Analytical Port Extensions: Moving Phase Reference Planes Precisely

If you perform a standard Short-Open-Load-Thru (SOLT) calibration at the end of your coaxial cables, your reference plane is locked right at the cable face. However, when using custom material fixtures, your sample is often located deep inside a transmission line structure. To resolve the true material parameters, you must analytically shift the calibration planes using mathematical port extensions.

The Mathematics of Phase Rotation

An empty transmission line between your cable end and your material sample introduces a predictable linear phase shift without adding major magnitude loss. Analytical port extension mathematical algorithms manipulate the measured scattering matrix by multiplying the phase factor across the complex S-parameters.

For a transmission lead of physical length l and guided phase constant β, the phase factor is θ = β × l. The corrected reflection coefficient is calculated by rotating the measured phase backward: S_{11_Corrected} = S_{11_Measured} × e^j2θ. The factor of 2 accounts for the round-trip travel path of the reflected wave.

Key Calibration Steps

• Account for Medium Dispersion: If your extension occurs inside an empty coaxial line, the velocity factor is constant. If the extension occurs inside a waveguide structure, the phase velocity varies non-linearly with frequency. The calculation must account for waveguide dispersion or the phase rotation will over-correct your high-frequency data.
• Perform Precision Physical Measurements: Keep a high-resolution digital caliper at your station. You must measure the exact distance from the physical cable mating face to the front edge of where the sample block rests. An error of just 0.1 mm will introduce noticeable slope artifacts into your high-frequency dielectric loss tangent data.