Analytical equivalent circuit modeling and analysis of complex BGA for 3D silicon-interposer packaging

Through analyzing different cross-sections of the complex ball grid array (CBGA) sandwiched by silicon interposers, various distributed capacitances of the CBGA and the significance of the CBGA's coplanar capacitances are revealed. From the analyses, the composition of the CBGA's capacitance is found to have similarity with that of the coplanar waveguide's capacitance. Based on the electric-field coupling mechanism of the coplanar waveguide, geometrical analysis of the CBGA, and analysis of multiconductor transmission lines, analytical equations are derived to allow accurate modeling the equivalent circuit of the CBGA for 3D silicon-interposer packaging. The modeled equivalent circuit of the CBGA is validated with the full wave electromagnetic simulation, and it is demonstrated to have a good accuracy up to 40 GHz. Moreover, the CBGA structure can be further optimized for impedance matching with the guidance of the derived analytical equations.