FWLec19

FWLec19 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #19 Waves in Lossy Media Objectives: Consider the effect of medium loss on electromagnetic fields. Develop expressions for the attenuation and phase constants in low-loss situations. Discuss the behavior of fields within good conductors. Introduce the skin depth associated with good conductors and discuss the skin effect approximation. Previous Notes have discussed electromagnetic plane waves in lossless environments. When the medium under consideration has conductivity, the conductivity is a source of loss — waves traveling through that medium will lose some of their power to heat. In the following, we consider various aspects of wave propagation in lossy media. Consider a medium with permeability m , permittivity e , and conductivity s . For such a medium, Maxwell’s curl equations can be expressed in their phasor form as —¥ =- Ej H wm (19.1) = + HE j E sw (19.2) where E and H are phasors with suppressed time dependence e jt w . The right-hand side of equation (19.2) can be re-written in the form jE E j j E jE c we w e += + Ê Ë Á ˆ ¯ ˜ = (19.3) where c represents an equivalent complex-valued permittivity ee c j j =+ =- (19.4) One advantage of using phasors for electromagnetic fields is that they permit the combination of the two terms on the right-hand side of (19.2), which reduces that equation to a form that appears essentially the same as it would in the lossless case: = Hj E c (19.5) Although the permittivity c in (19.5) is complex-valued, since we are already using complex-valued quantities for the phasors it is virtually no additional trouble to incorporate complex parameters for the other quantities.

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 The equivalent complex permittivity There are several different conventions that are in use for the equivalent complex permittivity e c . In these Notes, we will primarily use ee s w c j =- (19.6) However, in much of the literature it is common to express the permittivity as c j = ¢ - ¢¢ (19.7) where ¢ and ¢¢ are real valued. A third notation that is widely used is to express c as we ed c j j Ê Ë Á ˆ ¯ ˜ () 1 1 tan (19.8) where tan d = (19.9) is known as the loss tangent of the material. Frequently, manufacturers will specify dielectric materials by their real-valued permittivity and their loss tangent at a particular frequency. The complex wavenumber By combining (19.1) and (19.5), we obtain the vector Helmholtz equation —¥ —¥ = = ( ) Ej j E E kE c c c wm wme 2 2 (19.10) kj cc == - Ê Ë Á ˆ ¯ ˜ wme wme 1 (19.11)
A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 is complex valued, by virtue of the fact that e c is complex valued. Thus, in the presence of medium loss the electric field satisfies the same vector Helmholtz equation as in Note #18, but with a complex-valued wavenumber.

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FWLec19 - A F Peterson Notes on Electromagnetic Fields...

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