Characterization of Transmission Lines on Lossy or Dispersive Substrates With a Complete Model of a Series Resistor for Impedance Standard of Scattering Parameter Measurements

This article presents a complete model and process for obtaining a solution for a series resistor with frequency-dependent resistance and inductance to characterize planar transmission lines (TLs) after a one-tier multiline thru-reflect-line (TRL) calibration. The impedance of TL characterization actually is the reference impedance of scattering parameter (S-parameter) measurements and is not easy to determine for TLs on lossy or dispersive substrates. To account for parasitic effects at high frequencies, the series resistor is modeled as a lossy TL segment with a frequency-dependent resistance and inductance per unit length with additional discontinuities between the TL and the resistor. The TL characteristic impedance can then be evaluated based on the de-embedded S-parameters of series resistors with lengths
$l_{s}$
and
$2l_{s}$
after calibration, in which the TL propagation constant is determined as well to complete the TL characterization. The proposed method is examined by using coplanar waveguides (CPWs) fabricated with an integrated passive device (IPD) process on high-resistivity silicon and fabricated with standard CMOS technology to demonstrate the frequency dependence of the resistance of the series resistors at high frequencies. Independent calibration comparison measurements are conducted to validate the acquired TL characteristic impedances up to 110 GHz.