Symmetric-Cell EBG Theory and Its Applications to Vias Daisy Chain for Residual Stub Detection

This study employs a daisy chain topology by periodic back-drilled vias and transmission lines for the detection of the residual via stubs. There will be obvious stopbands due to the minor parasitic capacitances of the via stubs. An extended symmetric-cell electromagnetic bandgap (S-C EBG) theory is proposed for periodic structures with symmetric unit cells and finite periods. Theoretical formulas are derived for the three major EBG characteristics of daisy chain, as well as the detection capability of the second and fourth stopbands for residual via stubs with different length. Furthermore, approximate formulas are obtained so as to rapidly estimate the S-parameters versus the via stub’s capacitance and the characteristic impedance, delay time and attenuation constant of transmission lines. Through the simulation results, the detection charts of S
dd21,min
in the second and fourth stopbands versus via stub length are established to measure the residual stub length of the back-drilled vias.