The University of Delaware ELEG 648 Plane waves Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Email: mirotzni@ece.udel.edu

SUMMARY Time Domain Frequency Domain

Wave Equation Time Dependent Homogenous Wave Equation (E-Field) Vector Identity

Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field) Source Free

Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field) Source Free Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field) Lossless

Wave Equation: Time Harmonic Time Domain Frequency Domain Source Free Source Free Lossless Lossless “Helmholtz Equation”

General Solution Case: Time Harmonic Rectangular Coordinates Wave Number

Separation of Variable Solutions Assume Solution of the form:

Separation of Variable Solutions Assume Solution of the form:

Separation of Variable Solutions Assume Solution of the form: function of x function of y function of z constant

Separation of Variable Solutions function of x function of y function of z constant

Separation of Variable Solutions

Separation of Variable Solutions b purely real b purely imaginary b complex Traveling and standing waves Exponentially modulated traveling wave Evanescent waves or or

Wave Propagation and Polarization TEM: Transverse Electromagnetic Waves “A mode is a particular field configuration. For a given electromagnetic boundary value problem, many field configurations that satisfy the wave equation, Maxwell’s equations, and boundary conditions usually exits. A TEM mode is one whole field intensities, both E and H, at every point in space are contained in a local plane, referred to as equiphase plane, that is independent of time” E H Plane Waves “If the space orientation of the planes for a TEM mode are the same (equiphase planes are parallel) then the fields form a plane wave. E H E H E H E H Uniform Plane Waves “If in addition to having planar equiphases the field has equiamplitude (the amplitude of the field is the same over each plane) planar surfaces then it is called a uniform plane wave.”

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation Let’s begin by assuming the solution is only a function of z and has only the x component of electric field. Let’s also look at the term that represents a traveling wave moving in the +z direction

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation For uniform plane wave assume the solution is only a function of z and has only the x component of electric field. Lets find H

Several observations: Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation Several observations: E and H are orthogonal to each other and to the direction of energy propagation E and H are in phase with each other H is smaller in amplitude than E by the term (for a uniform plane wave) Wave Impedance

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation How fast does the wave move?

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation How much power does the wave carry?

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation How fast does the power flow?

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation Relationship between phase and group velocity

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

Example: Skin Depth in Sea Water Skin Depth, cm Frequency, GHz

Example: Skin Depth in Copper Skin Depth, microns Frequency, GHz

Example: Skin Depth in Teflon Skin Depth, meters Frequency, GHz

Polarization

Polarization

Polarization

Polarization

Polarization

Polarization

Uniform Plane Waves: Propagation in Any Arbitrary Direction z H f E q y x

Uniform Plane Waves: Propagation in Any Arbitrary Direction H y x z q f Since E and b are at right angles from each other. where and

Uniform Plane Waves: Propagation in Any Arbitrary Direction Summary and Observations: Frequency Domain Time Domain Observation 1. E, H and b vectors are pointing in orthogonal directions. Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance