Approximate Equivalent Circuits to Understand Tradeoffs in Geometry of On-Chip Inductors

We develop an approximate frequency-independent equivalent circuit for integrated inductors on semiconducting substrates. All values of LCR in the equivalent circuit can be calculated directly from inductor geometry. The equivalent circuit captures: the skin effect, the proximity effect, the substrate capacitive loss, and the distributed capacitance. With careful definition of inductor quality factor, we are able to calculate the contribution of substrate capacitive loss and benefits of a substrate shield. Our model was put to test by successfully predicting the published quality factors of various inductors and proving that their geometries are close to optimum.