S Matrix versus ABCD Chain Matrix Formulation in Probe-tip Calibrations

Abstract

We present some advantageous features of the chain ABCD matrix formulation in comparison with S-matrix formulation in characteristic impedance extraction with probetip calibrations. The procedure is based on the known measurement technique of two different on-wafer line standards [1]. An initial off-wafer LRM or TRL calibration with standard calibration substrate is assumed. The first feature is the formulation of the whole extraction problem in terms of the ABCD chain matrix. It allows us to omit one of the limitations of the true traveling waves based Smatrix [3] asymmetry of a general reciprocal transition between two different waveguides, in particular probe-tip-line junction. This limitation is a difference between complex characteristic impedances of both waveguides. On the contrary, the symmetry of its admittance matrix or equivalently the determinant AD-BC of its chain matrix is not influenced by this effect. If one models the probe-tip-line transition by use of only a shunt admittance or cascade of a shunt admittance and a series impedance, the A element of chain matrix is not influenced by these parasitics and is equal to one. This allows immediate characteristic impedance extraction without analyzing any of these transition models. Moreover, one is able to choose the valid model and verify if any of the possible equivalent parasitic elements can be neglected. The next feature is that the chain matrix formulation modified in a specific way allows the extraction of the equivalent position of the reference planes at the probe-tips. Equivalent means that their locations are equal to their physical locations in case of negligible influence of the series parasitic impedance (usually parasitic inductance). This is very interesting property if we take into consideration that the exact position of the probe tips and reference planes are not known. The last feature allows us to extract the characteristic impedance of the lines without modeling the transition structure under assumption that the condition A=D of its chain ABCD matrix (the model of a reciprocal structure is symmetric) is approximated with good accuracy. Such a formulation takes automatically into consideration even a distributed nature of the transition, which can be of importance at mm-wave frequencies. All the equivalent elements values of the models are extracted from analytical equations for every frequency point. Thus their frequency behavior can be investigated and possible validity of the model (constant equivalent element values) proved. I General Waveguide Theory With the increasing use of planar transmission lines, junctions between waveguides supporting lossy hybrid modes have become common. In this case the classical microwave circuit theories fail. The classical waveguide theory fails to