Calculation of self- and mutual impedances in planar sandwich inductors

High-frequency planar magnetic components, employing thin film and thick film technology, have become important components in applications, such as filters and switching converters, due to their ease of manufacture and reliability. In a previous paper, the authors established a frequency dependent impedance formula for planar coils on a magnetic substrate that is infinitely thick. In this paper, two new impedance models are described: the first is for planar coils on a magnetic substrate of finite thickness, and the second represents a planar coil sandwiched between two substrates. The models include the electrical conductivity of the magnetic material so that the effects of eddy currents, particularly at high frequencies, are taken into account. The eddy currents reduce the inductance and increase the losses associated with the device. The new impedance formulas are derived from Maxwell's equations. Simulations were carried out on a typical device, using finite element analysis, and the results validate the new formulas. This paper establishes the frequency limitations of lossy magnetic substrates.