A closed-form spatial Green´s function for the thick microstrip substrate

The spatial Green´s function for the open microstrip structure, especially with a thick substrate, is generally represented in the time-consuming Sommerfield integrals. Through the Sommerfield identity, a closed-form spatial Green´s function of a few terms is found from the quasi-dynamic images, the complex images, and the surface waves. With the numerical integration of the Sommerfeld integrals thus avoided, this closed-form Green´s function is computationally very efficient. Numerical examples show that the closed-form Green´s function gives less than 1% error for all substrates and source-to-field distances