Causal Characteristic Impedance Determination Using Calibration Comparison and Propagation Constant

It is common to determine the characteristic impedance
${Z}_{0}$
of a low-loss transmission line using the measured propagation constant
$\gamma$
through
${Z}_{0}=\gamma/({G}+\mathrm{j}\omega {C})$
, assuming that the parallel conductance
${G}=0$
and that the parallel capacitance
$C$
is constant. However, if the line is a coplanar waveguide or a microstrip, two dielectrics (substrate and air) are involved and its effective dielectric constant is frequency-dependent, which makes
$C$
also frequency-dependent. If so, the assumption
$G=0$
is inconsistent with causality constraints. This paper presents an attempt at determining
${Z}_{0}$
by augmenting the above method in a way consistent with causality. A method of calibration comparison with 2nd-tier multiline TRL is used to extract
${Z}_{0,\text{meas}}$
and
$\gamma_{\text{meas}}$
, following 1st-tier probe-tip calibration that is assumed to be valid at least up to 67 GHz. Then, a causal model,
${Y}_{\mathrm{p},\mathrm{model}}$
, of parallel admittance
${G}(\omega)+\mathrm{j}\omega {C}(\omega)$
is built from
${Y}_{\mathrm{p}}^{\prime}=\gamma_{\text{meas}}/{Z}_{0,\text{meas}}\ (\underset{\sim}{<} \ 67$
GHz) by network synthesis.
${Y}_{\mathrm{p},\mathrm{model}}$
and
$\gamma_{\text{meas}}$
are then used to determine
${Z}_{0}$
including
$\underset{\sim}{>} 67$
GHz. The method is applied to a CMOS transmission line from 20 MHz to 220 GHz. A well-built model gives well-behaved and plausible estimate of causal
${Z}_{0}$
over the entire measurement frequency range.