General Loss Model

This loss model includes the metalization surface resistance described  in Rdc/Rrf. It also includes the metalization reactance, composed of the surface reactance, Xdc and the surface inductance, Ls. Surface reactance, Xdc, is specified, in Ohms/square and uses the same reactance at all frequencies. The surface inductance, Ls, is specified in pH/square and models the reactance as being linearly proportional to frequency.

A superconductive effect known as “kinetic inductance” slows the velocity of the electrons with no loss of energy. This can be modeled as a surface inductance.

There are three recommended approaches to obtaining a value for Ls in superconductors. A first order approximation is to assume the metal is a perfect conductor.

RDC = 0    Rrf = 0     XDC = 0     Ls = 0

This model works well for moderate frequencies (less then 150 GHz) and moderate circuit dimensions which are much greater than the London depth of penetration.

The second approach is a model which is still valid at moderate frequencies, but includes effects due to kinetic inductance. The kinetic inductance is a function of temperature and can be approximated in the following manner:

The third model should be used to account for high frequency effects or effects due to small circuit dimensions. In these cases, the surface resistance’s proportionality to ω2 begins to dominate and the following model is suggested. The surface resistance is a function of frequency-squared, so you should use the equation below as the entry for RDC and use the FREQ function in the equation so that the value is valid for all analysis frequencies. Use the following equations for this model:1

The Surface Impedance, Zs, is modeled using the following equation:

where RDC and Ls are defined in the superconductor models given above.

1Shen, Z. Y., “High-Temperature Superconducting Microwave Circuits,” Boston, 1994, Artech House.