Rectangular Waveguide Dispersion: Compensating for Non-Linear Phase Velocity

Unlike coaxial cables, which propagate waves in a pure TEM mode where phase velocity remains constant across all frequencies, hollow rectangular waveguides are highly dispersive structures. Failing to account for waveguide dispersion during material extraction is a critical error that leads to catastrophic failures in complex permittivity calculations.

The Changing Nature of Guide Wavelength

In a rectangular waveguide, the phase velocity and guided wavelength (λ_g) are highly non-linear functions that vary with frequency. Near the waveguide's cutoff frequency, the guided wavelength stretches out toward infinity. As the operating frequency increases toward the top of the band, λ_g compresses and approaches the free-space wavelength (λ₀).

This means your material extraction code cannot rely on simple free-space math. The wave impedance inside an empty guide is always higher than 377 ohms, varying across the band according to the structural relationship: Z_Waveguide = Z₀ / √[1 - (f_c/f)²].

The De-Embedding Impact

If your software applies a standard linear phase shift to correct for a waveguide port extension, it will over-correct the lower frequencies and under-correct the higher frequencies. This leaves a severe, artificial bow or tilt across your broadband permittivity plot.

Best Practices for the Lab

• Input Exact Structural Dimensions: Never rely on generic nominal waveguide parameters. Measure the broad wall interior dimension (dimension 'a') of your specific physical sample holder using a bore gauge. Slight manufacturing variations in custom-milled fixtures will shift the localized cutoff frequency.
• Enforce Strict Band Bounds: Never process S-parameter data that falls outside the recommended operational frequency band of the specific waveguide size (e.g., 8.2 to 12.4 GHz for WR-90). Near the cutoff boundary, the dispersion equations become highly sensitive, amplifying tiny instrument errors into massive extraction artifacts.