Simple formulas for the input impedance, bandwidth, and radiation efficiency of a rectangular patch

Simple approximate formulas for the input impedance, bandwidth, and radiation efficiency of a resonant rectangular microstrip patch are derived. Although approximate, these formulas are derived from asymptotic approximations of a rigorous Sommerfeld solution. The formulas thus provide physical insight into the effects of the substrate parameters on the patch properties, in addition to providing approximate design formulas. The formulas become increasingly accurate for thinner substrates. Results are shown for a nonmagnetic substrate with f=3.0 GHz, b=0.25 cm (b/ lambda /sub 0/=0.025), and W/L=1.5.<>