Analytical expressions for inductances of 3D air core inductors for integrated power supply

This work presents analytical expressions for the DC inductance of 3D air core inductors with circular cross section pillars (CCSP) and rectangular cross section pillars (RCSP). We consider the following four types of inductor structures: (1) a toroid with CCSP; (2) a toroid with RCSP; (3) a solenoid with RCSP; and (4) a solenoid with CCSP. For each type, a unique analytical model is developed for obtaining DC inductance. High frequency (1-100 MHz) effects on inductance are also discussed. The inductance values predicted by the proposed analytical models of the first three types of inductor structures are in an acceptable agreement with numerical Finite Element Analysis (FEA) solutions, where the maximum difference is 7.3%. Also, our analytical model for the fourth type inductor reduces the error, when correlated with FEA inductance value, up to 6× compared to previously published models. A comparison of results using proposed analytical expressions with published measured values as well as our measurement data demonstrates error ranging from 0.5 to 16.2%, while conventional formulae show errors of up to 143%. The results of the proposed models could serve as a good initial estimate for power supply on chip (PwrSoC) and power supply in package (PSiP) applications.