Uniform Plane Waves in Lossy Materials. Skin Depth.
All real materials have loss, to one extent or another. We will now consider uniform plane waves that propagate through lossy materials. The only loss mechanism we’ll consider here is that due to conduction current, `J =s`E , everywhere in an infinite space. So, for example, perhaps we have a current sheet J producing (or radiating) electromagnetic plane waves in an infinite lossy space:
Wave Equation for Lossy Media
We’ll derive the wave equation for this lossy space. We begin with the curl forms of Faraday’s and Ampere’s laws:
Ñ×`E = - jww`H
Ñ×`H = jwe`E+ `J = jwe`E+ s`E = (s+ jwe)`E
Notice that we now have a `J (=s`E) in Ampere’s law (2) in contrast to a lossless media in the previous lecture. Proceeding as we did in the previous lecture, there is no spatial variation of the current sheet and the space in the x or y directions, so we expect the `E or `H fields not to vary in x or y either. Consequently, with ¶/¶ x ® 0 and ¶/¶ x ® 0 become, respectively,
To form the wave equation for `E in this lossy space, we first take d / dz of (3)
Next, we substitute for dHx/ dz and dHy/ dz from (4) to give
As we learned in the previous lecture, because ` Js is uniform and oriented only in the ax direction, then the `E fields produced by this current sheet will be oriented only in the ax direction as well. From here onwards, we’ll focus only on Ex and its associated `H (an `Hy).
From (9), we’ll define the propagation constant Ñ such that
Ñ2 = jww(s + jwÑ)
leading from (9) to
This is the phasor wave equation for Ex in a lossy space. Notice that this equation is identical in form to the wave equation for V on a lossy transmission line in Lecture 20.
Solutions to the Wave Equation in a Lossy Space
The solutions to the wave equation (11) are
Ex(z) = E0 + e-yz + E0-e+yz
where E+0 and E-0 are complex constants. Defining the real and imaginary parts of Ñ as
As we saw in Lecture 20, these two terms are the phasor forms of waves propagating in the +z and –z directions, respectively, with attenuation:
Associated with this time varying electric field is a corresponding magnetic field. To determine this `H , we could derive the wave equation for `H from (1) and (2), as we did for `E , and then solve for `H . Alternatively, and more simply, we can determine `H directly from Faraday’s law:
Using we find that
The factor jww/Ñ in (18) we expect to be an impedance quantity. Using (10) we find
This quantity Ñc is the complex intrinsic impedance of the lossy space. It is a complex quantity, just as the characteristic impedance of a lossy TL was found to be complex in Lecture 20. Notice in (19) that as s Ñ0, Ñc ÑÑ as expected for a lossless space, as we saw in the previous lecture. Using this Ñc, the magnetic field associated with Ex in (16) for this lossy space is
From (16) and (20), notice that
• The fields do not vary spatially in planes perpendicular to the direction of propagation (±z ). Even though the space is lossy, these fields are still uniform plane waves, as in the lossless case.
• Ratios of perpendicular components of `E and `H equal ±Ñc .
. For the +z propagating wave:
. For the -z propagating wave:
This is a result similar to lossless spaces, but Ñ c is now complex for a lossy space.
• For a lossy space, the wavelength and wave speed are defined exactly the same for lossless spaces as Ñ = 2Ñ/Ñ and u =w/Ñ , respectively. However, Ñ ÑwÖwÑ for lossy spaces. Rather, from (15) Ñ= Im[Ñ].
We’ve now learned that as a plane wave propagates through a lossy space, it attenuates in amplitude as it propagates. For a UPW propagating in the +z direction, for example, then from (16) and (20)
The distance this electromagnetic wave must travel for the amplitude to be reduced by the factor eÑ1 is called the skin depth, Ñ, of the material. We can derive an equation for Ñ beginning with the magnitude of the electric field in (21) At some arbitrary position z0, from (22) while at some position Ñ meters further away
For this ratio to equal eÑ1 requires
As an example, consider copper which has an electrical conductivity s = 5.8×107 S/m, Ñ ÑÑ0 , and w Ñ w0 . Using (13) to calculate Ñ and hence Ñ, then using (25):
f Ñ =1/ Ñ 60 Hz 8530 wm 1 MHz 66.1 wm 10 MHz 20.9 wm 100 MHz 6.6 wm 1 GHz 2.09 wm
This skin depth of an electromagnetic wave in a lossy space is directly related to the skin effect in a round wire that we discussed in Lecture 9