Compact Fractional-order Model of On-Chip Inductors with BCB on High Resistivity Silicon

Fractional-order calculus exhibits clear advantages over modeling the nonlinear and complex physics phenomena. In this article, fractional-order calculus is introduced to the field of on-chip inductor modeling. A novel fractional-order 1- $\pi $ inductor equivalent circuit model is proposed to characterize on-chip inductors based on silicon-benzocyclobutene (Si-BCB) technology. The proposed model, which contains six integer-order circuit elements and three fractional-order elements, is sufficient to describe frequency-dependent effects, such as the skin effect, proximity effect, and distributed effect. Three fractional-order elements (one fractional-order inductor and two fractional-order capacitors) introduced in the above model are major contributors to improving the model’s accuracy and bandwidth. The parameter extraction method is discussed in this article, and the direct simulation method is employed to solve fractional-order equations. The accuracy of the proposed fractional-order model is validated up to 40 GHz. An excellent agreement is observed between the model and the measurements.