# 3.3: Wave Equations for Lossy Regions

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- 24783

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The wave equations for electromagnetic propagation in lossless and source-free media, in differential phasor form, are:

\[\begin{align} \nabla^2\widetilde{\bf E} +\omega^2\mu\epsilon \widetilde{\bf E} &= 0 \label{m0128_eEL} \\ \nabla^2\widetilde{\bf H} +\omega^2\mu\epsilon \widetilde{\bf H} &= 0 \label{m0128_eHL}\end{align} \]

The constant \(\omega^2\mu\epsilon\) is labeled \(\beta^2\), and \(\beta\) turns out to be the phase propagation constant.

Now, we wish to upgrade these equations to account for the possibility of loss. First, let’s be clear on what we mean by “loss.” Specifically, we mean the possibility of conversion of energy from the propagating wave into current, and subsequently to heat. This mechanism is described by Ohm’s law:

\[\widetilde{\bf J} = \sigma\widetilde{\bf E} %~~~ \mbox{(phasor form)} \nonumber \]

where \(\sigma\) is conductivity and \(\widetilde{\bf J}\) is conduction current density (SI base units of A/m\(^2\)). In the lossless case, \(\sigma\) is presumed to be zero (or at least negligible), so \({\bf J}\) is presumed to be zero. To obtain wave equations for media exhibiting significant loss, we cannot assume \({\bf J}=0\).

To obtain equations that account for the possibility of significant conduction current, hence loss, we return to the phasor forms of Maxwell’s equations:

\[\begin{align} \nabla \cdot \widetilde{\bf E} &= \frac{\widetilde{\rho}_v}{\epsilon} \label{m0128_eMFE} \\ \nabla \times \widetilde{\bf E} &= -j\omega\mu\widetilde{\bf H} \label{m0128_eGL} \\ \nabla \cdot \widetilde{\bf H} &= 0 \label{m0128_eGM} \\ \nabla \times \widetilde{\bf H} &= \widetilde{\bf J} + j\omega\epsilon\widetilde{\bf E} \label{m0128_eACL}\end{align} \]

where \(\widetilde{\rho}_v\) is the charge density. If the region of interest is source-free, then \(\widetilde{\rho}_v=0\). However, we may not similarly suppress \(\widetilde{\bf J}\) since \(\sigma\) may be non-zero. To make progress, let us identify the possible contributions to \(\widetilde{\bf J}\) as follows:

\[\widetilde{\bf J} = \widetilde{\bf J}_{imp} + \widetilde{\bf J}_{ind} \label{m0128_eJimpJind} \]

where \(\widetilde{\bf J}_{imp}\) represents *impressed* sources of current and \(\widetilde{\bf J}_{ind}\) represents current which is *induced* by loss. An impressed current is one whose behavior is independent of other features, analogous to an independent current source in elementary circuit theory. In the absence of such sources, Equation \ref{m0128_eJimpJind} becomes:

\[\begin{aligned} \widetilde{\bf J} &= 0 + \sigma\widetilde{\bf E} \end{aligned} \nonumber \]

Equation \ref{m0128_eACL} may now be rewritten as:

\[\begin{aligned} \nabla \times \widetilde{\bf H} &= \sigma\widetilde{\bf E} + j\omega\epsilon\widetilde{\bf E} \\ &= \left(\sigma+j\omega\epsilon\right)\widetilde{\bf E} \\ &= j\omega\epsilon_c\widetilde{\bf E} \end{aligned} \nonumber \]

where we defined the new constant \(\epsilon_c\) as follows:

\[\epsilon_c \triangleq \epsilon - j\frac{\sigma}{\omega} \nonumber \]

This constant is known as *complex permittivity*. In the lossless case, \(\epsilon_c = \epsilon\); i.e., the imaginary part of \(\epsilon_c\rightarrow 0\) so there is no difference between the physical permittivity \(\epsilon\) and \(\epsilon_c\). The effect of the material’s loss is represented as a non-zero imaginary component of the permittivity.

It is also common to express \(\epsilon_c\) as follows:

\[\epsilon_c = \epsilon' - j\epsilon'' \label{m0128_eecdef} \]

where the real-valued constants \(\epsilon'\) and \(\epsilon''\) are in this case:

\[\begin{align} \epsilon' &= \epsilon \label{m0128_eepdef} \\ \epsilon'' &= \frac{\sigma}{\omega} \label{m0128_eeppdef}\end{align} \]

This alternative notation is useful for three reasons. First, some authors use the symbol \(\epsilon\) to refer to *both* physical permittivity *and* complex permittivity. In this case, the “\(\epsilon'-j\epsilon''\)” notation is helpful in mitigating confusion. Second, it is often more convenient to specify \(\epsilon''\) at a frequency than it is to specify \(\sigma\), which may also be a function of frequency. In fact, in some applications the loss of a material is most conveniently specified using the ratio \(\epsilon''/\epsilon'\) (known as *loss tangent*, for reasons explained elsewhere). Finally, it turns out that *nonlinearity* of permittivity can *also* be accommodated as an imaginary component of the permittivity. The “\(\epsilon''\)” notation allows us to accommodate both effects – nonlinearity and conductivity – using common notation. In this section, however, we remain focused exclusively on conductivity.

Complex permittivity \(\epsilon_c\) (SI base units of F/m) describes the combined effects of permittivity and conductivity. Conductivity is represented as an imaginary-valued component of the permittivity.

Returning to Equations \ref{m0128_eMFE}-\ref{m0128_eACL}, we obtain:

\[\begin{aligned} \nabla \cdot \widetilde{\bf E} &= 0 \\ \nabla \times \widetilde{\bf E} &= -j\omega\mu\widetilde{\bf H} \\ \nabla \cdot \widetilde{\bf H} &= 0 \\ \nabla \times \widetilde{\bf H} &= j\omega\epsilon_c\widetilde{\bf E} \end{aligned} \nonumber \]

These equations are *identical* to the corresponding equations for the lossless case, with the exception that \(\epsilon\) has been replaced by \(\epsilon_c\). Similarly, we may replace the factor \(\omega^2 \mu\epsilon\) in Equations \ref{m0128_eEL} and \ref{m0128_eHL}, yielding:

\[\begin{aligned} \nabla^2\widetilde{\bf E} +\omega^2\mu\epsilon_c \widetilde{\bf E} &= 0 \label{m0128_eE} \\ \nabla^2\widetilde{\bf H} +\omega^2\mu\epsilon_c \widetilde{\bf H} &= 0 \label{m0128_eH}\end{aligned} \]

In the lossless case, \(\omega^2\mu\epsilon_c \rightarrow \omega^2\mu\epsilon\), which is \(\beta^2\) as expected. For the general (i.e., possibly lossy) case, we shall make an analogous definition

\[\gamma^2 \triangleq -\omega^2\mu\epsilon_c \label{m0128_egammadef} \]

such that the wave equations may now be written as follows:

\[\boxed{ \nabla^2\widetilde{\bf E} -\gamma^2 \widetilde{\bf E} = 0 } \label{m0128_eEg} \]

\[\boxed{ \nabla^2\widetilde{\bf H} -\gamma^2 \widetilde{\bf H} = 0 } \label{m0128_eHg} \]

Note the minus sign in Equation \ref{m0128_egammadef}, and the associated changes of signs in Equations \ref{m0128_eEg} and \ref{m0128_eHg}. For the moment, this choice of sign may be viewed as arbitrary – we would have been equally justified in choosing the opposite sign for the definition of \(\gamma^2\). However, the choice we have made is customary, and yields some notational convenience that will become apparent later.

In the lossless case, \(\beta\) is the phase propagation constant, which determines the rate at which the phase of a wave progresses with increasing distance along the direction of propagation. Given the similarity of Equations \ref{m0128_eEg} and \ref{m0128_eHg} to Equations \ref{m0128_eEL} and \ref{m0128_eHL}, respectively, the constant \(\gamma\) must play a similar role. So, we are motivated to find an expression for \(\gamma\). At first glance, this is simple: \(\gamma=\sqrt{\gamma^2}\). However, recall that every number has two square roots. When one declares \(\beta=\sqrt{\beta^2}=\omega\sqrt{\mu\epsilon}\), there is no concern since \(\beta\) is, by definition, positive; therefore, one knows to take the positive-valued square root. In contrast, \(\gamma^2\) is complex-valued, and so the two possible values of \(\sqrt{\gamma^2}\) are both potentially complex-valued. We are left without clear guidance on which of these values is appropriate and physically-relevant. So we proceed with caution.

First, consider the special case that \(\gamma\) is purely imaginary; e.g., \(\gamma = j\gamma''\) where \(\gamma''\) is a real-valued constant. In this case, \(\gamma^2 = -\left(\gamma''\right)^2\) and the wave equations become

\[\begin{aligned} \nabla^2\widetilde{\bf E} +\left(\gamma''\right)^2 \widetilde{\bf E} &= 0 \\ \nabla^2\widetilde{\bf H} +\left(\gamma''\right)^2 \widetilde{\bf H} &= 0\end{aligned} \nonumber \]

Comparing to Equations \ref{m0128_eEL} and \ref{m0128_eHL}, we see \(\gamma''\) plays the exact same role as \(\beta\); i.e., \(\gamma''\) is the phase propagation constant for whatever wave we obtain as the solution to the above equations. Therefore, let us make that definition formally:

\[\beta \triangleq \mbox{Im}\left\{\gamma\right\} \nonumber \]

Be careful: Note that we are *not* claiming that \(\gamma''\) in the possibly-lossy case is equal to \(\omega^2\mu\epsilon\). Instead, we are asserting exactly the opposite; i.e., that there is a phase propagation constant in the general (possibly lossy) case, and we should find that this constant simplifies to \(\omega^2\mu\epsilon\) in the lossless case.

Now we make the following definition for the real component of \(\gamma\):

\[\alpha \triangleq \mbox{Re}\left\{\gamma\right\} \nonumber \]

Such that

\[\gamma = \alpha + j\beta \nonumber \]

where \(\alpha\) and \(\beta\) are real-valued constants.

Now, it is possible to determine \(\gamma\) explicitly in terms of \(\omega\) and the constitutive properties of the material. First, note:

\[\begin{align} \gamma^2 &= \left(\alpha + j\beta\right)^2 \nonumber \\ &= \alpha^2-\beta^2 + j2\alpha\beta \label{m0128_eg2a}\end{align} \]

Expanding Equation \ref{m0128_egammadef} using Equations \ref{m0128_eecdef}-\ref{m0128_eeppdef}, we obtain:

\[\begin{align} \gamma^2 &= -\omega^2\mu\left(\epsilon-j\frac{\sigma}{\omega}\right) \nonumber \\ &= -\omega^2\mu\epsilon +j\omega\mu\sigma \label{m0128_eg2b}\end{align} \]

The real and imaginary parts of Equations \ref{m0128_eg2a} and \ref{m0128_eg2b} must be equal. Enforcing this equality yields the following equations:

\[\alpha^2-\beta^2 = -\omega^2\mu\epsilon \label{m0128_ese1} \]

\[2\alpha\beta = \omega\mu\sigma \label{m0128_ese2} \]

Equations \ref{m0128_ese1} and \ref{m0128_ese2} are independent simultaneous equations that may be solved for \(\alpha\) and \(\beta\). Sparing the reader the remaining steps, which are purely mathematical, we find:

\[\boxed{ \alpha = \omega \left\{ \frac{\mu\epsilon'}{2} \left[ \sqrt{1+\left(\frac{\epsilon''}{\epsilon'}\right)^2}-1 \right] \right\}^{1/2} } \label{m0128_ealpha} \]

\[\boxed{ \beta = \omega \left\{ \frac{\mu\epsilon'}{2} \left[ \sqrt{1+\left(\frac{\epsilon''}{\epsilon'}\right)^2}+1 \right] \right\}^{1/2} } \label{m0128_ebeta} \]

Equations \ref{m0128_ealpha} and \ref{m0128_ebeta} can be verified by confirming that Equations \ref{m0128_ese1} and \ref{m0128_ese2} are satisfied. It is also useful to confirm that the expected results are obtained in the lossless case. In the lossless case, \(\sigma=0\), so \(\epsilon''=0\). Subsequently, Equation \ref{m0128_ealpha} yields \(\alpha=0\) and Equation \ref{m0128_ebeta} yields \(\beta=\omega\sqrt{\mu\epsilon}\), as expected.

The electromagnetic wave equations accounting for the possibility of lossy media are Equations \ref{m0128_eEg} and \ref{m0128_eHg} with \(\gamma=\alpha+j\beta\), where \(\alpha\) and \(\beta\) are the positive real-valued constants determined by Equations \ref{m0128_ealpha} and \ref{m0128_ebeta}, respectively.

We conclude this section by pointing out a very useful analogy to transmission line theory. In the section “Wave Propagation on a TEM Transmission Line,”^{1} we found that the potential and current along a transverse electromagnetic (TEM) transmission line satisfy the same wave equations that we have developed in this section, having a complex-valued propagation constant \(\gamma=\alpha+j\beta\), and the same physical interpretation of \(\beta\) as the phase propagation constant. As is explained in another section, \(\alpha\) too has the same interpretation in both applications – that is, as the *attenuation constant*.

**Additional Reading:**

- “Electromagnetic Wave Equation” on Wikipedia.

- This section may appear in a different volume, depending on the version of this book.↩