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SOLUTION OF BASIC EQUATIONS OF ELECTRODYNAMICS 193

Since at microwave frequencies the skin-depth is so tiny the surface electric current follows the
roughness profile, effectively increasing the current path and thus the Ohmic loss. The
simplified model of metal surface with the equal to the RMSH peak-to-valley distance, i.e. the
10
bump height ∆, is demonstrated in Figure 4.4.4a . Such simple approximation let predict quite
accurately the increase in attenuation due to surface roughness as [5]

2
= 1 + � ∙ tan −1 �1.4 �� ( − 1) (4.91)

Here the coefficient RF is the expected growth of conductivity current path on rough relative
to perfectly flat surface of copper conductor with = 5 ∙ 10 [S/m]. Typically this coefficient
7
9
is between 1 (no bumps) and 3. The plot in Figure 4.4.4b illustrates how the attenuation could
growth.

Figure 4.4.4 Attenuation factor vs. roughness RMS height
4.4.2 Surface Resistivity

The Ohms law in differential form tell us that an EM wave penetrating the metal surface should
2 is
induce there the electric current = [A/m ]. Meanwhile, the penetration depth
so tiny that this volume current is really could be regarded as the surface with the density =

= = [A/m]. Here the coefficient is called surface conductivity and

equal to
⁄
= ≅ 503�1 = 503� [1/Ω] (4.92)
⁄

It is more common in engineering practice to use the inverse constant measured in Ohms and
called surface resistivity
= 1 ≅ 0.002� [Ω] (4.93)
⁄
⁄

Pay attention that ~� meaning that the RF resistivity and dissipation in metals grows

proportional to square root of frequency.
4.4.3 Conclusion
We have derived directly from wave equations the radiation properties of three infinitesimal or
customary called elementary radiators: electrical dipole, magnetic dipole that can be realized
as electrically small loop, and Huygens’s wavefront radiator. In theory, we can follow the same

10 Courtesy of Dr. Yuriy Shlepnev [5].