Plane Waves in Lossy Media & Seawater — Accessible Edition (VeriQAI exemplar)

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2.6.4 Plane Waves in Lossy Media

In this section, we investigate wave propagation in a medium that exhibits loss. Losses in a material are accounted for by defining a complex quantity to describe the permittivity. The complex permittivity causes the wave vector to also be a complex quantity, which in turn causes the plane wave to decay as it is propagating. In addition, the wave impedance becomes a complex quantity, which means that now the electric and magnetic fields are out‑of‑phase. We next investigate the mathematical description of propagation in lossy media.

One mechanism that leads to loss in a material is conductive loss, which occurs in materials that can conduct electric current, such as metals. If there are conductive losses in a material, then the electric field induces a current density given by

\[ \overline{J} = \sigma\,\overline{E} \tag{2.73} \]

where \(\sigma\) is the conductivity in siemens per meter, or S/m (a siemen has the units of \(\Omega^{-1}\)). Substituting this current density into Ampere’s law (Equation 2.40) we obtain

\[ \nabla \times \overline{H} = j\omega\varepsilon\overline{E} + \sigma\overline{E} = j\omega\varepsilon\left(1 - j\frac{\sigma}{\omega\varepsilon}\right)\overline{E} \tag{2.74} \]

By defining a complex permittivity

\[ \varepsilon_c \equiv \varepsilon\left(1 - j\frac{\sigma}{\omega\varepsilon}\right) \tag{2.75} \]

then we can rewrite Equation 2.74 as

\[ \nabla \times \overline{H} = j\omega\varepsilon_c\overline{E} \tag{2.76} \]

Notice that this equation has exactly the same form as Ampere’s law. In fact, all the remaining equations hold by simply replacing the real quantity \(\varepsilon\) by the complex quantity \(\varepsilon_c\). This observation is extremely important! The complex wave impedance is

\[ \eta_c = \sqrt{\frac{\mu_o}{\varepsilon_c}} = |\eta_c|e^{j\phi} \tag{2.77} \]

where the impedance has explicitly been written in polar form to show that now there will be an amplitude difference and a phase difference between the electric and magnetic fields. The complex propagation constant is

\[ k_c = \omega\sqrt{\mu_o\varepsilon_c} \tag{2.78} \]

The complex wave vector is \(\overline{k}_c = k_c\hat{k}\), where \(\hat{k}\) is the unit vector in the direction of wave propagation. The time‑harmonic Maxwell’s equations in an infinite lossy medium are obtained by simply replacing \(\overline{k}\) with \(\overline{k}_c\):

\[ \overline{k}_c \times \overline{E} = \omega\mu_o\overline{H} \tag{2.79} \]

\[ \overline{k}_c \times \overline{H} = -\omega\varepsilon_c\overline{E} \tag{2.80} \]

\[ \overline{k}_c \cdot \overline{E} = 0 \tag{2.81} \]

\[ \overline{k}_c \cdot \overline{H} = 0 \tag{2.82} \]

which, with the exception of the complex wave vector, look identical to Equations 2.53–2.56. This means that the spatial relationships between the electromagnetic fields remain unchanged.

Using a similar derivation as in free space, it follows that for a homogeneous plane wave (one in which the wave is attenuating in the direction of propagation),

\[ \overline{H} = \frac{1}{\eta_c}\hat{k} \times \overline{E} \tag{2.83} \]

It also follows that the Helmholtz equation in a lossy medium is

\[ \nabla^2\overline{E} + k_c^2\,\overline{E} = 0 \tag{2.84} \]

One very important point to note when dealing with lossy materials is the following. For simplicity, let us assume that the direction of propagation is the z‑direction. (Note that we could assume the propagation in any direction, but this assumption greatly simplifies the following analysis.) For propagation in the forward z‑direction, the complex exponential can be written as

\[ e^{-jk_c z} = e^{-j\omega\sqrt{\mu_o\varepsilon_c}\,z} = e^{-j(\beta \pm j\alpha)z} = e^{-j\beta z}e^{\pm\alpha z} \tag{2.85} \]

if we define \(k_c = \omega\sqrt{\mu_o\varepsilon_c} = \beta \pm j\alpha\). Notice that there are two possible solutions since the square root of a complex number has two solutions. Which sign should we choose? Since the wave is propagating in the +z‑direction, then the wave must decay in the +z‑direction, and therefore we must choose \(k_c = \beta - j\alpha\) so that \(e^{-jk_c z} = e^{-j\beta z}e^{-\alpha z}\) and the wave exponentially decays in the +z‑direction. If the wrong sign is chosen, the wave will exponentially grow in the +z direction, which does not make physical sense.

Representing \(\varepsilon_c\) as a vector in the complex plane, the tangent of the angle that the vector makes with the real axis is \(\sigma/(\omega\varepsilon)\) (see Figure 2.8). This quantity is called the loss tangent:

\[ \operatorname{loss\ tangent} = \frac{\sigma}{\omega\varepsilon} \tag{2.86} \]

Argand diagram of the complex permittivity. A horizontal real axis labeled Re of epsilon-c and a vertical imaginary axis labeled Im of epsilon-c. A point epsilon-c sits in the fourth quadrant, at real part epsilon on the positive real axis and imaginary part minus sigma over omega below the axis. A dotted vector from the origin to epsilon-c makes an angle phi below the positive real axis. — Figure 2.8: Plotting the complex permittivity, \(\varepsilon_c\), as a vector in the complex plane, the loss tangent is the tangent of the angle that the vector makes with the real axis, i.e. \(\tan\phi = \sigma/(\omega\varepsilon)\).
Extended description
The figure is an Argand (complex‑plane) diagram. The horizontal axis is the real part, \(\operatorname{Re}(\varepsilon_c)\); the vertical axis is the imaginary part, \(\operatorname{Im}(\varepsilon_c)\). The complex permittivity \(\varepsilon_c\) is drawn as a point in the fourth quadrant: its real coordinate is \(\varepsilon\) (marked on the positive real axis) and its imaginary coordinate is \(-\sigma/\omega\) (marked below the origin on the vertical axis). A dotted line from the origin to \(\varepsilon_c\) represents the permittivity vector, and the angle \(\phi\) between this vector and the positive real axis opens downward (clockwise). Because the vector lies below the real axis, \(\varepsilon_c\) has a negative imaginary part, which represents loss; the steeper the angle \(\phi\), the larger the loss tangent \(\tan\phi = \sigma/(\omega\varepsilon)\).

The loss tangent describes the loss characteristics of a medium. In a lossless material, \(\sigma/(\omega\varepsilon) = 0\). If \(\sigma/(\omega\varepsilon) \ll 1\), the material is called a good dielectric. If \(\sigma/(\omega\varepsilon) \gg 1\), the material is called a good conductor, and if \(\sigma/(\omega\varepsilon) = \infty\) the material is a perfect conductor. The concept of a “perfect conductor” is more of a mathematical idealization than an actual physical reality. Even so, highly conducting materials are often approximated as perfect conductors to simplify the mathematics while providing valuable physical insight into a problem.

Other definitions and terminology pertaining to propagation in lossy media that you will need to be familiar with are the following:

Skin depth: The skin depth, \(\delta\), is the distance in which the wave decays by \(e^{-1}\). Since a wave propagating in a lossy medium decays as \(e^{-\alpha z}\), we have \(-1 = -\alpha\delta\), or \(\delta = 1/\alpha\).
Decibels: Defined in terms of the electric field amplitude as \(\operatorname{dB} \equiv 20\log_{10}(E_2/E_1)\) or in terms of power as \(\operatorname{dB} \equiv 10\log_{10}(P_2/P_1)\). For example, if a certain wave has an initial amplitude of 10 V/m, but decays to 5 V/m after propagating 1 meter, then we have: \(20\log_{10}(5/10) \approx -6\) dB/meter. Note that often the negative sign is dropped, and would simply be referred to as a “6 dB per meter loss.”
Nepers: Defined as \(\operatorname{Nepers} \equiv \log_e(E_1/E_2)\). In a lossy medium, the loss in Nepers/meter \(= \log_e(|E_1|/|E_1|e^{-\alpha}) = \log_e(e^{\alpha}) = \alpha\). Therefore, the attenuation constant \(\alpha\) has the units Nepers/meter (Np/m).

Example 2.4

Human brain tissue has a relative permittivity and conductivity of \(\varepsilon_r \approx 55\) and \(\sigma \approx 2.1\) S/m, respectively, at 2.45 GHz [2]. This is a frequency of interest since many cell phones (and microwave ovens!) operate near this frequency. Assume that a plane wave at 2.45 GHz is normally incident from free space onto a human brain. Find the complex permittivity, complex impedance, and loss tangent.

Solution

The complex permittivity of the brain tissue is

\[ \varepsilon_{\text{brain}} = \varepsilon_r\varepsilon_o\left(1 - j\frac{\sigma}{\omega\varepsilon_r\varepsilon_o}\right) \] \[ = (55)(8.854\times10^{-12})\left\{1 - j\frac{2.1}{(2\pi\times2.45\times10^{9})(55)(8.854\times10^{-12})}\right\} \] \[ = (4.87 - j1.36)\times10^{-10}\ \text{F/m} \]

From this we obtain the complex impedance

\[ \eta_{\text{brain}} = \sqrt{\frac{\mu_o}{\varepsilon_{\text{brain}}}} = \sqrt{\frac{4\pi\times10^{-7}}{(4.87 - j1.36)\times10^{-10}}} \] \[ = 49.4 + j6.77\ \Omega = 49.9\,e^{j0.136}\ \Omega \]

The loss tangent is

\[ \operatorname{loss\ tangent} = \frac{\sigma}{\omega\varepsilon_r\varepsilon_o} = \frac{2.1}{(2\pi\times2.45\times10^{9})(55)(8.854\times10^{-12})} = 0.280 \]

Example 2.5

Using the results from the previous example, find the phasor and time domain expressions for the electric and magnetic fields propagating in brain tissue at 2.45 GHz. Assume that the electric field vector points in the x‑direction, propagates in the z‑direction, and has a maximum amplitude of 10 V/m just below the surface of the brain tissue at \(z=0\). What is the skin depth at this frequency? After what depth has the field amplitude been attenuated by 100 dB?

Solution

Because the brain tissue is a lossy material, the wave vector will be complex and the electric field will have the general form

\[ \overline{E}(z) = E_o e^{-jk_{\text{brain}}z}\hat{x} = E_o e^{-j(\beta - j\alpha)z}\hat{x} = E_o e^{-j\beta z}e^{-\alpha z}\hat{x} \]

The radian frequency of the wave is \(\omega = 2\pi\times2.45\times10^{9} = 1.54\times10^{10}\) rad/s. The complex propagation constant can then be found from the complex permittivity obtained in the previous example:

\[ k_{\text{brain}} = \omega\sqrt{\mu_o\varepsilon_{\text{brain}}} = 1.54\times10^{10}\sqrt{(4\pi\times10^{-7})(4.87 - j1.36)\times10^{-10}} \] \[ = 384 - j52.7 = \beta - j\alpha \]

Using these values for \(\beta\) and \(\alpha\) we can write the phasor and time‑domain expressions for the electric field:

\[ \overline{E}(z) = 10\,e^{-j384z}e^{-52.7z}\hat{x}\ \text{V/m} \] \[ \overline{\mathcal{E}}(z,t) = 10\cos(1.54\times10^{10}t - 384z)\,e^{-52.7z}\hat{x}\ \text{V/m} \]

The corresponding magnetic field phasor is found using the complex impedance found in the previous example:

\[ \overline{H}(z) = \frac{1}{\eta_{\text{brain}}}\hat{z}\times\overline{E}(z) = \frac{\hat{z}\times\overline{E}(z)}{49.9\,e^{j0.136}} = \frac{10\,e^{-j384z}e^{-52.7z}\hat{y}}{49.9\,e^{j0.136}} \] \[ = 0.201\,e^{-j0.136}e^{-j384z}e^{-52.7z}\hat{y}\ \text{A/m} \]

Notice the phase shift (0.136 radians, or \(7.79^\circ\)) between the electric and magnetic fields. This phase shift is clearly apparent in the time domain form of the magnetic field:

\[ \overline{\mathcal{H}}(z,t) = 0.201\cos(1.54\times10^{10}t - 384z - 7.79^\circ)\,e^{-52.7z}\hat{y}\ \text{A/m} \]

The skin depth at 2.45 GHz is simply the inverse of the attenuation constant:

\[ \delta = \frac{1}{\alpha} = \frac{1}{52.7} = 19.0\ \text{mm} \]

To find the depth at which the field has been attenuated by 100 dB, first note that a decrease in 100 dB corresponds to a field amplitude that has decreased to \(1\times10^{-5}\) of its initial value. Therefore, we find

\[ 1\times10^{-5}E_o = E_o e^{-\alpha z} \;\;\rightarrow\;\; z = -\ln(1\times10^{-5})/\alpha = 21.8\ \text{cm} \]

Note that this requires a very large head!

2.6.5 Example of a Lossy Medium: Wave Propagation in Seawater

As an example of propagation in a lossy medium, let us look at a wave propagating in seawater at different frequencies. This example will show that one needs to be careful in making general conclusions about wave propagation in a lossy medium.

Seawater has a permittivity of \(\varepsilon \approx 79\varepsilon_o\) and an electrical conductivity of \(\sigma \approx 4\) S/m (siemens/meter; a siemen has the units of \(\Omega^{-1}\)). The loss tangent for seawater is plotted as function of frequency in Figure 2.9. For frequencies less than 100 MHz (\(10^{8}\) Hz), the loss tangent is greater than 10 and the seawater behaves like a good conductor. For frequencies greater than 10 GHz (\(10^{10}\) Hz), the loss tangent is less than 0.1 and seawater behaves like a good dielectric.

Log-log plot of the loss tangent of seawater versus frequency. The loss tangent falls as a straight line from about 10 to the 5th at 10 kilohertz down to about 10 to the minus 3 at 1 terahertz, crossing 1 near a few gigahertz. — Figure 2.9: Loss tangent of seawater.
Extended description
A log‑log graph. The horizontal axis is frequency in hertz, from \(10^{4}\) to \(10^{12}\) Hz; the vertical axis is the loss tangent \(\sigma/(\omega\varepsilon)\), from \(10^{-3}\) to \(10^{5}\). The curve is a straight, downward‑sloping line: the loss tangent decreases by one decade for every decade increase in frequency (an inverse relationship, because the loss tangent is proportional to \(1/\omega\)). It starts near \(10^{5}\) at \(10^{4}\) Hz and falls to about \(10^{-3}\) at \(10^{12}\) Hz, passing through a loss tangent of 10 near \(10^{8}\) Hz (100 MHz) and through 0.1 near \(10^{10}\) Hz (10 GHz). Above a loss tangent of 10 seawater acts as a good conductor; below 0.1 it acts as a good dielectric.

In Figure 2.10, the loss in decibels per meter (dB/m) is plotted for a wave in seawater as a function of frequency. Compare the results in Figure 2.9 and Figure 2.10 and note that even though seawater is a good conductor for \(f < 100\) MHz and a good dielectric for \(f > 10\) GHz, the loss per meter in the “good conducting” region is lower than in the “good dielectric” region! Therefore, we cannot make the generalization that propagation loss is directly proportional to the conductivity of the medium.

Log-log plot of propagation loss in decibels per meter for seawater versus frequency. The loss rises with frequency then levels off: it climbs from a few dB per meter at 10 kilohertz to a plateau of roughly 700 to 800 dB per meter above about 1 gigahertz. — Figure 2.10: Propagation loss per meter in seawater.
Extended description
A log‑log graph. The horizontal axis is frequency in hertz, from \(10^{4}\) to \(10^{12}\) Hz; the vertical axis is propagation loss in decibels per meter, from about 1 to \(10^{3}\). Unlike the loss tangent, the loss per meter *increases* with frequency: it rises roughly as a straight line (proportional to \(\sqrt{f}\)) from a few dB/m at \(10^{4}\) Hz through about \(10^{2}\) dB/m near \(10^{8}\) Hz, then bends over and saturates at a plateau of roughly 700–800 dB/m for frequencies above about \(10^{9}\) Hz. The key point: the plateau (the “good dielectric” region) has a *higher* loss per meter than the low‑frequency “good conductor” region, so a large loss tangent does not imply a large loss per meter.

References

G. Schmid, G. Neubauer, P. R. Mazal, F. Alesch, and U. M. Illievich, “Dielectric properties of brain tissue: measurements on humans,” presented at the 5th COST 281 MCM and Workshop, Budapest, Nov. 15–16, 2003.