Nicolson–Ross–Weir method is a measurement technique for determination of complex permittivities and permeabilities of material samples for microwave frequencies. The method is based on insertion of a material sample with a known thickness inside a waveguide, such as a coaxial cable or a rectangular waveguide, after which the dispersion data is extracted from the resulting scattering parameters. The method is named after A. M. Nicolson and G. F. Ross, and W. B. Weir, who developed the approach in 1970 and 1974, respectively. The technique is one of the most common procedures for material characterization in microwave engineering.
Method The method uses scattering parameters of a material sample embedded in a waveguide, namely S 11 {\displaystyle S_{11}} and S 21 {\displaystyle S_{21}} , to calculate permittivity and permeability data. S 11 {\displaystyle S_{11}} and S 21 {\displaystyle S_{21}} correspond to the cumulative reflection and transmission coefficient of the sample that are referenced to the each sample end, respectively: these parameters account for the multiple internal reflections inside the sample, which is considered to have a thickness of d {\displaystyle d} . The reflection coefficient of the bulk sample is:
Γ = X ± X 2 − 1 {\displaystyle \Gamma =X\pm {\sqrt {X^{2}-1}}}
where
X = 1 − ( S 21 2 − S 11 2 ) 2 S 11 {\displaystyle X={\frac {1-(S_{21}^{2}-S_{11}^{2})}{2S_{11}}}}
The sign of the root for the reflection coefficient is chosen appropriately to ensure its passivity ( | Γ | ≤ 1 {\displaystyle |\Gamma |\leq 1} ). Similarly, the transmission coefficient of the bulk sample can be written as:
T = S 11 + S 21 − Γ 1 − ( S 11 + S 21 ) Γ {\displaystyle T={\frac {S_{11}+S_{21}-\Gamma }{1-(S_{11}+S_{21})\Gamma }}}
Thus, the effective permeability ( μ ∗ {\displaystyle \mu ^{*}} ) and permittivity ( ε ∗ {\displaystyle \varepsilon ^{*}} ) of the material can be written as:
μ ∗ = λ 0 g Λ ( 1 + Γ 1 − Γ ) {\displaystyle \mu ^{*}={\frac {\lambda _{0g}}{\Lambda }}\left({\frac {1+\Gamma }{1-\Gamma }}\right)}
ε ∗ = λ 0 2 ( 1 Λ 2 + 1 λ c 2 ) μ ∗ {\displaystyle \varepsilon ^{*}={\frac {\lambda _{0}^{2}\left({\frac {1}{\Lambda ^{2}}}+{\frac {1}{\lambda _{c}^{2}}}\right)}{\mu ^{*}}}}
where
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