Closed-Form Formulas for Frequency-Dependent Per-Unit-Length Inductance and Resistance of Microstrip Transmission Lines That Provide Causal Response

Closed-form expressions for the approximation of frequency variations of per-unit-length resistance and inductance of microstrip transmission lines are presented. The provided expressions guarantee causality of the time-domain response. They are valid from dc up to the highest frequencies for which the quasi-static analysis of microstrip transmission lines is applicable. Additionally, the expressions take into account the finite width of the ground plane and can deal with a broad range of dimensions of microstrip lines. The correctness of the formulas is experimentally verified.