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1 EEE 498/598 Overview of Electrical Engineering Lecture 10: Uniform Plane Wave Solutions to Maxwell’s Equations

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Lecture 10 2 Lecture 10 Objectives To study uniform plane wave solutions to Maxwell’s equations: To study uniform plane wave solutions to Maxwell’s equations: In the time domain for a lossless medium. In the time domain for a lossless medium. In the frequency domain for a lossy medium. In the frequency domain for a lossy medium.

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Lecture 10 3 Overview of Waves A wave is a pattern of values in space that appear to move as time evolves. 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. 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. Examples of waves include water waves, sound waves, seismic waves, and voltage and current waves on transmission lines.

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Lecture 10 4 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. 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 Waves possess no mass no mass energy energy momentum momentum velocity velocity

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Lecture 10 5 Time-Domain Maxwell’s Equations in Differential Form

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Lecture 10 6 Time-Domain Maxwell’s Equations in Differential Form for a Simple Medium

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Lecture 10 7 Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, and Lossless Medium

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Lecture 10 8 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. 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. However, we are looking at the properties of waves in a region far from the source.

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Lecture 10 9 Derivation of Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium 0 0

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Lecture Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium The wave equations are not independent. Usually we solve the electric field wave equation and determine H from E using Faraday’s law.

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Lecture 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. 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 Let us examine a possible plane wave solution given by

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) The wave equation for this field simplifies to The wave equation for this field simplifies to The general solution to this wave equation is The general solution to this wave equation is

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) The functions p 1 (z-v p t) and p 2 (z+v p t) represent uniform waves propagating in the +z and -z directions respectively. The functions p 1 (z-v p t) and p 2 (z+v p t) 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. Once the electric field has been determined from the wave equation, the magnetic field must follow from Maxwell’s equations.

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) The velocity of propagation is determined solely by the medium: The velocity of propagation is determined solely by the medium: The functions p 1 and p 2 are determined by the source and the other boundary conditions. The functions p 1 and p 2 are determined by the source and the other boundary conditions.

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) Here we must have Here we must have where

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) is the intrinsic impedance of the medium given by 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. Like the velocity of propagation, the intrinsic impedance is independent of the source and is determined only by the properties of the medium.

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) In free space (vacuum): In free space (vacuum):

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) Strictly speaking, uniform plane waves can be produced only by sources of infinite extent. 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. 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. Thus, an understanding of plane waves is very important in the study of electromagnetics.

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) Assuming that the source is sinusoidal. We have Assuming that the source is sinusoidal. We have

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) The electric and magnetic fields are given by The electric and magnetic fields are given by

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Lecture 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: The argument of the cosine function is the called the instantaneous phase of the field:

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Lecture 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. 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.

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Lecture 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. 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.

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) The wavelength is also the distance between every other zero crossing of the sinusoid. The wavelength is also the distance between every other zero crossing of the sinusoid. Function vs. position at a fixed time

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) Relationship between wavelength and frequency in free space: Relationship between wavelength and frequency in free space: Relationship between wavelength and frequency in a material medium: Relationship between wavelength and frequency in a material medium:

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) is the phase constant and is given by is the phase constant and is given by rad/m

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Lecture Uniform Plane Wave Solutions in the Time Domain (Cont’d) In free space (vacuum): In free space (vacuum): free space wavenumber (rad/m)

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Lecture 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. 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 periodic, it can be represented using a Fourier series. If the waveform is not periodic, it can be represented using a Fourier transform. If the waveform is not periodic, it can be represented using a Fourier transform.

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Lecture Time-Harmonic Maxwell’s Equations in Differential Form for a Simple, Source-Free, Possibly Lossy Medium

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Lecture Derivation of Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium 0 0

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Lecture Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium The Helmholtz equations are not independent. Usually we solve the electric field equation and determine H from E using Faraday’s law.

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Lecture Uniform Plane Wave Solutions in the Frequency Domain Assuming a plane wave solution of the form Assuming a plane wave solution of the form The Helmholtz equation simplifies to The Helmholtz equation simplifies to

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) The propagation constant is a complex number that can be written as The propagation constant is a complex number that can be written as attenuation constant (Np/m) phase constant (rad/m) (m -1 )

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Lecture 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 attenuation constant and has units of nepers per meter (Np/m). is the phase constant and has units of radians per meter (rad/m). is the phase constant and has units of radians per meter (rad/m). Note that in general for a lossy medium Note that in general for a lossy medium

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Lecture The general solution to this wave equation is The general solution to this wave equation is Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) wave traveling in the +z-direction wave traveling in the -z-direction

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Converting the phasor representation of E back into the time domain, we have Converting the phasor representation of E back into the time domain, we have We have assumed that C 1 and C 2 are real.

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Lecture 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: The corresponding magnetic field for the uniform plane wave is obtained using Faraday’s law:

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Evaluating H we have Evaluating H we have

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Lecture We note that the intrinsic impedance is a complex number for lossy media. We note that the intrinsic impedance is a complex number for lossy media. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Converting the phasor representation of H back into the time domain, we have Converting the phasor representation of H back into the time domain, we have

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Lecture 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. 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 . The magnetic field lags the electric field by an angle of .

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Note that we have Note that we have These form a right- handed coordinate system These form a right- handed coordinate system Uniform plane waves are a type of transverse electromagnetic (TEM) wave.

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Relationships between the phasor representations of electric and magnetic fields in uniform plane waves: Relationships between the phasor representations of electric and magnetic fields in uniform plane waves: unit vector in direction of propagation

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Example: Example: Consider Consider

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Snapshot of E x + (z,t) at t = 0

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Properties of the wave determined by the source: Properties of the wave determined by the source: amplitude amplitude phase phase frequency frequency

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Lecture Uniform Plane Wave Solutions in the Frequency Domain (Cont’d) Properties of the wave determined by the medium are: Properties of the wave determined by the medium are: velocity of propagation ( v p ) velocity of propagation ( v p ) intrinsic impedance ( ) intrinsic impedance ( ) propagation constant constant ( = j ) propagation constant constant ( = j ) wavelength ( ) wavelength ( ) also depend on frequency

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Lecture 10 48Dispersion 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. 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. input signal output signal

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Lecture Plane Wave Propagation in Lossy Media Assume a wave propagating in the + z - direction: Assume a wave propagating in the + z - direction: We consider two special cases: We consider two special cases: Low-loss dielectric. Low-loss dielectric. Good (but not perfect) conductor. Good (but not perfect) conductor.

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Lecture 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. A lossy dielectric exhibits loss due to molecular forces that the electric field has to overcome in polarizing the material. We shall assume that We shall assume that

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Lecture 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: Assume that the material is a low-loss dielectric, i.e, the loss tangent of the material is small:

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Lecture Plane Waves in a Low-Loss Dielectric (Cont’d) Assuming that the loss tangent is small, approximate expressions for and can be developed. Assuming that the loss tangent is small, approximate expressions for and can be developed. wavenumber

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Lecture Plane Waves in a Low-Loss Dielectric (Cont’d) The phase velocity is given by The phase velocity is given by

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Lecture Plane Waves in a Low-Loss Dielectric (Cont’d) The intrinsic impedance is given by The intrinsic impedance is given by

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Lecture 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. 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. The approximate expression for is used to accurately compute the loss per unit length.

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Lecture Plane Waves in a Good Conductor In a perfect conductor, the electromagnetic field must vanish. In a perfect conductor, the electromagnetic field must vanish. In a good conductor, the electromagnetic field experiences significant attenuation as it propagates. 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. The properties of a good conductor are determined primarily by its conductivity.

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Lecture Plane Waves in a Good Conductor For a good conductor, For a good conductor, Hence, Hence,

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Lecture Plane Waves in a Good Conductor (Cont’d)

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Lecture Plane Waves in a Good Conductor (Cont’d) The phase velocity is given by The phase velocity is given by

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Lecture Plane Waves in a Good Conductor (Cont’d) The intrinsic impedance is given by The intrinsic impedance is given by

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Lecture 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. 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 We have

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Lecture Plane Waves in a Good Conductor (Cont’d) For a good conductor, we have For a good conductor, we have

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Lecture Wave Equations for Time-Harmonic Fields in Simple Medium

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