Universal closed-form expression based on magnetic flux density for the inductance of Tapered Through-Silicon Vias (T-TSVs)

In this paper, a universal closed-form expression for the parasitic inductance of tapered through-silicon vias (T-TSVs) with a frequency of up to 20 GHz is proposed. The expression, which considers skin and proximity effects, can be used to calculate the self-partial inductance and mutual-partial inductance, considering TSVs located in adjacent layers or in the same layer. When the slope angle is 90°, the obtained formulas can be reduced to the formulas of cylindrical TSVs. The comparison between the results of the proposed formulas and the results of the three-dimensional quasi-static field solver (Q3D) demonstrates that the proposed formulas are extremely accurate, with a maximum error of 2%.