Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The complex permittivity

Consider a homogeneous, linear medium with permittivity $ε$ , permeability $μ$ and conductivity $σ$ , free of impressed sources. In time-harmonic form (phasors with the time factor $e^{j ω t}$ suppressed, Section 5.6) the current in the medium is the conduction current of Ohm's law, $J = σ E$ (Section 4.2.6), and the generalised Ampère law reads

\nabla \times H = σ E + j ω ε E = j ω (ε - j \frac{σ}{ω}) E = j ω ε_{c} E . (6.2.1)

The conduction current and the displacement current are both proportional to $E$ , so the two can be gathered into one term with the complex permittivity

ε_{c} = ε - j \frac{σ}{ω} = ε (1 - j \frac{σ}{ω ε}) . (6.2.2)

With this substitution Eq. (6.2.1) has exactly the form of the lossless equation. Every result of the lossless plane wave of Section 6.1 therefore stays valid in the conducting medium if $ε$ is replaced by $ε_{c}$ . This is the whole trick of the section: we do not solve Maxwell's equations again, we only follow what the complex $ε_{c}$ does to the formulas. A dielectric with polarisation losses, $ε = ε^{'} - j ε^{''}$ , is included in the same way, since its $ω ε^{''}$ acts exactly like a conductivity.

The propagation constant

In the lossless medium the wave number is $k = ω μ ε$ and the wave $E = E_{0} e^{- j k \cdot r}$ . With $ε \to ε_{c}$ the wave number becomes complex,

k = ω^{2} μ ε_{c} = β - j α, (6.2.3)

and for a wave travelling along $+ z$ , $e^{- j k z} = e^{- α z} e^{- j β z}$ . It is customary to work instead with the propagation constant $γ = j k$ , the coefficient of $- z$ in the exponent:

γ^{2} γ = - k^{2} = - ω^{2} μ ε_{c} = j ω μ (σ + j ω ε), = j k = α + j β . (6.2.4)

Both notations describe the same wave. The lecture decks use $k = β - j α$ where they continue the lossless notation of Section 6.1, and $γ = α + j β$ everywhere else. In this course we use $γ$ , and write $k$ only where a problem is posed with it.

The plane wave in a lossy medium

In a medium $(ε, μ, σ)$ a uniform plane wave travelling along $+ z$ and polarised along $x$ is

E E (z, t) = a_{x} E_{0} e^{- γ z} = a_{x} E_{0} e^{- α z} e^{- j β z}, = a_{x} E_{0} e^{- α z} cos (ω t - β z), (6.2.5)

with the propagation constant

γ = α + j β = j ω μ (σ + j ω ε), (6.2.6)

where $α$ is the attenuation constant $[Np/m]$ and $β$ the phase constant $[rad/m]$ , both taken positive. The wavelength is $λ = 2 π / β$ and the phase velocity $v_{p} = ω / β$ .

In plain words

The wave still oscillates in space with the period $λ = 2 π / β$ and moves with $v_{p} = ω / β$ , but its amplitude shrinks by the same factor $e^{- α Δ z}$ over every distance $Δ z$ . Everything the medium does to the wave is written in the two numbers $α$ and $β$ .

Figure 6.2.1 shows the field at one instant. The envelope $\pm E_{0} e^{- α z}$ is fixed in space, while the oscillation inside it moves to the right with $v_{p}$ . The attenuation is usually quoted in decibels. Since the field ratio over a distance $z$ is $e^{- α z}$ and $20 lo g_{10} e = 8.686$ ,

A_{dB} = 20 lo g_{10} e^{α z} = 8.686 α z, (6.2.7)

so a field decaying as $e^{- α z}$ loses $8.686 α dB$ per metre. The unit neper (Np) is the natural-logarithm counterpart of the decibel: a field ratio $e^{- 1}$ is $1 Np$ , that is $8.686 dB$ .

The field of a plane wave in a lossy medium at t=0, drawn for \alpha\lambda=0.45. The oscillation has the period \lambda=2\pi/\beta in space and moves with the phase velocity, the envelope \pm E_0e^{-\alpha z} (dashed) stays where it is. Over one wavelength the amplitude falls by the factor e^{-\alpha\lambda}. — **Figure 6.2.1.** The field of a plane wave in a lossy medium at $t = 0$ , drawn for $α λ = 0.45$ . The oscillation has the period $λ = 2 π / β$ in space and moves with the phase velocity, the envelope $\pm E_{0} e^{- α z}$ (dashed) stays where it is. Over one wavelength the amplitude falls by the factor $e^{- α λ}$ .

Quick check

A field loses $20 dB$ over one metre of a medium. What is $α$ , and what fraction of the field amplitude is left after one metre?

$α = 20/8.686 = 2.30 Np/m$ , and the amplitude ratio is $1 0^{- 20/20} = 0.1$ , the same as $e^{- 2.30}$ .

6.2.1 Plane waves in a lossy medium

The complex permittivity

The propagation constant