6.2.4 Low-loss dielectrics and good conductors

The loss tangent

When a medium has some conductivity, the ratio $\sigma /\omega \epsilon$ decides how it behaves. It has a direct meaning in Ampère's law Eq. (6.2.1): it compares the two currents on the right-hand side, the conduction current $J_c=\sigma E$ and the displacement current $J_d=\mathrm{j}\omega \epsilon E$. In the complex plane (Figure 6.2.4a) the two are perpendicular, $J_c$ in phase with $E$ and $J_d$ leading it by $90^{\circ }$, and the angle ${\delta }_c$ by which their sum is tilted away from $J_d$ is the loss angle.

The loss tangent is a property of the medium at a frequency. Since $\sigma /\omega \epsilon$ falls as $1/f$ for a constant $\sigma$, every conductor becomes a dielectric at a high enough frequency and every lossy dielectric a conductor at a low enough one. The crossover $\sigma /\omega \epsilon =1$ lies at the frequency

$889\mathrm{MHz}$ for seawater, $9.00\mathrm{MHz}$ for moist sandy soil (Exercise 6.2.1) and far above any radio frequency for a metal. Figure 6.2.4b places some common cases.

Perfect dielectrics

For a dielectric material the conductivity is zero and the relative permeability one: $\epsilon ={\epsilon }_r{\epsilon }_0$, $\mu ={\mu }_0$, $\sigma =0$. Then $k=\sqrt{{\omega }^2{\epsilon }_r{\epsilon }_0{\mu }_0}$ is real, so that

Waves propagating in a dielectric do not decay. At the same frequency the wave is slowed and shortened by the refractive index $n=\sqrt{{\epsilon }_r}$, $\lambda ={\lambda }_0/\sqrt{{\epsilon }_r}$ (Figure 6.2.5), and for the same $|E|$ the magnetic field $|H|=|E|/\eta$ is $\sqrt{{\epsilon }_r}$ times larger than in air.

Low-loss dielectrics: the approximate formulas

Write Eq. (6.2.6) as $\gamma =\mathrm{j}\omega \sqrt{\mu \epsilon }(1-\mathrm{j}p{)}^{1/2}$ with $p=\sigma /\omega \epsilon \ll 1$, and expand with the binomial series $(1+x{)}^{1/2}\approx 1+\frac{x}{2}-\frac{x^2}{8}$ for $x=-\mathrm{j}p$:

In the same way $(1-\mathrm{j}p{)}^{-1/2}\approx 1+\mathrm{j}p/2$ in Eq. (6.2.12).

When the loss is a polarisation loss, $\sigma \to \omega {\epsilon }^{\prime \prime }$, the attenuation constant grows in proportion to $f$, $\alpha \approx \frac{\pi f}{c}\sqrt{{\epsilon }_r^{\prime }}\tan {\delta }_c$. A useful rule of thumb follows from $\alpha =\frac{\beta }{2}\tan {\delta }_c$ and $\lambda =2\pi /\beta$:

A cable with PTFE insulation ($\tan {\delta }_c\approx 3\times 10^{-4}$) therefore loses only $0.008\mathrm{dB}$ per wavelength in its dielectric. Most of its loss is in the copper, through the skin effect of the next subsection. Table 6.2.1 collects typical values.

Table 6.2.1. Relative permittivity and loss tangent of some dielectrics.

Material	${\epsilon }_r^{\prime }$	$\tan {\delta }_c$	Where used
PTFE (Teflon), 10 GHz	$2.1$	$\approx 3\times 10^{-4}$	microwave substrates, coaxial cables
Polyethylene, 10 GHz	$2.25$	$\approx 4\times 10^{-4}$	cable insulation
Alumina, 10 GHz	$9.8$	$\approx 2\times 10^{-4}$	thin-film circuits
FR-4 (PCB), 1 GHz	$4.4$	$\approx 0.02$	ordinary printed circuit boards
Water, 2.45 GHz	$78$	$\approx 0.13$	the microwave oven (Worked example 6.2.1)
Seawater, 30 GHz${}^{†}$	$81$	$0.030$

${}^{†}$ With the model $\sigma =4S/m$, ${\epsilon }_r=81$ of Problem 6.2.1.

Good conductors

For $\sigma /\omega \epsilon \gg 1$ the displacement current is negligible against the conduction current, and Eq. (6.2.6) becomes

since $\sqrt{\mathrm{j}}=e^{\mathrm{j}\pi /4}=(1+\mathrm{j})/\sqrt{2}$. The real and the imaginary parts are equal. Their common value defines the skin depth.

Figure 6.2.6 shows how fast this is: at $z=\lambda =2\pi \delta$ only $e^{-2\pi }=0.19\%$ of the field is left. The impedance is small and inductive. Copper ($\sigma =5.8\times 10^{7}S/m$) at $1\mathrm{MHz}$ has $|{\eta }_c|=0.37\mathrm{m}\Omega$ against $377\Omega$ in air, so in a metal $H$ is large and $E$ almost vanishes. That is why a metal surface looks like a short circuit to an incoming wave (Section 6.3).