How do I calculate the radiation loss from a microstrip bend or discontinuity?

Radiation loss from a microstrip bend or discontinuity occurs because the abrupt change in geometry causes the guided electromagnetic field to partially convert into radiated (unguided) waves. The radiation loss increases with frequency (proportional to f^2 for most discontinuities) and becomes a significant contributor to total circuit loss at millimeter-wave frequencies. For a right-angle (90-degree) microstrip bend: the radiation loss is approximately P_rad/P_inc = (2 pi h / lambda_0)^2 x (W/h)^2 x F(Er) where h is the substrate height, lambda_0 is the free-space wavelength, W is the trace width, and F(Er) is a function of the dielectric constant (approximately 1 for low Er, decreasing for high Er as more field is confined in the substrate). At 30 GHz on a typical PCB (h = 0.5 mm, W = 1 mm, Er = 3.5): the radiation loss from a single right-angle bend is approximately 0.2-0.5 dB. At 77 GHz on the same substrate: approximately 1-3 dB per bend. Mitigation techniques include: mitered (chamfered) bends (cutting the outer corner by 50-70% reduces the discontinuity and cuts radiation loss by approximately 50%), curved bends (a smooth radius bend with R > 3W has even lower radiation), thinner substrates (reducing h by 2x reduces radiation loss by 4x), and higher dielectric constant substrates (Er = 10 vs. Er = 3 reduces radiation by approximately 70% because more of the field is confined in the substrate).