What is the skin effect and how does it influence conductor loss at microwave frequencies?

The skin effect causes high-frequency currents to concentrate in a thin layer near the surface of a conductor, increasing effective resistance and power loss. At microwave frequencies, current penetrates only a few micrometers into the conductor surface. The skin depth delta = sqrt(2 / (omega × mu × sigma_conductor)) = sqrt(1 / (pi × f × mu_0 × sigma)), where f is frequency, mu_0 is the permeability of free space, and sigma is the conductor conductivity. For copper (sigma = 5.8×10^7 S/m): skin depth at 1 GHz = 2.09 μm, at 10 GHz = 0.66 μm, at 100 GHz = 0.21 μm. The surface resistance Rs = 1/(sigma × delta) = sqrt(pi × f × mu_0 / sigma), which increases as the square root of frequency. For copper: Rs = 0.026 ohms/square at 10 GHz. Conductor loss in a transmission line is proportional to Rs and to the current density distribution: alpha_c = Rs / (2 × Z_0 × w) for microstrip (approximate), where w is trace width and Z_0 is characteristic impedance. At 10 GHz, a 50-ohm microstrip on 0.254 mm substrate has conductor loss of approximately 0.02-0.03 dB/mm, making it the dominant loss mechanism (exceeding dielectric loss for most substrates). Surface roughness on PCB copper (typically 1-5 μm RMS) significantly increases conductor loss at frequencies where the roughness approaches the skin depth; at 30+ GHz, roughness can double or triple the smooth-conductor loss prediction.