A Closed-Form Electrical Model of Skin and Proximity Effects for Circular Microstrip Inductors

This paper develops a closed-form electrical model of skin and proximity effects using a three-ladder circuit and verifies it with measured results for a circular microstrip inductor. The complete equivalent electrical model of the inductor consists of the three-ladder circuit using RL elements, substrate capacitance, and adjacent metal capacitance of the inductor. Closed-form expressions are formulated for self-inductance, substrate capacitance and adjacent metal capacitance from the equivalent electrical model of the inductor. All the model parameters of the ladder circuit are accurately derived and calculated using layout and process parameters. Four planar circular spiral inductors are fabricated on FR4 substrate to verify the proposed models. The measured results, EM simulated, and analytically calculated results of the modelled three-ladder circuit are in close agreement with each other.