Modeling of planar inductors with non-uniform flux distribution and non-linear permeability for high-density integration

High-density integrated point-of-load (POL) converter is today´s industry trend for portable electronic applications. Low-profile planar inductors can be used as substrates for 3D integrated POL converters to achieve very high power density. However, most planar inductors have non-uniform flux distribution. Some planar inductors, such as low-temperature co-fired ceramics (LTCC) inductors, even have non-linear permeability. This means that the conventional inductance calculation method, which is based on uniform flux and constant permeability, cannot be used directly for these planar inductors. This paper proposes some simple but sufficiently accurate models (one numerical, the other analytical) to calculate inductance for planar inductors with non-uniform flux distribution and non-linear permeability. Based on these models, several low-profile LTCC inductor substrates are designed for 3D integrated POLs. The measured inductance closely matches the predicted result, which means the proposed models have very good accuracy.