Navigating Low-Frequency Breakdown in Nicolson-Ross-Weir Extraction

The Nicolson-Ross-Weir (NRW) method is widely celebrated for offering a closed-form algebraic solution that extracts permittivity and permeability simultaneously. However, if you push the NRW algorithm below 1 GHz on thin samples, you will face a harsh mathematical wall known as the low-frequency breakdown.

The Physics of Thin Electrical Lengths

At low frequencies, the physical thickness of your material sample is a microscopic fraction of the operating wavelength. For example, a 3 mm sample evaluated at 100 MHz represents an electrical length approaching zero. Because the material sample is so thin relative to the long wave, the magnitude of the reflection coefficient (S₁₁) drops into the noise floor of the instrument.

When S₁₁ approaches zero, the denominator in the classical NRW equations vanishes. The resulting algebraic equation tries to divide one tiny, noise-corrupted number by another number near zero, causing the output traces for permittivity and permeability to diverge into wild, non-physical oscillations.

Strategic Countermeasures

• Increase Physical Sample Thickness: The most direct physical solution is to make your low-frequency sample significantly thicker. Increasing the physical length from 3 mm to 30 mm increases the electrical phase delay, boosting S₁₁ well above the VNA's ambient noise floor.
• Deploy S21-Only Transmission Solvers: If your material is strictly non-magnetic (μ_r = 1), you can bypass the reflection metric entirely. Transitioning to a non-iterative, transmission-only algorithm allows you to evaluate permittivity using only S₂₁ phase and magnitude, completely avoiding low-frequency instabilities.