossy media. e 39 j Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Converting the phasor representation of H back into the time domain, we have H y z , t Re H y z e C1 jt e z cost z C2 e z cost z 40 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) We note that in a lossy medium, the electric field and the magnetic field are no longer in phase. The magnetic field lags the electric field by an angle of . 41 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Note that we have E H aˆ z These form a righthanded coordinate system Uniform plane waves are a type of transverse electromagnetic (TEM) wave. âE â z âH 42 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Relationships between the phasor representations of electric and magnetic fields in uniform plane waves: H 1 aˆ p E unit vector in direction of propagation E aˆ p H 43 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Example: 9 f 1 10 Hz 0 0.300 m 2.5 0 0 0.01 S/m Consider Ex z , t e α 1.191 Np/m 33.16 rad/m z cost z 44 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Snapshot of Ex+(z,t) at t = 0 1 0.8 e 0.6 z 0.4 x E+ (z,t) 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.5 1 1.5 2 2.5 z/0 45 3 3.5 4 4.5 5 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Properties of the wave determined by the source: amplitude phase frequency 46 Lecture 10 Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Properties of the wave determined by the medium are: • also depend on frequency velocity of propagation (vp) intrinsic impedance () propagation constant constant (=j) wavelength () vp f 47 2 Lecture 10 Dispersion For a signal (such as a pulse) comprising a band of frequencies, different frequency components propagate with different velocities causing distortion of the signal. This phenomenon is called dispersion. 25 input signal 20 15 10 output signal 5 0 -5 0 100 200 300 48 400 500 600 Lecture 10 Plane Wave Propagation in Lossy Media Assume a wave propagating in the +zdirection: Ex z , t Ex 0 e z cost z We consider two special cases: Low-loss dielectric. Good (but not perfect) conductor. 49 Lecture 10 Plane Waves in a Low-Loss Dielectric A lossy dielectric exhibits loss due to molecular forces that the electric field has to overcome in polarizing the material. We shall assume that r 0 j 1 j 1 j tan r 0 1 j tan 50 Lecture 10 Plane Waves in a Low-Loss Dielectric (Cont’d) Assume that the material is a low-loss dielectric, i.e, the loss tangent of the material is small: tan 1 51 Lecture 10 Plane Waves in a Low-Loss Dielectric (Cont’d) Assuming that the loss tangent is small, approximate expressions for and can be developed. j j 0 1 j tan 1 x 1/ 2 1 x 2 tan j 0 1 j j 2 wavenumber 0 r k 0 k tan k tan 0 2 2 52 Lecture 10 Plane Waves in a Low-Loss Dielectric (Cont’d) The phase velocity is given by c vp k r 53 Lecture 10 Plane Waves in a Low-Loss Dielectric (Cont’d) The intrinsic impedance is given by 0 r 1 / 2 1 j tan tan j tan 0 2 1 j e 2 r 1 x 1 / 2 x 1 2 1 x e x 54 Lecture 10 Plane Waves in a Low-Loss Dielectric (Cont’d) In most low-loss dielectrics, r is more or less independent of frequency. Hence, dispersion can usually be neglected. The approximate expression for is used to accurately compute the loss per unit length. 55 Lecture 10 Plane Waves in a Good Conductor In a perfect conductor, the electromagnetic field must vanish. In a good conductor, the electromagnetic field experiences significant attenuation as it propagates. The properties of a good conductor are determined primarily by its conductivity. 56 Lecture 10 Plane Waves in a Good Conductor For a good conductor, 1 Hence, j 57 Lecture 10 Plane Waves in a Good Conductor (Cont’d) j j j 1 j j 2 2 2 58 Lecture 10 Plane Waves in a Good Conductor (Cont’d) The phase velocity is given by 2 vp c 59 Lecture 10 Plane Waves in a Good Conductor (Cont’d) The intrinsic impedance is given by j j 1 j j 45 e 2 60 Lecture 10 Plane Waves in a Good Conductor (Cont’d) The skin depth of material is the depth to which a uniform plane wave can penetrate before it is attenuated by a factor of 1/e. We have e 1 61 1 Lecture 10 Plane Waves in a Good Conductor (Cont’d) For a good conductor, we have 1 2 62 Lecture 10 Wave Equations for Time-Harmonic Fields in Simple Medium 1 Ki 2 E k0 r E j 0 J i r r 1 Ji 2 E k0 r E j 0 K i r r k0 0 0 63 Lecture 10